# Nonlinear elliptic equations with variable exponents satisfying Cerami
condition

Omar Benslimane1∗ Ahmed Aberqi2 and Jaouad Bennouna1 1Sidi Mohamed Ben
Abdellah University, Faculty of Sciences Dhar Al Mahraz, Department of
Mathematics, B.P 1796 Atlas Fez, Morocco<EMAIL_ADDRESS><EMAIL_ADDRESS>2Sidi Mohamed Ben Abdellah University, National School of
Applied Sciences Fez, Morocco<EMAIL_ADDRESS>
(Date: Received: xxxxxx; Accepted: zzzzzz.
∗Corresponding author)

###### Abstract.

We are concerned with the study of the existence and multiplicity of solutions
for Dirichlet boundary value problems involving the $(p(x),\,q(x))-$ equation
and the nonlinearity is superlinear but does not satisfy the usual
Ambrossetti-Rabinowitz condition in the framework of Sobolev spaces with
variable exponents in Complete manifolds. The main results are established by
means of the mountain pass theorem and Fountain theorem with Cerami condition.
Moreover, we are giving an example of a $(p(x),\,q(x))$ equation that verifies
all our demonstrated results.

###### Key words and phrases:

Elliptic equation, weak solution, Cerami condition, Sobolev-Orlicz Riemannian
manifold with variable exponants

###### 2010 Mathematics Subject Classification:

Primary 35J60; Secondary 58J05.

##  1\. Introduction

Let $(M,g)$ be a complete non-compact Riemannian manifold, we consider the
following problem

$(\mathcal{P})\,\begin{cases}-\,\Delta_{p(x)}u(x)-\Delta_{q(x)}u(x)=\,f(x,u(x))&\text{in
\,\,M },\\\\[8.5359pt] \,u\,=\,0&\text{on \,$\partial$M },\end{cases}$

where the variable exponents $p,\,q\in\overline{M}\rightarrow(1,\,+\infty)$
are continuous functions, satisfy the following assumption:

$1<q^{-}\leq q^{+}<p^{-}\leq p^{+}<\infty,$ (1.1)

with

$q^{+}=\sup_{x\in\overline{M}}q(x),\,\,q^{-}=\inf_{x\in\overline{M}}q(x),\,\,p^{+}=\sup_{x\in\overline{M}}p(x),\,\,\mbox{and}\,\,p^{-}=\inf_{x\in\overline{M}}p(x).$

The operator $\Delta_{p(x)}u(x)=-\mbox{div}(|\,\nabla u(x)\,|^{p(x)-2}\nabla
u(x))$ and $\Delta_{q(x)}u(x)=\\\ -\mbox{div}(|\,\nabla u(x)\,|^{q(x)-2}\nabla
u(x))$ are called the $p(x)-$Laplacian and $q(x)-$Laplacian in $(M,\,g)$.
Before giving our hypotheses on the nonlinearity $f.$ We first define the
family of functions

$\mathcal{F}=\\{\,G_{\lambda}(x,t)=f(x,t)t-\lambda
F(x,t),\,\,\lambda\in[2q^{-},\,2p^{+}]\,\\},$

where $F(x,t)=\displaystyle\int_{0}^{t}f(x,s)\,\,ds$. Note that when $p(x)=p$
is constant, the family $\mathcal{F}$ consists of only one element, that is,

$\mathcal{F}=\\{\,G_{p}(x,t)=f(x,t)t-p\,F(x,t)\,\\}.$

Throughout this paper, we assume the following hypotheses on the Carathéodory
function $f:M\times\mathbb{R}\rightarrow\mathbb{R}$

  * ($f_{1})$:

$|\,f(x,t)\,|\leq c_{1}+c_{2}\,|\,t\,|^{r(x)-1}$ for all $(x,t)\in
M\times\mathbb{R}$ where $r:\overline{M}\rightarrow(1,\,+\infty)$ is a bounded
continuous function such that $r(x)<q^{*}(x)=\frac{N\,q(x)}{N-q(x)}$ for any
$x\in\overline{M}$.

  * ($f_{2})$:

$\displaystyle\lim_{|\,t\,|\rightarrow\infty}\frac{f(x,t)\,t}{|\,t\,|^{p^{+}}}=+\infty$
uniformly for a.e $x\in M$.

  * ($f_{3})$:

$f(x,t)=0(|\,t\,|^{p^{+}-1})$ as $t\rightarrow 0$ uniformly for $x\in M$.

  * ($f_{4})$:

There exists a constant $\theta\geq 1$ such that for any
$\beta\in[0,\,1],\,t\in\mathbb{R}$ and for all
$G_{\lambda}\in\mathcal{F},\,\eta\in[2q^{-},\,2p^{+}]$ the inequality

$\theta\,G_{\lambda}(x,t)\geq G_{\eta}(x,\beta t)\,\,\mbox{holds for
a.e}\,\,x\in M$

  * ($f_{5})$:

$f(x,-t)=-f(x,t)$ for all $(x,t)\in M\times\mathbb{R}.$

There are many papers done by authors in this context, and we start by a
pioneer work for multiplicity results with superlinear nonlinearities was
published in the classical Sobolev space when $p(x)=p=\mbox{cste}$ by Liu and
Li in [19] used the critical point theory with Cerami condition which is
weaker than the Palais-Smale condition. For more results, we refer the reader
to [1, 2, 4, 5, 6, 7] and the references therein.
In Sobolev space with variable exponent Zhang and Zhao in [25] proved the
existence of strong solutions of the following $p(x)-$Laplacian Dirichlet
problem via critical point theory:

$\begin{cases}-\,\mbox{div}(|\nabla u|^{p(x)-2}\nabla u=\,f(x,u(x))&\text{in
\,\,$\Omega$ },\\\\[8.5359pt] \,u\,=\,0&\text{on
\,$\partial\Omega$},\end{cases}$

and they give a new growth condition, under which, they used a new method to
check the Cerami compactness condition. Hence, they proved the existence of
strong solutions to the problem as above without the growth condition of the
well-known Ambrossetti-Rabinowitz type and also they give some results about
the multiplicity of the solutions.
In [16] and using the variational methods, the authors established the
existence and multiplicity of weak solutions for a general class of
quasilinear problems involving $p(.)-$Laplace type operators, with Dirichlet
boundary conditions involving variable exponents without Ambrossetti and
Rabinowitz type growth conditions, namely

$\begin{cases}-\,\mbox{div}(a(|\nabla u|^{p(x)})|\nabla u|^{p(x)-2}\nabla
u=\,\lambda\,f(x,u(x))&\text{in \,\,$\Omega$ },\\\\[8.5359pt]
\,u\,=\,0&\text{on \,$\partial\Omega$},\end{cases}$

and by different types of versions of the Mountain Pass Theorem with Cerami
condition, as well as, the Fountain and Dual Theorem with Cerami condition,
they obtained some existence of weak solutions for the above problem under
some considerations. Moreover, they have shown that the problem treated has at
least one non-trivial solution for any parameter $\lambda>0,$ small enough,
and also that the solution blows up, in the Sobolev norm, as
$\lambda\rightarrow 0^{+}.$ Finally, by suitable hypotheses on the
nonlinearity $f(x,.)$ they get the existence of infinitely many weak solutions
by using the Genus Theory introduced by Krasnoselskii. For a deeper
comprehension, we refer the reader to [20, 13, 23] and the references therein.
As applications, we cite for examples: the study of fluid filtration in porous
media, constrained heating, elastoplasticity, optimal control, financial
mathematics, and others, see [10, 14, 21, 24] and the references therein.

To our knowledge, the results presented here are new, and they complement and
improve the ones obtained in [23, 25, 16] because we are considering the
general framework of Sobolev spaces with variable exponents in Complete
manifolds and nonlinearities is superlinear but does not satisfy the usual
Ambrossetti-Rabinowitz condition. However, we address the challenges presented
by the fact that the $p(x)-$ laplacian and $q(x)-$ laplacian operators possess
more complicated nonlinearities thanthe p-laplacian and q-laplacian operator,
due to the fact that $\Delta_{p(x)}$ and $\Delta_{q(x)}$ are not homogeneous.
Moreover, we can not use Lagrange Multiplier Theorem in many problems
involving this operator, which shows that our problem has more difficulty than
the operators p-Laplace type.

The remainder of the paper is organized as follows. In section 2 we will
recall the definitions and some properties of Sobolev spaces with variable
exponents and Sobolev spaces with variable exponents in complete manifolds.
The readers can consult the following papers [3, 4, 5, 6, 7, 15, 22] for
details. In section 3, using the mountain pass theorem with Cerami condition,
we prove the existence of non-trivial weak solutions of problem
$(\mathcal{P}).$ Moreover, Using the Fountain theorem with Cerami condition,
we demonstrate that the problem $(\mathcal{P})$ has infinitely many (pairs) of
solutions with unbounded energy.

##  2\. Notations and Basic Properties

In order to discuss the problem $(\mathcal{P})$, we need some facts on spaces
$W_{0}^{1,q(x)}(\Omega)$ where $\Omega$ is an open subset of $\mathbb{R}^{N}$
and $W_{0}^{1,q(x)}(M)$ which are called the Sobolev spaces with variable
exponents and the Sobolev spaces with variable exponents in complete manifolds
setting. For this reason, we will recall some properties involving the above
spaces, which can be found in [3, 4, 11, 15, 13] and references therein.

###  2.1. Sobolev spaces with variable exponents

Let $\Omega$ be a bounded open subset of $\mathbb{R}^{N}\,(N\geq 2)$, we
define the Lebesgue space with variable exponent $L^{q(.)}(\Omega)$ as the set
of all measurable function $u:\Omega\longmapsto\mathbb{R}$ for which the
convex modular

$\rho_{q(.)}(u)=\int_{\Omega}|\,u(x)\,|^{q(x)}\,\,dx,$

is finite. If the exponent is bounded, i.e if
$q^{+}=ess\,sup\\{\,q(x)/x\in\Omega\,\\}<+\infty,$ then the expression

$||\,u\,||_{q(.)}=\inf\\{\,\lambda>0:\,\rho_{q(.)}\bigg{(}\frac{u}{\lambda}\bigg{)}\leq
1\,\\},$

defines a norm in $L^{q(.)}(\Omega)$, called the Luxemburg norm.
The space $(L^{q(.)}(\Omega),\,||\,.\,||_{q(.)})$ is a separable Banach space.
Moreover, if $1<q^{-}\leq q^{+}<+\infty,$ then $L^{q(.)}(\Omega)$ is uniformly
convex, where $q^{-}=ess\,inf\\{\,q(x)/x\in\Omega\,\\},$ hence reflexive, and
its dual space is isomorphic to $L^{q^{{}^{\prime}}(.)}(\Omega)$ where
$\frac{1}{q(x)}+\frac{1}{q^{{}^{\prime}}(x)}=1.$
Finally, we have the Hölder type inequality:

$\bigg{|}\,\int_{\Omega}u\,v\,\,dx\,\bigg{|}\leq\bigg{(}\,\frac{1}{q^{-}}+\frac{1}{(q^{{}^{\prime}})^{-}}\bigg{)}\,||\,u\,||_{q(.)}||\,v\,||_{q^{{}^{\prime}}(.)},$

for all $u\in L^{q(.)}(\Omega)$ and $v\in L^{q^{{}^{\prime}}(.)}(\Omega).$

We define the variable exponents Sobolev space by

$W^{1,q(x)}(\Omega)=\\{\,u\in L^{q(x)}(\Omega)\,\,\mbox{and}\,\,|\,\nabla
u\,|\in L^{q(x)}(\Omega)\,\\},$

with the norm

$||\,u\,||_{W^{1,q(x)}(\Omega)}=||\,u\,||_{L^{q(x)}(\Omega)}+||\,\nabla
u\||_{L^{q(x)}(\Omega)}\,\,\forall u\in W^{1,q(x)}(\Omega).$

We denote by $W^{1,q(x)}_{0}(\Omega)$ the closure of $C_{0}^{\infty}(\Omega)$
in $W^{1,q(x)}(\Omega),$ and we define the Sobolev exponent by
$q^{*}(x)=\frac{N\,q(x)}{N-q(x)}$ for $q(x)<N.$

###  2.2. Sobolev spaces with variable exponents in complete manifolds

######  Definition 2.1.

Let $(M,g)$ be a smooth Riemannain manifolds and let $\nabla$ be the Levi-
Civita connection. If $u$ is a smooth function on $M$, then $\nabla^{k}u$
denotes the $k-$th covariant derivative of $u$, and $|\,\nabla^{k}u\,|$ the
norm of $\nabla^{k}u$ defined in local coordinates by

$|\,\nabla^{k}u\,|^{2}=g^{i_{1}j_{1}}\cdots
g^{i_{k}j_{k}}\,(\nabla^{k}u)_{i_{1}\cdots i_{k}}\,(\nabla^{k}u)_{j_{1}\cdots
j_{k}}$

where Einstein’s convention is used.

######  Definition 2.2.

To define variable Sobolev spaces, given a variable exponent $q$ in
$\mathcal{P}(M)$ ( the set of all measurable functions
$p(.):M\rightarrow[1,\infty]$ ) and a natural number $k$, introduce

$C^{q(.)}_{k}(M)=\\{\,u\in C^{\infty}(M)\,\,\mbox{such that }\,\,\forall
j\,\,0\leq j\leq k\,\,|\,\nabla^{k}u\,|\in L^{q(.)}(M)\,\\}.$

On $C^{q(.)}_{k}(M)$ define the norm

$||\,u\,||_{L^{q(.)}_{k}}=\sum_{j=0}^{k}||\,\nabla^{j}u\,||_{L^{q(.)}}.$

######  Definition 2.3.

The Sobolev spaces $L_{k}^{q(.)}(M)$ is the completion of $C^{q(.)}_{k}(M)$
with respect to the norm $||\,u\,||_{L^{q(.)}_{k}}$.

######  Definition 2.4.

Given $(M,g)$ a smooth Riemannian manifold, and
$\gamma:\,[\,a,\,b\,]\longrightarrow M$ a curve of class $C^{1}$. The length
of $\gamma$ is

$l(\gamma)=\int_{a}^{b}\sqrt{g\,(\,\frac{d\gamma}{dt},\,\frac{d\gamma}{dt}\,)}\,\,dt,$

and for a pair of points $x,\,y\in M$, we define the distance $d_{g}(x,y)$
between $x$ and $y$ by

$d_{g}(x,y)=\inf\,\\{\,l(\gamma):\,\gamma:\,[\,a,\,b\,]\rightarrow
M\,\,\mbox{such that}\,\,\gamma(a)=x\,\,\mbox{and}\,\,\gamma(b)=y\,\\}.$

######  Definition 2.5.

A function $s:\,M\longrightarrow\mathbb{R}$ is log-Hölder continuous if there
exists a constant $c$ such that for every pair of points $\\{x,\,y\\}$ in $M$
we have

$|\,s(x)-s(y)\,|\leq\frac{c}{log(e+\frac{1}{d_{g}(x,y)}\,)}.$

We note by $\mathcal{P}^{log}(M)$ the set of log-Hölder continuous variable
exponents. The relation between $\mathcal{P}^{log}(M)$ and
$\mathcal{P}^{log}(\mathbb{R}^{N})$ is the following:

######  Proposition 2.6.

[3, 11] Let $q\in\mathcal{P}^{log}(M)$, and let $(\Omega,\phi)$ be a chart
such that

$\frac{1}{2}\delta_{ij}\leq g_{ij}\leq 2\,\delta_{ij}$

as bilinear forms, where $\delta_{ij}$ is the delta Kronecker symbol. Then
$qo\phi^{-1}\in\mathcal{P}^{log}(\phi(\Omega)).$

######  Definition 2.7.

We say that the n-manifold $(M,g)$ has property $B_{vol}(\lambda,v)$ if its
geometry is bounded in the following sense:
$\hskip 28.45274pt\bullet\,\,\mbox{The Ricci tensor of g noted by Rc ( g )
verify,}\,\,Rc(g)\geq\lambda(n-1)\,g$ for some $\lambda.$
$\hskip 28.45274pt\bullet$ There exists some $v>0$ such that
$|\,B_{1}(x)\,|_{g}\geq v\,\,\forall x\in M,$ where $B_{1}(x)$ are the balls
of radius 1 centered at some point $x$ in terms of the volume of smaller
concentric balls.

######  Proposition 2.8.

[3, 15] Let $(M,g)$ be a complete Riemannian n-manifold. Then, if the
embedding $L^{1}_{1}(M)\hookrightarrow L^{\frac{n}{n-1}}(M)$ holds, then
whenever the real numbers $q$ and $p$ satisfy

$1\leq q<n,$

and

$q\leq p\leq q*=\frac{nq}{n-q},$

the embedding $L^{q}_{1}(M)\hookrightarrow L^{p}(M)$ also holds.

######  Proposition 2.9.

[3, 15] Assume that the complete n-manifold $(M,g)$ has property
$B_{vol}(\lambda,v)$ for some $(\lambda,v).$ Then there exist positive
constants $\delta_{0}=\delta_{0}(n,\,\lambda,\,v)$ and $A=A(n,\,\lambda,\,v)$,
we have, if $R\leq\delta_{0}$, if $x\in M$ if $1\leq q\leq n$, and if $u\in
L^{q}_{1,0}(\,B_{R}(x)\,)$ the estimate

$||\,u\,||_{L^{p}}\leq A\,p\,||\,\nabla u\,||_{L^{q}},$

where $\frac{1}{p}=\frac{1}{q}-\frac{1}{n}.$

######  Proposition 2.10.

[3, 15, 11] Assume that for some $(\lambda,v)$ the complete n-manifold $(M,g)$
has property $B_{vol}(\lambda,v)$. Let $p\in\mathcal{P}(M)$ be uniformly
continuous with $q^{+}<n.$ Then $L^{q(.)}_{1}(M)\hookrightarrow
L^{p(.)}(M)\,\,\forall q\in\mathcal{P}(M)$ such that $q\ll p\ll
q*=\frac{nq}{n-q}.$ In fact, for $||\,u\,||_{L^{q(.)}_{1}}$ sufficiently small
we have the estimate

$\rho_{p(.)}(u)\leq G\,(\,\rho_{q(.)}(u)+\rho_{q(.)}(|\,\nabla u\,|)\,),$

where the positive constant $G$ depend on $n,\,\lambda,\,v,\,q$ and $p$.

######  Proposition 2.11.

Let $u\in L^{q(x)}(M),\,\\{\,u_{k}\,\\}\subset L^{q(x)}(M),\,k\in\mathbb{N},$
then we have

  1. (i)

$||u||_{q(x)}<1\,\,\mbox{( resp. = 1, $>$ 1 )}\iff\rho_{q(x)}(u)<1\,\,\mbox{(
resp. = 1, $>$ 1 )},$

  2. (ii)

For $u\in
L^{q(x)}(M)\backslash\\{0\\},\,\,||u||_{q(x)}=\lambda\Longleftrightarrow\rho_{q(x)}\big{(}\frac{u}{\lambda}\big{)}=1.$

  3. (iii)

$||u||_{q(x)}<1\Rightarrow||u||_{q(x)}^{q^{+}}\leq\rho_{q(x)}(u)\leq||u||_{q(x)}^{q^{-}},$

  4. (iv)

$||u||_{q(x)}>1\Rightarrow||u||_{q(x)}^{q^{-}}\leq\rho_{q(x)}(u)\leq||u||_{q(x)}^{q^{+}},$

  5. (v)

$\lim_{k\rightarrow+\infty}||u_{k}-u||_{q(x)}=0\iff\lim_{k\rightarrow+\infty}\rho_{q(x)}(u_{k}-u)=0.$

######  Definition 2.12.

[13] The Sobolev space $W^{1,q(x)}(M)$ consists of such functions $u\in
L^{q(x)}(M)$ for which $\nabla^{k}u\in L^{q(x)}(M)$ $k=1,2,\cdots,n.$ The norm
is defined by

$||\,u\,||_{W^{1,q(x)}(M)}=||\,u\,||_{L^{q(x)}(M)}+\sum_{k=1}^{n}||\,\nabla^{k}u\,||_{L^{q(x)}(M)}.$

The space $W_{0}^{1,q(x)}(M)$ is defined as the closure of $C^{\infty}(M)$ in
$W^{1,q(x)}(M).$

######  Theorem 2.13.

[13] Let $M$ be a compact Riemannian manifold with a smooth boundary or
without boundary and $q(x),\,p(x)\in C(\overline{M})\cap L^{\infty}(M).$
Assume that

$q(x)<N,\hskip
14.22636ptp(x)<\frac{N\,q(x)}{N-q(x)}\,\,\mbox{for}\,\,x\in\overline{M}.$

Then,

$W^{1,q(x)}(M)\hookrightarrow L^{p(x)}(M),$

is a continuous and compact embedding.

######  Theorem 2.14.

Let $M$ be a compact Riemannian manifold with a smooth boundary or without
boundary and $q(x),\,r(x)\in C(\overline{M})\cap L^{\infty}(M).$ Assume that

$q(x)<N,\hskip
14.22636ptr(x)<\frac{N\,q(x)}{N-q(x)}\,\,\mbox{for}\,\,x\in\overline{M}.$

Then,

$W^{1,q(x)}(M)\hookrightarrow L^{r(x)}(M),$

is a continuous and compact embedding.

###### Proof.

The demonstration of this theorem is the same as the previous one. ∎

######  Proposition 2.15.

[3] If $(M,g)$ is complete, then $W^{1,q(x)}(M)=W^{1,q(x)}_{0}(M).$

Let $D\Psi=DJ-DI:\,L^{q(x)}_{1}(M)\cap
L^{p(x)}_{1}(M)\rightarrow\mbox{Hom}(L^{q(x)}_{1}(M)\cap
L^{p(x)}_{1}(M),\,\mathbb{R})$ the differential of $\Psi=J-I$ with the
functional $J:X\rightarrow\mathbb{R}$ defined by:

$J(u)=\int_{M}\frac{1}{p(x)}|\,\nabla
u(x)\,|^{p(x)}\,\,dv_{g}(x)+\int_{M}\frac{1}{q(x)}|\,\nabla
u(x)\,|^{q(x)}\,\,dv_{g}(x)$

and

$I(u)=\int_{M}F(x,u(x))\,\,dv_{g}(x).$

Where $X=W_{0}^{1,q(x)}(M)\,\cap\,W_{0}^{1,p(x)}(M),$ with the norm
$||u||_{X}=||u||_{q(x)}+||u||_{p(x)}\,\,\mbox{for all}\,\,x\in M.$ Then, for
all $\varphi\in D(M)$ we have

$\langle\,J(u),\,\varphi\,\rangle=\int_{M}|\,\nabla
u(x)\,|^{p(x)-2}u(x)\,\varphi(x)\,\,dv_{g}(x)+\int_{M}|\,\nabla
u(x)\,|^{q(x)-2}u(x)\,\varphi(x)\,\,dv_{g}(x),$

where $\langle.,\,.\rangle$ denotes the usual duality between $X$ and its dual
space.

######  Lemma 2.16.

The following assumptions hold:

  * i/

$J$ is a continuous, bounded homeomorphism and strictly monotone operator.

  * ii/

$J$ is a mapping of type $(S_{+})$, that is, if $u_{n}\rightharpoonup u$ and

$\lim\,\sup_{n\rightarrow+\infty}\langle\,J(u_{n})-J(u),\,u_{n}-u\,\rangle\,\leq
0,$

then, $u_{n}\rightarrow u.$

Next, we recall the definition of Cerami condition $(C)$ which is introduced
by G. Cerami in [9].

######  Definition 2.17.

Let $(E,\,||\,.\,||)$ be a Banach space and $\Phi\in C^{1}(E,\,\mathbb{R}).$
Given $c\in\mathbb{R},$ we say that $\Phi$ satisfies the Cerami $C$ condition
( we denote condition $(C_{c})$), if:

  * $(C_{1})$:

Any bounded sequence ${\,u_{n}\,}\subset E$ such that $\Phi(u_{n})\rightarrow
c$ and $\Phi^{\prime}(u_{n})\rightarrow 0$ has a convergent subsequence.

  * $(C_{2})$ :

There exist constant $\delta,\,R,\,\beta>0$ such that

$||\,\Psi^{\prime}(u)\,||_{E^{*}}\,||\,u\,||\geq\beta\,\,\mbox{for
all}\,\,u\in\Psi^{-1}([\,c-\delta,\,c+\delta\,])\,\,\mbox{with}\,\,||\,u\,||\geq
R.$

If $\Psi\in C^{1}(X,\,\mathbb{R})$ satisfies condition $(C_{c})$ for every
$c\in\mathbb{R},$ we say that $\Psi$ satisfies condition $(C)$.
Let us recall the following version of mountain pass Lemma with Cerami
condition with will be used in the sequel.

######  Proposition 2.18.

Let $(E,\,||.||)$ a Banach space, $\Psi\in C^{1}(X,\,\mathbb{R}),\,\,u_{0}\in
E$ and $\upsilon>0,$ be such that $||u_{0}||>\upsilon$ and

$b=\inf_{||u_{0}||=\upsilon}\Psi(u)>\Psi(0)\geq\Psi(u_{0}).$

If $\Psi$ satisfies the condition $(C_{c})$ with

$c=\inf_{\gamma\in\Gamma}\max_{t\in[0,\,1]}\Psi(\gamma(t)),\,\,\,\Gamma=\\{\gamma\in
C([0,\,1],\,X)\,\mbox{such as}\,\,\gamma(0)=0,\,\gamma(1)=u_{0}\,\\}.$

Then $c$ is a critical value of $\Psi.$

######  Remark 2.19.

Since $X$ be a reflexive and separable Banach space, there exist
$\\{e_{n}\\}_{n=1}^{\infty}\subset X$ and
$\\{e^{*}_{n}\\}_{n=1}^{\infty}\subset X^{*}$ such that

$\langle e^{*}_{n},\,e_{m}\rangle\,=\delta_{n,\,m}=\begin{cases}1&\text{ if
$n=m$ },\\\\[8.5359pt] 0&\text{ in $n\neq m$ }.\end{cases}$

Hence,

$X=\overline{\mbox{span}}\,\\{e_{n},\,n\geq
1\,\\}\,\,\mbox{and}\,\,X^{*}=\overline{\mbox{span}}\,\\{e^{*}_{n},\,n\geq
1\,\\}.$

For $k\geq 1,$ denote
$X^{k}=\mbox{span}\,\\{\,e_{k}\,\\},\,\,Y_{k}=\displaystyle\bigoplus_{i=0}^{k}X^{k},\,\,Z_{k}=\displaystyle\overline{\bigoplus_{i=k}^{\infty}X^{k}}.$

Next, as in [26] we introduce the Fountain Theorem with the condition $(C)$ as
follow:

######  Theorem 2.20.

Assume that $X$ is a separable Banach space, $\Psi\in C^{1}(X,\,\mathbb{R})$
is an even functional satisfying the Cerami condition $(C).$ Moreover, for
each $k\geq 1$ there exist $D_{k}>d_{k}>0$ such as

  * $(A_{1})$:

$\displaystyle\inf_{\\{u\in
Z_{k}:||u||=d_{k}\\}}\Psi(u)\rightarrow+\infty\,\,\mbox{as}\,\,k\rightarrow+\infty.$

  * $(A_{2})$:

$\displaystyle\max_{\\{u\in Y_{k}:||u||=D_{k}\\}}\Psi(u)\leq 0.$

Then, $\Psi$ has a sequence of critical values which tends to $+\infty.$

##  3\. Existence and multiplicity results:

In this section, we state our main results and we note by $D(M)$ the space of
$C^{\infty}$ functions with compact support in $M$.

######  Definition 3.1.

$u\in X$ is said to be a non-trivial solution of the problem $(\mathcal{P})$
if for every $\phi\in D(M)$ we have

$\displaystyle\int_{M}|\nabla u(x)|^{p(x)-2}g(\nabla
u(x),\,\nabla\phi(x))\,\,dv_{g}(x)+\int_{M}|\nabla u(x)|^{q(x)-2}g(\nabla
u(x),\nabla\phi(x))\,\,dv_{g}(x)$
$\displaystyle-\int_{M}f(x,u(x))\,.\,\phi(x)\,\,dv_{g}(x)=0.$

Considering the energy function $\Psi:X\rightarrow\mathbb{R}$ associated to
problem $(\mathcal{P})$ defined by

$\displaystyle\Psi(u)=$ $\displaystyle\int_{M}\frac{1}{p(x)}|\nabla
u(x)|^{p(x)}\,\,dv_{g}(x)+\int_{M}\frac{1}{q(x)}|\nabla
u(x)|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle-\int_{M}F(x,u(x))\,\,dv_{g}(x).$

By condition $(f_{1})$ and Theorems 2.13 and 2.14, the functional $\Psi\in
C^{1}(X,\,\mathbb{R})$ is well defined.
Moreover, for all $\phi\in D(M)$ we have

$\displaystyle\langle\Psi^{\prime}(u),\phi\rangle\,=$
$\displaystyle\int_{M}|\nabla u(x)|^{p(x)-2}g(\nabla
u(x),\,\nabla\phi(x))\,\,dv_{g}(x)$ $\displaystyle+\int_{M}|\nabla
u(x)|^{q(x)-2}g(\nabla u(x),\,\nabla\phi(x))\,\,dv_{g}(x)$
$\displaystyle-\int_{M}f(x,u(x))\,.\,\phi(x)\,\,dv_{g}(x)\hskip
28.45274pt\forall u\in X$

######  Lemma 3.2.

Assume that the assumptions $(f_{1}),\,(f_{2}),$ and $(f_{4})$ are satisfied.
Then the functional $\Psi$ satisfies the Cerami condition $(C).$

###### Proof.

For any $c\in\mathbb{R},$ we first show that $\Psi$ satisfies the assertion
$(C_{1})$ of Cerami condition $(C)$. In fact, let $\\{u_{m}\\}\subset X$ be a
bounded sequence such as

$\Psi(u_{m})\xrightarrow{m\rightarrow+\infty}c\hskip
14.22636pt\mbox{and}\hskip
14.22636pt\Psi^{\prime}(u_{m})\xrightarrow{m\rightarrow+\infty}0,$ (3.1)

without loss of generality, we assume that $u_{m}\rightharpoonup u$ as
$m\rightarrow+\infty.$ By (3.1) we have

$\langle\Psi^{\prime}(u_{m}),u_{m}-u\rangle\,\xrightarrow{m\rightarrow+\infty}0,$

that is

$\displaystyle\langle\Psi^{\prime}(u_{m}),\,u_{m}-u\rangle\,=$
$\displaystyle\int_{M}|\nabla u_{m}|^{p(x)-2}\nabla u_{m}\,(\nabla
u_{m}-\nabla u)\,\,dv_{g}(x)$ $\displaystyle+\int_{M}|\nabla
u_{m}|^{q(x)-2}\nabla u_{m}\,(\nabla u_{m}-\nabla u)\,\,dv_{g}(x)$
$\displaystyle-\int_{M}f(x,u_{m}(x))\,(u_{m}-u)\,\,dv_{g}(x)\rightarrow 0,$
(3.2)

as $m\rightarrow+\infty.$
On the other hand, using $(f_{1})$ and the Hölder inequality, we obtain

$\int_{M}f(x,u_{m}(x))\,(u_{m}-u)\,\,dv_{g}(x)\xrightarrow{m\rightarrow+\infty}0.$
(3.3)

Combining (3) and (3.3), we get

$\displaystyle\int_{M}|\nabla u_{m}(x)|^{p(x)-2}\,\nabla u_{m}\,(\nabla
u_{m}-\nabla u)\,\,dv_{g}(x)$ $\displaystyle+\int_{M}|\nabla
u_{m}|^{q(x)-2}\,\nabla u_{m}\,(\nabla u_{m}-\nabla u)\,\,dv_{g}(x)\rightarrow
0\,\,\mbox{as}\,\,m\rightarrow+\infty.$

That is

$\langle J(u_{m}),\,u_{m}-u\,\rangle\rightarrow
0\,\,\mbox{as}\,\,m\rightarrow+\infty.$ (3.4)

Furthermore, since $u_{m}\rightharpoonup u$ as $m\rightarrow+\infty,$ from
(3.1) we have

$\langle\Psi^{\prime}(u_{m}),\,u_{m}-u\rangle\rightarrow
0\,\,\mbox{as}\,\,m\rightarrow+\infty.$

Using the same technique as before, we deduce that

$\langle J(u),\,u_{m}-u\rangle\rightarrow
0\,\,\mbox{as}\,\,m\rightarrow+\infty.$ (3.5)

Hence, according to (3.4) and (3.5) we deduce that

$\lim_{m\rightarrow+\infty}\sup\,\langle
J(u_{m})-J(u),\,u_{m}-u\,\rangle\,\leq 0.$

Then, as $u_{m}\rightharpoonup u$ in $X$ and since $J$ is a mapping of type
$(S_{+}),$ we conclude that $u_{m}\xrightarrow{m\rightarrow+\infty}u$ in $X$.
Now, we check that $\psi$ satisfies the assertion $(C_{2})$ of Cerami
condition $(C)$. Arguing by contradiction, there exist $c\in\mathbb{R}$ and
$\\{u_{m}\\}\subset X$ satisfying:

$\Psi(u_{m})\xrightarrow{n\rightarrow+\infty}c,\hskip
5.69046pt||u_{m}||_{X}\xrightarrow{m\rightarrow+\infty}+\infty,\hskip
5.69046pt||\Psi^{\prime}(u_{m})||_{X^{*}}\,||u_{m}||_{X}\xrightarrow{m\rightarrow+\infty}0.$
(3.6)

Let

$\alpha_{m}=\frac{\displaystyle\int_{M}\big{(}|\nabla u_{m}|^{p(x)}+|\nabla
u_{m}|^{q(x)}\,\big{)}\,\,dv_{g}(x)}{J^{\prime}(u_{m})},$

choosing $||u_{m}||_{X}>1,$ for $m\in\mathbb{N},$ then we have

$\displaystyle c$
$\displaystyle=\lim_{m\rightarrow+\infty}\\{\Psi(u_{m})-\frac{1}{\alpha_{m}}\,\langle\Psi^{\prime}(u_{m}),\,u_{m}\,\rangle\,\\}$
$\displaystyle=\lim_{m\rightarrow+\infty}\\{\,\frac{1}{\alpha_{m}}\,\int_{M}f(x,u_{m}(x))\,.\,u_{m}\,\,dv_{g}(x)-\int_{M}F(x,u_{m}(x))\,\,dv_{g}(x)\,\\}.$
(3.7)

Denote $w_{m}=\frac{u_{m}}{||u_{m}||},$ so $||w_{m}||_{X}=1,$ which implies
that $\\{u_{m}\\}$ is bounded in $X$.
Hence, for a subsequence of $\\{u_{m}\\}$ still denoted by $\\{w_{m}\\},$ and
$w\in X,$ we obtain

$w_{m}\rightharpoonup w\hskip 8.5359pt\mbox{in}\,\,\,X,$ (3.8)
$w_{m}\rightarrow w\hskip 8.5359pt\mbox{in}\,\,\,L^{r(x)}(M)$ (3.9)
$w_{m}(x)\rightarrow w(x)\hskip 8.5359pt\mbox{a.e in}\,\,\,M$ (3.10)

Step 1: If $w=0$: As in [17] (Lemma 3.6) we can define a sequence
$\\{t_{m}\\}\subset[0,\,1]$ such as

$\psi(t_{m}u_{m})=\max_{t\in[0,\,1]}\psi(tu_{m}).$ (3.11)

If for $m\in\mathbb{N},\,t_{m}$ satisfying (3.11) is not unique, then we
choose the smaller positive value. For that, we fix $A>\frac{1}{2p^{+}},$ let
$\tilde{w}_{m}=(2p^{+}A)^{\frac{1}{p^{-}}},$ and according to (3.9) we have
that

$\tilde{w}_{m}\rightarrow 0\,\,\mbox{in}\,\,L^{r(x)}(M),$

and by $(f_{1}),$ we have

$|\,F(x,t)\,|\leq c\,(1+|\,t\,|^{r(x)}).$

Hence, from the continuity of the Nemitskii operator, we get

$F(.,\tilde{w}_{m})\rightarrow
0\,\,\mbox{in}\,\,L^{1}(M)\,\,\mbox{as}\,\,m\rightarrow+\infty.$

Therefore,

$\lim_{m\rightarrow+\infty}\int_{M}F(x,\tilde{w}_{m})\,\,dv_{g}(x)=0.$ (3.12)

Then, for $m$ large enough,

$\frac{(2p^{+}A)^{\frac{1}{p^{-}}}}{||u_{m}||_{X}}\in(0,\,1),$

and

$\displaystyle\Psi(t_{m}u_{m})$ $\displaystyle\geq\,\Psi(\tilde{w}_{m})$
$\displaystyle\geq\int_{M}\frac{1}{p(x)}|\nabla\tilde{w}_{m}|^{p(x)}\,\,dv_{g}(x)+\int_{M}\frac{1}{q(x)}|\nabla\tilde{w}_{m}|^{q(x)}\,\,dv_{g}(x)$
$\displaystyle\hskip 14.22636pt-\int_{M}F(x,\tilde{w}_{m})\,\,dv_{g}(x)$
$\displaystyle\geq\frac{1}{p^{+}}\,\int_{M}(2p^{+}A)\,|\nabla
w_{m}|^{p(x)}\,\,dv_{g}(x)+\frac{1}{q^{+}}\int_{M}(2p^{+}A)\,|\nabla
w_{m}|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
14.22636pt-\int_{M}F(x,\tilde{w}_{m})\,\,dv_{g}(x)$ $\displaystyle\geq
2A\int_{M}|\nabla
w_{m}|^{p(x)}\,\,dv_{g}(x)+\frac{2Ap^{+}}{q^{+}}\int_{M}|\nabla
w_{m}|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
14.22636pt-\int_{M}F(x,\tilde{w}_{m})\,\,dv_{g}(x)$ $\displaystyle\geq
2Ac||w_{m}||^{p^{+}}+\frac{2A}{\eta\,q^{+}}||w_{m}||^{p^{+}}-\int_{M}F(x,\tilde{w}_{m})\,\,dv_{g}(x)$
$\displaystyle\geq A.$

That is

$\Psi(t_{m}u_{m})\rightarrow+\infty.$ (3.13)

As $\Psi(0)=0$ and $\Psi(u_{m})\xrightarrow{m\rightarrow+\infty}c,$ then, we
have $t_{m}\in(0,\,1)$ for $m$ large enough, and

$\displaystyle\int_{M}|\nabla(t_{m}u_{m})|^{p(x)}\,\,dv_{g}(x)+\int_{M}|\nabla
t_{m}u_{m})|^{q(x)}\,\,dv_{g}(x)-\int_{M}f(x,t_{m}u_{m})\,\,dv_{g}(x)$
$\displaystyle=\langle\Psi^{\prime}(t_{m}u_{m}),\,t_{m}u_{m}\,\rangle=t_{m}\,\frac{\mbox{d}}{\mbox{dt}}\bigg{|}_{t=t_{m}}\Psi(tu_{m})=0.$
(3.14)

Thus, from (3.13) and (3), we get

$\displaystyle\int_{M}\bigg{(}\,\frac{1}{\alpha_{t_{m}}}f(x,\,t_{m}u_{m})\,t_{m}u_{m}-F(x,\,t_{m}u_{m})\,\bigg{)}\,\,dv_{g}(x)$
$\displaystyle=\frac{1}{\alpha_{t_{m}}}\int_{M}|\nabla
t_{m}u_{m}|^{p(x)}\,\,dv_{g}(x)+\frac{1}{\alpha_{t_{m}}}\int_{M}|\nabla(t_{m}u_{m})|^{p(x)}\,\,dv_{g}(x)$
$\displaystyle\hskip
8.5359pt+\frac{1}{\alpha_{t_{m}}}\int_{M}|\nabla(t_{m}u_{m})|^{q(x)}\,\,dv_{g}(x)-\int_{M}F(x,t_{m}u_{m})\,\,dv_{g}(x)$
$\displaystyle=\Psi(t_{m}u_{m})\xrightarrow{m\rightarrow+\infty}+\infty,$

where,

$\alpha_{t_{m}}=\frac{\displaystyle\int_{M}\bigg{(}\,|\nabla(t_{m}u_{m})|^{p(x)}+|\nabla(t_{m}u_{m})|^{q(x)}\,\bigg{)}\,\,dv_{g}(x)}{J^{\prime}(t_{m}u_{m})}.$

From the definition of $\alpha_{m}$ and $\alpha_{t_{m}}$, we have
$\alpha_{m},\,\alpha_{t_{m}}\in[2q^{-},\,2p^{+}].$ Hence,
$G_{\alpha_{m}},\,G_{\alpha_{t_{m}}}\in\mathcal{F}.$ Then, according to
$(f_{4})$ and the fact that

$\frac{\alpha_{t_{m}}}{\theta\,\alpha_{m}}>0,$

we deduce that

$\displaystyle\int_{M}\bigg{(}\,\frac{1}{\alpha_{m}}f(x,u_{m})\,u_{m}-F(x,u_{m})\,\bigg{)}\,\,dv_{g}(x)$
$\displaystyle=\frac{1}{\alpha_{m}}\,\int_{M}G_{\alpha_{m}}(x,u_{m})\,\,dv_{g}(x)$
$\displaystyle\geq\frac{1}{\theta\,\alpha_{m}}\int_{M}G_{\alpha_{t_{m}}}(x,t_{m}u_{m})\,\,dv_{g}(x)$
$\displaystyle=\frac{\alpha_{t_{m}}}{\theta\alpha_{m}}\int_{M}\bigg{(}\frac{1}{\alpha_{t_{m}}}f(x,t_{m}u_{m})\,t_{m}u_{m}-F(x,t_{m}u_{m})\,\bigg{)}\,\,dv_{g}(x)\longrightarrow+\infty,$

which contradicts (3).

Step 2: If $w\neq 0$: From (3.6) we write

$\displaystyle\int_{M}|\nabla u_{m}|^{p(x)}\,\,dv_{g}(x)+\int_{M}|\nabla
u_{m}|^{q(x)}\,\,dv_{g}(x)-\int_{M}f(x,u_{m})\,u_{m}\,\,dv_{g}(x)$
$\displaystyle=\langle\Psi^{\prime}(u_{m}),\,u_{m}\rangle=o(1)\,||u_{m}||_{X},$
(3.15)

then,

$\displaystyle 1-o(1)$
$\displaystyle=\int_{M}\frac{f(x,u_{m})\,.\,u_{m}}{\displaystyle\int_{M}|\nabla
u_{m}|^{p(x)}\,\,dv_{g}(x)+\displaystyle\int_{M}|\nabla
u_{m}|^{q(x)}\,\,dv_{g}(x)}\,\,dv_{g}(x)$
$\displaystyle\geq\int_{M}\frac{f(x,u_{m})\,u_{m}}{||u_{m}||^{p^{+}}}\,\,dv_{g}(x)$
$\displaystyle=\int_{M}\frac{f(x,u_{m})\,.\,u_{m}}{|u_{m}|^{p^{+}}}\,.\,|w_{m}|^{p^{+}}\,\,dv_{g}(x).$
(3.16)

Next, we define the set $\mathcal{B}=\\{\,x\in M;\,w(x)=0\,\\}.$ So for any
$x\in\mathcal{B}\backslash\mathcal{B}_{0}=\\{\,x\in M;\,w(x)\neq 0\,\\},$ we
have

$|u_{m}(x)|\longrightarrow+\infty\,\,\mbox{as}\,\,m\rightarrow+\infty.$

Then, by $(f_{4})$ we have

$\frac{f(x,u_{m}(x))\,.\,u_{m}(x)}{|u_{m}(x)|^{p^{+}}}\,.\,|w_{m}(x)|^{p^{+}}\longrightarrow+\infty\,\,\mbox{as}\,\,m\rightarrow+\infty.$
(3.17)

Since, $|\mathcal{B}\backslash\mathcal{B}_{0}|>0,$ we deduce via the Fatou’s
Lemma that

$\int_{\mathcal{B}\backslash\mathcal{B}_{0}}\frac{f(x,\,u_{m})\,.\,u_{m}}{|u_{m}|^{p^{+}}}\,.\,|w_{m}|^{p^{+}}\,\,dv_{g}(x)\longrightarrow+\infty\,\,\mbox{as}\,\,m\rightarrow+\infty.$
(3.18)

On the other hand, from $(f_{1})$ and $(f_{4}),$ there exists $l>-\infty$ such
as

$\frac{f(x,t)\,.\,t}{|t|^{p^{+}}}\geq
l\,\,\mbox{for}\,\,t\in\mathbb{R}\,\,\mbox{and a.e}\,\,x\in M.$

Moreover, it is easy to see that

$\int_{\mathcal{B}_{0}}|w_{m}(x)|^{p^{+}}\,\,dv_{g}(x)\longrightarrow 0.$

Thus, there exists $j>-\infty$ such as

$\int_{\mathcal{B}_{0}}\frac{f(x,u_{m})\,.\,u_{m}}{|u_{m}|^{p^{+}}}\,.\,|w_{m}|^{p^{+}}\,\,dv_{g}(x)\geq
l\,\int_{\mathcal{B}_{0}}|w_{m}|^{p^{+}}\,\,dv_{g}(x)\geq j>-\infty.$ (3.19)

Combining (3.17) - (3.19) we get a contradiction. Thus, the functional $\psi$
satisfies the assertion $(C_{2})$ of Cerami condition $(C).$ This completes
the proof of Lemma 3.2. ∎

Now, we demonstrate our first existence result.

######  Theorem 3.3.

Suppose that $(f_{1})-(f_{4})$ are satisfied, and we assume that the complete
n-manifold $(M,\,g)$ has property $B_{vol}(\lambda,\,v).$ If $q^{+}<p^{-},$
then the problem $(\mathcal{P})$ has at least one non-trivial solution.

###### Proof.

By Lemma 3.2, $\psi$ satisfies the Cerami condition $(C)$ on $X$. Firstly, we
show that the functional $\Psi$ has a geometrical structure, in order to apply
Proposition 2.18. For that, we claim that there exists $\mu,\,\nu>0$ such as

$\Psi(u)\geq\mu>0\,\,\mbox{for any}\,\,u\in
X\,\,\mbox{with}\,\,||u||_{X}=\nu.$

Let $||u||_{X}<1.$ Then by Propositions 2.8, 2.11 and the fact that
$q^{+}<p^{+}$we get

$\displaystyle\Psi(u)$
$\displaystyle\geq\frac{c}{p^{+}}||u||^{p^{+}}+\frac{1}{Apq^{+}}||u||^{q^{+}}-\int_{M}F(x,u)\,\,dv_{g}(x)$
$\displaystyle\geq c^{*}||u||^{q^{+}}_{X}-\int_{M}F(x,u(x))\,\,dv_{g},$ (3.20)

with $c^{*}=\min\,\\{\frac{c}{p^{+}},\,\frac{1}{Apq^{+}}\,\\}.$ According to
theorems 2.13 and 2.14, there exist two positive constants $c_{1},\,c_{2}>0$
such as

$|u|_{p^{+}}\leq
c_{1}||u||_{X}\,\,\mbox{and}\,\,|u|_{r(x)}\leq||u||_{X}\,\,\mbox{for
all}\,\,u\in X.$

Let $\epsilon>0$ be small enough, such as

$\epsilon\,c_{1}^{p^{+}}<\frac{c^{*}}{2}.$

According to $(f_{1})$ and $(f_{2})$, we have

$F(x,t)\leq\epsilon\,|t|^{p^{+}}+c_{\epsilon}\,|t|^{r(x)}\,\,\mbox{for
all}\,\,(x,t)\in M\times\mathbb{R},$

for $||u||\leq 1,$ we get

$\displaystyle\Psi(u)$ $\displaystyle\geq
c^{*}\,||u||^{q^{+}}_{X}-\epsilon\,\int_{M}|u|^{p^{+}}\,\,dv_{g}(x)-c_{\epsilon}\,\int_{M}|u(x)|^{r(x)}\,\,dv_{g}(x)$
$\displaystyle\geq
c^{*}\,||u||_{X}^{q^{+}}-\epsilon\,||u||_{p^{+}}^{p^{+}}-c_{\epsilon}\,||u||_{r(x)}^{r^{-}}$
$\displaystyle\geq
c^{*}\,||u||_{X}^{q^{+}}-\epsilon\,c_{1}^{p^{+}}\,||u||_{X}^{p^{+}}-c_{\epsilon}\,c_{2}^{r^{-}}\,||u||_{X}^{r^{-}}.$

Since, $q^{+}<p^{+}<r^{-},$ then there exist two positive real number $\mu$
and $\nu$ such as

$\Psi(u)\geq\mu>0\,\,\mbox{for all}\,\,u\in
X\,\,\mbox{with}\,\,||u||_{X}=\nu.$

On the other hand, we affirm that there exists $u_{0}\in
X\backslash\overline{\mathcal{B}_{0}(\nu)}$ such as

$\Psi(u)<0.$ (3.21)

Let $\phi_{0}\in X\backslash\\{0\\},$ by $(f_{4})$ we can choose a constant

$\delta>\frac{\displaystyle\frac{1}{p^{-}}\,\int_{M}|\nabla\phi_{0}|^{p(x)}\,\,dv_{g}(x)-\frac{1}{q^{-}}\int_{M}|\nabla\phi_{0}|^{q(x)}\,\,dv_{g}(x)}{\displaystyle\int_{M}|\phi_{0}|^{p^{+}}\,\,dv_{g}(x)},$

and a constant $c_{\delta}>0$ depending on $\delta$ such as

$F(x,t)\geq\delta\,|t|^{p^{+}}\,\,\mbox{for
all}\,\,|t|>c_{\delta}\,\,\mbox{and uniformly in}\,\,M.$

Let $k>1$ be large enough, we have

$\displaystyle\Psi(k\,\phi_{0})$
$\displaystyle=\int_{M}\frac{1}{p(x)}|\nabla(k\,\phi_{0})|^{p(x)}\,\,dv_{g}(x)+\int_{M}\frac{1}{q(x)}|\nabla(k\,\phi_{0})|^{q(x)}\,\,dv_{g}(x)$
$\displaystyle\hskip 8.5359pt-\int_{M}F(x,k\,\phi_{0})\,\,dv_{g}(x)$
$\displaystyle\geq\frac{k^{P^{+}}}{p^{-}}\int_{M}|\nabla\phi_{0}|^{p(x)}\,\,dv_{g}(x)+\frac{k^{q^{+}}}{q^{-}}\int_{M}|\nabla\phi_{0}|^{q(x)}\,\,dv_{g}(x)$
$\displaystyle\hskip
8.5359pt-\int_{\\{\,|k\phi_{0}|>c_{\delta}\,\\}}F(x,k\phi_{0})\,\,dv_{g}(x)-\int_{\\{\,|k\phi_{0}|<c_{\delta}\,\\}}F(x,k\phi_{0})\,\,dv_{g}(x)$
$\displaystyle\geq\frac{k^{P^{+}}}{p^{-}}\int_{M}|\nabla\phi_{0}|^{p(x)}\,\,dv_{g}(x)+\frac{k^{q^{+}}}{q^{-}}\int_{M}|\nabla\phi_{0}|^{q(x)}\,\,dv_{g}(x)$
$\displaystyle\hskip 8.5359pt-\int_{\\{\,|k\phi_{0}|\leq
c_{\delta}\,\\}}F(x,k\phi_{0})\,\,dv_{g}(x)-\delta\,k^{p^{+}}\int_{M}|\phi_{0}|^{p^{+}}\,\,dv_{g}(x)$
$\displaystyle\hskip 8.5359pt+\delta\,\int_{\\{|k\phi_{0}|\leq
c_{\delta}\\}}|k\phi_{0}|^{p^{+}}\,\,dv_{g}(x)$
$\displaystyle\geq\frac{k^{P^{+}}}{p^{-}}\int_{M}|\nabla\phi_{0}|^{p(x)}\,\,dv_{g}(x)+\frac{k^{q^{+}}}{q^{-}}\int_{M}|\nabla\phi_{0}|^{q(x)}\,\,dv_{g}(x)$
$\displaystyle\hskip
8.5359pt-\delta\,k^{p^{+}}\int_{M}|\phi_{0}|^{p^{+}}\,\,dv_{g}(x)+c_{5},$

which implies that

$\Psi(k\phi_{0})\longrightarrow-\infty\,\,\mbox{as}\,\,k\rightarrow+\infty.$

Then, there exists $k_{0}>0$ and $u_{0}=k_{0}\,\phi_{0}\in
X_{0}\backslash\overline{\mathcal{B}_{\nu}(0)}$ such as (3) hold true.
Thereby, proposition 2.18 guarantees that problem $(\mathcal{P})$ has at least
a non-trivial weak solution. This completes the proof. ∎

######  Theorem 3.4.

Assume that $(f_{1}),\,(f_{3}),\,(f_{4}),\,(g)$ hold and we assume that the
complete n-manifold $(M,\,g)$ has property $B_{vol}(\lambda,\,v).$ If
$q^{-}>p^{+},$ then the problem $(\mathcal{P})$ has a sequence of weak
solutions with unbounded energy.

###### Proof.

we will divide the proof of this theorem into two steps. In the first step, we
will demonstrate that the problem $(\mathcal{P})$ acquires a sequence of weak
solutions $\\{\pm u_{m}\\}_{m=1}^{\infty}$ such as

$\Psi(\pm u_{m})\longrightarrow+\infty\,\,\mbox{as}\,\,m\rightarrow+\infty.$

In the second step, we will prove that if $k$ is large enough, then there
exist $D_{k}>d_{k}>0$ such as the assertions $(A_{1})$ and $(A_{2})$ are
satisfied.
Step 1: The proof is based on the Fountain Theorem (given by Theorem 2.20).
Indeed, from $(f_{3}),\,\Psi$ is an even functional. And from Lemma 3.2,
$\Psi$ satisfies the condition $(C).$
For that, we will use the mean value theorem in the following form: For every
$\beta\in C_{+}(\overline{M})=\\{\beta\in
C(\overline{M}),\,\beta(x)>1\,\,\forall x\in M\\}$ and $u\in L^{\beta(x)}(M),$
there exist $\zeta\in M$ such that

$\int_{M}|u(x)|^{\beta(x)}\,\,dv_{g}(x)=|u|^{\beta(\zeta)}_{\beta(x)}.$ (3.22)

Indeed, on the one hand, by proposition 2.11, it is easy to see that

$\rho_{\beta(x)}\bigg{(}\frac{u}{||u||_{\beta(x)}}\bigg{)}=\int_{M}\bigg{(}\frac{|u|}{||u||_{\beta(x)}}\bigg{)}^{\beta(x)}\,\,dv_{g}(x)=1.$

On the other hand, by the mean value theorem for integrals, there exists a
positive constant $\bar{\beta}\in[\beta^{-},\,\beta^{+}]$ depending on $\beta$
such as

$\int_{M}\bigg{(}\frac{|u|}{||u||_{\beta(x)}}\bigg{)}^{\beta(x)}\,\,dv_{g}(x)=\bigg{(}\frac{1}{||u||_{\beta(x)}}\bigg{)}^{\bar{\beta}}\,\int_{M}|u|^{\beta(x)}\,\,dv_{g}(x).$

Moreover, the continuity of $\beta$ ensures that there exists $\zeta\in M$
such as $\beta(\zeta)=\bar{\beta}.$ Combining this fact with the above
inequalities, we get (3.22).
Step 2: $(A_{1}):$ For all $u\in Z_{k}$ such as $||u||_{X}=d_{k}$ ( $d_{k}$
will be specified below ), by $(f_{1}),$(3.22) and Proposition 2.8 we obtain

$\displaystyle\Psi(u)$ $\displaystyle=\int_{M}\frac{1}{p(x)}\,|\nabla
u(x)|^{p(x)}\,\,dv_{g}(x)+\int_{M}\frac{1}{q(x)}|\nabla
u(x)|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
8.5359pt-\int_{M}F(x,u(x))\,\,dv_{g}(x)$
$\displaystyle\geq\bigg{(}\frac{1}{p^{+}}+\frac{1}{Apq^{+}}\bigg{)}\,||u||_{X}^{p^{-}}-c_{5}\,||u||_{r(x)}^{r(\zeta)}-c_{6}\,||u||_{X}\,\,\,\mbox{where}\,\,\,\zeta\in
M$
$\displaystyle\geq\begin{cases}\big{(}\frac{1}{p^{+}}+\frac{1}{Apq^{+}}\big{)}\,||u||_{X}^{p^{-}}-c_{5}-c_{6}\,||u||_{X}&\text{if
\, $||u||_{r(x)}\leq 1$}\\\\[8.5359pt]
\big{(}\frac{1}{p^{+}}+\frac{1}{Apq^{+}}\big{)}\,||u||_{X}^{p^{-}}-c_{5}(\eta_{k}\,||u||_{X})^{r^{+}}-c_{6}\,||u||_{X}&\text{if
\, $||u||_{r(x)}>1$}\end{cases}$
$\displaystyle\geq\bigg{(}\frac{1}{p^{+}}+\frac{1}{Apq^{+}}\bigg{)}\,||u||_{X}^{p^{-}}-c_{5}(\eta_{k}\,||u||_{X})^{r^{+}}-c_{6}\,||u||_{X}-c_{5}$
$\displaystyle\geq
d_{k}^{p^{-}}\,\bigg{(}\frac{1}{p^{+}}+\frac{1}{Apq^{+}}-c_{5}\eta_{k}^{r^{+}}d_{k}^{r^{+}-p^{-}}\,\bigg{)}-c_{6}\,d_{k}-c_{5}.$

We fix $d_{k}$ as follows

$d_{k}=(r^{+}\,c_{5}\,\eta_{k}^{r^{+}})^{\frac{1}{p^{-}-r^{+}}}.$

Then,

$\Psi(u)\geq
d_{k}^{p^{-}}\,\bigg{(}\frac{1}{p^{+}}+\frac{1}{Apq^{+}}-\frac{1}{r^{+}}\,\bigg{)}-c_{6}\,d_{k}-c_{5}.$

According to Lemma 3.4 in [25]. We know that
$\displaystyle\lim_{k\rightarrow+\infty}\theta(x)=0.$ Then, since
$1<q^{+}<p^{-}\leq p^{+}<r^{+},$ we conclude that

$d_{k}\longrightarrow+\infty\,\,\,\mbox{as}\,\,\,k\rightarrow+\infty.$

Thus,

$\Psi(u)\longrightarrow+\infty\,\,\,\mbox{as}\,\,\,||u||\rightarrow+\infty\,\,\,\mbox{with}\,\,\,u\in
Z_{k}.$

Which means that the assertion $(A_{1})$ is verified.
$(A_{2}):$ According to Remark 2.19, we have $\mbox{dim}Y_{k}<+\infty,$ and
since all norms are equivalent in finite dimensional space, then there exists
$h_{k}>0,$ for all $u\in Y_{k}$ with $||u||_{X}$ is big enough, we obtain

$\displaystyle J(u)$ $\displaystyle\leq\frac{1}{p^{-}}\int_{M}|\nabla
u(x)|^{p(x)}\,\,dv_{g}(x)+\frac{1}{q^{-}}\int_{M}|\nabla
u(x)|^{q(x)}\,\,dv_{g}(x)$
$\displaystyle\leq\frac{1}{p^{-}}||u||_{p(x)}^{p^{+}}+\frac{1}{q^{-}}||u||_{q(x)}^{q^{+}}$
$\displaystyle\leq\frac{c_{7}}{p^{-}}||u||_{X}^{p^{+}}+\frac{c_{8}}{q^{-}}||u||_{X}^{q^{+}},$

since $q^{+}<p^{+},$ we get

$J(u)\leq\bigg{(}\frac{c_{7}}{p^{-}}+\frac{c_{8}}{q^{-}}\bigg{)}||u||_{X}^{p^{+}}=h_{k}\,||u||_{p^{+}}^{p^{+}}.$
(3.23)

Next, according to $(f_{2})$, there exists $B_{k}>0$ such as for any $|t|\geq
B_{k},$ we have

$F(x,t)\geq 2\,h_{k}\,|t|^{p^{+}}\,\,\,\mbox{for any}\,\,\,x\in M.$

Moreover, by $(f_{1}),$ there exist a positive $L_{k}$ such as

$F(x,t)\leq L_{k}\,\,\,\mbox{for all}\,\,\,(x,t)\in M\times[-B_{k},\,B_{k}].$

Then, for every $(x,t)\in M\times\mathbb{R}$ we conclude that

$F(x,t)\geq 2\,h_{k}\,|t|^{p^{+}}-L_{k}.$ (3.24)

Combining (3.23) and (3.24), for all $y\in Y_{k}$ such as
$||u||_{X}=D_{k}>d_{k}$ we infer that

$\displaystyle\Psi(u)$ $\displaystyle=J(u)-I(u)$ $\displaystyle\leq
h_{k}\,||u||_{p^{+}}^{p^{+}}-2\,h_{k}\,||u||_{p^{+}}^{p^{+}}+L_{k}$
$\displaystyle\leq-h_{k}\,c_{9}\,||u||_{X}^{p^{+}}+L_{k}\,|M|,$

where $|M|$ is the measure of $M$ and $|M|<\infty.$
Hence, for $D_{k}$ large enough $(D_{k}>d_{k}),$ we obtain that

$\max_{\\{u\in Y_{k}:\,||u||_{X}=D_{k}\\}}\Psi(u)=0.$

which implies that the assertion $(A_{2})$ holds. Then by applying the
Fountain Theorem, we achieve the proof of Theorem 3.4. ∎

We will end this section with a suitable example, that checks all the above
conditions and Theorems,

######  Example 3.5.

For $f(x,t)=\displaystyle\frac{t\,|t|^{\alpha(x)-2}}{\mbox{Log}\,(1+|t|)},$ if
$t\neq 0$ and $f(x,0)=0$ with $2<p^{+}\leq\alpha(x)<r(x)$ for all
$x\in\overline{M}.$ So it is easy to see that the function$f$ does not satisfy
the Ambrosetti-Rabinowitz condition, but it satisfies our assumptions
$(f_{1})-(f_{5})$. Then, our problem $(\mathcal{P})$ becomes

$(\mathcal{P})\,\begin{cases}-\,\Delta_{p(x)}u(x)-\Delta_{q(x)}u(x)=\,\frac{t\,|t|^{\alpha(x)-2}}{\mbox{Log}\,(1+|t|)}&\text{in
\,\,M },\\\\[8.5359pt] \,u\,=\,0&\text{on \,$\partial$M },\end{cases}$

Consequently, the results corresponding to Theorems 3.3 and 3.4 can be
achieved and still be true for the problem $(\mathcal{P}).$

## Acknowledgments

The authors would like to thank the anonymous referees for the valuable
suggestions and comments which improved the quality of the presentation.

## References

  * [1] A. Aberqi and J. Bennouna and M. Elmassoudi and M. Hammoumi,  Existence and uniqueness of a renormalized solution of parabolic problems in Orlicz spaces,  _Monatshefte für Mathematik_ , 189 (2019), 195–219.
  * [2] A. Aberqi and J. Bennouna and M. Mekkour and H. Redwane,  Nonlinear parabolic inequalities with lower order terms,  _Applicable Analysis_ , 96 (2017), 2102–2117.
  * [3] (MR1301779) [10.1007/978-1-4612-0873-0]  T. Aubin,  _Nonlinear analysis on manifolds. Monge-Ampere equations_ ,  Springer Science & Business Media, 1982.
  * [4] (MR1301779) [10.1007/978-1-4612-0873-0]  O. Benslimane and A. Aberqi and J. Bennouna,  _Existence and Uniqueness of Weak solution of $p(x)$-laplacian in Sobolev spaces with variable exponents in complete manifolds_,  _arXiv preprint arXiv:2006.04763_ , (2020).
  * [5] O. Benslimane and A. Aberqi and J. Bennouna,  The existence and uniqueness of an entropy solution to unilateral Orlicz anisotropic equations in an unbounded domain,  _Axioms_ , 9 (2020), 109.
  * [6] O. Benslimane and A. Aberqi and J. Bennouna,  Existence and uniqueness of entropy solution of a nonlinear elliptic equation in anisotropic Sobolev–Orlicz space,  _Rendiconti del Circolo Matematico di Palermo Series 2_ , (2020), 1–30.
  * [7] O. Benslimane and A. Aberqi and J. Bennouna,  On some nonlinear anisotropic elliptic equations in anisotropic Orlicz space,  _Arab Journal of Mathematical Sciences_ , _DOI: 10.1108/AJMS-12-2020-0133_ , (2021).
  * [8] L. Boccardo and T. Gallouët and J.L. Vazquez,  Nonlinear elliptic equations in $\,\mathbb{R}^{N}\,$ without growth restrictions on the data,  _Journal of Differential Equations_ , 105 (1993), 334–363.
  * [9] G. Cerami,  An existence criterion for the critical points on unbounded manifolds,  _Istit. Lombardo Accad. Sci. Lett. Rend. A_ , 112 (1978), 332–336.
  * [10] Y. Chen and S. Levine and M. Rao,  Variable exponent, linear growth functionals in image restoration,  _SIAM journal on Applied Mathematics_ , 66 (2006), 1383–1406.
  * [11] M. Gaczkowski and P. Górka and D. J. Pon,  Sobolev spaces with variable exponents on complete manifolds,  _Journal of Functional Analysis_ , 270 (2016), 1379–1415.
  * [12] A.K. Gushchin,  The Dirichlet problem for a second-order elliptic equation with an $L^{p}$ boundary function,  _Sbornik: Mathematics_ , 203 (2012), 1.
  * [13] L. Guo,  The Dirichlrt Problems For Nonlinear Elliptic Equations With Variable Exponents On Riemannian Manifolds,  _Journal of Applied Analysis & Computation_, 5 (2015), 562–569.
  * [14] P. Gwiazda and A. Świerczewska-Gwiazda,  On non-Newtonian fluids with a property of rapid thickening under different stimulus,  _Mathematical Models and Methods in Applied Sciences_ , 18 (2008), 1073–1092.
  * [15] (MR1301779) [10.1007/978-1-4612-0873-0]  E. Hebey,  _Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities: Sobolev Spaces and Inequalities_ ,  American Mathematical Soc., 2000.
  * [16] E. J. Hurtado and O. H. Miyagaki and R. S. Rodrigues,  Existence and multiplicity of solutions for a class of elliptic equations without Ambrosetti-Rabinowitz type conditions,  _Journal of Dynamics and Differential Equations_ , 30 (2018), 405–432.
  * [17] L. Jeanjean,  On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on $\mathbb{R}^{N}$,  _Proceedings of the Royal Society of Edinburgh Section A: Mathematics_ , 129 (1999), 787–809.
  * [18] GI. Laptev,  Existence of solutions of certain quasilinear elliptic equations in $\,\mathbb{R}^{N}\,$ without conditions at infinity,  _Journal of Mathematical Sciences_ , 150 (2008), 2384–2394.
  * [19] S. B. Liu and S. J. Li,  Infinitely many solutions for a superlinear elliptic equation,  _Acta Mathematica Sinica_ , 46 (2003), 625–630.
  * [20] M. A. Ragusa and A. Tachikawa,  Regularity for minimizers for functionals of double phase with variable exponents,  _Advances in Nonlinear Analysis_ , 9 (2019), 710–728.
  * [21] M. Růžička,  Modeling, mathematical and numerical analysis of electrorheological fluids,  _Applications of Mathematics_ , 49 (2004), 565–609.
  * [22] N. S. Trudinger,  Remarks concerning the conformal deformation of Riemannian structures on compact manifolds,  _Annali della Scuola Normale Superiore di Pisa-Classe di Scienze_ , 22 (1968), 265–274.
  * [23] A. Zang,  $p(x)$-Laplacian equations satisfying Cerami condition,  _Journal of mathematical analysis and applications_ , 337 (2008), 547–555.
  * [24] V. V. Zhikov,  On density of smooth functions in Sobolev–Orlich spaces,  _Zapiski Nauchnykh Seminarov POMI_ , 310 (2004), 67-81.
  * [25] Q. Zhang, and Ch. Zhao,  Existence of strong solutions of a $p(x)$-Laplacian Dirichlet problem without the Ambrosetti–Rabinowitz condition,  _Computers & Mathematics with Applications_, 69 (2015), 1-12.
  * [26] W. Zou,  Variant fountain theorems and their applications,  _Manuscripta Mathematica_ , 104 (2001), 343–358.


# Universality and thermoelectric transport properties of quantum dot systems

D. F. Aranguren-Quintero1, E. Ramos 2, J. Silva-Valencia 1, M. S. Figueira 3,
L. N. Oliveira4 and R. Franco1 1 Departamento de Física, Universidad Nacional
de Colombia (UNAL), A. A. 5997, Bogotá, Colombia.
2 División de Ciencias Básicas, Fundación Universidad de America, Bogotá,
Colombia.
3 Instituto de Física-Universidade Federal Fluminense (IF-UFF), Av. Litorânea
s/n, CEP:24210-346, Niterói, Rio de Janeiro, Brazil.
4 Instituto de Física de São Carlos-Universidade de São Paulo (IFSC-USP), 369
São Carlos, São Paulo, Brazil.

([)

###### Abstract

We discuss the temperature-dependent thermoelectric transport properties of
semiconductor nanostructures comprising a quantum dot coupled to quantum
wires: the thermal dependence of the electrical conductance, thermal
conductance, and thermopower. We explore the universality of the
thermoelectric properties in the temperature range associated with the Kondo
crossover. In this thermal range, general arguments indicate that any
equilibrium property’s temperature dependence should be a universal function
of the ratio $T^{*}=T/T_{K}$, where $T_{K}$ is the Kondo temperature.

Considering the particle-hole symmetric, spin-degenerate Anderson model, the
zero-bias electrical conductance has already been shown to map linearly onto a
universal conductance through a quantum dot embedded or side-coupled to a
quantum wire. Employing rigorous renormalization-group arguments, we calculate
universal thermoelectric transport coefficients that allow us to extend this
result to the thermopower and the thermal conductance. We present numerical
renormalization-group results to illustrate the physics in our findings.

Applying the universal thermoelectric coefficients to recent experimental
results of the electrical conductance and thermo-voltages versus $V_{gate}$,
at different temperatures in the Kondo regime, we calculate all the
thermoelectric properties and obtain simple analytical fitting functions that
can be used to predict the experimental results of these properties. However,
we cannot check all of them, due to the lack of available experimental results
over a broad temperature range.

###### pacs:

72.15.Jf, 73.21.La., 73.63.Kv, 73.50.Lw, 73.23

Date: ]

##  I Introduction

The discovery of the Seebeck and Peltier effects in different metal junctions
at the beginning of the $19^{th}$ century gives rise to a branch of physics
cakked “Thermoelectric" (TE) Sánchez and López (2016). Seebeck observed that
when two different metals joined together (thermocouple) with the junctions
maintained at different temperatures, a voltage difference is generated
proportional to the temperature variation between the couple’s ends. Some
times later, Peltier observed that when an electric current flows through the
Seebeck device, heat is either absorbed or rejected depending on the direction
of the current along the circuit. Today, the Peltier effect is the basis for
many TE refrigeration devices, and the Seebeck effect is the basis for TE
power generation devices Tritt (2002).

Ioffe’s prediction in the fifties that doped semiconductors could exhibit
relatively large thermoelectric effects Joffe and Stil (1959) had a strong
impact on the area of thermoelectric materials. It was followed step by step
with the discovery that a thermo-junction between $p-$type $Bi_{2}Te_{3}$ and
bismuth exhibits the maximum temperature difference of $26^{\circ}$C and
$40^{\circ}$C between ($p-$ and $n-$) types $Bi_{2}Te_{3}$ Wright (1958). This
compound has dominated the whole field of thermoelectric materials; more
specifically, the alloys of $Bi_{2}Te_{3}$ with $Sb_{2}Te_{3}$ for $p-$type
and $Bi_{2}Se_{3}$ for $n-$type compounds have the highest $ZT$ (see Eq. 7),
at around room temperatures, when compared to any other known material Witting
_et al._ (2019), and up until now, it is the working material for most Peltier
cooling devices and Seebeck thermoelectric generators.

Most state-of-the-art thermoelectric materials have their dimensionless
thermoelectric figure of merit, $ZT$, in the interval $ZT\simeq[1-2.5]$ (see
Fig. 2 of the reference He and Tritt (2017)), which is well below the Carnot
efficiency Benenti  _et al._ (2017). However, the advent of nanotechnology
opens up new possibilities for increasing $ZT$, mainly due to the level
quantization and the Coulomb interaction, leading to essential changes in the
system’s thermoelectric properties. Some promising compounds are the
topological insulators (TIs) and Weyl and Dirac’s semi-metals, characterized
by nontrivial topological orders. The new characteristic of the TIs is that
besides the conventional semiconductor bulk band structure, they also exhibit
topological surface conduction states. Some of the best thermoelectric
materials are also three-dimensional topological insulators, such as
$Bi_{2}Te_{3}$, $Bi_{2}Se_{3}$, and $Sb_{2}Te_{3}$ Xu  _et al._ (2017); Gooth
_et al._ (2018).

Thermoelectric devices must have at least a $ZT>3$ in order to attain
industrial and household spread; this efficiency has been improved over the
years, but it was not attained until now He and Tritt (2017). This is the
reason why thermoelectric generators or thermoelectric refrigerators are not
part of our daily technology. They are used in particular fields like the
satellite and aerospace industry, where the advantages of not having movable
parts and not requiring maintenance overshadow the low efficiency He and Tritt
(2017). One example of this is the radioisotope thermoelectric generator (RTG)
Zoui  _et al._ (2020), a nuclear electric generator that exploits a
radioactive atom’s natural decay, usually plutonium dioxide
${{}^{238}}PuO_{2}$, converting, via the Seebeck effect, the heat released by
the disintegrated atoms into electricity. Furthermore, in the context of
today’s climate change, research on new thermoelectric materials that improve
thermal efficiency is essential as part of our efforts to obtain
environmentally clean sources of energy.

In this investigation, we focus on studying the semiconductor single-electron
transistor (SET), which is the experimental realization of the single impurity
Anderson model (SIAM) for finite electronic correlation $U$. The SIAM was
experimentally realized by the Goldhaber-Gordon group Goldhaber-Gordon  _et
al._ (1998a), with complete control over all of the model’s parameters. They
measured the electric conductance of a SET and showed its universal character.
Recently, interest in studying the thermoelectric properties of the SET has
greatly increased and has given rise to several papers that have discuss it,
both from the theoretical side Yoshida and Oliveira (2009); Costi and Zlatic
(2010); Hershfield  _et al._ (2013); Donsa  _et al._ (2014); Talbo  _et al._
(2017); Costi (2019a, b); Kleeorin  _et al._ (2019); Eckern and Wysokiński
(2020) and from the experimental one Heremans  _et al._ (2004); Scheibner  _et
al._ (2007); Hoffmann  _et al._ (2009); Hartman  _et al._ (2018); Svilans  _et
al._ (2018); Dutta  _et al._ (2017, 2019). A useful review can be found in the
references He and Tritt (2017); Sánchez and López (2016); Benenti  _et al._
(2017).

Universal relations for the thermal dependence of the thermodynamic
properties, in the Kondo regime, for the SIAM are well known, and a didactic
discussion can be found in Hewson’s book Hewson (1993). Costi $et$ $al$. Costi
_et al._ (1994); Costi and Zlatic (2010), showed that in the Kondo limit of
the SIAM, the thermoelectric transport coefficients (TTC) are only functions
of $T^{*}=\left(\frac{T}{T_{K}}\right)$, with the temperature $T$ normalized
by the Kondo temperature ($T_{K}$). For simplicity, we only use
$\left(\frac{T}{T_{K}}\right)$ in the figures, in the text, we use $T^{*}$.
They also showed that by employing the NRG, the electric conductance, the
temperature normalized thermal conductance, and the thermopower exhibit a
universal behavior in the Kondo regime.

The universal behavior of $G\left(T^{*}\right)$ in semiconductor
nanostructures was studied in earlier papers Yoshida  _et al._ (2009);
Seridonio  _et al._ (2009a). The authors derived an analytical expression that
maps the thermal dependence of $G\left(T^{*}\right)$, of a Kondo-asymmetric
condition of the SIAM to a universal conductance function
$G_{S}\left(T^{*}\right)$. The corresponding TTC $L_{0}(T^{*})$ (Eq. 27), was
calculated by employing the NRG. This analytical mapping is parametrized by
$T_{K}$ and the ground state phase shift $\delta=\frac{\pi n_{d}}{2}$, which
is related to the Friedel’s sum rule Langreth (1966); Hewson (1993). For
brevity, we call this quantity parameter $\delta$. In this investigation, we
derive similar analytical expressions associated with the other TTCs, allowing
us to obtain both the thermopower and the thermal conductance in a Kondo
asymmetric situation, employing only universal functions derived from the
thermal dependence of the symmetric SIAM. Again those analytical functions are
parametrized by the Kondo temperature and the parameter $\delta$.

In Sec. II, we define the problem and the model employed in the study of the
SET. In Sec. III, we define the thermoelectric properties and their relation
with the TTCs. In Sec. IV, we develop the calculations of a mapping between
the thermal dependence of the TTCs coefficients in the symmetric limit and the
asymmetric condition in the quantum dot’s Kondo regime. In Sec. V, we present
the results of our research associated with the TTCs and discuss their
physical consequences. In Sec. VI, we make a prospect of comparison between
our results and the available experimental one. In Sec. VII, we present a
summary and the conclusions of this investigation. In Appendix A, we develop a
methodology to apply universal TTCs to experimental thermoelectric properties;
we obtained simple analytical fitting functions that can be used to predict
these properties’ observed behavior.

##  II The Hamiltonian for the SET

In this investigation, we extend the previous results obtained for the
electrical conductance $G\left(T^{*}\right)$, associated with the TTC
$L_{0}\left(T^{*}\right)$ Costi and Zlatic (2010); Yoshida and Oliveira
(2009); Yoshida  _et al._ (2009), to all the other TTCs:
$L_{1}\left(T^{*}\right)$ and $L_{2}\left(T^{*}\right)$. We map the thermal
dependence of the TTCs for a Kondo-asymmetric situation as a function of the
TTCs for a Kondo-symmetric condition. These mappings are obtained in terms of
the renormalized Kondo temperature $T^{*}$ and the parameter $\delta$.

Figure 1: Schematic picture of a semiconductor electron transistor (SET): a
quantum dot embedded into conduction leads.

The standard Hamiltonian for the SET studied in this investigation can be
written as

$H=\sum_{\mathbf{k},\sigma}\sum_{\alpha=L,R}E^{\alpha}_{\mathbf{k},\sigma}c^{\alpha\dagger}_{\mathbf{k},\sigma}c^{\alpha}_{\mathbf{k},\sigma}+\left(E_{d}n_{d}+Un_{d\uparrow}n_{d\downarrow}\right)\\\
$
$+\sum_{\alpha=L,R}\sum_{\mathbf{k},\sigma}\frac{V_{\alpha}}{\sqrt{2N}}\left(c_{d\sigma}^{\dagger}c^{\alpha}_{\mathbf{k},\sigma}+c^{\alpha\dagger}_{\mathbf{k},\sigma}c_{d\sigma}\right),$
(1)

where the first term represents the left ($\alpha=L$) and right ($\alpha=R$)
leads, characterized by hot and cold free conduction electron (c-electrons)
reservoirs, respectively. The quantum dot is embedded in the leads as visually
represented in Fig. 1. The second term describes the QD characterized by the
local dot energy $E_{d}$, and $U$ represents the onsite Coulomb repulsion
between the electrons of the QD Grobis  _et al._ (2008); Kretinin  _et al._
(2011); Parks  _et al._ (2010). The third term corresponds to the tunneling
between the immersed dot and the left (L) and right (R) semi-infinite leads.
The amplitude $V_{\alpha}$ is responsible for the tunneling between the QD and
the lead $\alpha$. For simplicity, we assume symmetric junctions (i.e.
$V_{\alpha}=V_{L}=V_{R}=V$) and identical leads (i.e.
$E^{L}_{\mathbf{k},\sigma}=E^{R}_{\mathbf{k},\sigma}=E_{\mathbf{k},\sigma}$)
connecting the QD to the quantum wire. The $L(R)$ semi-infinite leads comprise
$N$ states $c^{L}_{\mathbf{k},\sigma}$ ($c^{R}_{\mathbf{k},\sigma}$) with
energies defined by the linear dispersion relation
$E_{\mathbf{k},\sigma}=\left(k-k_{F}\right)v_{F}$ ($0\leq k\leq 2k_{F}$) so
that the bandwidth is $2D=2v_{F}k_{F}$, with $D$ being the half-width of the
conduction band. In all the numerical calculations, we consider the unit of
energy to be $D=1$.

##  III Thermoelectric properties

We calculated the electrical conductance $G(T)$, the thermal conductance
$\kappa_{e}(T)$ and the thermopower $S(T)$ (Seebeck effect) in terms of the
transport coefficients, following the standard textbook derivation Mahan
(1990); Ziman (1999), and the results are

$G(T)=e^{2}L_{o}(T),$ (2)
$\kappa_{e}(T)=\frac{1}{T}\left(L_{2}(T)-\frac{L_{1}^{2}(T)}{L_{o}(T)}\right),$
(3)

and

$S=\left(\frac{-1}{eT}\right)\frac{L_{1}(T)}{L_{o}(T)}.$ (4)

To calculate the transport coefficients $L_{o}(T)$, $L_{1}(T)$, and
$L_{2}(T)$, we employed the results derived by Dong and X. L. Lei Dong and Lei
(2002). They considered the particle current and thermal flux formulas,
through an interacting QD connected to the leads at different temperatures,
within the Keldysh non-equilibrium Green’s functions (GF) formalism. The
electric and thermoelectric transport coefficients were obtained in the
presence of the chemical potential and temperature gradients, with the Onsager
relation in the linear regime being automatically satisfied. The transport
coefficients are given by

$L_{n}(T)=\frac{2}{h}\int{\left(-\frac{\partial
n_{F}(\omega,T)}{\partial\omega}\right)\omega^{n}\tau(\omega,T)d\omega},$ (5)

where $n_{F}(\epsilon,T)=1/(1+e^{(\epsilon-\mu)/k_{B}T})$ is the Fermi-Dirac
distribution, with $\mu$ being the chemical potential and the transmittance
$\tau(\omega,T)$ is given by

$\tau(\omega,T)=\pi\Gamma\rho_{d}(\omega,T),$ (6)

where $\rho_{d}(\omega,T)$ is the spectral density of the QD and
$\Gamma=\pi\rho_{c}(\mu=0)V^{2}$ is the Anderson parameter, which is a measure
of the $d-$level width, and $\rho_{c}(\omega)=\frac{1}{2D}$ is the flat
conduction density of states of the leads.

A useful quantity that indicates the system performance is the thermoelectric
dimensionless figure of merit $ZT$ Joffe and Stil (1959), which is given by

$ZT=\frac{S^{2}TG}{\kappa_{e}}.$ (7)

##  IV Universal mapping: Transmission coefficient and thermoelectric
coefficients

Following the reference Yoshida  _et al._ (2009), we introduce the normalized
even ($a_{\mathbf{k},\sigma}$) and odd ($b_{\mathbf{k},\sigma}$) operators, in
order to exploit the inversion symmetry of the system

$a_{\mathbf{k},\sigma}=\frac{1}{\sqrt{2}}\left(c^{L}_{\mathbf{k},\sigma}+c^{R}_{\mathbf{k},\sigma}\right),$
(8)
$b_{\mathbf{k},\sigma}=\frac{1}{\sqrt{2}}\left(c^{L}_{\mathbf{k},\sigma}-c^{R}_{\mathbf{k},\sigma}\right).$
(9)

It is convenient to write the Hamiltonian of Eq. 1 on the basis of the new
operators ($a_{\mathbf{k},\sigma}$ and $b_{\mathbf{k},\sigma}$), to “split” it
into two decoupled pieces $H=H_{A}+H_{B}$, with

$\displaystyle H_{A}$ $\displaystyle=$
$\displaystyle\sum_{\mathbf{k},\sigma}E_{\mathbf{k},\sigma}a_{\mathbf{k},\sigma}^{\dagger}a_{\mathbf{k},\sigma}+\sum_{\sigma}V\left(f_{0}^{\dagger}c_{d,\sigma}+h.c\right)$
$\displaystyle+\left(E_{d}n_{d}+Un_{d\uparrow}n_{d\downarrow}\right),$

where
$f_{0}=\sum_{\mathbf{k},\sigma}\left(\frac{a_{\mathbf{k},\sigma}}{\sqrt{N}}\right)$,
is the traditional NRG shorthand notation, and $H_{B}$ is given by

$H_{B}=\sum_{\mathbf{k},\sigma}E_{\mathbf{k},\sigma}b_{\mathbf{k},\sigma}^{\dagger}b_{\mathbf{k},\sigma}.$
(11)

The Hamiltonian $H_{B}$ is quadratic and can be exactly diagonalized and
decoupled from the quantum dot. On the contrary, the Hamiltonian $H_{A}$
“carries” all the correlation effects of the quantum dot and the coupling
between it and the conduction band. Due to this, the Hamiltonian $H_{A}$ is
the only one relevant for obtaining the transmittance and the spectral density
of states for the quantum dot, and can be written as Yoshida  _et al._ (2009)

$\displaystyle\rho_{d}(\omega,T)$ $\displaystyle=$
$\displaystyle\frac{1}{f(\omega,T)}\sum_{mn,\sigma}\frac{e^{-\beta
E_{m}}}{Z(T)}$
$\displaystyle|<n|c_{d,\sigma}^{\dagger}|m>|^{2}\delta(E_{mn}-\hbar\omega).$

Here $|m>$ and $|n>$ are the eigenstates of $H_{A}$, with eigenvalues $E_{m}$
and $E_{n}$, respectively, $E_{mn}=E_{m}-E_{n}$, and $Z(T)$ is the partition
function for the Hamiltonian $H_{A}$. The Hamiltonian $H_{B}$ is not dependent
on $c_{d,\sigma}$, $\left[H_{B},c_{d,\sigma}\right]=0$, and only the
eigenvalues and eigenvectors of $H_{A}$ are required to obtain
$\rho_{d}(\omega,T)$. On the other hand, to calculate the matrix elements
$<n|c_{d,\sigma}^{\dagger}|m>$ in Eq. $16$, we evaluate the commutator
$\left[H_{A},a_{\mathbf{q},\sigma}\right]$

$\left[H_{A},a_{\mathbf{q},\sigma}\right]=E_{q,\sigma}a_{\mathbf{q},\sigma}^{\dagger}+\frac{V}{\sqrt{N}}c_{d,\sigma},$
(13)

and performing the summation over $q$ and $\sigma$ we obtain

$\left[H_{A},f_{0}^{\dagger}\right]=\frac{1}{\sqrt{3}}f_{1}^{\dagger}+Vc_{d}^{\dagger},$
(14)

where

$f_{1}=\sqrt{\frac{3}{N}}\sum_{q}\left(\frac{E_{q}}{D}\right)a_{q},$ (15)

define a new NRG shorthand notation operator.

Equation 14 permits to relate the matrix elements
$<n|c_{d,\sigma}^{\dagger}|m>$ in Eq. $16$ with the same matrix elements of
the operators $f_{0}$ and $f_{1}$,

$V<m|c_{d}^{\dagger}|n>=E_{mn}<m|f_{0}^{\dagger}|n>-\sqrt{3}D<m|f_{1}^{\dagger}|n>,$
(16)

and a Schrieffer-Wolff transformation of the Hamiltonian $H_{A}$, allows us to
write it in the Kondo form Yoshida  _et al._ (2009)

$H_{J}=\sum_{k}E_{l}g_{l}^{\dagger}g_{l}+J_{W}\sum_{\mu,\nu}\Phi_{0\mu}^{\dagger}\mathbf{\sigma}_{\mu\nu}\Phi_{0\nu}\cdot\mathbf{S},$
(17)

where the $g_{l}$ operators are the eigenoperators of the fixed-point
Hamiltonian, associated with the unstable local moment (LM) condition of the
Anderson impurity model (see ref. Yoshida  _et al._ (2009) for details). Here,
$J_{W}=4D\frac{\Gamma U}{\pi|E_{d}|\left(E_{d}+U\right)}cos^{2}\delta_{LM}$,
where $\delta_{LM}$ is the quantum scattering phase shift, associated with the
LM fixed point, and

$\Phi_{0}=\frac{1}{\sqrt{N}}\sum_{l}g_{l},$ (18)

where in the symmetric condition $\delta_{LM}=0$ and $\Phi_{0}=f_{0}$.

The second term in the Eq. 17 is responsible, in the Kondo regime, for the
evolution from the LM fixed point to a Fermi liquid (FL) fixed point,
associated with an antiferromagnetic $J_{W}$ coupling, characteristic of the
Kondo effect. We can define the operator

$\Phi_{1}=\sqrt{\frac{3}{N}}\sum_{l}\left(\frac{E_{l}}{D}\right)g_{l},$ (19)

in an way analogous to the $f_{1}$ operator’s definition (Eq. 15). In the
symmetric condition $\Phi_{1}=f_{1}$, something similar happens with
$\Phi_{0}=f_{0}$.

Eqs. 13 and 16 show the universal character of the product
$V<m|c_{d}^{\dagger}|n>$ in the symmetric point (remember $\Phi_{0}=f_{0}$ and
$\Phi_{1}=f_{1}$ in this condition). In order to explore what happens in the
asymmetric condition, it is necessary to relate the operators $f_{0}$ and
$f_{1}$ to $\Phi_{0}$ and $\Phi_{1}$, (see Appendix A2 of reference Costi  _et
al._ (1994)). Substituting Eq. $(A21)$ of the reference Costi  _et al._ (1994)
in Eq. 16 we obtain

$\sqrt{\pi\rho\Gamma}<m|c_{d}^{\dagger}|n>=\alpha_{0}<m|\Phi_{0}^{\dagger}|n>+\alpha_{1}<m|\Phi_{1}^{\dagger}|n>.$
(20)

Performing the substitution of Eq. 20 in Eq. 13, we obtain the localized QD
spectral density $\rho_{d}(\omega,T)$, which can be written as

$\pi\rho\Gamma\rho_{d}(\omega,T)=\alpha_{0}^{2}\rho_{0}(\omega,T)+\alpha_{1}^{2}\rho_{1}(\omega,T)+\alpha_{0}\alpha_{1}\rho_{(01)}(\omega,T),$
(21)

where $\rho_{0}(\omega,T)$, $\rho_{1}(\omega,T)$ and $\rho_{(01)}(\omega,T)$
are universal expressions of the Kondo regime and are given by

$\displaystyle\rho_{j}(\omega,T)$ $\displaystyle=$
$\displaystyle\sum_{mn}\frac{e^{-\beta
E_{m}}}{Z(T)f(\omega,T)}|<n|\Phi_{j}|m>|^{2}\times$
$\displaystyle\delta(E_{mn}-\hbar\omega),\hskip 28.45274pt(j=0,1),$

and

$\displaystyle\rho_{(01)}(\omega,T)$ $\displaystyle=$
$\displaystyle\sum_{mn}\frac{e^{-\beta
E_{m}}}{Z(T)f(\omega,T)}(<n|\Phi_{0}|m>\times$
$\displaystyle<n|\Phi_{1}|m>+c.c.)\delta(E_{mn}-\hbar\omega).$

Substituting Eq. 21 in Eq. 6, the transmittance at energy
$\epsilon=\frac{h\omega}{2\pi}$ and temperature $T$ is given by

$\tau(\omega,T)=\frac{\alpha_{0}^{2}}{\rho}\rho_{0}(\omega,T)++\frac{\alpha_{1}^{2}}{\rho}\rho_{1}(\omega,T)+\frac{\alpha_{0}\alpha_{1}}{\rho}\rho_{(01)}(\omega,T).$
(24)

The universal expressions $\rho_{0}(\omega,T)$, $\rho_{1}(\omega,T)$ and
$\rho_{(01)}(\omega,T)$ “carry” the thermal dependence of $\tau(\omega,T)$.
The important point that should be stressed here is that the transmittance is
the key physical quantity that enters the calculations of all the
thermoelectric coefficients given by Eq. 5. All the dependence of the
parameters of the model is taken into account through the coefficients
$\alpha_{0}$ and $\alpha_{1}$, given by the reference Yoshida  _et al._ (2009)

$\alpha_{0}^{2}=cos^{2}(\delta),$ (25)
$\alpha_{1}^{2}=\frac{6}{\pi^{2}}sin^{2}(\delta).$ (26)

Taking into account the result of Eqs. 5 and 24, it is possible to compute the
thermal dependence of the linear TTCs as a function of ($T^{*}$)

$L_{0}\left(T^{*}\right)-\frac{1}{h}=-\left(L_{0}^{S}\left(T^{*}\right)-\frac{1}{h}\right)cos(2\delta),$
(27)

where $L_{0}^{S}\left(T^{*}\right)$ is the universal coefficient $L_{0}$ in
the electron-hole symmetric condition of the model, when $E_{d}=\frac{-U}{2}$
and $\delta=\frac{\pi n_{d}}{2}$, with $n_{d}=1$. All the thermal dependence
of $L_{0}\left(T^{*}\right)$ is contained in the universal function
$L_{0}^{S}\left(T^{*}\right)$. The function $cos(2\delta)$ carries all the
parameter dependence apart from temperature $T$, and is characteristic of the
asymmetric conditions for the model ($\delta\neq\frac{\pi}{2}$).

Taking into account that ${\cal{G}}_{2}(T)=e^{2}L_{0}(T)$, we can write the
Eq. 27 in the same form as a result previously obtained in reference Yoshida
_et al._ (2009)

$G_{2}\left(T^{*}\right)-{\cal{G}}_{2}=-\left({\cal{G}}_{2}^{S}\left(T^{*}\right)-{\cal{G}}_{2}\right)cos(2\delta),$
(28)

where we also should observe that the $\rho_{(01)}(\omega,T)$ term in Eq. 24,
due to particle-hole symmetry arguments, makes no contribution to the
thermoelectric coefficients $L_{0}$ Seridonio  _et al._ (2009b) and $L_{2}$,
but contributes to $L_{1}$, as indicated in Eqs. 29 and 30.

The evaluation of the $L_{1}$ coefficient, employing the result for the
transmission coefficient (Eq. 24) in Eq. 5 ($n=1$), and taking into account
the parity condition of the integrand, give us

$L_{1}\left(T^{*}\right)=L_{(01)}\left(T^{*}\right)cos(\delta)sin(\delta),$
(29)

where

$L_{(01)}(T)=\frac{\sqrt{2}\pi}{h\rho}\int_{-D}^{D}{\epsilon\rho_{(01)}(\epsilon,T)\left(-\frac{\partial
f(\epsilon,T)}{\partial\epsilon}\right)d\epsilon},$ (30)

which is an universal function of $\left(T^{*}\right)$ in the symmetric Kondo
condition, and contains all the thermal dependence of
$L_{1}\left(T^{*}\right)$.

Finally, employing the result for the transmittance (Eq.24) in Eq. 5 ($n=2$),
we obtain the $L_{2}\left(T^{*}\right)$ coefficient. Again, we take into
account the parity of the integrand

$\frac{L_{2}\left(T^{*}\right)}{\left(\frac{k_{B}T}{T_{K}}\right)^{2}}-\frac{\pi^{2}}{6}=-cos(2\delta)\left(\frac{L_{2}^{S}\left(T^{*}\right)}{\left(\frac{k_{B}T}{T_{K}}\right)^{2}}-\frac{\pi^{2}}{6}\right).$
(31)

The quantity
$\frac{L_{2}^{S}\left(T^{*}\right)}{\left(\frac{k_{B}T}{T_{k}}\right)^{2}}$ is
a universal function of $\left(T^{*}\right)$, obtained in terms of
$L_{2}^{S}(T)$, the coefficient for the symmetric condition of the model. As
in the previous cases, all the thermal dependence of Eq. 31 in any asymmetric
condition of the model is contained in
$\frac{L_{2}^{S}\left(T^{*}\right)}{\left(\frac{k_{B}T}{T_{K}}\right)^{2}}$,
and all the dependence on the parameters of the model is taken into account
through the scattering phase shift factor $\delta$.

##  V Results and Discussion: Universal Mapping

In previous papers, Seridonio  _et al._ (2009b); Yoshida  _et al._ (2009);
Seridonio  _et al._ (2009a); Oliveira  _et al._ (2010) one of us (L. N.
Oliveira), argued that it is possible to employ experimental data of
electrical conductance $G\left(T^{*}\right)$ to obtain the parameter $\delta$,
and with it "check" the validity of the Eq. 27 for the SET. Computations for
the case of a side-coupled quantum dot were also considered in references
Seridonio  _et al._ (2009b, a); Oliveira  _et al._ (2010), including a
comparison with experimental results Seridonio  _et al._ (2009b). In all those
previous papers, the numerical calculations were done employing NRG, including
the computation of the parameter $\delta$.

The NRG logarithmic discretization parameter employed in the simulations of
this work was $\Lambda=2.25$ and the chemical potential, $\mu=0.0D$. The Kondo
temperature, $T_{K}$, for each case was obtained by computing the value of the
temperature, where the electrical conductance attains the value
$G(T_{K})=\frac{G_{o}}{2}=\frac{e^{2}}{h}$ Goldhaber-Gordon  _et al._ (1998b).

Figure 2: (Color online) The electrical conductance $G(T)$ vs. $T^{*}$,
corresponding to the Kondo regime. In the inset we show another situation,
closer to the crossover between the Kondo and intermediate valence regime.

Fig. 2 shows the results obtained for the electrical conductance $G(T)$ vs.
$T^{*}$, in units of $G_{o}=\frac{2e^{2}}{h}$, in an asymmetric situation
employing the results of the symmetrical limit as shown in Fig. 11 of Appendix
A. In the main panel, we plot results corresponding to the Kondo regime,
employing the following parameters: $E_{f}=-5.0D$, $U=30.0D$, with the Kondo
temperature being $T_{K}=9.422.10^{-4}D$. The agreement between the calculated
NRG asymmetric results and those obtained employing the NRG symmetric one, the
Eq. 27, is notable. The parameter $\delta$ computed by the NRG for the
asymmetric case is $\delta=0.9490\frac{\pi}{2}$, which confirms the validity
of Eq. 27 for the SET, previously obtained in reference Yoshida  _et al._
(2009). In the inset, we plot a situation closer to the crossover transition
between the Kondo and intermediate valence regime, with $U=50.0D$,
$E_{f}=-2.5D$, $T_{K}=1.88310^{-2}D$, and $\delta=0.882\frac{\pi}{2}$. The
agreement obtained is notable for temperatures below $T\simeq 10T_{K}$, but
for temperatures above this value, the results show a small departure from
each other, due to the rising of charge fluctuations not being well described
by the present treatment.

Figure 3: (Color online) Universal thermoelectric coefficient
$\left(\frac{L_{1}}{T}\right)$, expressed in $\frac{G_{o}k_{B}}{e^{2}}$ units,
vs. $T^{*}$ for the asymmetric Kondo limit. In the inset, we show the same
results for an asymmetric case, in the crossover from an intermediate valence
situation to the Kondo limit.

In Fig. 3, in the main panel, we show the results for
$\left(\frac{L_{1}}{T}\right)$ vs. $T^{*}$ for the asymmetric Kondo limit,
with $E_{QD}=-10.0D$, $U=30.0D$ and $T_{K}=1.517.10^{-5}D$. The direct NRG
computations are shown by the continuous red line, whereas the results
obtained when employing the NRG calculations for the particle-hole symmetric
case of the SIAM and Eq. 29 are shown by the black curve. Again, the parameter
$\delta$ was computed following the same procedure described in the Appendix
A. We obtained excellent agreement for a large range of temperatures, between
($10^{-4}T_{K}\leq T\leq 10^{3}T_{K}$). In the inset, we show the same
results, but now for a set of parameters closer to the crossover transition
region, between the Kondo and the intermediate valence regimes, with
$E_{QD}=-5.0D$, $U=30.0D$, and $T_{K}=9.422.10^{-4}D$.

Figure 4: (Color online) Thermopower $S(T^{*})$, in $\frac{K_{B}}{e}$ units
vs. $T^{*}$. The inset shows the crossover’s results from the intermediate
valence to the Kondo regime.

In Fig. 4, we plot the thermopower $S(T^{*})$ vs. $T^{*}$ for $E_{QD}=-10.0D$,
$U=30.0D$, and $T_{K}=1.517.10^{-5}D$. Employing Eqs. 4, 27, and 29 (red
curve), we obtain an excellent agreement between the asymmetric direct NRG
results (black curve) and the one employing the symmetric universal TTCs. The
minimum at the Kondo temperature manifests the Kondo effect in the thermopower
$S(T)$ Costi and Zlatic (2010). There is excellent agreement between both
curves up to $T\leq 10^{2}T_{K}$, when charge fluctuations dominate the
process. In the inset, we represent a crossover from intermediate valence to
the Kondo regime $E_{QD}=-5.0D$, $U=30.0D$, and $T_{K}=9.42210^{-4}D$. Below
$T\geq T_{K}$, the agreement between the two curves is excellent, but above
$T_{K}$, there is a visible difference between the two results at higher
temperatures. We attribute this difference to the intermediate valence
region’s proximity, because the present treatment does not describe charge
fluctuations well.

Figure 5: (Color online) Universal thermoelectric coefficient
$\left(\frac{L_{2}}{T^{2}}\right)$, in $G_{o}L_{N}$ units, vs. $T^{*}$ for the
asymmetric Kondo limit. The inset shows the temperature-normalized electronic
contribution to the thermal conductance $\left(\frac{\kappa}{T}\right)$ vs.
$T^{*}$.

In Fig. 5, we plot the results for the universal thermoelectric coefficient
for the asymmetric Kondo limit, $\left(\frac{L_{2}}{T^{2}}\right)$ vs. $T^{*}$
in $G_{o}L_{N}$ units, where $L_{N}=(\frac{\pi^{2}}{3})(\frac{k_{B}}{e})^{2}$
is the Lorenz number, with $E_{QD}=-5.0D$, $U=30.0D$, and
$T_{K}=9.422.10^{-4}D$. Again the agreement obtained between the direct
asymmetric NRG results and those achieved employing Eq. 31 and particle-hole
symmetric NRG results (fitting presented in Fig. 13) is excellent. In the
inset, we show the temperature-normalized electronic contribution to the
thermal conductance $\left(\frac{\kappa}{T}\right)$ vs. $T^{*}$. In this case,
some small differences appear above $T\geq 30.0T_{K}$, which is a
manifestation of the charge fluctuation process, present in this range of
temperatures.

Figure 6: (Color online) Thermal dependence of universal quantities in the
symmetric limit. Temperature normalized electronic contribution to the thermal
conductance, $\left(\frac{\kappa(T)}{T}\right)(G_{o}L_{N}$ and the electric
conductance $G(T)$ vs. $T^{*}$.

Fig. 6 shows the thermal dependence of the universal quantities in the
symmetric limit of the SIAM, employing the following parameters: $U=30.0D$,
$E_{f}=-15.0D$. We plot the temperature normalized electronic contribution to
the thermal conductance, $\left(\frac{\kappa(T)}{T}\right)$ and the electric
conductance $G(T)$ vs. $T^{*}$. The striking similarity of both curves at low
temperatures is associated with the Fermi-liquid character of the system and
the validity of the Wiedemann-Franz law in this temperature range Costi and
Zlatic (2010); Yoshida and Oliveira (2009), which leads to the relation
$\frac{\kappa(T)}{T}=G(T)$. However, besides the relative closeness of the
curves well below and above the Kondo temperature, the two properties are not
equal, once the Kondo temperature rules the electrical conductance, whereas
the thermal conductance is ruled by a different Kondo scale $T_{K}^{\theta}$,
as defined in reference Costi and Zlatic (2010)

$\frac{\kappa(T=T_{K}^{\theta})}{T_{K}^{\theta}}=\frac{\tilde{\alpha}}{2},$
(32)

where

$\tilde{\alpha}=\lim_{T\rightarrow 0}\frac{\kappa(T)}{T}.$ (33)

##  VI Comparison with experimental results

In this section, we discuss how to use the methodology employing the universal
TTCs to calculate the thermoelectric properties from experimental
measurements. In Appendix A, we present a discussion and some examples of
applying the universal TTCs methodology to experimental thermoelectric data.

Unfortunately, we did not find experimental SET works in the literature that
performed measurements of the electric and thermal conductances and the
thermopower in a broad range of temperatures. On the other hand, several
papers measured the gate dependence $V_{gate}=V$ of some of these properties
for a fixed temperature, $T$ Scheibner  _et al._ (2005); Hoffmann  _et al._
(2009); Svensson  _et al._ (2013); Svilans  _et al._ (2018); Dutta  _et al._
(2019). We focus on applying the universal TTC methodology to the experimental
results of the Artis $et$ $al.$ paper Svilans  _et al._ (2018), because they
performed several high-quality thermoelectric measurements of Kondo correlated
quantum dots (QDs), both below and above the Kondo temperature. They measured
the electric conductance $G(T)$, thermocurrent $I_{Th}$ normalized by $\Delta
T$ (under closed-circuit conditions), and the thermovoltage $V_{th}$ (under
open-circuit conditions) as a function of the gate voltage $V$ for a fixed
temperature.

Figure 7: (Color online) $G(T^{*})$ vs. $G^{S}(T^{*})$, where the experimental
data were obtained from the Artis $et$ $al.$ experimental paper Svilans  _et
al._ (2018) with $V_{gate}=1.044V$ (see Eq. 28). Figure 8: (Color online)
Thermoelectric properties as temperature function in Kelvin degrees. The
experimental data were obtained from the Artis $et$ $al.$ experimental paper
Svilans  _et al._ (2018) with $V_{gate}=1.044V$. Figure 9: (Color online)
$G(T^{*})$ vs. $G^{S}(T^{*})$, where the experimental data were obtained from
the Artis $et$ $al.$ experimental paper Svilans  _et al._ (2018) with
$V_{gate}=1.054V$ (see Eq. 28). Figure 10: (Color online) Thermoelectric
properties as a function of temperature in Kelvin degrees. The experimental
data were obtained from the Artis $et$ $al.$ experimental paper Svilans  _et
al._ (2018) with $V_{gate}=1.054V$.

Considering an ohmic dependence between the thermovoltage and the
thermocurrent in an experimental device, the relation
$SG=\alpha\frac{I_{Th}}{\Delta T}$ is valid, where $\Delta T$ is the
difference of temperatures associated with the Seebeck effect, and where
$\alpha$ must be a dimensionless constant for a fixed $V_{gate}$, but it could
be a temperature function. If we assume that at low temperatures $\alpha=\eta
T$, with $\eta$ being a constant that has the inverse of temperature units, we
expected that

$\frac{I_{Th}}{\Delta T}\simeq\frac{SG}{T}.$ (34)

To explore the validity of our predictions, we employed the following results
of the Artis $et$ $al.$ paper Svilans  _et al._ (2018): the data of figures
$2b$ and $3(a,b)$ for the electrical conductance $G(T)$ at different gate
voltages $V_{gate}$, and the figures $2c$ and $4a$ that present the results of
$\frac{I_{Th}}{\Delta T}$ as $V_{gate}$ function for different temperatures.

In Figs. 7 and 9, we show the results of the slope and the intercept of the
linear figures corresponding to the electrical conductance $G(T)$ of the Artis
$et$ $al.$ Svilans  _et al._ (2018) experimental data. We obtained for
$V_{gate}=1.044V$: $\delta=0.895012\pi/2$ and $T_{K}=1.1905K$ and for
$V_{gate}=1.054V$: $\delta=0.937832\pi/2$ and $T_{K}=1.0K$. These Kondo
temperatures agree well with the experimental results indicated in Fig. (3b)
of the reference Svilans  _et al._ (2018).

Employing the $\delta$ and $T_{K}$ values obtained, the universal relations
for the Onsager coefficients (Eqs. 27-31) and Eqs. 2, 3 and 4, we compute the
thermal dependence of the thermoelectric properties: $G(T)$, $S(T)$,
$\frac{\kappa}{T}$, and $GS/T$.

In Fig. 8, we show the results of those properties, corresponding to
$V_{gate}=1.044V$. Initially, we adjust the universal electrical conductance
$G_{Universal}$ to the corresponding experimental one, $G-Exp.$, and we
compute $\frac{SG}{T}$ and compare it with the experimental data of
$\frac{I_{Th}}{\Delta T}$. Unfortunately, the number of experimental data for
the $\frac{I_{Th}}{\Delta T}$ in the Artis $et$ $al.$ paper is limited, but
the agreement of both properties with the available experimental data is
excellent. Although there are no available experimental results for the
temperature-normalized thermal conductance $\kappa/T$ and the thermopower
$S(T)$, we calculated these properties and obtained fair, reliable results:
the $\kappa/T$ exhibits behavior similar to Fig. 6, and $S(t)$ goes to zero at
low temperatures, indicating that the experimental measurements were performed
in a Kondo situation nearer the symmetric limit.

In Fig. 10, we show the same results of Fig. 8, corresponding to
$V_{gate}=1.054V$, and the results show the same overall behavior as the
previous one.

##  VII Conclusions

In the present investigation, we employ the NRG treatment to compute the
thermal dependence of the TTCs in the Kondo regime. From Eqs. 27-31, we can
obtain the thermal dependence of the thermoelectric transport coefficients
$L_{0}$, $L_{1}$ and $L_{2}$ in asymmetric conditions in terms of the Kondo
temperature and the parameter $\delta$. All of the thermal dependence is
“carried" through the symmetric thermoelectric coefficients’ universal
functions of $T^{*}$, and all of the dependence on the parameters of the model
is taken into account through the parameter $\delta$. We also derived simple
universal fitting formulas for the TTCs, given by Eqs. 37, 38, and 39,
discussed in the Appendix A, that can be used to predict the thermoelectric
properties.

In practical terms, knowledge of the experimental results of the electrical
conductance or the thermopower in the Kondo regime at temperature function
$\left\\{G(T_{i}),T_{i}\right\\}$ or $\left\\{S(T_{i},T_{i}\right\\}$, allows
the determination of the Kondo temperature $T_{K}$ and the parameter $\delta$,
and employing the TTCs, we can calculate all the other thermoelectric
properties. The ideal situation to “check” our procedure is to obtain all the
thermoelectric properties from the experimental measurements, but this
requires a consistent and complete set of experimental data for $G(T)$,
$S(T)$, and $\kappa(T)$, in a broad temperature range below and above the
Kondo temperature, for the same $V_{gate}$. Unfortunately, we did not find
such experimental measurements in the literature, but several papers report
measured the gate dependence $V_{gate}=V$ of those properties, for a fixed
temperature, $T$ Scheibner  _et al._ (2005); Hoffmann  _et al._ (2009);
Svensson  _et al._ (2013); Svilans  _et al._ (2018); Dutta  _et al._ (2019).

We focused on applying the universal TTC methodology to the experimental
results of the Artis $et$ $al.$ paper Svilans  _et al._ (2018), which measured
the electric conductance $G(T)$, thermocurrent $I_{Th}$ normalized by $\Delta
T$, and the thermovoltage $V_{th}$ as a function of the gate voltage $V$ for a
fixed temperature. We adjusted the experimental results of the $G(T)$ and
$I_{Th}/\Delta T$ employing the universal TTCs, obtaining excellent agreement.
Although the Artis group did not measure the temperature-normalized thermal
conductance $\kappa/T$ and the thermopower $S(T)$, we calculated those
properties, obtaining reliable results.

We expect that this investigation will motivate researchers to carry out
experimental work in this direction, in order to compare the procedure
expounded here to experimental testing.

###### Acknowledgements.

We are thankful for the financial support of the Research Division of the
Colombia National University, Bogotá (DIB) and the Colombian Scientific Agency
- COLCIENCIAS, the São Paulo State Research Foundation (FAPESP), the Brazilian
National Research Council (CNPq) and Coordination of Superior Level Staff
Improvement (CAPES). E. Ramos acknowledge support from COLCIENCIAS-COLFUTURO
doctoral scholarship “Convocatoria Doctorados Nacionales No. 617, 2014-2”. R.
Franco is grateful for the hospitality of the IFSC-USP-São Carlos and the IF-
UFF- Niterói, where part of this work was done.

##  Appendix A Application of the universal TTCs methodology to experimental
data

Some earlier papers Seridonio  _et al._ (2009b); Yoshida  _et al._ (2009);
Seridonio  _et al._ (2009a); Oliveira  _et al._ (2010) discussed how to employ
experimental data of the thermal dependence of the electric conductance
$G(T)$, to calculate the parameter $\delta$ to check the validity of Eq. 27.
In particular, an almost perfect fit of $G(T)$ with experimental results was
found in reference Merker  _et al._ (2013). From the theoretical point of
view, it is possible to adjust the experimental results by employing the TTCs
in the symmetrical limit of the SIAM obtained from the NRG calculations.
Nevertheless, for practical purposes, we can also employ the fitting formulas
obtained in Eqs. 37, 38 and 39.

Essentially, the procedure is the following: Given a set of experimental data
$\left[G_{i}\left(T_{i}\right),T_{i}\right]$, choose a trial Kondo temperature
$T_{K}$, and a new data set
$\left[G_{i}\left(T_{i}^{*}\right),T_{i}^{*}\right]$, with
$T_{i}^{*}=\frac{T_{i}}{T_{K}}$ is generated. Since the universal curve for
the electric conductance in the symmetric limit of the SIAM, $G^{S}(T^{*})$
vs. $T^{*}$ is known for the fitting Eq. 37 (Fig. 11), it is possible to
obtain the value $G^{S}_{i}(T_{i}^{*})$ for each experimental data set of
$T_{i}^{*}$, and plot $G_{i}\left(T_{i}^{*}\right)$ vs.
$G^{S}_{i}\left(T_{i}^{*}\right)$. If the plot followed a straight line, the
correct Kondo temperature value $T_{K}$ was attained, and the corresponding
parameter $\delta$ could be obtained by the slope and the intercept of the
straight line (see Eq. 28 and Figs. 7 and 9). On the contrary, if the obtained
plot does not follow a straight line, a new trial Kondo temperature $T_{K}$
must be employed, until a straight line is obtained. Employing Eq. 27 and the
fit shown in Eq. 37, it is also possible to compute $L_{0}(T^{*})$.

The same procedure can be performed using the thermopower. Employing Eqs. 4,
27 and 29, it is possible to write the thermopower $S$ as

$\frac{e}{h}S\left(T^{*}\right)=\frac{sin(2\delta)\left(\frac{L_{01}^{S}}{T}\right)\left(T^{*}\right)}{-4cos^{2}(\delta)+2hcos(2\delta)L_{0}^{S}\left(T^{*}\right)},$
(35)

which is equivalent to the equation

$\frac{h}{e}\frac{\left(\frac{L_{01}^{S}}{T}\right)\left(T^{*}\right)}{S\left(T^{*}\right)}=-2cot(\delta)+2hcot(2\delta)L_{0}^{S}\left(T^{*}\right).$
(36)

Given a set of temperatures and a thermopower experimental data set $T_{i}$
and $S(T_{i})$ ($i=1,...,N$) ($\left[S(T_{i}),T_{i}\right]$), we can choose a
tentative Kondo temperature $T_{K}$, compute $T_{i}^{*}$, and obtain
$\left[S_{i}(T_{i}^{*}),T_{i}^{*}\right]$. Since we know the universal
functions $L_{0}^{S}(T_{i}^{*})$ (Fitting of $G^{S}_{0}(T_{i}^{*})$\- Eq. 37
associated with Fig. 11) and $\frac{L_{01}^{S}}{T}(T_{i}^{*})$, it is then a
simple matter to compute the fraction on the left-hand side and plot it as a
function of $L_{0}^{S}(T_{i}^{*})$. If the plot is a straight line, the Kondo
temperature has been found. If not, we continue the process until the correct
value is attained.

Once the correct parameters $\delta$ and $T_{K}$ are obtained, it is possible
to compute $\frac{L_{1}}{T}(T^{*})$, employing the universal function
$\frac{L_{(10)}}{T}(T^{*})$, given by Eq. 29 or the adjusted Eq. 38 of the
results shown in Fig. 12. Additionally, it is possible to compute the quantity
$\frac{L_{2}\left(T^{*}\right)}{\left(k_{B}T^{*}\right)^{2}}$ by employing Eq.
31, or the fit Eq. 39 of the results shown in Fig. 13. Finally, using Eqs. 2,
3, and 4, we can calculate $G(T^{*})$, $S(T^{*})$ and $\kappa(T^{*})$, and
with these, other quantities, such as $ZT$ and the Wiedemman-Franz law.

Figure 11: (Color online) NRG Universal electrical conductance $G^{S}(G_{o})$
in the symmetrical limit of the SIAM vs. $\left(T^{*}\right)$, and its fit to
Eq. 37.

Figure 12: Thermal dependence of the universal function
$\left(\frac{L_{(01)}}{T}\right)$ vs. $T^{*}$, in the symmetric limit of the
SIAM and its fitting to the Eq. 38

.

Figure 13: (Color online) The universal quantity
$\frac{L_{2}^{S}\left(T^{*}\right)}{\left(\frac{k_{B}T}{T_{K}}\right)^{2}}$
vs. $T^{*}$, obtained by NRG and its fit to Eq. 39. Figure 14: (Color online)
Electrical and thermal conductances and thermopower vs. temperature (Kelvin)
for a side-coupled SET, obtained from the experimental electrical conductance
at $T_{K}=0.85$K Sato  _et al._ (2005). Figure 15: (Color online) Electrical
and thermal conductances and thermopower vs. temperature (Kelvin) for a side-
coupled SET, obtained from the experimental electrical conductance at
$T_{K}=1.23$K Sato  _et al._ (2005).

In Fig. 11, we plot the NRG universal result Costi  _et al._ (1994) for the
electrical conductance in the symmetrical limit of the Kondo regime
$G^{S}(T^{*})$, employing the following parameters: $U=30.0D$ and
$E_{f}=-15.0D$. The red line is the fit of the NRG data using a one-parameter
equation employed in the reference Goldhaber-Gordon  _et al._ (1998b),

$G(T^{*})=\frac{Go}{[(T^{*})^{2}(2^{1/\xi}-1)+1]^{\xi}},$ (37)

to adjust the electrical conductance. The parameter $\xi$ determines the
steepness of the decrease in conductance with increasing temperature and
provides a good fit to the numerical renormalization group (NRG) results for
the Kondo regime. In our case, $\xi=0.226022$ and the correlation
coefficient$=0.999986$. The agreement achieved is excellent. Eq. 37,
associated with this fit, allows us to compute the universal TTC
$L^{S}_{0}(T^{*})$ for any $T^{*}$ value in the range of temperatures
presented.

In Fig. 12, we show the thermal dependence of the universal function
$\left(\frac{L_{(01)}}{T}\right)(T^{*})$ vs. $T^{*}$ in the symmetric limit of
the SIAM, employing the following parameters: $U=30.0D$ and $E_{f}=-15.0D$. We
obtain good agreement with the universal TTC
$\left(\frac{L_{(01)}}{T}\right)(T^{*})$ (red line), employing a three-
parameter fit expression similar of Eq. 37

$\left(\frac{L_{(01)}}{T}\right)\left(T^{*}\right)=A_{0}\left(\frac{(T^{*})^{\frac{A_{1}}{\xi}}}{(T^{*})^{2}\left[A_{2}^{1/\xi}-1\right]+1}\right)^{\xi},$
(38)

where $A_{0}=11.945007$, $A_{1}=0.860404$, $A_{2}=63.2865$ and $\xi=0.674506$.

In Fig. 13, we plot the NRG universal results obtained for the TTC
$\frac{L_{2}^{S}\left(T^{*}\right)}{\left(k_{B}T^{*}\right)^{2}}$ in the
symmetric limit of the SIAM, employing the following parameters: $U=30.0D$ and
$E_{f}=-15.0D$. The red line is the fit of the NRG numerical data to Eq. 39;
the agreement achieved is excellent. Again, the expression associated with
this fit has a form similar to that of Eq. 37

$\frac{L_{2}^{S}}{(T^{*})^{2}}=\frac{A_{0}}{[(T^{*})^{2}(2^{1/\xi}+A_{1})+1]^{\xi}},$
(39)

where, $A_{0}=3.29776$, $\xi=0.238365$ and $A_{1}=43.6995$. This formula
reduces to Eq. 37 if $A_{0}=G_{0}$ and $A_{1}=-1.0$. It also permits computing
$\frac{L_{2}^{S}\left(T^{*}\right)}{\left(k_{B}T^{*}\right)^{2}}$ for any
$T^{*}$ value in the temperature range indicated in the figure.

For completeness, we also repeat earlier calculations employed in the paper
Seridonio  _et al._ (2009b), which obtained the electrical conductance and the
Kondo temperature using the experimental results of the electrical conductance
SET in a side-coupled geometry Sato  _et al._ (2005). We show the results of
those calculations in Figs. 14 and 15. The fit of the electrical conductance
is excellent. We also calculate $S(T)$ and $\kappa(T)/T$, but we cannot check
the reliability of these results, due to the absence of experimental
measurements.

## References

  * Sánchez and López (2016) D. Sánchez and R. López, Comptes Rendus Physique 17, 1060 (2016).
  * Tritt (2002) T. Tritt, in  _Encyclopedia of Materials: Science and Technology_ , edited by K. J. Buschow, R. W. Cahn, M. C. Flemings, B. Ilschner, E. J. Kramer, S. Mahajan, and P. Veyssiere (Elsevier, Oxford, 2002) p. 1.
  * Joffe and Stil (1959) A. F. Joffe and L. S. Stil, Reports on Progress in Physics 22, 167 (1959).
  * Wright (1958) D. A. Wright, Nature 181, 834 (1958).
  * Witting  _et al._ (2019) I. T. Witting, T. C. Chasapis, F. Ricci, M. Peters, N. A. Heinz, G. Hautier, and G. J. Snyder, Advanced Electronic Materials 5, 1800904 (2019).
  * He and Tritt (2017) J. He and T. M. Tritt, Science 357 (2017).
  * Benenti  _et al._ (2017) G. Benenti, G. Casati, K. Saito, and R. Whitney, Physics Reports 694, 1 (2017).
  * Xu  _et al._ (2017) N. Xu, Y. Xu, and J. Zhu, npj Quantum Materials 2, 51 (2017).
  * Gooth  _et al._ (2018) J. Gooth, G. Schierning, C. Felser, and K. Nielsch, MRS Bulletin 43, 187 (2018).
  * Zoui  _et al._ (2020) M. Zoui, B. S., S. J.G., and B. M., Energies 13, 3606 (2020).
  * Goldhaber-Gordon  _et al._ (1998a) D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-Magder, U. Meirav, and M. A. Kastner, Nature 391, 156 (1998a).
  * Yoshida and Oliveira (2009) M. Yoshida and L. Oliveira, Physica B: Condensed Matter 404, 3312 (2009).
  * Costi and Zlatic (2010) T. A. Costi and V. Zlatic, Phys. Rev. B 81, 235127 (2010).
  * Hershfield  _et al._ (2013) S. Hershfield, K. A. Muttalib, and B. J. Nartowt, Phys. Rev. B 88, 085426 (2013).
  * Donsa  _et al._ (2014) S. Donsa, S. Andergassen, and K. Held, Phys. Rev. B 89, 125103 (2014).
  * Talbo  _et al._ (2017) V. Talbo, J. Saint-Martin, S. Retailleau, and P. Dollfus, Scientific Reports 7, 14783 (2017).
  * Costi (2019a) T. A. Costi, Phys. Rev. B 100, 161106 (2019a).
  * Costi (2019b) T. A. Costi, Phys. Rev. B 100, 155126 (2019b).
  * Kleeorin  _et al._ (2019) Y. Kleeorin, H. Thierschmann, H. Buhmann, A. Georges, L. W. Molenkamp, and Y. Meir, Nature Communications 10, 5801 (2019).
  * Eckern and Wysokiński (2020) U. Eckern and K. I. Wysokiński, New Journal of Physics 22, 013045 (2020).
  * Heremans  _et al._ (2004) J. P. Heremans, C. M. Thrush, and D. T. Morelli, Phys. Rev. B 70, 115334 (2004).
  * Scheibner  _et al._ (2007) R. Scheibner, E. G. Novik, T. Borzenko, M. König, D. Reuter, A. D. Wieck, H. Buhmann, and L. W. Molenkamp, Phys. Rev. B 75, 041301 (2007).
  * Hoffmann  _et al._ (2009) E. A. Hoffmann, H. A. Nilsson, J. E. Matthews, N. Nakpathomkun, A. I. Persson, L. Samuelson, and H. Linke, Nano Letters 9, 779 (2009).
  * Hartman  _et al._ (2018) N. Hartman, C. Olsen, S. Lüscher, M. Samani, S. Fallahi, G. C. Gardner, M. Manfra, and J. Folk, Nature Physics 14, 1083 (2018).
  * Svilans  _et al._ (2018) A. Svilans, M. Josefsson, A. M. Burke, S. Fahlvik, C. Thelander, H. Linke, and M. Leijnse, Phys. Rev. Lett. 121, 206801 (2018).
  * Dutta  _et al._ (2017) B. Dutta, J. T. Peltonen, D. S. Antonenko, M. Meschke, M. A. Skvortsov, B. Kubala, J. König, C. B. Winkelmann, H. Courtois, and J. P. Pekola, Phys. Rev. Lett. 119, 077701 (2017).
  * Dutta  _et al._ (2019) B. Dutta, D. Majidi, A. García Corral, P. A. Erdman, S. Florens, T. A. Costi, H. Courtois, and C. B. Winkelmann, Nano Letters 19, 506 (2019).
  * Hewson (1993) A. C. Hewson, The Kondo problem to Heavy Fermions - Cambridge University Press (1993).
  * Costi  _et al._ (1994) T. A. Costi, A. C. Hewson, and V. Zlatic, Journal of Physics: Condensed Matter 6, 2519 (1994).
  * Yoshida  _et al._ (2009) M. Yoshida, A. C. Seridonio, and L. N. Oliveira, Phys. Rev. B 80, 235317 (2009).
  * Seridonio  _et al._ (2009a) A. C. Seridonio, M. Yoshida, and L. N. Oliveira, Phys. Rev. B 80, 235318 (2009a).
  * Langreth (1966) D. C. Langreth, Phys. Rev. 150, 516 (1966).
  * Grobis  _et al._ (2008) M. Grobis, I. G. Rau, R. M. Potok, H. Shtrikman, and D. Goldhaber-Gordon, Phys. Rev. Lett. 100, 246601 (2008).
  * Kretinin  _et al._ (2011) A. V. Kretinin, H. Shtrikman, D. Goldhaber-Gordon, M. Hanl, A. Weichselbaum, J. von Delft, T. Costi, and D. Mahalu, Phys. Rev. B 84, 245316 (2011).
  * Parks  _et al._ (2010) J. J. Parks, A. R. Champagne, T. A. Costi, W. W. Shum, A. N. Pasupathy, E. Neuscamman, S. Flores-Torres, P. S. Cornaglia, A. A. Aligia, C. A. Balseiro, G. K.-L. Chan, H. D. Abruña, and D. C. Ralph, Science 328, 1370 (2010).
  * Mahan (1990) G. D. Mahan, Many-Particle Physics - Springer , 227 (1990).
  * Ziman (1999) J. M. Ziman, Principles of the Theory of Solids - Cambridge University Press , 229 (1999).
  * Dong and Lei (2002) B. Dong and X. L. Lei, Journal of Physics: Condensed Matter 14, 11747 (2002).
  * Seridonio  _et al._ (2009b) A. C. Seridonio, M. Yoshida, and L. N. Oliveira, EPL (Europhysics Letters) 86, 67006 (2009b).
  * Oliveira  _et al._ (2010) L. N. Oliveira, M. Yoshida, and A. C. Seridonio, Journal of Physics: Conference Series 200, 052020 (2010).
  * Goldhaber-Gordon  _et al._ (1998b) D. Goldhaber-Gordon, J. Göres, M. A. Kastner, H. Shtrikman, D. Mahalu, and U. Meirav, Phys. Rev. Lett. 81, 5225 (1998b).
  * Scheibner  _et al._ (2005) R. Scheibner, H. Buhmann, D. Reuter, M. N. Kiselev, and L. W. Molenkamp, Phys. Rev. Lett. 95, 176602 (2005).
  * Svensson  _et al._ (2013) S. F. Svensson, E. A. Hoffmann, N. Nakpathomkun, P. M. Wu, H. Q. Xu, H. A. Nilsson, D. Sánchez, V. Kashcheyevs, and H. Linke, New Journal of Physics 15, 105011 (2013).
  * Merker  _et al._ (2013) L. Merker, S. Kirchner, E. Muñoz, and T. A. Costi, Phys. Rev. B 87, 165132 (2013).
  * Sato  _et al._ (2005) M. Sato, H. Aikawa, K. Kobayashi, S. Katsumoto, and Y. Iye, Phys. Rev. Lett. 95, 066801 (2005).