# Existence and Hölder regularity of infinitely many solutions to a
$p$-Kirchhoff type problem involving a singular nonlinearity without the
Ambrosetti-Rabinowitz (AR) condition

Debajyoti Choudhuri‡
‡Department of Mathematics, National Institute of Technology Rourkela, India
Emails<EMAIL_ADDRESS>
###### Abstract

We carry out an investigation of the existence of infinitely many solutions to
a fractional $p$-Kirchhoff type problem with a singularity and a superlinear
nonlinearity with a homogeneous Dirichlet boundary condition. Further the
solution(s) will be proved to be bounded and a weak comparison principle has
also been proved. A ‘$C^{1}$ versus $W_{0}^{s,p}$’ analysis has also been
discussed.

Keywords: singularity, non-Ambrosetti-Rabinowitz condition, Cerami condition,
multiplicity, symmetric Mountain-Pass theorem.

AMS Classification: 35J35, 35J60.

##  1 Introduction

Off late, the problems involving a nonlocal and fractional operators have
become hugely popular area of investigation owing to its manifold
applications, viz. stratified materials, population dynamics, continuum
mechanics, water waves, minimal surface problems etc. Interested readers may
refer to [3, 6, 11, 16, 18, 30, 43] and the references therein.
The problem addressed in this article is as follows.

$\displaystyle\left(a+b\int_{\mathbb{R}^{2N}}|u(x)-u(y)|^{p}K(x-y)dxdy\right)\mathfrak{L}_{p}^{s}u-\lambda
g(x)|u|^{p-2}u=$
$\displaystyle\mu\frac{h(x)u}{|u|^{\gamma+1}}+f(x,u),~{}\text{in}~{}\Omega$
$\displaystyle u=$ $\displaystyle
0,~{}\text{in}~{}\mathbb{R}^{N}\setminus\Omega$ (1.1)

where $\lambda,\mu>0$, $g,h\geq 0$ are functions that are defined and bounded
over $\Omega$. Here $\Omega\subset\mathbb{R}^{N}$ is a bounded domain with
Lipschitz boundary $\partial\Omega$ for $N\geq 2$, $a>0,b\geq 0$,
$0<\gamma,s<1<p<\infty$, $sp<N$. The function $f$ is a Carathéodory function
and the operator $\mathfrak{L}_{p}^{s}$ is defined as

$\mathfrak{L}_{p}^{s}u=2\underset{\epsilon\rightarrow
0^{+}}{\lim}\int_{\mathbb{R}^{N}\setminus
B_{\epsilon}(x)}|u(x)-u(y)|^{p-2}(u(x)-u(y))K(x-y)dy$

for all $x\in\mathbb{R}^{N}$ where $B_{\epsilon}(x)=\\{y:|y-x|<\epsilon\\}$.
The function $K:\mathbb{R}^{N}\setminus\\{0\\}\rightarrow(0,\infty)$ is
measurable with the following properties:

$\displaystyle\begin{split}&(i)~{}\mathfrak{\rho}K\in
L^{1}(\mathbb{R}^{N})~{}\text{where}~{}\mathfrak{\rho}(x)=\min\\{|x|^{2},1\\}\\\
&(ii)~{}\text{There exists}~{}\delta>0~{}\text{such
that}~{}K(x)\geq\delta|x|^{-N-ps}~{}\text{for all}~{}x\in\mathbb{R}^{N}\\\
&(iii)~{}K(x)=K(-x)~{}\text{for
all}~{}x\in\mathbb{R}^{N}\setminus\\{0\\}.\end{split}$ (1.2)

One can retrieve the fractional $p$-Laplace operator if the ‘kernel’ $K$ is
chosen to be $K(x)=|x-y|^{-N-ps}$. The discussion in Section 4.1 uses
following condition $(P)$.

$(P^{\prime}):~{}\delta_{1}|x|^{-N-ps}\geq K(x)\geq\delta_{2}|x|^{-N-ps}$

for all $x\in\mathbb{R}^{N}$ where $\delta_{1},\delta_{2}>0$. In general, it
is a practice to denote the Kirchhoff function as $\mathfrak{M}$. In the
current case $\mathfrak{M}(t)=a+bt^{p}$ where $a,b\geq 0$ with $a+b>0$. Such a
Kirchhoff function is termed as the nondegenerate ones. Therefore when
$\mathfrak{M}(t)\equiv 1$, $\lambda=0$, $p=2$, $g(x)=1$, $h(x)=1$ a.e. in
$\Omega$, we reduce to the problem in (1) to

$\displaystyle\begin{split}(-\Delta)^{s}u&=\mu
u^{-\gamma}+f(x,u),~{}\text{in}~{}\Omega\\\ u&>0,~{}\text{in}~{}\Omega\\\
u&=0,~{}\text{in}~{}\mathbb{R}^{N}\setminus\Omega.\end{split}$ (1.3)

For further details on the problem in (1.3) one may refer to [43]. The problem
in (1.3) with $\mu=0$ has been addressed in [42]. The authors have used a
variational technique to guarantee the existence of multiple solutions.
Further results on existence of multiple solutions can be found in [5, 46]. In
most of these studies, the authors obtained two distinct weak solutions.
Readers may refer to [2, 27, 28, 35, 47, 53] for ideas and techniques
developed in order to guarantee the existence of infinitely many solutions to
nonlocal elliptic problems driven by a singularity.
Meanwhile, we direct the readers to a variety of forms for the function
$\mathfrak{M}$ [12, 13, 17, 22, 24, 48, 49, 50, 54]. With the advent of these
references and with the help of fountain theorem, the authors in [23] have
proved the existence of infinitely many solutions for a fractional
$p$-Kirchhoff problem. A use of the fountain theorem has also been made in
[52] to guarantee the existence of infintely many solutions for a fractional
Kirchhoff type problem. In [37], the authors showed the existence and
multiplicity of solutions to a degenerate fractional $p$-Kirchhoff problem.
Fiscella et al in [4] dealt with Kirchhoff type equation over $\mathbb{R}^{N}$
involving Hardy-Sobolev nonlinearities with critical exponent. We also refer
to [25, 33, 40] for a related problem. Recently, Ghosh [45] has proved the
existence of infinitely many solutions to a system of fractional Laplacian
Kirchhoff type problem with a sublinear growth. For articles that addressed
the problem of $p$-Kirchhoff system, one may refer to [39] and the references
therein. A futuristic research in this direction is to consider the parabolic
counterparts to these problems. The readers may refer to the work due to [36]
and the references therein. Motivated from the work due to Ren et al [55] we
will show the existence of infinitely many solutions to a $p$-Kirchhoff type
problem with a superlinear growth without the AR condition. It will also be
proved that the solution (if exists) is in $L^{\infty}(\Omega)$. A weak
comparison principle has also been proved. A little bit of history about the
AR condition - this condition was first introduced by Ambrosetti and
Rabinowitz (refer [1]) in 1973. Thereafter this condition formed a formidable
tool in the analysis of elliptic PDEs, especially to prove the boundedness of
the Palais-Smale (PS) sequences for the associated energy functional to the
problem. To our knowledge there is no evidence in the literature that
considered a nonlocal $p$-Kirchhoff type problem with a singular nonlinearity
without the AR condition. Therefore the problem considered and the results
obtained here are new.

##  2 A simple physical motivation

This section is devoted to a physical motivation to the problem considered in
this article. The explanation is physically heuristic but nevertheless gives a
strong mathematical motivation to take up this problem. We will restrict to
the one dimensional case of the model of an elastic string of finite length
fixed at both the ends. The vertical displacement of the string will be
represented by $u:[-1,1]\times[0,\infty)\rightarrow\mathbb{R}$. Then,
mathematically, the end point constraints can be expressed as

$u(0,t)=u(2,t)=0$

for all $t\geq 0$. In order to identify this finite string with an infinite
string one can consider

$u(x,t)=0$

for all $x\in\mathbb{R}\setminus[0,2]$, $t\geq 0$. Thus, the acceleration
$\frac{\partial^{2}u}{\partial t^{2}}$ of the vertical displacement $u$ of the
vibrating string must be balanced (thanks to Newton’s laws) by the elastic
force of the string and by the external force field $f$. Therefore we have

$\frac{\partial^{2}u}{\partial t^{2}}=m\frac{\partial^{2}u}{\partial
x^{2}}+f,~{}\text{for all}~{}x~{}\text{in}~{}[0,2],~{}t\geq 0.$

When the steady case is considered, we have

$m\frac{\partial^{2}u(x)}{\partial x^{2}}=f(x),~{}\text{in}~{}[0,2].$

We quote here G.F. Carrier [15] \- it is well known that the classical
linearized analysis of the vibrating string can lead to results which are
reasonably accurate only when the minimum (rest position) tension and the
displacements are of such magnitude that the relative change in tension during
the motion is small. Taking this into account one can suppose that the tension
due to small deformation is linear in form, then we have the following
expression.

$\mathfrak{M}(l)=m_{0}+2Cl$

where $l$ is the increment in the length of the string with respect to its
mean position, i.e.

$l=\int_{0}^{2}\sqrt{1+\left(\frac{\partial u}{\partial x}\right)^{2}}dx-2,$

$C>0$ is a constant of proportionality. Thus for small deformations we have

$\sqrt{1+\left(\frac{\partial u}{\partial
x}\right)^{2}}=1+\frac{1}{2}\left(\frac{\partial u}{\partial x}\right)^{2}.$

Hence,

$l=\frac{1}{2}\left(\frac{\partial u}{\partial x}\right)^{2}.$

Therefore, the problem now boils down to a Kirchhoff type problem

$\displaystyle\left(m_{0}+C\frac{1}{2}\left(\frac{\partial u}{\partial
x}\right)^{2}\right)\frac{\partial^{2}u(x)}{\partial x^{2}}$ $\displaystyle=$
$\displaystyle f(x);~{}\text{for all}~{}x~{}\text{in}~{}[0,2],$ $\displaystyle
u(x)$ $\displaystyle=$ $\displaystyle
0,~{}\text{in}~{}\mathbb{R}\setminus[0,2].$ (2.1)

In other words it models the vibration of a string.

##  3 Technical preliminaries and functional analytic set up

We begin by giving the conditions of the function $f$ which is assumed to have
a superlinear growth. We quickly will recall the AR condition, which is as
follows.

$\displaystyle(AR):~{}$ $\displaystyle\text{there exist
constants}~{}r>0,1<\eta<\theta~{}\text{such that}$ $\displaystyle 0<\theta
F(x,t)\leq tf(x,t)~{}\text{for
any}~{}x\in\Omega,t\in\mathbb{R}~{}\text{and}~{}|t|\geq r.$ (3.1)

Here $F(x,t)=\int_{0}^{t}f(x,t)dt$. From (3) we have that $F(x,t)\geq
c_{1}|t|^{\theta}-c_{2}$, for any $(x,t)\in\Omega\times\mathbb{R}$, where
$\theta>\eta$ for $c_{1},c_{2}>0$ constants. However, there are still many
functions which are superlinear at infinity yet not satisfying the AR
condition. We make note of another form which is given by

$\displaystyle\lim_{|t|\rightarrow\infty}\frac{F(x,t)}{|t|^{\eta}}=\infty~{}\text{uniformly
for}~{}x\in\Omega.$ (3.2)

A closer observation shows that the nonlinearity $f$, under the condition
(3.2), is also superlinear at infinity. Clearly, for example, the function
$f(x,t)=t^{\eta-1}\log(1+t)$ satisfies (3.2) but does not satisfy (3). Some
important works that has proved the existence of infinitely many solutions for
nonlocal equations of fractional Laplacian type driven by a superlinear term
and with homogeneous Dirichlet boundary data, but without the AR condition can
be found in [38]. We now give the assumptions made on the function
$f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$.

$\displaystyle(f_{1})$ $\displaystyle\exists
C>0~{}\text{and}~{}q\in(p,p_{s}^{*})~{}\text{such that}~{}|f(x,t)|\leq
C(1+|t|^{q-1})$ $\displaystyle(f_{2})$ $\displaystyle
f(x,-t)=-f(x,t)\forall(x,t)\in\Omega\times\mathbb{R}~{}\text{and}~{}f\in
C(\Omega\times\mathbb{R},\mathbb{R})$ $\displaystyle(f_{3})$
$\displaystyle\lim_{|t|\rightarrow\infty}\frac{F(x,t)}{|t|^{2p}}=\infty~{}\text{uniformly
for all}~{}x\in\Omega$ $\displaystyle(f_{4})$
$\displaystyle\lim_{|t|\rightarrow
0}\frac{f(x,t)}{|t|^{p-1}}=0~{}\text{uniformly for all}~{}x\in\Omega$
$\displaystyle(f_{5})$ $\displaystyle\exists\overline{t}>0~{}\text{such
that}~{}t\mapsto\frac{f(x,t)}{t^{2p-1}}~{}\text{is decreasing
if}~{}t\leq-\overline{t}<0$ $\displaystyle~{}\text{and increasing
if}~{}t\geq\overline{t}>0\forall x\in\Omega$ $\displaystyle(f_{6})$
$\displaystyle\exists\sigma\geq 1~{}\text{and}~{}T\in
L^{1}(\Omega)~{}\text{satisfying}~{}T(x)\geq 0~{}\text{such
that}~{}\mathfrak{G}(x,s)\leq\sigma\mathfrak{G}(x,t)+T(x)$
$\displaystyle\forall
x\in\Omega~{}\text{and}~{}0\leq|s|\leq|t|,~{}\text{where}~{}\mathfrak{G}(x,t)=\frac{1}{2p}tf(x,t)-F(x,t).$

######  Remark 3.1.

The condition $(f_{6})$ was assumed by Jeanjean [31]. When $\sigma=1$ one can
see that the conditions $(f_{5})$ and $(f_{6})$ are equivalent. In general,
there are functions (for example $f(x,t)=2p|t|^{2p-2}t\ln(1+t^{2p})+p\sin t$)
that satisfy $(f_{6})$ but not $(f_{5})$.

We assign $Q=\mathbb{R}^{2N}\setminus C(\Omega)\times C(\Omega)$ where
$C(\Omega)=\mathbb{R}^{N}\setminus\Omega$. The space $X$ will denote the space
of Lebesgue measurable functions from $\mathbb{R}^{N}$ to $\mathbb{R}$ such
that its restriction to $\Omega$ of any function $u$ in $X$ belongs to
$L^{p}(\Omega)$ and

$\int_{Q}|u(x)-u(y)|^{p}K(x-y)dxdy<\infty.$

The space $X$ is equipped with the norm

$\displaystyle\|u\|_{X}$ $\displaystyle=$
$\displaystyle\|u\|_{L^{p}(\Omega)}+\left(\int_{Q}|u(x)-u(y)|^{p}K(x-y)dxdy\right)^{\frac{1}{p}}.$

We define the subspace $X_{0}$ of $X$ as

$X_{0}=\\{u\in X:u=0~{}\text{a.e. in}~{}\mathbb{R}^{N}\setminus\Omega\\}$

equipped with the norm

$\|u\|=\left(\int_{Q}|u(x)-u(y)|^{p}K(x-y)dxdy\right)^{\frac{1}{p}}.$

The space $X_{0}$ is a Banach and a reflexive space (refer Lemma 2.4 of [50]
and Theorem 1.2 of [41]). We will denote the usual fractional Sobolev space by
$W_{0}^{s,p}(\Omega)$ equipped with norm

$\|u\|_{W^{s,p}(\Omega)}=\|u\|_{L^{p}(\Omega)}+\left(\int_{\Omega\times\Omega}|u(x)-u(y)|^{p}K(x-y)dxdy\right)^{\frac{1}{p}}.$

Note that the norms $\|\cdot\|_{X}$ and $\|\cdot\|_{W^{s,p}}$ are not
equivalent when $K(x)=|x|^{-N-ps}$ as $\Omega\times\Omega$ is strictly
contained in $Q$. This makes the space $X_{0}$ different from the usual
fractional Sobolev space. Thus the fractional Sobolev space is insufficient
for dealing with our problem from the variational method point of view. For a
general kernel $K$ satisfying (1.2) we have $X_{0}\subset\\{u\in
W^{s,p}(\mathbb{R}^{N}):u(x)=0~{}\text{a.e.
in}~{}\mathbb{R}^{N}\setminus\Omega\\}$. For $K(x)=\frac{1}{|x|^{N+sp}}$, we
have $X_{0}=\\{u\in W^{s,p}(\mathbb{R}^{N}):u(x)=0~{}\text{a.e.
in}~{}\mathbb{R}^{N}\setminus\Omega\\}=W_{0}^{s,p}(\mathbb{R}^{N})$.
We now define the definition of a solution to (1) in a weaker sense.

######  Definition 3.2.

A function $u\in X_{0}$ is a weak solution to (1) if $h|u|^{-\gamma}\phi\in
L^{1}(\Omega)$ and

$\displaystyle(a+b\|u\|^{p})\int_{Q}|u(x)-u(y)|^{p-2}(u(x)-u(y))(\phi(x)-\phi(y))K(x-y)dxdy$
$\displaystyle-\lambda\int_{\Omega}g(x)|u|^{p-2}u\phi
dx-\mu\int_{\Omega}h(x)|u|^{-\gamma-1}u\phi dx-\int_{\Omega}F(x,u)dx=0$

for all $\phi\in X_{0}$.

######  Remark 3.3.

We will sometimes denote

$\langle u,v\rangle=\int_{Q}|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))K(x-y)dxdy$

######  Remark 3.4.

Henceforth, we will mean a weak solution whenever we use the term solution.

######  Remark 3.5.

Throughout the article any constant will be denoted by alphabets $C$ with a
prefix/suffix.

With these developements, we are now in a position to state our main
result(s).

######  Theorem 3.6.

Let $K:\mathbb{R}^{N}\setminus\\{0\\}\rightarrow(0,\infty)$ be a function as
above in (1.2); $(i)-(iii)$, and let conditions $(f_{1})-(f_{5})$ hold. Then,
for any $\lambda>0$ and for small enough $\mu_{0}>0$ such that for any
$\mu\in(0,\mu_{0})$, the problem in (1) has infinitely many solutions in
$X_{0}$ with unbounded energy.

######  Theorem 3.7.

Let $K:\mathbb{R}^{N}\setminus\\{0\\}\rightarrow(0,\infty)$ be a function as
above in (1.2); $(i)-(iii)$ and let conditions $(f_{1})-(f_{4})$ hold. If
condition $(f_{6})$ is considered instead of $(f_{5})$ then also the
conclusion of the Theorem 3.6 holds.

The next theorem stated is a $C^{1}$ versus $W_{0}^{s,p}$ analysis when the
kernel $K(x)=|x|^{-N-sp}$ is considered. The functional will still be denoted
by $I$.

######  Theorem 3.8.

Let $\Omega\subset\mathbb{R}^{N}$ be a bounded domain with a
$C^{1,1}$-boundary and let $u_{0}\in C^{1}(\bar{\Omega})$ which satisfies

$u_{0}\geq C\mbox{d}(x,\partial\Omega)^{s}\mbox{ for some }C>0$ (3.3)

be a local minimizer of $I$ (defined later in this section) in
$C^{1}(\bar{\Omega})$ topology; that is, there exists $\epsilon>0$ such that,
$u\in C^{1}(\bar{\Omega}),\;\|u-u_{0}\|_{C^{1}(\bar{\Omega})}<\epsilon$
implies $I(u_{0})\leq I(u)$. Then, $u_{0}$ is a local minimum of $I$ in
$W^{s,p}_{0}(\Omega)$ as well.

We quickly recall the following eigenvalue problem [55].

$\displaystyle\mathfrak{L}_{p}^{s}u$ $\displaystyle=$
$\displaystyle\lambda|u|^{p-2}u~{}\text{in}~{}\Omega$ $\displaystyle u$
$\displaystyle=$ $\displaystyle
0~{}\text{in}~{}\mathbb{R}^{N}\setminus\Omega.$ (3.4)

This has a divergent sequence of positive eigenvalues

$0<\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{n}\leq\cdots$

which has eigenvectors say $(e_{n})_{n\in\mathbb{N}}$. Refer to Proposition
$9$ of [44] where it has been shown that this sequence
$(e_{n})_{n\in\mathbb{N}}$ can be so chosen that it provides an orthonormal
basis in $L^{p}(\Omega)$ and an orthogonal basis in $X_{0}$.
We first define $I:X_{0}\rightarrow\mathbb{R}$ as

$\displaystyle I(u)$ $\displaystyle=$ $\displaystyle A(u)-B(u)-C(u)$ (3.5)

where

$\displaystyle\begin{split}A(u)&=\frac{a}{p}\|u\|^{p}+\frac{b}{2p}\|u\|^{2p}\\\
B(u)&=\frac{\lambda}{p}\int_{\Omega}g(x)|u|^{p}dx\\\
C(u)&=\frac{\mu}{1-\gamma}\int_{\Omega}h(x)|u|^{1-\gamma}dx+\int_{\Omega}F(x,u)dx\\\
\text{and}~{}F(x,u)&=\int_{0}^{u}f(x,s)ds.\end{split}$ (3.6)

From here onwards we will denote $\|\cdot\|_{L^{p}(\Omega)}$ as
$\|\cdot\|_{p}$. To our dissatisfaction, the functional $I$ is not a $C^{1}$
functional. However we redefine this functional $I$ as follows. We define

$\bar{f}(x,t)=\begin{cases}\mu
h(x)|t|^{-\gamma-1}t+\tilde{f}(x,t),&~{}\text{if}~{}|t|>\underline{u}_{\mu}\\\
\mu
h(x)\underline{u}_{\mu}^{-\gamma}+\tilde{f}(x,\underline{u}_{\mu}),&~{}\text{if}~{}|t|\leq\underline{u}_{\mu}\end{cases}$

where $\tilde{f}(x,t)=\lambda g(x)|t|^{p}+f(x,t)$ and $\underline{u}_{\mu}$ is
a solution to

$\displaystyle\left(a+b\int_{\mathbb{R}^{2N}}|u(x)-u(y)|^{p}K(x-y)dxdy\right)\mathfrak{L}_{p}^{s}u$
$\displaystyle=$ $\displaystyle\mu h(x)u^{-\gamma},~{}\text{in}~{}\Omega$
$\displaystyle u$ $\displaystyle>$ $\displaystyle 0,~{}\text{in}~{}\Omega$
$\displaystyle u$ $\displaystyle=$ $\displaystyle
0,~{}\text{in}~{}\mathbb{R}^{N}\setminus\Omega$

whose existence can be guaranteed from Lemma 5.2. Let
$\bar{F}(x,s)=\int_{0}^{s}\bar{f}(x,t)ds$. We now define the functional as
follows.

$\displaystyle\bar{I}(u)$ $\displaystyle=$
$\displaystyle\frac{a}{p}\|u\|^{p}+\frac{b}{2p}\|u\|^{2p}-\int_{\Omega}\bar{F}(x,u)dx.$
(3.8)

The way the functional has been defined, it is easy to see that the critical
points of the functional in (3.8) are also the critical points of the
functional (3.5). Most importantly, the functional $\bar{I}$ which is defined
in (3.8) is $C^{1}$ which allows us to use the variational methods. Further

$\displaystyle\langle\bar{I}^{\prime}(u),\phi\rangle$ $\displaystyle=$
$\displaystyle(a+b\|u\|^{p})\int_{Q}|u(x)-u(y)|^{p-2}(u(x)-u(y))(\phi(x)-\phi(y))K(x-y)dxdy$
(3.9) $\displaystyle-\int_{\Omega}\bar{f}(x,u)\phi dx$

for all $\phi\in X_{0}$.

######  Remark 3.9.

It can easily seen that if $u$ is a weak solution to (3) with
$u>\underline{u}_{\lambda}$, then $u$ is also a weak solution to (1).

The following lemma will be used in this work.

######  Lemma 3.10.

(Refer Lemma 2.4 [50]) Let the kernel $K$ be as above in (1.2). We then have
the following.

  1. 1.

For any $r\in[1,p_{s}^{*})$, the embedding $X_{0}\hookrightarrow
L^{r}(\Omega)$ is compact when $\Omega$ is bounded with smooth enough
boundary.

  2. 2.

For all $r\in[1,p_{s}^{*}]$, the embedding $X_{0}\hookrightarrow
L^{r}(\Omega)$ is continuous.

Following is the definition of Cerami condition (Definition 2.1 [10]).

######  Definition 3.11.

[Cerami condition] Let $B$ be an infinite dimensional Banach space with respet
to $\|\cdot\|_{B}$ and $I$ be a $C^{1}(B,\mathbb{R})$ functional. $I$ is said
to satisfy the $(Ce)_{c}$ at level $c\in\mathbb{R}$, if any sequence
$(u_{n})\subset B$ for which $I(u_{n})\rightarrow c$ in $B$,
$(1+\|u_{n}\|_{B})I^{\prime}(u_{n})\rightarrow 0$ in $B^{\prime}$, the dual
space of $B$, as $n\rightarrow\infty$, then there exists a strongly convergent
subsequence of $(u_{n_{k}})$ of $(u_{n})$ in $B$.

######  Remark 3.12.

Henceforth, a subsequence of any sequence, say $(v_{n})$, will also be denoted
by $(v_{n})$.

We now give the symmetric mountain pass theorem [14].

######  Theorem 3.13.

(Symmetric mountain pass theorem) Let $B$ be as in Definition 3.11 and $Y$ be
a finite dimensional Banach space such $B=Y\bigoplus Z$. For any $c>0$ if
$I\in C^{1}(B,\mathbb{R})$ satisfies $(Ce)_{c}$ and

  1. 1.

$I$ is even and $I(0)=0$ for all $u\in B$

  2. 2.

There exists $r>0$ such that $I(u)\geq R$ for all $u\in B_{r}(0)=\\{u\in
B:\|u\|_{B}\leq r\\}$

  3. 3.

For any finite dimensional subspace $\bar{B}\subset B$, there exists
$r_{0}=r(\bar{B})>0$ such that $I(u)\leq 0$ on $\bar{B}\setminus
B_{r_{0}}(0_{\bar{B}})$, where $0_{\bar{B}}$ is the null vector in $\bar{B}$

then there exists an unbounded sequence of critical values of $I$
characterized by a minimax argument.

##  4 Auxiliary and main result

We begin this section by proving a few auxiliary lemmas.

######  Lemma 4.1.

Let $(f_{1})$ hold. Then any bounded sequence $(u_{n})$ in $X_{0}$ which
satisfies $(1+\|u_{n}\|)\bar{I}^{\prime}(u_{n})\rightarrow 0$ as
$n\rightarrow\infty$ possesses a strongly convergent subsequence in $X_{0}$.

###### Proof.

Let $(u_{n})$ be a bounded sequence in $X_{0}$ with
$|u_{n}|>\underline{u}_{\mu}$. Since $X_{0}$ is reflexive we have

$\displaystyle\begin{split}&u_{n}\rightharpoonup u~{}\text{in}~{}X_{0},\\\
&u_{n}\rightarrow u~{}\text{in}~{}L^{r}(\Omega),1\leq r<p_{s}^{*},\\\
&u_{n}\rightarrow u~{}\text{a.e. in}~{}\Omega.\end{split}$ (4.1)

All we need to prove is that $u_{n}\rightarrow u$ strongly in $X_{0}$. By the
Hölder’s inequality and $(f_{1})$ we have

$\displaystyle\begin{split}0\leq\int_{\Omega}|f(x,u_{n})|(u_{n}-u)dx&\leq\int_{\Omega}C(1+|u_{n}|^{q-1})(u_{n}-u)dx\\\
&\leq
C(|\Omega|^{\frac{q-1}{q}}+\|u_{n}\|_{q}^{q-1})\|u_{n}-u\|_{q}.\end{split}$
(4.2)

By (4.1) we obtain

$\underset{n\rightarrow\infty}{\lim}\int_{\Omega}|f(x,u_{n})|(u_{n}-u)dx=0.$

Further we have from the Hölder’s inequality that

$\int_{\Omega}|u_{n}|^{p-2}u_{n}(u_{n}-u)dx\leq\|u_{n}\|_{p}^{p-1}\|u_{n}-u\|_{p}.$

Therefore again from (4.1) we have

$\underset{n\rightarrow\infty}{\lim}\int_{\Omega}|u_{n}|^{p-2}u_{n}(u_{n}-u)dx=0.$

We observe that $u$ cannot be zero over a subset of $\Omega$ of non-zero
measure. For if it is so, then
$\langle\bar{I}^{\prime}(u_{n}),\phi\rangle\rightarrow-\infty$ for a suitable
$\phi\in X_{0}$. But then we have from the given condition that
$(1+\|u_{n}\|)\bar{I}^{\prime}(u_{n})=o(1)$ as $n\rightarrow\infty$ and this
gives rise to an absurdity of $0=-\infty$.
Note that since $|u_{n}|>\underline{u}_{\mu}$, we have

$\displaystyle\underset{n\rightarrow\infty}{\lim}||u_{n}|^{-\gamma-1}u_{n}(u_{n}-u)|$
$\displaystyle=$
$\displaystyle\underset{n\rightarrow\infty}{\lim}||u_{n}|^{1-\gamma}-u|u_{n}|^{-\gamma-1}u_{n}|=0$
(4.3)

and hence

$\underset{n\rightarrow\infty}{\lim}\int_{\Omega}|u_{n}|^{-\gamma-1}u_{n}(u_{n}-u)dx=0.$

We now define a linear functional

$J_{v}(u)=\int_{\mathbb{R}^{N}}|v(x)-v(y)|^{p-2}(v(x)-v(y))(u(x)-u(y))K(x-y)dxdy.$

The Hölder inequality yields

$|J_{v}(u)|\leq\|v\|^{p-1}\|u\|$

for every $u\in X_{0}$ thereby implying that $J$ is a continuous linear
functional on $X_{0}$. Therefore

$\underset{n\rightarrow\infty}{\lim}J_{v}(u_{n}-u)=0.$

From the discussion so far in the proof of this theorem and since
$u_{n}\rightharpoonup u$ in $X_{0}$, we conclude that
$\langle\bar{I}^{\prime}(u_{n}),u_{n}-u\rangle\rightarrow 0$ as
$n\rightarrow\infty$. Also
$(1+\|u_{n}\|_{X_{0}})\bar{I}^{\prime}(u_{n})\rightarrow 0$ in
$X_{0}^{\prime}$, the dual space of $X_{0}$. Therefore, we have

$\displaystyle\begin{split}o(1)=&\langle\bar{I}^{\prime}(u_{n}),u_{n}-u\rangle-\lambda\int_{\Omega}g(x)|u_{n}|^{p-2}u_{n}(u_{n}-u)dx\\\
&-\mu\int_{\Omega}h(x)|u_{n}|^{-\gamma-1}u_{n}(u_{n}-u)dx-\int_{\Omega}f(x,u_{n})(u_{n}-u)dx\\\
=&(a+b\|u_{n}\|^{p})J_{u_{n}}(u_{n}-u)+o(1)~{}\text{as}~{}n\rightarrow\infty.\end{split}$
(4.4)

Hence, by the boundedness of $(u_{n})$ in $X_{0}$ and
$\underset{n\rightarrow\infty}{\lim}J_{v}(u_{n}-u)=0$ we have

$\displaystyle\underset{n\rightarrow\infty}{\lim}(J_{u_{n}}(u_{n}-u)-J_{u}(u_{n}-u))$
$\displaystyle=$ $\displaystyle 0.$ (4.5)

Recall the Simon inequalities which is as follows.

$\displaystyle\begin{split}|U-V|^{p}&\leq
C_{p}(|U|^{p-2}U-|V|^{p-2}V).(U-V),~{}p\geq 2\\\ |U-V|^{p}&\leq
C_{p}^{\prime}(|U|^{p-2}U-|V|^{p-2}V)^{\frac{p}{2}}.(|U|^{p}+|V|^{p})^{\frac{2-p}{2}},~{}1<p<2\end{split}$
(4.6)

for all $U,V\in\mathbb{R}^{N}$, $C_{p},C_{p}^{\prime}>0$ are constants.
When $p\geq 2$, we have

$\displaystyle\begin{split}\|u_{n}-u\|^{p}\leq&C_{p}\int_{\mathbb{R}^{2N}}[|u_{n}(x)-u_{n}(y)|^{p-2}(u_{n}(x)-u_{n}(y))-|u(x)-u(y)|^{p-2}(u(x)-u(y))]\\\
&\times[(u_{n}(x)-u_{n}(y))-(u(x)-u(y))]K(x-y)dxdy\\\
=&C_{p}[J_{u_{n}}(u_{n}-u)-J_{u}(u_{n}-u)]\rightarrow
0~{}\text{as}~{}n\rightarrow\infty.\end{split}$ (4.7)

When $1<p<2$

$\displaystyle\begin{split}\|u_{n}-u\|^{p}\leq&C_{p}^{\prime}[J_{u_{n}}(u_{n}-u)-J_{u}(u_{n}-u)]^{\frac{p}{2}}(\|u_{n}\|^{p}+\|u\|^{p})^{\frac{2-p}{2}}\\\
\leq&C_{p}^{\prime}[J_{u_{n}}(u_{n}-u)-J_{u}(u_{n}-u)]^{\frac{p}{2}}(\|u_{n}\|^{\frac{2-p}{2}}+\|u\|^{\frac{2-p}{2}})\rightarrow
0~{}\text{as}~{}n\rightarrow\infty.\end{split}$ (4.8)

Thus $u_{n}\rightarrow u$ strongly in $X_{0}$ as $n\rightarrow\infty$. ∎

######  Lemma 4.2.

Let $(f_{1}),(f_{3}),(f_{5})$ hold. Then the functional $\bar{I}$ satisfies
the $(Ce)_{c}$ condition.

###### Proof.

Let $(f_{5})$ hold. From the monotonicity of $t\mapsto\frac{f(x,t)}{t^{2p-1}}$
there exists $C_{1}>0$ such that

$\displaystyle\mathfrak{G}(x,s)$ $\displaystyle\leq$
$\displaystyle\mathfrak{G}(x,t)+C,$ (4.9)

for all $x\in\Omega$ and $0\leq|s|\leq|t|$. Refer $(f_{6})$ for the definition
of $\mathfrak{G}(.,.)$. Let $(u_{n})$ be a Cerami sequence in $X_{0}$. Thus
for $c\in\mathbb{R}$

$\displaystyle\begin{split}\bar{I}(u_{n})\rightarrow c~{}\text{in}~{}X_{0}\\\
(1+\|u_{n}\|)\bar{I}^{\prime}(u_{n})\rightarrow
0~{}\text{in}~{}X_{0}^{\prime}\end{split}$ (4.10)

as $n\rightarrow\infty$. All we need to show is that the sequence $(u_{n})$ is
bounded in $X_{0}$ and the conclusion will follow from the Lemma 4.1.
Suppose not, i.e. there exists a subsequence such that
$\|u_{n}\|_{X_{0}}\rightarrow\infty$. By the second condition of (4.10) we
have

$\bar{I}^{\prime}(u_{n})\rightarrow 0$

as $n\rightarrow\infty$. Hence

$\|u_{n}\|\left\langle\bar{I}^{\prime}(u_{n}),\frac{u_{n}}{\|u_{n}\|}\right\rangle\rightarrow
0$

as $n\rightarrow\infty$. We define $\xi_{n}=\frac{u_{n}}{\|u_{n}\|}$ so that
$\|\xi_{n}\|=1$. Thus $(\xi_{n})$ is a bounded sequence and hence

$\displaystyle\begin{split}\xi_{n}\rightarrow\xi~{}\text{in}~{}L^{p}(\Omega)\\\
\xi_{n}\rightarrow\xi~{}\text{in}~{}L^{q}(\Omega)\\\
\xi_{n}\rightarrow\xi~{}\text{a.e. in}~{}\Omega.\end{split}$ (4.11)

Further, from Lemma A.1 of [51] there exists a function $\alpha(x)$ such that

$|\xi_{n}(x)|\leq\alpha(x)~{}\text{in}~{}\mathbb{R}^{N}.$

This leads to the consideration of two cases, viz. $\xi=0$ and $\xi\neq 0$.
Since $\bar{I}$ is a $C^{1}$ functional, therefore
$\bar{I}(\alpha_{n}u_{n})=\underset{\alpha\in[0,1]}{\max}\bar{I}(\alpha
u_{n})$ makes sense.
Let $\xi=0$ and define $h_{T}=\left(\frac{4pT}{b}\right)^{\frac{1}{2p}}$ such
that $\frac{h_{T}}{\|u_{n}\|}\in(0,1)$ for $T\in\mathbb{N}$ and sufficiently
large $n$, say for $n\geq n(T)$. Since $\xi=0$ and by (4.11) we have

$\displaystyle\int_{\Omega}|h_{T}\xi_{n}|^{p}dx\rightarrow 0$ (4.12)

as $n\rightarrow\infty$. By the continuity of $F$ we obtain

$\displaystyle F(x,h_{T}\xi_{n}(x))\rightarrow
F(x,h_{T}\xi(x))=F(x,0)~{}\text{in}~{}\Omega~{}\text{as}~{}n\rightarrow\infty.$
(4.13)

From $(f_{1})$ and $|\xi_{n}(x)|\leq\alpha(x)~{}\text{in}~{}\mathbb{R}^{N}$ in
combination with the Hölder inequality we get

$\displaystyle|F(x,h_{T}\xi_{n})|$ $\displaystyle\leq$ $\displaystyle
C|h_{T}\alpha(x)|+\frac{C}{q}|h_{T}\alpha(x)|^{q}~{}\in L^{1}(\Omega)$ (4.14)

for any $n,T\in\mathbb{N}$. Therefore from the Lebesgue dominated convergence
theorem we get

$\displaystyle F(.,h_{T}\xi_{n}(.))\rightarrow
F(.,h_{T}\xi(.))~{}\text{in}~{}L^{1}(\Omega)~{}\text{as}~{}n\rightarrow\infty.$
(4.15)

for any $T\in\mathbb{N}$. Therefore, since $F(x,0)=0$ for each $x\in\Omega$,
$\xi=0$ and by (4.15), we have

$\int_{\Omega}F(x,h_{T}\xi_{n}(x))dx\rightarrow
0~{}\text{as}~{}n\rightarrow\infty.$

We further have

$\displaystyle\bar{I}(\alpha_{n}u_{n})$ $\displaystyle\geq$
$\displaystyle\bar{I}\left(\frac{h_{T}}{\|u_{n}\|}u_{n}\right)$ (4.16)
$\displaystyle=$ $\displaystyle\bar{I}(h_{T}\xi_{n})$ $\displaystyle\geq$
$\displaystyle\frac{a}{p}\|h_{T}\xi_{n}\|^{p}+\frac{b}{2p}\|h_{T}\xi_{n}\|^{2p}-\frac{\lambda\|g\|_{\infty}}{p}\int_{\Omega}|h_{T}\xi_{n}|^{p}dx$
$\displaystyle-\frac{\mu\|h\|_{\infty}}{1-\gamma}\int_{\Omega}|h_{T}\xi_{n}|^{1-\gamma}dx-\int_{\Omega}F(x,h_{T}\xi_{n})dx$
$\displaystyle\geq$ $\displaystyle\frac{b}{2p}\|h_{T}\xi_{n}\|^{2p}+o(1)=2T$

as $n\rightarrow\infty$ and for any $T\in\mathbb{N}$. Therefore

$\displaystyle\bar{I}(\alpha_{n}u_{n})\rightarrow\infty$ (4.17)

as $n\rightarrow\infty$. We now show that

$\underset{n\rightarrow\infty}{\lim}\sup\bar{I}(\alpha_{n}u_{n})\leq\beta_{0}$

for some $\beta_{0}>0$. Since
$\int_{\Omega}\frac{|u_{n}|^{p}}{\|u_{n}\|^{p}}dx\leq C$ and
$\frac{1}{\|u_{n}\|^{p}}\leq C^{\prime}$ for all $n\in\mathbb{N}$ we have that
$\int_{\Omega}|u_{n}|^{p}dx\leq C^{\prime\prime}$ for all $n\in\mathbb{N}$.
Furthermore, since the boundary $\partial\Omega$ is Lipschitz continuous we
have that

$\displaystyle\begin{split}&\left|\frac{\lambda}{2p}\int_{\Omega}g(x)|\alpha_{n}u_{n}|^{p}dx\right|\leq
C(\lambda,p)\|h\|_{\infty}<\infty\\\
&\left|\mu\left(\frac{1}{1-\gamma}-\frac{1}{2p}\right)\int_{\Omega}h(x)(\alpha_{n}u_{n})^{1-\gamma}dx\right|\leq
C(\mu,\gamma,p)\|h\|_{\infty}<\infty.\end{split}$ (4.18)

Since $\frac{d}{d\alpha}|_{\alpha=\alpha_{n}}\bar{I}(\alpha u_{n})=0$ for all
$n$ we have

$\displaystyle\langle\bar{I}^{\prime}(\alpha_{n}u_{n}),\alpha_{n}u_{n}\rangle$
$\displaystyle=$
$\displaystyle\alpha_{n}\frac{d}{d\alpha}|_{\alpha=\alpha_{n}}\bar{I}(\alpha
u_{n})=0.$ (4.19)

We further have

$\displaystyle\bar{I}(\alpha_{n}u_{n})$ $\displaystyle=$
$\displaystyle\bar{I}(\alpha_{n}u_{n})-\frac{1}{2p}\langle\bar{I}^{\prime}(\alpha_{n}u_{n}),\alpha_{n}u_{n}\rangle$
$\displaystyle=$
$\displaystyle\frac{a}{2p}\|\alpha_{n}u_{n}\|^{p}-\frac{\lambda}{2p}\int_{\Omega}g(x)|\alpha_{n}u_{n}|^{p}dx-\mu\left(\frac{1}{1-\gamma}-\frac{1}{2p}\right)\int_{\Omega}h(x)(\alpha_{n}u_{n})^{1-\gamma}dx$
$\displaystyle-\int_{\Omega}F(x,\alpha_{n}u_{n})dx+\frac{1}{2p}\int_{\Omega}f(x,\alpha_{n}u_{n})\alpha_{n}u_{n}dx$
$\displaystyle\leq$
$\displaystyle\frac{a}{2p}\|\alpha_{n}u_{n}\|^{p}+C(\lambda,p)+C(\mu,\gamma,p)+\int_{\Omega}\mathfrak{G}(x,\alpha_{n}u_{n})dx$
$\displaystyle\leq$
$\displaystyle\frac{a}{2p}\|\alpha_{n}u_{n}\|^{p}+C(\lambda,p)+C(\mu,\gamma,p)+\int_{\Omega}\mathfrak{G}(x,u_{n})dx+C|\Omega|$
$\displaystyle=$
$\displaystyle\frac{a}{2p}\|\alpha_{n}u_{n}\|^{p}+C(\lambda,p)+C(\mu,\gamma,p)+\int_{\Omega}\mathfrak{G}(x,u_{n})dx+C|\Omega|$
(4.20)
$\displaystyle-\frac{\lambda}{2p}\int_{\Omega}g(x)|\alpha_{n}u_{n}|^{p}dx+\frac{\lambda}{2p}\int_{\Omega}g(x)|\alpha_{n}u_{n}|^{p}dx$
$\displaystyle-\mu\left(\frac{1}{1-\gamma}-\frac{1}{2p}\right)\int_{\Omega}h(x)(\alpha_{n}u_{n})^{1-\gamma}dx+\mu\left(\frac{1}{1-\gamma}-\frac{1}{2p}\right)\int_{\Omega}h(x)(\alpha_{n}u_{n})^{1-\gamma}dx$
$\displaystyle+C|\Omega|$ $\displaystyle\leq$
$\displaystyle\overline{I}(\alpha_{n}u_{n})-\frac{1}{2p}\langle\overline{I}^{\prime}(\alpha_{n}u_{n}),\alpha_{n}u_{n}\rangle+2C(\lambda,p)+2C(\mu,\gamma,p)+C|\Omega|$
$\displaystyle=$ $\displaystyle
c+o(1)+2C(\lambda,p)+2C(\mu,\gamma,p)+C|\Omega|<\infty~{}\text{as}~{}n\rightarrow\infty.$

where $|\Omega|$ is the Lebesgue measure of $\Omega$. This is a contradiction
to (4.17). Thus $(u_{n})$ is bounded in $X_{0}$.
Let $\xi\neq 0$. Define

$A=\\{x\in\Omega:\xi(x)\neq 0\\}.$

Therefore we have

$\displaystyle|u_{n}(x)|$ $\displaystyle=$
$\displaystyle|\xi_{n}(x)|\|u_{n}\|\rightarrow\infty~{}\text{in}~{}A~{}\text{as}~{}n\rightarrow\infty.$
(4.21)

Further from $(f_{3})$

$\displaystyle\frac{F(x,u_{n}(x))}{\|u_{n}\|^{2p}}$ $\displaystyle=$
$\displaystyle\frac{F(x,u_{n}(x))}{|u_{n}(x)|^{2p}}\frac{|u_{n}(x)|^{2p}}{\|u_{n}\|^{2p}}$
(4.22) $\displaystyle=$
$\displaystyle\frac{F(x,u_{n}(x))}{|u_{n}(x)|^{2p}}|\xi_{n}|^{2p}\rightarrow\infty~{}\text{in}~{}A~{}\text{as}~{}n\rightarrow\infty.$

By the Fatou’s lemma

$\displaystyle\int_{A}\frac{F(x,u_{n}(x))}{\|u_{n}\|^{2p}}dx\rightarrow\infty.$
(4.23)

Let us now analyse the case over $\Omega\setminus A$. From $(f_{3})$ again we
have

$\underset{|t|\rightarrow\infty}{\lim}F(x,t)=\infty$

for $x\in\Omega$. Therefore for arbitrary $M>0$ there exists $t^{\prime}$ such
that

$\displaystyle F(x,t)\geq M~{}\text{whenever}~{}|t|\geq
t^{\prime},x\in\Omega.$ (4.24)

Hence

$\displaystyle F(x,t)$
$\displaystyle\geq\min\left\\{M,\underset{(x,t)\in\Omega\times[-t^{\prime},t^{\prime}]}{\min}\\{F(x,t)\\}\right\\}=M^{\prime}.$
(4.25)

So we get

$\displaystyle\underset{|t|\rightarrow\infty}{\lim}\int_{\Omega\setminus
A}\frac{F(x,u_{n}(x))}{\|u_{n}\|^{2p}}dx\geq 0.$ (4.26)

Also

$\displaystyle
0\leq\frac{\mu}{1-\gamma}\int_{\Omega}h(x)\frac{|u_{n}|^{1-\gamma}}{\|u_{n}\|^{2p}}dx$
$\displaystyle\leq$
$\displaystyle\frac{\mu\|h\|_{\infty}}{1-\gamma}\|u_{n}\|^{1-\gamma-2p}=o(1).$
(4.27)

By the variational characterization (refer [55]) of $\lambda_{j}$, the $j$-th
eigenvalue of $\mathfrak{L}_{p}^{s}$ is

$\lambda_{j}=\underset{u\in
X_{0}\setminus\\{0\\}}{\min}\left\\{\frac{\int_{\mathbb{R}^{2N}}|u(x)-u(y)|^{p}K(x-y)dxdy}{\int_{\Omega}|u(x)|^{p}dx}\right\\}.$

Now since

$\displaystyle\begin{split}o(1)&=\frac{\overline{I}(u_{n})}{\|u_{n}\|^{2p}}\\\
=&\frac{a}{p\|u_{n}\|^{p}}+\frac{b}{2p}-\frac{\lambda}{p}\int_{\Omega}g(x)\frac{|u_{n}(x)|^{p}}{\|u_{n}\|^{2p}}dx\\\
&-\frac{\mu}{1-\gamma}\int_{\Omega}h(x)\frac{|u_{n}|^{1-\gamma}}{\|u_{n}\|^{2p}}dx-\int_{A}\frac{F(x,u_{n})}{\|u_{n}\|^{2p}}dx-\int_{\Omega\setminus
A}\frac{F(x,u_{n})}{\|u_{n}\|^{2p}}dx\\\
\leq&o(1)+\frac{b}{2p}+\frac{\lambda\|g\|_{\infty}}{p\lambda_{j}}\frac{1}{\|u_{n}\|^{p}}-\frac{\mu}{1-\gamma}\int_{\Omega}h(x)\frac{|u_{n}|^{1-\gamma}}{\|u_{n}\|^{2p}}dx-\int_{A}\frac{F(x,u_{n})}{\|u_{n}\|^{2p}}dx-\int_{\Omega\setminus
A}\frac{F(x,u_{n})}{\|u_{n}\|^{2p}}dx\\\ \leq&-\infty\end{split}$ (4.28)

where the last step is due to (4.23), (4.26), (4.27). This is again an
absurdity. Thus the sequence $(u_{n})$ is bounded in $X_{0}$ and hence by the
Lemma 4.1 we conclude that $(u_{n})$ possesses a strongly convergent
subsequence in $X_{0}$. ∎

We now prove the results stated in the Theorems 3.6 and 3.7. For this let us
develope some prerequisites. It is well known that the space $X_{0}$ is a
Banach space and we have that

$X_{0}=\underset{i\geq 1}{\bigoplus}X_{i}$

where $X_{i}=\text{span}\\{e_{j}\\}_{j\geq i}$. Define

$Y_{m}=\underset{1\leq j\leq m}{\bigoplus}X_{j}$ $Z_{m}=\underset{j\geq
m}{\bigoplus}X_{j}.$

Clearly, $Y_{m}$ is a finite dimensional subspace of $X_{0}$, for each $m$.

######  Theorem 4.3.

Let $\kappa\in[1,p_{s}^{*})$. We have

$\zeta_{m}(\kappa)=\sup\\{\|u\|_{\kappa}:u\in Z_{m},\|u\|=1\\}\rightarrow 0$

as $m\rightarrow\infty$.

###### Proof.

From the definition of $(Z_{m})$ we have that $Z_{m+1}\subset Z_{m}$ and thus
$0\leq\zeta_{m+1}\leq\zeta_{m}$. This implies that
$\zeta_{m}\rightarrow\zeta\geq 0$ as $m\rightarrow\infty$. Further by the
definition of supremum for every $m$ there exists $u_{m}\in Z_{m}$ such that
$\|u_{m}\|=1$ and $\|u_{m}\|_{\kappa}>\frac{\zeta_{m}}{2}$. By the reflexivity
of $X_{0}$ and $\|u_{m}\|=1$ we have that $u_{m}\rightharpoonup 0$ in $X_{0}$.
By the embedding results from Lemma 3.10 we have that $u_{m}\rightarrow 0$ as
$m\rightarrow\infty$ in $L^{\kappa}(\Omega)$ for any $\kappa\in[1,p_{s}^{*})$.
Thus we conclude that $\zeta=0$. ∎

Proof of Theorem 3.6: Since $Y_{m}$ is a finite dimensional space, hence the
norms $\|\cdot\|$ and $\|\cdot\|_{\kappa}$ are equivalent for
$\kappa\in[1,p_{s}^{*})$. Mathematically this means that there exists
$C_{1},C_{2}>0$ such that

$\displaystyle C_{1}\|u\|_{X_{0}}\leq\|u\|_{\kappa}\leq
C_{2}\|u\|_{X_{0}},~{}\text{for any}~{}\kappa\in[1,p_{s}^{*}).$ (4.29)

Observe that $\bar{I}(u)=0$. If $\frac{\lambda}{a}>\lambda_{1}$, then from the
definition of $\lambda_{j}$ given in the previous theorem, for any
$\lambda,\mu\in\mathbb{R}^{+}$ $\exists j>1$ such that $\lambda_{j}$ such that
$\frac{\lambda}{a}\in[\lambda_{j-1},\lambda_{j})$,
$\frac{\mu}{a}\in[\lambda_{k-1},\lambda_{k})$. Further, from $(f_{1})$ and
$(f_{4})$, for every $\epsilon>0$ there exists $C_{\epsilon}>0$ such that

$\displaystyle F(x,t)$ $\displaystyle\leq$
$\displaystyle\frac{\epsilon}{p}|t|^{p}+\frac{C_{\epsilon}}{q}|t|^{q},$ (4.30)

for any $(x,t)\in\Omega\times\mathbb{R}$. By the definition of $\zeta_{m}$ in
Lemma 4.3, fix $\epsilon>0$ and choose $m^{\prime}\geq 1$ such that

$\displaystyle\begin{split}\|u\|_{p}^{p}&\leq\frac{a-\frac{\lambda}{\lambda_{j}}}{2\epsilon}\|u\|^{p}\\\
\|u\|_{q}^{q}&\leq\frac{q(a-\frac{\lambda}{\lambda_{j}})}{2pC_{\epsilon}}\|u\|^{q}~{}\text{for
every}~{}u\in Z_{m^{\prime}}.\end{split}$ (4.31)

Choose $\|u\|=r<1$ sufficiently small, in Theorem 3.13. Since $q>p$, we have

$\displaystyle\begin{split}\bar{I}(u)&=\frac{a}{p}\|u\|^{p}+\frac{b}{2p}\|u\|^{2p}-\frac{\lambda}{p}\int_{\Omega}g(x)|u(x)|^{p}dx-\frac{\mu}{1-\gamma}\int_{\Omega}h(x)u^{1-\gamma}dx-\int_{\Omega}F(x,u)dx\\\
&\geq\frac{a}{p}\|u\|^{p}-\frac{\lambda\|g\|_{\infty}}{p\lambda_{j}}\|u\|^{p}-\frac{C\mu\|h\|_{\infty}}{1-\gamma}\|u\|^{1-\gamma}-\frac{\epsilon}{p}\|u\|_{p}^{p}-\frac{C_{\epsilon}}{q}\|u\|_{q}^{q}\end{split}$
$\displaystyle\begin{split}&\geq\left(\frac{a}{p}-\frac{\lambda\|g\|_{\infty}}{p\lambda_{j}}\right)\|u\|^{p}-\frac{C\mu\|h\|_{\infty}}{1-\gamma}\|u\|^{1-\gamma}-\frac{\left(a-\frac{\lambda}{\lambda_{j}}\right)}{2p}\|u\|^{p}-\frac{\left(a-\frac{\lambda}{\lambda_{j}}\right)}{2p}\|u\|^{q}\\\
&\geq\frac{a-\frac{\lambda}{\lambda_{j}}}{2p}\|u\|^{p}-\frac{a-\frac{\lambda}{\lambda_{j}}}{2p}\|u\|^{q}-\frac{C\mu\|g\|_{\infty}}{1-\gamma}\|u\|^{1-\gamma}+\frac{\lambda}{p\lambda_{j}}(1-\|g\|_{\infty})\|u\|^{p}\\\
&=\frac{a-\frac{\lambda}{\lambda_{j}}}{2p}(r^{p}-r^{q})-\frac{C\mu\|h\|_{\infty}}{1-\gamma}r^{1-\gamma}+\frac{\lambda}{p\lambda_{j}}(1-\|g\|_{\infty})r^{p}=R>0\end{split}$
(4.32)

for sufficiently small $\mu>0$ (the choice of $r$ tackles the case of
$\|g\|_{\infty}>1$). Finally, from $(f_{3})$, there exists
$C_{0}>\frac{b}{2pC^{2p}}$ (a possible choice of $C$, as we shall see later,
is a Sobolev constant), $C_{1}>0$ such that

$\displaystyle F(x,t)$ $\displaystyle\geq$ $\displaystyle C_{0}|t|^{2p}$
(4.33)

for any $x\in\Omega$ and $|t|>C_{1}$. From $(f_{1})$ we have

$\displaystyle|F(x,t)|$ $\displaystyle\leq$ $\displaystyle
C(1+C_{1}^{q-1})|t|,~{}\text{for every}~{}x\in\Omega~{}\text{and}~{}|t|\leq
C_{1}.$ (4.34)

Let $C^{\prime}=C(1+C_{2}^{q-1})>0$. Then we get

$\displaystyle F(x,t)$ $\displaystyle\geq$ $\displaystyle
C_{0}|t|^{2p}-C^{\prime}|t|,~{}\text{for}~{}(x,t)\in\Omega\times\mathbb{R}.$
(4.35)

By the equivalence of norm in $Y_{m}$ and (4.35), we have

$\displaystyle\begin{split}\bar{I}(u)=&\frac{a}{p}\|u\|^{p}+\frac{b}{2p}\|u\|^{2p}-\frac{\lambda}{p}\int_{\Omega}g(x)|u|^{p}dx-\frac{\mu}{1-\gamma}\int_{\Omega}h(x)|u|^{1-\gamma}dx\\\
&-\int_{\Omega}F(x,u)dx\\\
\leq&\frac{a}{p}\|u\|^{p}+\frac{b}{2p}\|u\|^{2p}-\frac{\lambda}{p}\int_{\Omega}g(x)|u|^{p}dx-\int_{\Omega}F(x,u)dx\\\
\leq&\frac{a}{p}\|u\|^{p}+\frac{b}{2p}\|u\|^{2p}-\frac{\lambda}{p\lambda_{j}}\|u\|^{p}-C_{0}\|u\|_{2p}^{2p}+C^{\prime}\|u\|_{1}\\\
\leq&\frac{a}{p}\|u\|^{p}+\left(\frac{b}{2p}-C_{0}C^{2p}\right)\|u\|^{2p}+C_{2}C^{\prime}\|u\|.\end{split}$
(4.36)

Thus for a sufficiently large $r_{0}=r(\bar{X})$, we have $\bar{I}(u)\leq 0$
whenever $\|u\|\geq r_{0}$. Hence, by the Theorem 3.13, there exists an
unbounded sequence of critical values of $\bar{I}$ characterized by a minimax
argument. In other words, from the Remark 3.9 the problem in (3) has
infinitely many solutions and hence the problem (1) also has infinitely many
solutions.

Proof of Theorem 3.7: Suppose now $(f_{6})$ holds. The proof follows verbatim
of the Theorem 3.6, except that we need to prove the inequality in (4.20).
Thus we have

$\displaystyle\begin{split}\frac{1}{\sigma}\bar{I}(\alpha_{n}u_{n})=&\frac{1}{\sigma}\left(\bar{I}(\alpha_{n}u_{n})-\frac{1}{2p}\langle\bar{I}^{\prime}(\alpha_{n}u_{n}),\alpha_{n}u_{n}\rangle\right)\\\
=&\frac{1}{\sigma}\left[\left(\frac{a}{2p}\right)\|\alpha_{n}u_{n}\|^{p}+\frac{\lambda}{2p}\int_{\Omega}g(x)|\alpha_{n}u_{n}|^{p}dx-\mu\left(\frac{1}{1-\gamma}-\frac{1}{2p}\right)\int_{\Omega}h(x)(\alpha_{n}u_{n})^{1-\gamma}dx\right.\\\
&\left.-\int_{\Omega}F(x,\alpha_{n}u_{n})dx+\frac{1}{2p}\int_{\Omega}f(x,\alpha_{n}u_{n})\alpha_{n}u_{n}dx\right]\\\
\leq&\frac{1}{\sigma}\left[\left(\frac{a}{2p}\right)\|\alpha_{n}u_{n}\|^{p}+\int_{\Omega}\mathfrak{G}(x,\alpha_{n}u_{n})dx\right]+C(\lambda,p)\|g\|_{\infty}\\\
\leq&\left(\frac{a}{2p}\right)\|\alpha_{n}u_{n}\|^{p}+\int_{\Omega}\mathfrak{G}(x,\alpha_{n}u_{n})dx+\frac{1}{\sigma}\int_{\Omega}T(x)dx+C(\lambda,p)\|g\|_{\infty}\\\
\leq&\left(\frac{a}{2p}\right)\|\alpha_{n}u_{n}\|^{p}+\int_{\Omega}\mathfrak{G}(x,\alpha_{n}u_{n})dx+\frac{1}{\sigma}\int_{\Omega}T(x)dx+C(\lambda,p)\|g\|_{\infty}\\\
&-\frac{\lambda}{2p}\int_{\Omega}|u_{n}|^{p}dx+\frac{\lambda}{2p}\int_{\Omega}|u_{n}|^{p}dx\end{split}$
$\displaystyle\begin{split}\leq&\bar{I}_{u_{n}}-\frac{1}{2p}\langle\bar{I}^{\prime}(\alpha_{n}u_{n}),\alpha_{n}u_{n}\rangle+2C(\lambda,p)\|g\|_{\infty}+\frac{1}{\sigma}\int_{\Omega}T(x)dx\\\
\leq&c+o(1)+2C(\lambda,p)\|g\|_{\infty}+\frac{1}{\sigma}\int_{\Omega}T(x)dx<\infty.\end{split}$
(4.37)

This completes the proof.

We will now show that the solution $u$ to (1) is in $L^{\infty}(\Omega)$, i.e.
bounded in $\Omega$ for $K(x)=|x|^{-N-sp}$. Firstly, we will prove the
following elementary inequality needed for the proof of the $L^{\infty}$
estimate.

######  Lemma 4.4.

(Lemma 5.1 in [29]) For all $a$, $b\in\mathbb{R}$, $\rho\geq p$, $p\geq 1$,
$k>0$ we have

$\displaystyle\begin{split}\frac{p^{p}(\rho+1-p)}{\rho^{p}}&(a|a|_{k}^{\frac{\rho}{p}-1}-b|b|_{k}^{\frac{\rho}{p}-1})^{p}\\\
&\leq(a|a|_{k}^{\rho-1}-b|b|_{k}^{\rho-1})(a-b)^{p-1}\end{split}$

with the assumption that $a\geq b$.

###### Proof.

Define

$m(t)=\begin{cases}\text{sgn}(t)|t|^{\frac{\rho}{p}-1},&|t|<k\\\
\frac{p}{\rho}\text{sgn}(t)k^{\frac{\rho}{p}-1},&|t|\geq k.\end{cases}$

Observe that

$\displaystyle\begin{split}\int_{b}^{a}m(t)dt&=\frac{p}{\rho}(a|a|_{k}^{\frac{\rho}{p}-1}-b|b|_{k}^{\frac{\rho}{p}-1}).\end{split}$

Similarly,

$\displaystyle\begin{split}\int_{b}^{a}m(t)^{p}dt&\leq\frac{1}{\rho+1-p}(a|a|_{k}^{\rho-p}-b|b|_{k}^{\rho-p}).\end{split}$

On using the Cauchy-Schwartz inequality we obtain

$\displaystyle\begin{split}\left(\int_{b}^{a}m(t)dt\right)^{p}&\leq(a-b)^{p-1}\int_{b}^{a}h(t)^{p}dt.\end{split}$

Thus

$\displaystyle\begin{split}&\frac{p^{p}}{\rho^{p}}(a|a|_{k}^{\frac{\rho}{p}-1}-b|b|_{k}^{\frac{\rho}{p}-1})^{p}\\\
&=\left(\int_{b}^{a}m(t)dt\right)^{p}\\\
&\leq(a-b)^{p-1}\int_{b}^{a}m(t)^{p}dt\\\
&\leq\frac{(a-b)^{p-1}}{\rho+1-p}(a|a|_{k}^{\rho-p}-b|b|_{k}^{\rho-p}).\end{split}$

∎

We now prove that a solution to (1) is bounded in $\Omega$.

######  Theorem 4.5.

Let $f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ be as defined in
$(f_{1})$, then for any weak solution $u\in X_{0}$, we have $u\in
L^{\infty}(\Omega)$.

###### Proof.

Let $1\leq q<p_{s}^{*}$ and let $u$ be any weak solution to the given problem
in (1) and let
$\tilde{\alpha}=\left(\frac{p_{s}^{*}}{p}\right)^{\frac{1}{p}}$. For every
$\rho\geq p(p-1)$, $k>0$, the mapping $t\mapsto t|t|_{k}^{r-p}$ is Lipschitz
in $\mathbb{R}$. Therefore, $u|u|_{k}^{\rho-p}\in X_{0}$. In general for any
$t$ in $\mathbb{R}$ and $k>0$, we have defined
$t_{k}=\text{sgn}(t)\min\\{|t|,k\\}$. Note that, for a fixed solution of (1),
say $u$, we have that $1/\left(a+b\|u\|^{p}\right)$ is finite. We apply the
embedding results due to Theorem 3.10, the previous lemma 4.4, test with the
test function $u|u|_{k}^{\rho-p}$ and on using the growth condition of $f$
which is given in $(f_{1})$ we get

$\displaystyle\begin{split}\|u|u|_{k}^{\frac{\rho}{p}-1}\|_{p_{s}^{*}}^{p}\leq&C\|u|u|_{k}^{\frac{\rho}{p}-1}\|^{p}\\\
\leq&C\frac{\rho^{p}}{\rho+1-p}\langle u,u|u|_{k}^{\rho-p}\rangle\\\
\leq&\rho^{p}C^{\prime}\frac{1}{\left(a+b\|u\|^{p}\right)}\left(\lambda\int_{\Omega}g(x)|u|^{p-1}(|u||u|_{k}^{\rho-p})dx+\mu\int_{\Omega}h(x)|u|^{1-\gamma}(|u|_{k}^{\rho-p})dx\right.\\\
&\left.+\int_{\Omega}|f(x,u)||u||u|_{k}^{\rho-p}dx\right)\\\
\leq&C^{\prime\prime}\rho^{p}\int_{\Omega}\left(g(x)|u|^{p}|u|_{k}^{\rho-p}+h(x)|u|^{1-\gamma}|u|_{k}^{\rho-p}+|u||u|_{k}^{\rho-p}+|u|^{q}|u|_{k}^{\rho-p}\right)dx\\\
\leq&C^{\prime\prime}\rho^{p}\\{\|g\|_{\infty}\int_{\Omega}|u|^{p}|u|_{k}^{\rho-p}dx+\|h\|_{\infty}\int_{\Omega}|u|^{1-\gamma}|u|_{k}^{\rho-p}dx\\\
&+\int_{\Omega}(|u||u|_{k}^{\rho-p}+|u|^{q}|u|_{k}^{\rho-p})dx\\}\\\
\leq&C^{\prime\prime\prime}\rho^{p}\\{\int_{\Omega}|u|^{p}|u|_{k}^{\rho-p}dx+\int_{\Omega}|u|^{1-\gamma}|u|_{k}^{\rho-p}dx+\int_{\Omega}(|u||u|_{k}^{\rho-p}+|u|^{q}|u|_{k}^{\rho-p})dx\\}\end{split}$
(4.38)

for some $C^{\prime\prime\prime}>0$ independent of $\rho\geq p$ and $k>0$; but
dependent on $g,h$ that have been considered in the problem. On applying the
Fatou’s lemma as $k\rightarrow\infty$ gives

$\displaystyle\begin{split}\|u\|_{\tilde{\alpha}^{p}\rho}&\leq
C^{\prime\prime\prime}\rho^{\frac{p}{\rho}}\left\\{\int_{\Omega}(|u|^{\rho}+|u|^{\rho-(p-1)}+|u|^{\rho+q-p}\right.\\\
&+\left.|u|^{\rho-p-\gamma+1})dx\right\\}^{1/\rho}.\end{split}$ (4.39)

The idea is to try and develop an argument to guarantee that $u\in
L^{p_{1}}(\Omega)$ for all $p_{1}\geq 1$. Therefore define a recursive
sequence $(\rho_{n})$ by setting $\rho_{0}=p_{s}^{*}+p-q$,
$\rho_{n+1}=\tilde{\alpha}^{p}\rho_{n}+q-p$. By the choice of $\rho_{0}$, we
have $u\in L^{\rho+q-p}(\Omega)$. Therefore, the choice $\rho=\rho_{0}$ in
(4.39) yields a finite right hand side and so $u\in
L^{\tilde{\alpha}^{p}\rho}(\Omega)=L^{\rho_{1}+q-p}(\Omega)$.

Repeating this argument and using the fact $\rho\mapsto\rho^{1/\rho}$ is
bounded in $[2,\infty)$ for all $n$, we have $u\in
L^{\tilde{\alpha}^{p}\rho_{n}}(\Omega)$. We further have

$\displaystyle\begin{split}\|u\|_{\tilde{\alpha}^{p}\rho_{n}}&\leq
G(n,\|u\|_{p_{s}^{*}}).\end{split}$ (4.40)

Arguments from Iannizzotto [7], guarantees

$\displaystyle\begin{split}\|u\|_{p_{1}}&\leq
G(p_{1},\|u\|_{p_{s}^{*}}),p_{1}\geq 1.\end{split}$ (4.41)

We now improve the estimate in (4.41) by making the function $G$ independent
of $p_{1}$. Set
$\tilde{\alpha}^{\prime}=\frac{\tilde{\alpha}}{\tilde{\alpha}-1}$. Thus by
(4.41) and Hölder’s inequality we have

$\displaystyle\begin{split}\||u|+|u|^{p}+|u|^{q}\|_{\tilde{\alpha}^{\prime}}&\leq
G(\|u\|_{p_{s}^{*}})\end{split}$

Therefore for $\rho\geq p(p-1)$ we have

$\displaystyle\begin{split}&\||u|^{\rho-(p-1)}+|u|^{\rho-p+p}+|u|^{r+q-p}+|u|^{\rho-p-\gamma+1}\|_{\tilde{\alpha}^{\prime}}\\\
&\leq\||u|+|u|^{p}+|u|^{q}+|u|^{1-\gamma}\|_{\alpha^{\prime}}\||u|^{\rho-p}\|_{\tilde{\alpha}}\\\
&\leq G(\|u\|_{p_{s}^{*}})\|u\|_{\tilde{\alpha}(\rho-p)}^{\rho-p}\\\ &\leq
G(\|u\|_{p_{s}^{*}})\|u\|_{\tilde{\alpha}^{p-1}(\rho-p)}^{\rho-p}\\\ &\leq
G(\|u\|_{p_{s}^{*}})|\Omega|^{\frac{1}{\tilde{\alpha}^{p-1}\rho}}\|u\|_{\tilde{\alpha}^{p-1}\rho}^{\rho-p}\end{split}$

We note that $t\mapsto|\Omega|^{p/(\tilde{\alpha}^{p-1}t)}$ is a bounded map
in $[p,\infty)$ and hence

$\displaystyle\begin{split}&\||u|^{\rho-(p-1)}+|u|^{\rho-p+p}+|u|^{\rho+q-p}+|u|^{\rho-p-\gamma+1}\|_{\tilde{\alpha}^{\prime}}\\\
&\leq
G(\|u\|_{p_{s}^{*}})\|u\|_{\tilde{\alpha}^{p-1}\rho}^{\rho-p}\end{split}$
(4.42)

For a sufficiently large $n$ we define $\rho=\tilde{\alpha}^{n-1}>>p$ and
further set $v=\frac{u}{H(\|u\|_{p_{s}^{*}})^{1/p}}$. Using these choices in
(4.39) and the recursive formula we obtain we get

$\displaystyle\|u\|_{\tilde{\alpha}^{n}+p-1}^{\tilde{\alpha}^{n-1}}$
$\displaystyle\leq
G(\|u\|_{p_{s}^{*}})\|u\|_{\tilde{\alpha}^{n-p+2}}^{\tilde{\alpha}^{n-1}-p}.$
(4.43)

On using the definition of $v$ and iterating we get,

$\displaystyle\begin{split}\|v\|_{\tilde{\alpha}^{n+p-1}}&\leq\|v\|_{\tilde{\alpha}^{n+p-2}}^{1-p\tilde{\alpha}^{1-n}}\\\
&\leq\|v\|_{\tilde{\alpha}^{n+p-3}}^{(1-p\tilde{\alpha}^{1-n})(1-(p-1)\tilde{\alpha}^{2-n})}\\\
&\cdots\\\
&\leq\|v\|_{\tilde{\alpha}^{p}}^{\prod_{i=1}^{n-1}[1-p\tilde{\alpha}^{i-n}]}\end{split}$

It is easy to see that the product
$\prod_{i=1}^{n-1}[1-p\tilde{\alpha}^{i-n}]$ is bounded in $\mathbb{R}$ and
hence for all $n$ we have

$\displaystyle\begin{split}\|v\|_{\tilde{\alpha}^{n+p-1}}&\leq\|v\|_{\tilde{\alpha}^{p}}^{\prod_{i=1}^{n-1}[1-p\tilde{\alpha}^{i-n}]}<\infty.\end{split}$

Reverting back to $u$ and recalling the fact that
$\tilde{\alpha}^{n-1}\rightarrow\infty$ as $n\rightarrow\infty$, we find that
there exists $H\in C(\mathbb{R}^{+})$ such that $\|u\|_{p_{1}}\leq
H(\|u\|_{p_{s}^{*}})$ for all $p_{1}\geq 1$. The function $H$ here has been
obtained from the function $G$ which was shown previously. Therefore, we have
$\|u\|_{\infty}<\infty$.

∎

######  Lemma 4.6 (Weak Comparison Principle).

Let $u,v\in X_{0}$. Suppose,
$(a+b\|v\|^{p})\mathfrak{L}_{p}^{s}v-h(x)\frac{\mu}{v^{\gamma}}\geq(a+b\|u\|^{p})\mathfrak{L}_{p}^{s}u-h(x)\frac{\mu}{u^{\gamma}}$
weakly with $v=u=0$ in $\mathbb{R}^{N}\setminus\Omega$. Then $v\geq u$ in
$\mathbb{R}^{N}.$

###### Proof.

Since,
$(a+b\|v\|^{p})\mathfrak{L}_{p}^{s}v-h(x)\frac{\mu}{v^{\gamma}}\geq(a+b\|u\|^{p})\mathfrak{L}_{p}^{s}u-h(x)\frac{\mu}{u^{\gamma}}$
weakly with $u=v=0$ in $\mathbb{R}^{N}\setminus\Omega$, we have

$\displaystyle\langle(a+b\|v\|^{p})\mathfrak{L}_{p}^{s}v,\phi\rangle-\int_{\Omega}h(x)\frac{\mu\phi}{v^{\gamma}}dx$
$\displaystyle\geq\langle(a+b\|u\|^{p})\mathfrak{L}_{p}^{s}u,\phi\rangle-\int_{\Omega}h(x)\frac{\mu\phi}{u^{\gamma}}dx$
(4.44)

$\forall{\phi\geq 0\in X_{0}}$.
Suppose $S=\\{x\in\Omega:u(x)>v(x)\\}$ is a set of non-zero measure. Over this
set $S$, we have

$\displaystyle(a+b\|v\|^{p})\mathfrak{L}_{p}^{s}v-(a+b\|u\|^{p})\mathfrak{L}_{p}^{s}u\geq\mu
h(x)\left(\frac{1}{v^{\gamma}}-\frac{1}{u^{\gamma}}\right)\geq 0.$ (4.45)

Define $\mathfrak{m}(t)=(a+bt^{p})\geq a>0$ for $t\geq 0$ and

$\mathfrak{M}(t)=\int_{0}^{t}\mathfrak{m}(t)dt.$

We will now show that the operator
$\mathfrak{M}(\cdot)\mathfrak{L}_{p}^{s}(\cdot)$ is a monotone operator. By
the Cauchy-Schwartz inequality we have

$\displaystyle|(u(x)-u(y))(v(x)-v(y))|$ $\displaystyle=$
$\displaystyle|u(x)-u(y)||v(x)-v(y)|$ (4.46) $\displaystyle\leq$
$\displaystyle\frac{|u(x)-u(y)|^{2}+|v(x)-v(y)|^{2}}{2}.$

Consider
$I_{1}=\langle\mathfrak{m}(u)\mathfrak{L}_{p}^{s}u,u\rangle-\langle\mathfrak{m}(u)\mathfrak{L}_{p}^{s}u,v\rangle-\langle\mathfrak{m}(v)\mathfrak{L}_{p}^{s}v,u\rangle+\langle\mathfrak{m}(v)\mathfrak{L}_{p}^{s}v,v\rangle$
and let $|u(x)-u(y)|\geq|v(x)-v(y)|$. Therefore using (4.46) we get

$\displaystyle I_{1}$ $\displaystyle=$ $\displaystyle
p\mathfrak{m}(\|u\|^{p})\left(\int_{Q}|u(x)-u(y)|^{p-2}\\{|u(x)-u(y)|^{2}\right.$
$\displaystyle\left.-(u(x)-u(y))(v(x)-v(y))\\}dxdy\right)$
$\displaystyle+p\mathfrak{m}(\|v\|^{p})\left(\int_{Q}|v(x)-v(y)|^{p-2}\\{|v(x)-v(y)|^{2}\right.$
$\displaystyle\left.-(u(x)-u(y))(v(x)-v(y))\\}dxdy\right)$ $\displaystyle\geq$
$\displaystyle\frac{p}{2}\mathfrak{m}(\|u\|^{p})\left(\int_{Q}|u(x)-u(y)|^{p-2}\\{|u(x)-u(y)|^{2}\right.$
(4.47) $\displaystyle\left.-|v(x)-v(y)|^{2}\\}dxdy\right)$
$\displaystyle+\frac{p}{2}\mathfrak{m}(\|v\|^{p})\left(\int_{Q}|v(x)-v(y)|^{p-2}\\{|v(x)-v(y)|^{2}\right.$
$\displaystyle\left.-|u(x)-u(y)|^{2}\\}dxdy\right)$ $\displaystyle\geq$
$\displaystyle\frac{p}{2}\mathfrak{m}(\|u\|^{p})\left(\int_{Q}(|u(x)-u(y)|^{p-2}-|v(x)-v(y)|^{p-2})(|u(x)-u(y)|^{2}-|v(x)-v(y)|^{2})dx\right).$
$\displaystyle\geq$
$\displaystyle\frac{p}{2}a\left(\int_{Q}(|u(x)-u(y)|^{p-2}-|v(x)-v(y)|^{p-2})(|u(x)-u(y)|^{2}-|v(x)-v(y)|^{2})dx\right).$

When $|u(x)-u(y)|\leq|v(x)-v(y)|$, we interchange the roles of $u$, $v$ to get

$\displaystyle I_{1}$ $\displaystyle\geq$ $\displaystyle
pa\left(\int_{Q}(|u(x)-u(y)|^{p-2}-|v(x)-v(y)|^{p-2})(|u(x)-u(y)|^{2}-|v(x)-v(y)|^{2})dx\right).$

Thus

$\displaystyle\langle\mathfrak{m}(u)\mathfrak{L}_{p}^{s}u-\mathfrak{m}(v)\mathfrak{L}_{p}^{s}v,u-v\rangle$
$\displaystyle=$ $\displaystyle I_{1}\geq 0.$ (4.49)

Thus $\mathfrak{m}(\cdot)\mathfrak{L}_{p}^{s}(\cdot)$ is a monotone operator.
This monotonicity is sufficient for our work.
Coming back to (4.45), by the monotonicity of
$\mathfrak{m}(\cdot)\mathfrak{L}_{p}^{s}(\cdot)$ thus proved implies that
$v\geq u$ in $S$. Therefore $u=v$ in $S$ and hence $u\geq v$ a.e. in $\Omega$.
∎

###  4.1 $C^{1}$ versus $W^{s,p}$ local minimizers of the energy

This section is devoted towards discussing ‘$C^{1}$ versus $W^{s,p}$’ analysis
of a solution to (1) for a particular class of Kernel $K(x)=|x|^{-N-sp}$. Some
motivation has been drawn from the works of [20, 26, 29]. Let us begin with
some well-known results and prove a few lemmas towards which a geometrical
property of a general bounded domain $\Omega$ with $C^{1,1}$ boundary is
stated and is as follows.

######  Lemma 4.7 (Lemma 3.5, Iannizzotto [8]).

Let $\Omega\subset\mathbb{R}^{N}$ be a bounded domain with a $C^{1,1}$
boundary $\partial\Omega$. Then, there exist $\rho>0$ such that for all
$x_{0}\in\partial\Omega$ there exist $x_{1},x_{2}\in\mathbb{R}^{N}$ on the
normal line to $\partial\Omega$ at $x_{0}$, with the following properties

  1. (i)

$B_{r}(x_{1})\subset\Omega$, $B_{r}(x_{2})\subset\Omega^{c}$;

  2. (ii)

$\bar{B}_{r}(x_{1})\cap\bar{B}_{r}(x_{2})=\\{x_{0}\\}$;

  3. (iii)

$d(x)=|x-x_{0}|$ for all $x\in[x_{0},x_{1}]$.

Using the Lemma 4.7, we generalize two of the results from Iannizzotto [8].
Before that, we set $\forall~{}R>0$, $x_{0}\in\mathbb{R}^{N}$

$\displaystyle\begin{split}Q(u;x_{0},R)&=\|u\|_{L^{\infty}(B_{R}(x_{0}))}+\text{Tail}(u;x_{0},R)\\\
Q(u,R)&=Q(u;0,R)\\\
d(x,\partial\Omega)&=\inf_{y\in\Omega}\\{d(x,y)\\}.\end{split}$ (4.50)

We now state the following Lemmas from [29] which is applicable to the present
work as well. The proofs follow verbatim of lemma $5.4,5.6,5.8$ of [29].

######  Lemma 4.8.

For any $r>0$, there exists $C^{\prime\prime}>0$ such that
$|(a+b\|u\|^{p})\mathfrak{L}_{p}^{s}u|\leq C^{\prime\prime}$ in $B_{r}(x)$,
where $u$ is a weak solution to the problem (1).

######  Lemma 4.9.

There exists $0<\delta\leq s$ such that any weak solution $u$ to the problem
(1) we have $[u/d^{s}]_{C^{\delta}(\overline{\Omega})}\leq K$.

######  Remark 4.10.

From [9] we can say that $u\in C^{1}(\bar{\Omega})$.

######  Lemma 4.11.

There exists $0<\delta\leq s$ such that for any weak solution $u$ of the
problem (1) we have $[Du]_{C^{\delta}(\overline{\Omega})}\leq C$.

We will now prove the Theorem 3.8. The main tool to prove this result requires
an application of the Lagrange multiplier rule which is given in the form of
Theorem $3.1$ from [34].

######  Theorem 4.12.

Let $L$ and $J$ be real $C^{1}$ functionals on a real Banach space say $X$. If
$z_{0}\in X$ satisfies the following problem:

$\text{minimizing}~{}L(z)~{}\text{under the constratint}~{}J(z)=0.$

Then there exists $\Lambda\in\mathbb{R}$ such that $L^{\prime}(z_{0})=\Lambda
J^{\prime}(z_{0})$.

For a more generalized version of the result, one may refer to [21] and the
references therein.
Proof of Theorem 3.8: Let $\Omega^{\prime}\Subset\Omega$. We will work with
the functional $\bar{I}$ since, a critical point of $\bar{I}$ is also a
critical point of $I$ whenever $u>\underline{u}_{\mu}$. We will only consider
the subcritical case i.e. when $q<p_{s}^{*}-1$. We prove by contradiction,
i.e. suppose $u_{0}$ is not a local minimizer. Let $r\in(q,p_{s}^{*}-1)$ and
define

$\displaystyle\begin{split}J(w)&=\frac{1}{r+1}\int_{\Omega^{\prime}}|w-u_{0}|^{r+1}dx,(w\in
W^{s,p}(\Omega^{\prime})).\end{split}$ (4.51)

Case i: Let $J(v_{\epsilon})<\epsilon$.
Define $S_{\epsilon}=\\{v\in W_{0}^{s,p}(\Omega):0\leq J(v)\leq\epsilon\\}$.
Consider the problem $I_{\epsilon}=\underset{v\in
S_{\epsilon}}{\min}\\{\bar{I}(v)\\}$. The infimum exists since the set
$S_{\epsilon}$ is bounded and the functional $\bar{I}$ is $C^{1}$.
Furthermore, $\bar{I}$ is also weakly lower semicontinuous and $S_{\epsilon}$
is closed, convex. Thus $I_{\epsilon}$ is actually attained, at say
$v_{\epsilon}\in S_{\epsilon}$, and
$I_{\epsilon}=\bar{I}(v_{\epsilon})<\bar{I}(u_{0})$.
Claim: We now show that $\exists\eta>0$ such that
$v_{\epsilon}\geq\eta\phi_{1}$, where $\phi_{1}$ is the eigenvector
corresponding to the first eigenvalue, say $\tilde{\lambda}_{1}$, of the
operator $(a+b\|\cdot\|^{p})\mathfrak{L}_{p}^{s}(\cdot)$.
Proof: We begin by observing that the existence of $\phi_{1}$ can be proved to
exist from the Lemma 5.2. Define $v_{\eta}=(\eta\phi_{1}-v_{\epsilon})^{+}$.
We prove the claim by contradiction, i.e. $\forall\eta>0$ let
$|\Omega_{\eta}|=|supp\\{(\eta\phi_{1}-v_{\epsilon})^{+}\\}|>0$. For $0<t<1$,
define $\xi(t)=\bar{I}(v_{\epsilon}+tv_{\eta})$. Thus

$\displaystyle\begin{split}\xi^{\prime}(t)&=\langle\bar{I}^{\prime}(v_{\epsilon}+tv_{\eta}),v_{\eta}\rangle\\\
&=\langle(a+b\|u\|^{p})\mathfrak{L}_{p}^{s}(v_{\epsilon}+tv_{\eta})-\lambda
g(x)(v_{\epsilon}+tv_{\eta})^{p-1}-\mu
h(x)(v_{\epsilon}+tv_{\eta})^{-\gamma}-f(x,v_{\epsilon}+tv_{\eta}),v_{\eta}\rangle.\end{split}$
(4.52)

Similarly,

$\displaystyle\begin{split}\xi^{\prime}(1)=&\langle\bar{I}^{\prime}(v_{\epsilon}+v_{\eta}),v_{\eta}\rangle\\\
=&\langle\bar{I}^{\prime}(\eta\phi_{1}),v_{\eta}\rangle\\\
=&\langle(a+b\|\eta\phi_{1}\|^{p})\mathfrak{L}_{p}^{s}(\eta\phi_{1})-\lambda
g(x)(\eta\phi_{1})^{p-1}-\mu h(x)(\eta\phi_{1})^{-\gamma}\\\
&-f(x,\eta\phi_{1}),v_{\eta}\rangle<0\end{split}$ (4.53)

for sufficiently small $\eta>0$. Moreover,

$\displaystyle\begin{split}-\xi^{\prime}(1)+\xi^{\prime}(t)=&\langle(a+b\|u\|^{p})\mathfrak{L}_{p}^{s}(v_{\epsilon}+tv_{\eta})-(a+b\|u\|^{p})\mathfrak{L}_{p}^{s}(v_{\epsilon}+v_{\eta})\\\
&+\lambda
g(x)((v_{\epsilon}+v_{\eta})^{p-1}-(v_{\epsilon}+tv_{\eta})^{p-1})\\\ &+\mu
h(x)((v_{\epsilon}+v_{\eta})^{-\gamma}-(v_{\epsilon}+tv_{\eta})^{-\gamma})\\\
&+(f(x,v_{\epsilon}+v_{\eta})-f(v_{\epsilon}+tv_{\eta}),v_{\eta}\rangle\leq
0\end{split}$ (4.54)

since $\lambda g(x)t^{p-1}+\mu h(x)t^{-\gamma}+f(.,t)$ is a uniformly
nonincreasing function with respect to, sufficiently small, $t>0$ for
$x\in\Omega$. From the monotonicity of
$(a+b\|\cdot\|^{p})\mathfrak{L}_{p}^{s}(\cdot)$ (refer Theorem 4.6) we have
that, for sufficiently small $\eta>0$, $0\leq\xi^{\prime}(1)-\xi^{\prime}(t)$.
From the Taylor series expansion and the fact that $J(v_{\epsilon})<\epsilon$,
$\exists~{}0<\theta<1$ such that

$\displaystyle\begin{split}0&\leq\bar{I}(v_{\epsilon}+v_{\eta})-\bar{I}(v_{\epsilon})\\\
&=\langle\bar{I}^{\prime}(v_{\epsilon}+\theta v_{\eta}),v_{\eta}\rangle\\\
&=\xi^{\prime}(\theta).\end{split}$ (4.55)

Thus for $t=\theta$ we get $\xi^{\prime}(\theta)\geq 0$ which is a
contradiction to $\xi^{\prime}(\theta)\leq\xi^{\prime}(1)<0$ as obtained above
in (4.53) and (4.54). Thus $v_{\epsilon}\geq\eta\phi_{1}$ for some $\eta>0$.
In fact, from the Lemmas 4.9 and 4.11 we have
$\underset{\epsilon\in(0,1]}{\sup}\\{\|v_{\epsilon}\|_{C^{1,\delta}(\bar{\Omega})}\\}\leq
C$. By the compact embedding $C^{1,\delta}(\bar{\Omega})\hookrightarrow
C^{1,\kappa}\overline{\Omega})$, for any $\kappa<\delta$, we have
$v_{\epsilon}\rightarrow u_{0}$ which contradicts the assumption made. Hence
$\bar{I}$ attains its minimum at $u_{0}$.
Case ii: $J(v_{\epsilon})=\epsilon$.
Let $v_{\eta}=(\eta\phi_{1}-v_{\epsilon})^{+}$ and
$\xi(t)=\bar{I}(v_{\epsilon}+tv_{\eta})$. Then by arguments as in Case i, we
have that $\xi$ is decreasing. This implies that
$\bar{I}(v_{\epsilon})>\bar{I}(v_{\epsilon}+tv_{\eta})$. Since the functionals
$\bar{I}$, $J$ are $C^{1}$, hence in this case from the Lagrange multiplier
rule (refer Theorem 4.12) there exists $\Lambda_{\epsilon}\in\mathbb{R}$ such
that
$\bar{I}^{\prime}(v_{\epsilon})=\Lambda_{\epsilon}J^{\prime}(v_{\epsilon})$.
We will first show that $\Lambda_{\epsilon}\leq 0$. Suppose
$\Lambda_{\epsilon}>0$, then $\exists~{}\phi\in X_{0}$ such that

$\displaystyle\begin{split}\langle\bar{I}^{\prime}(v_{\epsilon}),\phi\rangle<0~{}\text{and}~{}\langle
J^{\prime}(v_{\epsilon}),\phi\rangle<0.\end{split}$

Then for small $t>0$ we have

$\displaystyle\begin{split}\bar{I}(v_{\epsilon}+t\phi)&<\bar{I}(v_{\epsilon})\\\
J(v_{\epsilon}+t\phi)&<J(v_{\epsilon})=\epsilon\end{split}$

which is a contradiction to $v_{\epsilon}$ being a minimizer of $\bar{I}$ in
$S_{\epsilon}$.
We now consider the following two cases.
Case a: ($\Lambda_{\epsilon}\in(-l,0)$ where $l>-\infty$).
Now consider the sequence of problems

$\displaystyle\begin{split}(P_{\epsilon}):~{}(a+b\|u\|^{p})\mathfrak{L}_{p}^{s}u&=\lambda
g(x)u^{p-1}+\mu
h(x)u^{-\gamma}+f(x,u)+\Lambda_{\epsilon}|u-u_{0}|^{r-1}(u-u_{0})\end{split}$
(4.56)

Observe that $u_{0}$ is a weak solution to ($P_{\epsilon}$). From the weak
comparison principle (Theorem 4.6) we have $v_{\epsilon}\geq\eta\phi_{1}$ for
some $\eta>0$ small enough, independent of $\epsilon$ since $\eta\phi_{1}$ is
a strict subsolution to $(P_{\epsilon})$. Further, since
$-l\leq\Lambda_{\epsilon}\leq 0$, there exist $M$, $c$ such that

$\displaystyle\begin{split}(a+b\|(v_{\epsilon}-1)^{+}\|^{p})\mathfrak{L}_{p}^{s}(v_{\epsilon}-1)^{+}&\leq
M+c((v_{\epsilon}-1)^{+})^{r}.\end{split}$ (4.57)

Using the Moser iteration technique as in Theorem 4.5 we obtain
$\|v_{\epsilon}\|_{\infty}\leq C^{\prime}$. Therefore $\exists L>0$ such that
$\eta\phi_{1}\leq v_{\epsilon}\leq L\phi_{1}$. By using the arguments
previously used in Case i, we end up getting
$|v_{\epsilon}|_{C^{1,\delta}(\bar{\Omega})}\leq C^{\prime}$ for every
$\epsilon>0$. The conclusion follows as in the previous case of
$J(v_{\epsilon})<\epsilon$. Hence $\bar{I}$ attains its minimum at $u_{0}$.
Case b: $\underset{\epsilon>0}{\inf}\\{\Lambda_{\epsilon}\\}=-\infty$
Let us assume $\Lambda_{\epsilon}\leq-1$. As above, we can similarly obtain
$v_{\epsilon}\geq\eta\phi_{1}$ for $\eta>0$ small enough and independent of
$\epsilon$. Further, there exists a constant $M>0$ such that $\lambda
g(x)t^{p-1}+\mu h(x)t^{-\gamma}+f(x,t)+\tau|t-u_{0}(x)|^{r-1}(t-u_{0}(x))<0$,
$\forall(\tau,x,t)\in(-\infty,-1]\times\Omega\times(M,\infty)$.
From the weak comparison principle on
$(a+b\|\cdot\|^{p})\mathfrak{L}_{p}^{s}(\cdot)$, we get $v_{\epsilon}\leq M$
for $\epsilon>0$ sufficiently small. Since $u_{0}$ is a local $C^{1}$ \-
minimizer, $u_{0}$ is a weak solution to (1) and hence

$\displaystyle\begin{split}\langle(a+b\|u_{0}\|^{p})\mathfrak{L}_{p}^{s}u_{0},\phi\rangle&=\lambda\int_{\Omega}g(x)u_{0}^{p-1}\phi
dx+\mu\int_{\Omega}h(x)u_{0}^{-\gamma}\phi dx+\int_{\Omega}f(x,u_{0})\phi
dx\end{split}$ (4.58)

$\forall\phi\in C_{c}^{\infty}(\Omega)$. Also, $u_{0}$ satisfies

$\displaystyle\begin{split}\langle(a+b\|u_{0}\|^{p})\mathfrak{L}_{p}^{s}u_{0},w\rangle&=\lambda\int_{\Omega}g(x)u_{0}^{p-1}w+\mu\int_{\Omega}h(x)u_{0}^{-\gamma}wdx+\int_{\Omega}f(x,u_{0})wdx.\end{split}$
(4.59)

Similarly,

$\displaystyle\begin{split}\langle(a+b\|v_{\epsilon}\|^{p})\mathfrak{L}_{p}^{s}v_{\epsilon},w\rangle&=\lambda\int_{\Omega}g(x)v_{\epsilon}^{p-1}w+\mu\int_{\Omega}h(x)v_{\epsilon}^{-\gamma}wdx+\int_{\Omega}f(x,v_{\epsilon})wdx.\end{split}$
(4.60)

On subtracting (4.59) from (4.60) and testing with
$w=|v_{\epsilon}-u_{0}|^{\beta-1}(v_{\epsilon}-u_{0})$, where $\beta\geq 1$,
we obtain

$\displaystyle\begin{split}0=&\beta\langle(a+b\|u_{\epsilon}\|^{p})\mathfrak{L}_{p}^{s}v_{\epsilon}-(a+b\|u_{0}\|^{p})\mathfrak{L}_{p}^{s}u_{0},|v_{\epsilon}-u_{0}|^{\beta-1}(v_{\epsilon}-u_{0})\rangle\\\
&-\lambda\int_{\Omega}g(x)(v_{\epsilon}^{p-1}-u_{0}^{p-1})|v_{\epsilon}-u_{0}|^{\beta-1}(v_{\epsilon}-u_{0})dx\\\
&-\mu\int_{\Omega}h(x)(v_{\epsilon}^{-\gamma}-u_{0}^{-\gamma})|v_{\epsilon}-u_{0}|^{\beta-1}(v_{\epsilon}-u_{0})dx\\\
=&\int_{\Omega}(f(x,v_{\epsilon})-f(x,u_{0}))|v_{\epsilon}-u_{0}|^{\beta-1}(v_{\epsilon}-u_{0})dx\\\
&+\Lambda_{\epsilon}\int_{\Omega}|v_{\epsilon}-u_{0}|^{\beta+r}dx.\end{split}$
(4.61)

By the Hölder’s inequality and the bounds of $v_{\epsilon}$, $u_{0}$ we obtain

$\displaystyle\begin{split}-\Lambda_{\epsilon}\|v_{\epsilon}-u_{0}\|_{\beta+r}^{r}&\leq
C|\Omega|^{\frac{r}{\beta+r}}.\end{split}$ (4.62)

Here $C$ is independent of $\epsilon$ and $\beta$. On passing the limit
$\beta\rightarrow\infty$ we get
$-\Lambda_{\epsilon}\|v_{\epsilon}-u_{0}\|_{\infty}\leq C$. Working on similar
lines we end up getting $v_{\epsilon}$ is bounded in
$C^{1,\delta}(\bar{\Omega})$ independent of $\epsilon$ and the conclusion
follows.

##  5 Appendix

The appendix will address a few results that have been used in this article.
Lemma 5.2 guarantees the existence of a positive solution to (3), Lemma 5.5
will establish that the functional $\bar{I}$ verifies the mountain pass
geometry, whereas Lemma 5.4 guarantees that a solution to (1) is greater than
or equal to the solution to (3).

######  Remark 5.1.

By saying “$u>0$ in $\Omega$” we will mean $\underset{V}{\text{ess}\inf}u>0$
for any compact set $V\subset\Omega$.

######  Lemma 5.2.

Let $0<\gamma<1$, $\lambda,\mu>0$. Then the following problem

$\displaystyle\left(a+b\int_{\mathbb{R}^{N}}|u(x)-u(y)|^{p}K(x-y)dxdy\right)\mathfrak{L}_{p}^{s}u$
$\displaystyle=$ $\displaystyle\mu h(x)u^{-\gamma},~{}\text{in}~{}\Omega$
$\displaystyle u$ $\displaystyle>$ $\displaystyle 0,~{}\text{in}~{}\Omega$
$\displaystyle u$ $\displaystyle=$ $\displaystyle
0,~{}\text{in}~{}\mathbb{R}^{N}\setminus\Omega$

has a unique weak solution in $X_{0}$. This solution is denoted by
$\underline{u}_{\mu}$, satisfies $\underline{u}_{\mu}\geq\epsilon_{\mu}v_{0}$
a.e. in $\Omega$, where $\epsilon_{\mu}>0$ is a constant.

###### Proof.

We follow the proof in [19]. Firstly, we note that an energy functional on
$X_{0}$ formally corresponding to (5.2) can be defined as follows.

$\displaystyle E(u)$
$\displaystyle=\frac{a}{p}\|u\|^{p}+\frac{b}{2p}\|u\|^{2p}-\frac{\mu}{1-\gamma}\int_{\Omega}h(x)(u^{+})^{1-\gamma}dx$
(5.2)

for $u\in X_{0}$. By the Poincaré inequality, this functional is coercive and
continuous on $X_{0}$. It follows that $E$ possesses a global minimizer
$u_{0}\in X_{0}$. Clearly, $u_{0}\neq 0$ since $E(0)=0>E(\epsilon v_{0})$ for
sufficiently small $\epsilon$ and some $v_{0}>0$ in $\Omega$.
Secondly, we have the decomposition $u=u^{+}-u^{-}$. Thus if $u_{0}$ is a
global minimizer for $E$, then so is $|u_{0}|$, by $E(|u_{0}|)\leq E(u_{0})$.
Clearly enough, the equality holds iff $u_{0}^{-}=0$ a.e. in $\Omega$. In
other words we need to have $u_{0}\geq 0$, i.e. $u_{0}\in X_{0}$ where

$X_{0}^{+}=\\{u\in X_{0}:u\geq 0~{}\text{a.e. in}~{}\Omega\\}$

is the positive cone in $X_{0}$.
Third, we will show that $u_{0}\geq\epsilon v_{0}>0$ holds a.e. in $\Omega$
for small enough $\epsilon$. Observe that,

$\displaystyle\begin{split}E^{\prime}(tv_{0})|_{t=\epsilon}=&a\epsilon^{p-1}\|v_{0}\|^{p}+b\epsilon^{2p-1}\|v_{0}\|^{2p}-\mu\epsilon^{-\gamma}\int_{\Omega}h(x)v_{0}^{1-\gamma}dx<0\end{split}$
(5.3)

whenever $0<\epsilon\leq\epsilon_{\mu}$ for some sufficiently small
$\epsilon_{\mu}$. We now show that $u_{0}\geq\epsilon_{\mu}v_{0}$. On the
contrary, suppose $w=(\epsilon_{\mu}v_{0}-u_{0})^{+}$ does not vanish
identically in $\Omega$. Denote

$\Omega^{+}=\\{x\in\Omega:w(x)>0\\}.$

We will analyse the function $\zeta(t)=E(u_{0}+tw)$ of $t\geq 0$. This
function is convex owing to its definition over $X_{0}^{+}$ being convex.
Further $\zeta^{\prime}(t)=\langle E^{\prime}(u_{0}+tw),w\rangle$ is
nonnegative and nondecreasing for $t>0$. Consequently for $0<t<1$ we have

$\displaystyle\begin{split}0\leq\zeta^{\prime}(1)-\zeta^{\prime}(t)&=\langle
E^{\prime}(u_{0}+w)-E^{\prime}(u_{0}+tw),w\rangle\\\
&=\int_{\Omega^{+}}E^{\prime}(u_{0}+w)dx-\zeta^{\prime}(t)\\\ &<0\end{split}$
(5.4)

by inequality (5.3) and $\zeta^{\prime}(t)\geq 0$ with $\zeta^{\prime}(t)$
being nondecreasing for every $t>0$, which is a contradiction. Therefore $w=0$
in $\Omega$ and hence $u_{0}\geq\epsilon_{\mu}v_{0}$ a.e. in $\Omega$.
Finally, the functional $E$ being strictly convex on $X_{0}^{+}$, we conclude
that $u_{0}$ is the only critical point of $E$ in $X_{0}^{+}$ with the
property $\underset{V}{\text{ess}\inf}u_{0}>0$ for any compact subset
$V\subset\Omega$. Therefore we choose $\underline{u}_{\mu}=u_{0}$ in the
cutoff functional. ∎

######  Remark 5.3.

We now perform an apriori analysis on a solution (if it exists). Suppose $u$
is a solution to (1), then we observe the following

  1. 1.

$I(u)=I(|u|)$. This implies that $u^{-}=0$ a.e. in $\Omega$.

  2. 2.

In fact a solution to (1) can be considered to be positive, i.e. $u>0$ a.e. in
$\Omega$ due to the presence of the singular term.

Thus without loss of generality, we assume that the solution is positive.

Precisely, we now have the following result.

######  Lemma 5.4 (Apriori analysis).

Fix a $\mu\in(0,\mu_{0})$. Then a solution of (1), say $u>0$, is such that
$u\geq\underline{u}_{\lambda}$ a.e. in $\Omega$.

###### Proof.

Fix $\mu\in(0,\mu_{0})$ and let $u\in X_{0}$ be a positive solution to (1) and
$\underline{u}_{\lambda}>0$ be a solution to (5.2). We will show that
$u\geq\underline{u}_{\lambda}$ a.e. in $\Omega$. Thus, we let
$\underline{\Omega}=\\{x\in\Omega:u(x)<\underline{u}_{\lambda}(x)\\}$ and from
the equation satisfied by $u$, $\underline{u}_{\lambda}$, we have

$\displaystyle 0\leq$
$\displaystyle\langle(a+b\|\underline{u}_{\lambda}\|^{p})\mathfrak{L}_{p}^{s}\underline{u}_{\lambda}-(a+b\|u\|^{p})\mathfrak{L}_{p}^{s}u,\underline{u}_{\lambda}-u\rangle_{\underline{\Omega}}+\lambda\int_{\underline{\Omega}}g(x)u^{p-1}(\underline{u}_{\lambda}-u)dx$
$\displaystyle\leq$
$\displaystyle\mu\int_{\underline{\Omega}}h(x)(\underline{u}_{\lambda}^{-\gamma}-u^{-\gamma})(\underline{u}_{\lambda}-u)dx\leq
0.$ (5.5)

Further we have

$\displaystyle\langle(a+b\|\underline{u}_{\lambda}\|^{p})\mathfrak{L}_{p}^{s}\underline{u}_{\lambda}-(a+b\|u\|^{p})\mathfrak{L}_{p}^{s}u,\underline{u}_{\lambda}-u\rangle_{\underline{\Omega}}$
$\displaystyle\geq 0.$ (5.6)

Hence, from (5) and (5.6), we obtain $u\geq\underline{u}_{\lambda}$ a.e. in
$\Omega^{c}$. ∎

######  Lemma 5.5.

The redefined functional $\bar{I}$ given in (3.8) verifies the mountain pass
geometry for $\mu\in(0,\mu_{0})$ with $\mu_{0}<\infty$.

###### Proof.

By the Sobolev embedding we obtain

$\bar{I}(u)\geq\frac{a}{p}\|u\|^{p}+\frac{b}{2p}\|u\|^{2p}-\frac{\lambda\|g\|_{\infty}C_{1}}{p}\|u\|^{p}-\frac{\mu\|h\|_{\infty}C_{2}}{1-\gamma}\|u\|^{1-\gamma}-\int_{\Omega}F(x,u)dx.$

where $C_{1},C_{2}>0$ are uniform constants that are independent of the choice
of $u$ and $F(x,t)=\int_{0}^{t}f(x,\omega)d\omega$. Now for a pair $(\mu,r)$,
sufficiently small $\mu>0$ say $\mu_{0}$, we have that
$\frac{a}{p}\|u\|^{p}+\frac{b}{2p}\|u\|^{2p}-\frac{\mu\|h\|_{\infty}C_{2}}{1-\gamma}\|u\|^{1-\gamma}>0$
for each $\mu\in(0,\mu_{0})$ and $\|u\|=r$ sufficiently small. Define
$a(r)=\frac{a}{p}r^{p}+\frac{b}{2p}r^{2p}-\frac{\mu\|h\|_{\infty}C_{2}}{1-\gamma}r^{1-\gamma}$.
Therefore, to sum it up we have

$\bar{I}(u)\geq a(r)>0$

for any $\mu\in(0,\mu_{0})$ and for every $u$ such that $\|u\|=r$. On the
other hand, taking $u\in X_{0}$ and $t\geq 0$ we have
$\bar{I}(tu)\rightarrow-\infty$ as $t\rightarrow\infty$. This verifies the
second condition of the Mountain pass theorem. ∎

## Conclusions

Existence of infinitely many solutions to the problem in (1) has been proved.
In addition, a weak comparison result has been proved. It has also been shown
that the solutions are in $L^{\infty}(\Omega)$ when $K(x)=|x|^{-N-sp}$.
Further for this particular kernel $K$, it has also been proved that the
$C^{1}$ minimizers are the $W_{0}^{s,p}$ minimizers as well. Some future scope
of work on this line would be to prove that the $C^{1}$ minimizers are the
$W_{0}^{s,p}$ minimizers as well for a general Kernel satisfying (1.2).

## Acknowledgement

The author thanks S. Ghosh for the numerous discussion sessions and the
constructive criticisms on the article. Thanks are due to the anonymous
reviewers for their constructive comments that led to the improvement of this
manuscript. The author also dedicates this article to thousands of labourers
and workers of India who, during this COVID19 pandemic, have lost their lives
travelling on foot for thousands of kilometres to reach their respective
homes.

## References

  * [1] A. Ambrosetti, P.H. Rabinowitz,  Dual variational methods in critical point theory and applications,  Journal of Functional Analysis, 14, 349-381, 1973.
  * [2] A. Daoues, A. Hammami, K. Saoudi,  Multiple positive solutions for a nonlocal PDE with critical Sobolev-Hardy and singular nonlinearities via perturbation method,  Fract. Calc. Appl. Anal., 23 (3), 837-860, 2020.
  * [3] A. Fiscella, E. Valdinoci,  A critical Kirchhoff type problem involving a nonlocal operator,  Nonlinear Analysis , 94, 156-170 , 2014.
  * [4] A. Fiscella, P. Pucci,  $p$-fractional Kirchhoff equations involving critical nonlinearities,  Nonlinear Analysis: Real World Applications, 35, 350-378 , 2017.
  * [5] A. Ghanmi and K. Saoudi,  The Nehari manifold for a singular elliptic equation involving the fractional Laplace operator,  Fractional Differential Calculus, 6(2), 201-217, 2016.
  * [6] A. Iannizzotto, M. Squassina,  $\frac{1}{2}$-Laplacian problems with exponential nonlinearity,  J. Math. Anal. Appl., 414(1), 372-385, 2014.
  * [7] A. Iannizzotto, S. Mosconi and M. Squassina,  $H^{s}$ versus $C^{0}$-weighted minimizers,  Nonlinear Differential Equations and Applications NoDEA, 22(3), 477-497, 2015.
  * [8] A. Iannizzotto, S. Mosconi and M. Squassina,  Global Hölder regularity for the fractional $p$-Laplacian,  Revista Matemática Iberoamericana, 32(4), 1353-1392, 2016.
  * [9] A. Iannizzotto., S. Mosconi and M. Squassina,  Fine boundary regularity for the degenerate fractional $p$-Laplacian,  Journal Functional Analysis, 279(8), 108659 1-54, 2020.
  * [10] A. Soni, D. Choudhuri,  Existence of multiple solutions to an elliptic problem with measure data,  J Elliptic Parabol. Equ., 4(2), 369-388, 2018.
  * [11] B. Barrios, E, Colorado, A. De Pablo, U. Sánchez,  On some critical problems for the fractional Laplacian operator,  Journal of Differential Equations, 252(11), 6133-6162, 2012.
  * [12] B. Zhang, G. Molica Bisci and R. Servadei,  Superlinear nonlocal fractional problems with infinitely many solutions,  Nonlinearity, 28(7), 2247-2264, 2015.
  * [13] C.O. Alves, F.J.S.A. Corr$\hat{e}a$ and T.F. Ma,  Positive solutions for a quasilinear elliptic equation of Kirchhoff type,  Computers & Mathematics with Applications, 49(1), 85-93, 2005.
  * [14] F. Colasuonno, P. Pucci,  Multiplicity of solutions for $p(x)$-polyharmonic elliptic Kirchhoff equations,  Nonlinear Analysis: Theory, Method and Applications, 74(17), 5962-5974, 2011.
  * [15] G.F. Carrier,  On the non-linear vibration problem of the elastic string,  Quart. Appl. Math., 3, 157-165, 1945.
  * [16] G. Molica Bisci and B.A. Pansera,  Three weak solutions for nonlocal fractional equations,  Advanced Nonlinear Studies, 14(3), 619-629, 2014.
  * [17] G. Molica Bisci,  Sequences of weak solutions for fractional equations,  Math. Res. Lett., 21(2), 241-253, 2014.
  * [18] G. Molica Bisci,  Fractional equations with bounded primitive,  Applied Math. Lett., 27, 53-58, 2014.
  * [19] J. Giacomoni, I, Schindler, P. Takáč,  Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation,  Ann. Scuola Norm. Sup. Pisa Cl. Sci. $(5)$, 6(2), 117-158, 2007.
  * [20] J. Giacomoni, K. Saoudi,  $W_{0}^{1,p}$ versus $C^{1}$ local minimizers for a singular and critical functional,  J. Math. Anal. and Appl., 363 (2), 697-710, 2010.
  * [21] J. Zowe, S. Kurcyusz,  Regularity and stability for the mathematical programming problem in Banach spaces,  Appl. Math. Optim., 5(1), 49-62, 1979.
  * [22] J. Zuo, T. An, W. Liu,  A variational inequality of Kirchhoff type in $\mathbb{R}^{N}$,  Journal of Inequalities and Appl., 329, 1-9, 2018.
  * [23] J. Zuo, T. An, M. Li,  Superlinear Kirchhoff-type problems of the fractional $p$-Laplacian without the $(AR)$ condition,  Boundary Value Problems, 2018, p. 180, 2018.
  * [24] J. Zuo, T. An, L. Yang, X. Ren,  The Nehari manifold for a fractional $p$-Kirchhoff system involving sign-changing weight function and concave-convex nonlinearities,  J. of Function Spaces, Art. ID 7624373, 9 Pages, 2019.
  * [25] J. Zuo, T. An, X. Li, Y. Ma,  A fractional $p$-Kirchhoff type problem involving a parameter,  Journal of Nonlinear Functional Analysis, 2019, p. 32, 1-14, 2019.
  * [26] K. Saoudi,  On $W^{s,p}$ vs. $C^{1}$ local minimizers for a critical functional related to fractional $p$-Laplacian,  Applicable Analysis, 96(9), 1586-1595, 2017.
  * [27] K. Saoudi,  $W^{1,N}$ versus $C^{1}$ local minimizer for a singular functional with Neumann boundary condition,  Bol. Soc. Paran. Mat. (3), 37 (1), 71-86, 2019.
  * [28] K. Saoudi,  A critical fractional elliptic equation with singular nonlinearities, Fract. Calc. Appl. Anal., 20(6), 1507-1530, 2017.
  * [29] K. Saoudi, S. Ghosh, D. Choudhuri,  Multiplicity and Hölder regularity of solutions for a nonlocal elliptic PDE involving singularity,  J. Math. Phys., 60, 101509 1-28, 2019.
  * [30] K. Teng,  Two nontrivial solutions for hemivariational inequalities driven by nonlocal elliptic operators,  Nonlinear Analysis: Real World Applications, 14(1), 867-874, 2013.
  * [31] 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 Edingurgh, Section A: Mathematics, 129(4), 787-809, 1999.
  * [32] L. Wang, K. Xie, B. Zhang,  Existence and multiplicity of solutions for critical Kirchhoff-type $p$-Laplacian problems,  J. Math. Anal. Appl., 458(1), 361-378, 2018.
  * [33] M. Caponi, P. Pucci,  Existence theorems for entire solutions of stationary Kirchhoff fractional $p$-Laplacian equations,  Annali di Matematica Pura ed Applicata(4), 195(6), 2099-2129, 2016.
  * [34] M. Khaled, M. Rhoudaf, H. Sabiki,  Lagrange multiplier rule to a nonlinear eigenvalue problem in Musielak-Orlicz spaces,  Numerical Functional Analysis and Optimization, 41(2), 134-157, 2020.
  * [35] M. Kratou,  Ground state solutions of $p$-Laplacian singular Kirchhoff problem involving a Riemann-Liouville fractional derivative,  Filomat, 33 (7), 2073-2088, 2019.
  * [36] N. Pan, B. Zhang, J. Cao,  Degenerate Kirchhoff-type diffusion problems involving the fractional $p$-Laplacian,  Nonlinear Analysis: Real World Applications, 37, 56-70, 2017.
  * [37] N. Nyamoradi, L.I. Zaidan,  Existence of solutions for degenerate Kirchhoff type problems with fractional $p$-Laplacian,  EJDE, 2017, p. 115, 1-13, 2017.
  * [38] N. V. Thin,  Nontrivial solutions of some fractional problems,  Nonlinear Analysis: Real World Applications, 38, 146-170, 2017.
  * [39] P.K. Mishra, K. Sreenadh,  Fractional $p$-Kirchhoff system with sign-changing nonlinearities,  Revista de la Real Academia de Ciencias Exactas Fisicas y Naturales. Serie A Matematicas, 111(1), 281-296, 2017.
  * [40] P. Pucci, S. Saldi,  Critical stationary Kirchhoff equations in $\mathbb{R}^{N}$ involving nonlocal operators,  Revista Mat. Iberoamericana, 32(1), 1-22, 2016.
  * [41] R.A. Adams, J.J. Fournier,  Sobolev spaces,  Academic press, NY, USA, second edition, 2003.
  * [42] R. Servadei , E. Valdinoci,  Mountain pass solutions for non-local elliptic operators,  J. Math. Anal. Appl., 389(2), 887-898, 2012.
  * [43] R. Servadei, E. Valdinoci,  Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators,  Revista Mat. Iberoamericana, 29(3), 1091-1126, 2013.
  * [44] R. Servadei, E. Valdinoci,  Variational methods for nonlocal operators of elliptic type,  Discrete and continuous dynamical systems - Series A, 33(5), 2105-2137, 2013.
  * [45] S. Ghosh,  An existence result for singular nonlocal fractional Kirchhoff-Schrödinger-Poisson system,  arXiv:1909.13350
  * [46] T. Mukherjee, K. Sreenadh,  On Dirichlet problem for fractional $p$-Laplacian with singular non-linearity,  Advances in Nonlinear Analysis, 8(1), 52-72, 2019.
  * [47] X. Mingqi, G. M. Bisci, G. Tian, B. Zhang,  Infinitely many solutions for the stationary Kirchhoff problems involving the fractional $p$-Laplacian,  Nonlinearity, 29(2), 357-374, 2016.
  * [48] X. Mingqi, V.D. Rădulescu, B. Zhang,  Combined effects for fractional Schrödinger-Kirchhoff systems with critical nonlinearities,  ESAIM: Control, Optimisation and Calculus of Variations, 24(3), 1249-1273, 2018.
  * [49] X. Mingqi, V.D. Rădulescu, B. Zhang,  Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions,  Nonlinearity, 31(7), 3228-3250, 2018.
  * [50] X. Mingqi, B. Zhang and M. Ferrara,  Existence of solutions for Kirchhoff type problem involving the non-local fractional $p$-Laplacian,  J. Math. Anal. Appl., 424(2), 1021-1041, 2015.
  * [51] X. Mingqi, B. Zhang,  Degenerate Kirchhoff problems involving the fractional $p$-Laplacian without the $(AR)$ condition,  Complex variables and elliptic Equ., 60(9), 1277-1287, 2015.
  * [52] X. Mingqi, B. Zhang, X. Guo,  Infinitely many solutions for a fractional Kirchhoff type problem via fountain theorem,  Nonlinear Analysis: Theory, Methods and Applications, 120, 299-313, 2015.
  * [53] X. Mingqi, B. Zhang, V.D. Rădulescu,  Existence of solutions for a bi-nonlocal fractional $p$-Kirchhoff type problem,  Computers & Mathematics with Applications, 71(1), 255-266, 2016.
  * [54] X. Mingqi, B. Zhang, H. Qiu,  Existence of solutions for a critical fractional Kirchhoff type problem in $\mathbb{R}^{N}$,  Science China Mathematics, 60(9), 1647-1660, 2017.
  * [55] X. Ren, J. Zuo, Z. Qiao, L. Zhu,  Infinitely many solutions for a superlinear fractional $p$-Kirchhoff-type problem without the $(AR)$ condition,  Advances in Mathematical Physics, 1353961-1-10, 2019.


* Rader and Läuchli [2018] M. Rader and A. M. Läuchli, Finite Correlation Length Scaling in Lorentz-Invariant Gapless iPEPS Wave Functions, Phys. Rev. X 8, 031030 (2018).
  * Corboz  _et al._ [2018] P. Corboz, P. Czarnik, G. Kapteijns, and L. Tagliacozzo, Finite correlation length scaling with infinite projected entangled-pair states, Phys. Rev. X 8, 031031 (2018).
  * Czarnik and Corboz [2019] P. Czarnik and P. Corboz, Finite correlation length scaling with infinite projected entangled pair states at finite temperature, Phys. Rev. B 99, 245107 (2019).
  * Vanhecke  _et al._ [2022] B. Vanhecke, J. Hasik, F. Verstraete, and L. Vanderstraeten, Scaling hypothesis for projected entangled-pair states, Phys. Rev. Lett. 129, 200601 (2022).
  * Ponsioen  _et al._ [2023] B. Ponsioen, J. Hasik, and P. Corboz, Improved summations of $n$-point correlation functions of projected entangled-pair states (2023), arXiv:2306.13327 .
  * Lukin and Sotnikov [2024] I. V. Lukin and A. G. Sotnikov, Corner transfer matrix renormalization group approach in the zoo of archimedean lattices (2024), arXiv:2401.07274 [cond-mat.stat-mech] .
  * Kitaev [2006] A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321, 2 (2006).
  * Xie  _et al._ [2014] Z. Y. Xie, J. Chen, J. F. Yu, X. Kong, B. Normand, and T. Xiang, Tensor renormalization of quantum many-body systems using projected entangled simplex states, Phys. Rev. X 4, 011025 (2014).
  * Schmoll  _et al._ [2020] P. Schmoll, S. S. Jahromi, M. Hörmann, M. Mühlhauser, K. P. Schmidt, and R. Orús, Fine grained tensor network methods, Phys. Rev. Lett. 124, 200603 (2020).
  * Anderson [1951] P. W. Anderson, Limits on the energy of the antiferromagnetic ground state, Phys. Rev. 83, 1260 (1951).
  * Lukin and Sotnikov [2023c] I. V. Lukin and A. G. Sotnikov, Variational optimization of tensor-network states with the honeycomb-lattice corner transfer matrix, Phys. Rev. B 107, 054424 (2023c).
  * Farnell  _et al._ [2014] D. J. J. Farnell, O. Götze, J. Richter, R. F. Bishop, and P. H. Y. Li, Quantum spin-$\frac{1}{2}$ antiferromagnets on Archimedean lattices: The route from semiclassical magnetic order to nonmagnetic quantum states, Phys. Rev. B 89, 184407 (2014).
  * Jiang [2012] F. J. Jiang, High precision determination of the low-energy constants for the two-dimensional quantum heisenberg model on the honeycomb lattice, Europ. Phys. J. B 85, 402 (2012).
  * Ganesh  _et al._ [2013] R. Ganesh, J. van den Brink, and S. Nishimoto, Deconfined criticality in the frustrated heisenberg honeycomb antiferromagnet, Phys. Rev. Lett. 110, 127203 (2013).
  * Gong  _et al._ [2013] S.-S. Gong, D. N. Sheng, O. I. Motrunich, and M. P. A. Fisher, Phase diagram of the spin-$\frac{1}{2}$ ${J}_{1}$-${J}_{2}$ Heisenberg model on a honeycomb lattice, Phys. Rev. B 88, 165138 (2013).
  * Liao  _et al._ [2017] H. J. Liao, Z. Y. Xie, J. Chen, Z. Y. Liu, H. D. Xie, R. Z. Huang, B. Normand, and T. Xiang, Gapless spin-liquid ground state in the $s=1/2$ kagome antiferromagnet, Phys. Rev. Lett. 118, 137202 (2017).
  * Läuchli  _et al._ [2019] A. M. Läuchli, J. Sudan, and R. Moessner, Spin-$\frac{1}{2}$ kagome Heisenberg antiferromagnet revisited, Phys. Rev. B 100, 155142 (2019).
  * Yan  _et al._ [2011] S. Yan, D. A. Huse, and S. R. White, Spin-Liquid Ground State of the S = 1/2 Kagome Heisenberg Antiferromagnet, Science 332, 1173 (2011).
  * Depenbrock  _et al._ [2012] S. Depenbrock, I. P. McCulloch, and U. Schollwöck, Nature of the spin-liquid ground state of the $s=1/2$ heisenberg model on the kagome lattice, Phys. Rev. Lett. 109, 067201 (2012).
  * He  _et al._ [2017] Y.-C. He, M. P. Zaletel, M. Oshikawa, and F. Pollmann, Signatures of Dirac Cones in a DMRG Study of the Kagome Heisenberg Model, Phys. Rev. X 7, 031020 (2017).
  * Astrakhantsev  _et al._ [2021] N. Astrakhantsev, F. Ferrari, N. Niggemann, T. Müller, A. Chauhan, A. Kshetrimayum, P. Ghosh, N. Regnault, R. Thomale, J. Reuther, T. Neupert, and Y. Iqbal, Pinwheel valence bond crystal ground state of the spin-$\frac{1}{2}$ heisenberg antiferromagnet on the shuriken lattice, Phys. Rev. B 104, L220408 (2021).
  * Schmoll  _et al._ [2023] P. Schmoll, A. Kshetrimayum, J. Naumann, J. Eisert, and Y. Iqbal, Tensor network study of the spin-$\frac{1}{2}$ Heisenberg antiferromagnet on the shuriken lattice, Phys. Rev. B 107, 064406 (2023).
  * Fujihala  _et al._ [2020] M. Fujihala, K. Morita, R. Mole, S. Mitsuda, T. Tohyama, S.-I. Yano, D. Yu, S. Sota, T. Kuwai, A. Koda, H. Okabe, H. Lee, S. Itoh, T. Hawai, T. Masuda, H. Sagayama, A. Matsuo, K. Kindo, S. Ohira-Kawamura, and K. Nakajima, Gapless spin liquid in a square-kagome lattice antiferromagnet, Nature Comm. 11, 3429 (2020).
  * Li  _et al._ [2022] Q. Li, H. Li, J. Zhao, H.-G. Luo, and Z. Y. Xie, Magnetization of the spin-$\frac{1}{2}$ heisenberg antiferromagnet on the triangular lattice, Phys. Rev. B 105, 184418 (2022).
  * Huse and Elser [1988] D. A. Huse and V. Elser, Simple variational wave functions for two-dimensional heisenberg spin-½ antiferromagnets, Phys. Rev. Lett. 60, 2531 (1988).
  * White and Chernyshev [2007] S. R. White and A. L. Chernyshev, Neél order in square and triangular lattice heisenberg models, Phys. Rev. Lett. 99, 127004 (2007).
  * Yunoki and Sorella [2006] S. Yunoki and S. Sorella, Two spin liquid phases in the spatially anisotropic triangular heisenberg model, Phys. Rev. B 74, 014408 (2006).
  * Zauner  _et al._ [2015] V. Zauner, D. Draxler, L. Vanderstraeten, M. Degroote, J. Haegeman, M. M. Rams, V. Stojevic, N. Schuch, and F. Verstraete, Transfer matrices and excitations with matrix product states, New J. Phys. 17, 053002 (2015).
  * Vanderstraeten  _et al._ [2015] L. Vanderstraeten, M. Mariën, F. Verstraete, and J. Haegeman, Excitations and the tangent space of projected entangled-pair states, Phys. Rev. B 92, 201111 (2015).
  * Vanderstraeten  _et al._ [2019] L. Vanderstraeten, J. Haegeman, and F. Verstraete, Simulating excitation spectra with projected entangled-pair states, Phys. Rev. B 99, 165121 (2019).
  * Ponsioen and Corboz [2020] B. Ponsioen and P. Corboz, Excitations with projected entangled pair states using the corner transfer matrix method, Phys. Rev. B 101, 195109 (2020).
  * Tu  _et al._ [2023] W.-L. Tu, L. Vanderstraeten, N. Schuch, H.-Y. Lee, N. Kawashima, and J.-Y. Chen, Generating function for projected entangled-pair states (2023), arXiv:2307.08083 .
  * Barthel  _et al._ [2009] T. Barthel, C. Pineda, and J. Eisert, Contraction of fermionic operator circuits and the simulation of strongly correlated fermions, Phys. Rev. A 80, 042333 (2009).
  * Corboz  _et al._ [2010b] P. Corboz, G. Evenbly, F. Verstraete, and G. Vidal, Simulation of interacting fermions with entanglement renormalization, Phys. Rev. A 81, 010303 (2010b).
  * Wille  _et al._ [2017] C. Wille, O. Buerschaper, and J. Eisert, Fermionic topological quantum states as tensor networks, Phys. Rev. B 95, 245127 (2017).
  * Bultinck  _et al._ [2017] N. Bultinck, D. J. Williamson, J. Haegeman, and F. Verstraete, Fermionic projected entangled-pair states and topological phases, J. Phys. A 51, 025202 (2017).
  * Corboz  _et al._ [2010c] P. Corboz, R. Orús, B. Bauer, and G. Vidal, Simulation of strongly correlated fermions in two spatial dimensions with fermionic projected entangled-pair states, Phys. Rev. B 81, 165104 (2010c).
  * Pižorn and Verstraete [2010] I. Pižorn and F. Verstraete, Fermionic implementation of projected entangled pair states algorithm, Phys. Rev. B 81, 245110 (2010).
  * Sweke  _et al._ [2022] R. Sweke, P. Boes, N. Ng, C. Sparaciari, J. Eisert, and M. Goihl, Transparent Reporting of Research-Related Greenhouse Gas Emissions Through the Scientific CO2nduct Initiative, Comm. Phys. 5, 150 (2022).
  * Centre [2021] J. S. Centre, Jureca: Data centric and booster modules implementing the modular supercomputing architecture at jülich supercomputing centre, J. Large-Scale Res. Fac. 7, A182 (2021).
  * [128] J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, and Q. Zhang, JAX: composable transformations of Python+NumPy programs, GitHub.
  * Frostig  _et al._ [2018] R. Frostig, M. Johnson, and C. Leary, Compiling machine learning programs via high-level tracing (2018).
  * Jutho  _et al._ [2024] Jutho, Lukas, M. Hauru, ho oto, maartenvd, Gertian, J. TagBot, S. Carlström, and Xiaoyu, Jutho/TensorKit.jl: v0.12.2, Zenodo (2024).
  * [131] Quantum Group UGent, Open-source software and notebooks.
  * [132] Zygote contributors, Zygote: Source-to-source automatic differentiation (AD) in Julia, and the next-gen AD system for the Flux differentiable programming framework., GitHub.
  * Giles [2008] M. B. Giles, An extended collection of matrix derivative results for forward and reverse mode algorithmic differentiation (2008).
  * Xie  _et al._ [2020] H. Xie, J.-G. Liu, and L. Wang, Automatic differentiation of dominant eigensolver and its applications in quantum physics, Phys. Rev. B 101, 245139 (2020).
  * Wan and Zhang [2019] Z.-Q. Wan and S.-X. Zhang, Automatic differentiation for complex valued SVD (2019), arXiv:1909.02659 .
  * Christianson [1994] B. Christianson, Reverse accumulation and attractive fixed points, Opt. Meth. Soft. 3, 311 (1994).
  * Kolter  _et al._ [2020] Z. Kolter, D. Duvenaud, and M. Johnson, Deep implicit layers - Neural ODEs, deep equilibrium models, and beyond (2020).
  * Johnson and The JAX Authors [2020] M. J. Johnson and The JAX Authors, Custom derivative rules for JAX-transformable Python functions (2020).