# Mountain Pass Solutions for an entire semipositone problem involving the
Grushin Subelliptic Operator

Giovanni Molica Bisci Partially supported by GNAMPA 2024: "Aspetti geometrici
e analitici di alcuni problemi locali e non-locali in mancanza di compattezza"
Paolo Malanchini  Simone Secchi Partially supported by GNAMPA 2024: "Aspetti
geometrici e analitici di alcuni problemi locali e non-locali in mancanza di
compattezza"

###### Abstract

Abstract. For $N\geq 3$ we study the following semipositone problem

$-\Delta_{\gamma}u=g(z)f_{a}(u)\quad\hbox{in $\mathbb{R}^{N}$},$

where $\Delta_{\gamma}$ is the Grushin operator

$\Delta_{\gamma}u(z)=\Delta_{x}u(z)+\lvert
x\rvert^{2\gamma}\Delta_{y}u(z)\quad(\gamma\geq 0),$

$g\in L^{1}(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})$ is a positive
function, $a>0$ is a parameter and $f_{a}$ is a continuous function on
$\mathbb{R}$ that coincides with $f(t)-a$ for $t\in\mathbb{R}^{+}$, where $f$
is a continuous function with subcritical and Ambrosetti-Rabinowitz type
growth and which satisfies $f(0)=0$. Depending on the range of $a$, we obtain
the existence of positive mountain pass solutions in
$D_{\gamma}(\mathbb{R}^{N})$, extending the results found in [1, 7, 8].

##  1 Introduction

The term _semipositone_ is usually associated to a semilinear elliptic
equation of the form

$-\Delta u=f(u)\quad\hbox{in $\Omega$}$

in which the nonlinearity $f$ is such that $f(0)<0$, see [14]. In particular,
$u=0$ is not a subsolution of the equation, and it was remarked in [28] that
this fact may introduce several difficulties in the analysis of the problem.
When the domain $\Omega$ is unbounded the condition $f(0)<0$ is also a source
of concern from the viewpoint of variational methods, since non-zero constants
are not integrable on $\Omega$.

In literature, second order semipositone problems have been widely studied. In
the seminal paper [11], Brown and Shivaji considered the perturbed bifurcation
problem $-\Delta u=\lambda(u-u^{3})-\varepsilon$ on a bounded set
$\Omega\subset\mathbb{R}^{N}$ with Dirichlet boundary conditions and
$\varepsilon,\lambda>0$.

Some years later, Castro and Shivaji (see [13]) studied non-negative solutions
for the boundary value problem

$\begin{cases}-u^{\prime\prime}(x)=\lambda f(u(x)),\leavevmode\nobreak\
x\in(0,1)\\\ u(0)=0=u(1),\end{cases}$

where $\lambda>0$ is a constant and $f\in C^{2}(\mathbb{R})$ satisfies
$f(0)<0$.

Subsequently, in [12] the authors considered positive solutions for a class of
multiparameter semipositone problem

$\begin{cases}-\Delta u=\lambda g(u)+\mu f(u)\leavevmode\nobreak\ &\hbox{in
$\Omega$},\\\ u=0\leavevmode\nobreak\ &\hbox{on $\partial\Omega$},\end{cases}$

with $\lambda>0$, $\mu>0$ and $\Omega$ is a smooth bounded domain in
$\mathbb{R}^{N}$, $N\geq 2$. They assume $g(0)>0$ and superlinear while
$f(0)<0$ sublinear, and eventually strictly positive. For fixed $\mu$, they
establish existence and multiplicity for $\lambda$ small (when the problem
becomes semipositone), and nonexistence for $\lambda$ large.

To the best of our knowledge Alves, Holanda, and Santos in [1] were the first
to study semipositone problems in the entire space $\mathbb{R}^{N}$, $N\geq
3$, for the classical Laplace operator, described by the equation

$\begin{cases}-\Delta u=h(x)\left(f(u)-a\right)\leavevmode\nobreak\ &\hbox{in
$\mathbb{R}^{N}$, }\\\ u>0\leavevmode\nobreak\ &\hbox{in
$\mathbb{R}^{N}$},\end{cases}$

with $f(0)=0$ and $a>0$. The function $f\in C(\mathbb{R}^{+})$ is a locally
Lipschitz function with subcritical growth which satisfies the Ambrosetti-
Rabinowitz condition, and the weight $h$ is a continuous positive function
bounded by a radial function $P\in L^{1}(\mathbb{R}^{N})\cap
L^{\infty}(\mathbb{R}^{N})$.

Subsequently, many authors extended their result to other operators. In [9],
Biswas, Das and Sarkar examined the same problem for a fourth-order operator.
Additionally, in [7], Biswas investigated this problem for the fractional
Laplacian. In the very recent preprint [8], a semipositone problem with the
fractional $p$-Laplace operator was investigated.

In this paper we will be concerned with the following semipositone problem
associated with the Grushin operator

$-\Delta_{\gamma}u=g(z)f_{a}(u)\leavevmode\nobreak\ \hbox{in
$\mathbb{R}^{N}$},$ (SP)

where the function $g\in L^{1}(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})$
is positive, $a>0$ is a parameter and the function
$f_{a}\colon\mathbb{R}\to\mathbb{R}$ is defined as:

$f_{a}(t)=\begin{cases}f(t)-a&\hbox{if $t\geq 0$},\\\ -a(t+1)&\hbox{if
$t\in(-1,0)$},\\\ 0&\hbox{if $t\leq-1$},\end{cases}$

where $f$ is a contiuous function defined on $[0,+\infty)$ is such that
$f(0)=0$ and

  1. (f1)

$\lim_{t\to 0^{+}}\frac{f(t)}{t}=0$,
$\lim_{t\to\infty}\frac{f(t)}{t^{\alpha-1}}<+\infty$ for some
$\alpha\in(2,2^{*}_{\gamma})$,

  2. (f2)

(Ambrosetti-Rabinowitz) there exist $\theta>2$ and $t_{0}>0$ such that

$0<\theta F(t)\leq tf(t)$

for all $t>t_{0}$, where $F(t)=\int_{0}^{t}f(s)\,\mathrm{d}s$.

For the reader’s convenience, we briefly recall the basic construction of the
Baouendi-Grushin operator. We split the euclidean space $\mathbb{R}^{N}$ as
$\mathbb{R}^{m}\times\mathbb{R}^{\ell}$, where $m\geq 1$, $\ell\geq 1$ and
$m+\ell=N\geq 3$. A generic point $z\in\mathbb{R}^{N}$ can therefore be
written as

$z=\left(x,y\right)=\left(x_{1},\ldots,x_{m},y_{1},\ldots,y_{\ell}\right),$

where $x\in\mathbb{R}^{m}$ and $y\in\mathbb{R}^{\ell}$. For a differentiable
function $u$ we define the Grushin gradient

$\nabla_{\gamma}u(z)=(\nabla_{x}u(z),\lvert
x\rvert^{\gamma}\nabla_{y}u(z))=(\partial_{x_{1}}u(z),\dots,\partial_{x_{m}}u(z),\lvert
x\rvert^{\gamma}\partial_{y_{1}}u(z),\dots,\lvert
x\rvert^{\gamma}\partial_{y_{\ell}}u(z)).$

The Grushin operator $\Delta_{\gamma}$ is defined by111Some authors use the
slightly different definition
$\Delta_{\gamma}=\Delta_{x}+\left(1+\gamma\right)^{2}|x|^{2\gamma}\Delta_{y}$.

$\Delta_{\gamma}u(z)=\Delta_{x}u(z)+\lvert x\rvert^{2\gamma}\Delta_{y}u(z),$

where $\Delta_{x}$ and $\Delta_{y}$ are the Laplace operators in the variables
$x$ and $y$, respectively. A crucial property of the Grushin operator is that
it is not uniformly elliptic in the space $\mathbb{R}^{N}$, since it is
degenerate on the subspace $\\{0\\}\times\mathbb{R}^{\ell}$.

For a generic $\gamma\geq 0$, it is possible to embed $\Delta_{\gamma}$ into a
family of operators of the form

$\Delta_{\lambda}=\sum_{i=1}^{N}\partial_{x_{i}}\left(\lambda_{i}^{2}\partial_{x_{i}}\right),$

which have been studied from a geometrical viewpoint in [19, 20, 21]. Here
$\lambda_{1},\ldots,\lambda_{N}$ are given functions which satisfy suitable
conditions.

Moreover, the Grushin operator falls into the class of $X$-elliptic operators
studied in [22, 27]. In fact, the Grushin operator is uniformly $X$-elliptic
with respect to the family of vector fields $X=(X_{1},\dots,X_{N})$ defined as

$\displaystyle X_{i}$ $\displaystyle=\frac{\partial}{\partial
x_{i}}\quad\hbox{for $i=1,\ldots,m$}$ $\displaystyle X_{m+j}$
$\displaystyle=|x|^{\gamma}\frac{\partial}{\partial y_{j}}\quad\hbox{for
$j=1,\ldots,\ell$}.$

With the same notation we can write the Grushin operator $\Delta_{\gamma}$ as
sum of the vector fields

$\Delta_{\gamma}=\sum_{j=0}^{N}X_{j}^{2}.$

Note that, when $\gamma$ is an even integer, the $C^{\infty}$-field
$X=\left(X_{1},\ldots,X_{N}\right)$ satisfies the Hörmander’s finite rank
condition, see [24].

We will look for solutions to (SP) in the function space

$D_{\gamma}(\mathbb{R}^{N})=\left\\{u\in
L^{2^{*}_{\gamma}}(\mathbb{R}^{N}):\lVert u\rVert_{\gamma}<\infty\right\\},$
(1)

where

$\lVert
u\rVert_{\gamma}=\left(\int_{\mathbb{R}^{N}}\lvert\nabla_{\gamma}u\rvert^{2}\,{\mathrm{d}z}\right)^{1/2}.$

By definition, a function $u\in D_{\gamma}(\mathbb{R}^{N})$ is a weak solution
to (SP) if it satisfies the following variational identity:

$\forall\varphi\in
D_{\gamma}(\mathbb{R}^{N}):\quad\int_{\mathbb{R}^{N}}\nabla_{\gamma}u\nabla_{\gamma}\varphi\,{\mathrm{d}z}=\int_{\mathbb{R}^{N}}g(z)f_{a}(u)\varphi\,{\mathrm{d}z}.$

Two functionals $K_{a}$ and $I_{a}$ are defined on
$D_{\gamma}(\mathbb{R}^{N})$ by

$\displaystyle K_{a}(u)$
$\displaystyle=\int_{\mathbb{R}^{N}}g(z)F_{a}(u(z))\,{\mathrm{d}z}$
$\displaystyle I_{a}(u)$ $\displaystyle=\frac{1}{2}\lVert
u\rVert_{\gamma}^{2}-K_{a}(u).$

It is standard to prove that $K_{a}$ and $I_{a}$ are of class $C^{1}$, and the
corresponding Fréchet derivatives are given by

$\displaystyle DK_{a}(u)\colon
v\to\int_{\mathbb{R}^{N}}g(z)f_{a}(u)v\,{\mathrm{d}z}\leavevmode\nobreak\ $
$\displaystyle DI_{a}(u)\colon
v\to\int_{\mathbb{R}^{N}}\nabla_{\gamma}u\nabla_{\gamma}v\,{\mathrm{d}z}.$

As a consequence, weak solutions of (SP) are precisely the critical points of
$I_{a}$ on $D_{\gamma}(\mathbb{R}^{N})$.

The magnitude of the constant $a$ and the interplay with the function $g$ are
crucial. Specifically, we can prove the existence of positive solutions to
(SP) when the parameter $a$ is sufficiently small.

We will construct a solution by a mountain pass argument. The positivity of
the solution is obtained using the fundamental solution for the Grushin
operator.

Our main result can be stated as follows. The distance $d(z)$ appearing in the
statement is defined in (31) below.

######  Theorem 1.1.

Let $f$ satisfy (f1)-(f2) and let $g\in L^{1}(\mathbb{R}^{N})\cap
L^{\infty}(\mathbb{R}^{N})$ be positive. Then there exists $a_{1}>0$ such that
for each $a\in(0,a_{1})$ (SP) admits a weak solution $u_{a}\in
C(\mathbb{R}^{N})\cap D_{\gamma}(\mathbb{R}^{N})$ such that $u_{a}\geq 0$ a.e.
in $\\{0\\}\times\mathbb{R}^{\ell}$. If, in addition,

  1. (f3)

$f$ is locally Lipschitz,

  1. (g1)

for all $z\in(\\{0\\}\times\mathbb{R}^{\ell})\setminus\\{(0,0)\\}$,

$d(z)^{N_{\gamma}-2}\int_{\mathbb{R}^{N}}\frac{g(\xi)}{d(z-\xi)^{N_{\gamma}-2}}\mathrm{d}\xi<\infty,$

then there exists $\tilde{a}\in(0,a_{1})$ such that $u_{a}>0$ a.e. in
$\\{0\\}\times\mathbb{R}^{\ell}$ for every $a\in(0,\tilde{a})$.

######  Remark 1.2.

Actually, standard regularity results (see for instance [31] and the
references there-in for the case of the Heisenberg group, but the proof
extends to our setting) ensure that $u\in C^{1,\beta}(\mathbb{R}^{N})$ with
$0<\beta<1$.

To the best of our knowledge, there are no results in the literature about
semipositone problems for degenerate equations of Baouendi-Grushin type.
Before concluding this introduction, we would like to comment about the
positivity in the subspace $\\{0\\}\times\mathbb{R}^{\ell}$. In some sense
this is an expected result, since the restriction of $\Delta_{\gamma}$ to this
subspace is a uniformly elliptic operator for which all the standard tools are
available. Globally, the Grushin operator lacks a strong maximum principle for
weak solutions and $W^{2,p}$-estimates which are used in order to derive
additional regularity for weak solutions. We refer to Remark 6.8 for more
details.

##  2 Some inequalities for $f$

######  Remark 2.1.

In the rest of the paper we will denote by $C(\ldots)$ a positive constant
which depends on the variables between brackets.

Let us now identify the upper and lower bounds of $f$, $f_{a}$ and their
primitives. Clearly, the function $f_{a}$ is continuous and the function
$F_{a}$ defined by

$F_{a}(t)=\begin{cases}F(t)-at\leavevmode\nobreak\ &\hbox{for $t\geq 0$},\\\
-\frac{at^{2}}{2}-at\leavevmode\nobreak\ &\hbox{for $t\in(-1,0)$},\\\
\frac{a}{2}\leavevmode\nobreak\ &\hbox{for $t\leq-1$}\end{cases}$

satisfies $F_{a}^{\prime}=f_{a}$ and $F_{a}(0)=0$. The assumption

$\displaystyle\lim_{t\to 0^{+}}\frac{f(t)}{t}=0$

implies that for every $\varepsilon>0$, there exists $t_{1}(\varepsilon)>0$
such that $f(t)<\varepsilon t$ for $t\in(0,t_{1}(\varepsilon))$. Similarly,
assumption

$\lim_{t\to\infty}\frac{f(t)}{t^{\alpha-1}}<+\infty$

implies

$f(t)\leq C(f,t_{1}(\varepsilon))t^{\alpha-1}\quad\hbox{for $t\geq
t_{1}(\varepsilon)$}$

for some positive constant $C(f,t_{1}(\varepsilon))$. Hence, we have the
following bounds for every $\alpha\in(2,2^{*}_{\gamma}]$:

$\lvert f_{a}(t)\rvert\leq\varepsilon\lvert
t\rvert+C(f,t_{1}(\varepsilon))\lvert t\rvert^{\alpha-1}+a,\quad
F_{a}(t)\leq\varepsilon\lvert t\rvert^{2}+C(f,t_{1}(\varepsilon))\lvert
t\rvert^{\alpha}+a\lvert t\rvert\quad\hbox{for $t\in\mathbb{R}$}.$ (2)

By (f2), the function $t\mapsto\frac{F(t)}{t^{\theta}}$ is non decreasing for
$t>t_{0}$, so there exist two positive constants $M_{1}$ and $M_{2}$ such that

$F(t)\geq M_{1}t^{\theta}-M_{2}\quad\hbox{for all $t\in\mathbb{R}$}.$ (3)

######  Remark 2.2 (Ambrosetti-Rabinowitz condition on $f_{a}$).

For $t>t_{0}$, assumption (f2) yields

$\theta F_{a}(t)=\theta F(t)-\theta at\leq f(t)t-at=tf_{a}(t).$

For $t\in[0,t_{0}]$, by continuity of $F$, there exists a constant $M>0$
independent of $a$, such that

$\theta F_{a}(t)=\theta F(t)-\theta at\leq M-at\leq M-at+f(t)t\leq
M+tf_{a}(t)$

For $t\in[-1,0]$,

$\theta
F_{a}(t)-tf_{a}(t)=\left(-\frac{a\theta}{2}+a\right)t^{2}-\left(\theta-1\right)at\leq
ca$

where
$c=\max_{t\in[-1,0]}\left\\{\left(-\frac{\theta}{2}+1\right)t^{2}-\left(\theta-1\right)t\right\\}$
does not depend on $a$.

For $t<-1$,

$\theta F_{a}(t)=\frac{\theta a}{2}\leq\frac{\theta a}{2}+f_{a}(t)t$

Choosing $C=\max\left\\{c,\frac{\theta}{2}\right\\}$ we conclude that the
following Ambrosetti-Rabinowitz condition with correction terms holds for
$f_{a}$:

$\theta F_{a}(t)\leq tf_{a}(t)+Ca+M,\quad\hbox{for all $t\in\mathbb{R}$ and
$a>0$}.$ (4)

##  3 A functional setting for the Grushin operator

The space $D_{\gamma}(\mathbb{R}^{N})$ defined in (1) can be also
characterized as the completion of $C_{0}^{\infty}(\mathbb{R}^{N})$ with
respect to the norm

$\lVert
u\rVert_{\gamma}=\left(\int_{\mathbb{R}^{N}}\lvert\nabla_{\gamma}u\rvert^{2}\,{\mathrm{d}z}\right)^{1/2},$

where

$\displaystyle 2_{\gamma}^{*}=\frac{2N_{\gamma}}{N_{\gamma}-2}$

is the Sobolev critical exponent associated to $N_{\gamma}=m+(1+\gamma)\ell$,
the homogeneous dimension associated to the decomposition $N=m+\ell$. Since
the Sobolev-Gagliardo-Niremberg inequality proved in [29, Lemma 5.1]

$\lVert u\rVert_{2^{*}_{\gamma}}\leq C\lVert u\rVert_{\gamma},$

holds for all $u\in D_{\gamma}(\mathbb{R}^{N})$ for some positive constant
$C=C(m,k,\gamma)$, we have that the embedding

$D_{\gamma}(\mathbb{R}^{N})\hookrightarrow L^{2^{*}_{\gamma}}(\mathbb{R}^{N})$
(5)

is continuous. The homogeneous space $D_{\gamma}(\mathbb{R}^{N})$ is also a
Hilbert space with the inner product222For the sake of simplicity, we omit any
scalar multiplication sign between the two vectors $\nabla_{\gamma}u$ and
$\nabla_{\gamma}v$.

$\braket{u,v}_{\gamma}=\int_{\mathbb{R}^{N}}\nabla_{\gamma}u\nabla_{\gamma}v\,\,{\mathrm{d}z}.$

Given $q\in[1,\infty)$, with $L^{q}(\mathbb{R}^{N},\lvert g\rvert)$ we denote
the class of real-valued Lebesgue measurable functions $u$ such that

$\int_{\mathbb{R}^{N}}\lvert g(z)\rvert\lvert
u(z)\rvert^{q}\,{\mathrm{d}z}<+\infty.$

By the assumptions on $g$ and (5), we have that the embedding
$D_{\gamma}(\mathbb{R}^{N})\hookrightarrow L^{q}(\mathbb{R}^{N},\lvert
g\rvert)$ is continuous for $q\in[1,2^{*}_{\gamma}]$.

The following lemma states a compact embedding of the solution space for (SP)
into the weighted Lebesgue space $L^{q}(\mathbb{R},\lvert g\rvert)$.

######  Lemma 3.1.

Let $q\in[1,2^{*}_{\gamma})$. The embedding
$D_{\gamma}(\mathbb{R}^{N})\hookrightarrow L^{q}(\mathbb{R}^{N},\lvert
g\rvert)$ is compact.

###### Proof.

Let $\\{u_{j}\\}$ be a bounded sequence in $D_{\gamma}(\mathbb{R}^{N})$. We
can suppose that $u_{j}\rightharpoonup 0$. For each $R>0$, we have the
continuous embedding
$D_{\gamma}(\mathbb{R}^{N})\hookrightarrow\mathscr{H}_{\gamma}(B_{R}(0))$333For
an open bounded set $\Omega\subset\mathbb{R}^{N}$, we denote by
$\mathscr{H}_{\gamma}(\Omega)$ the completion of $C^{1}_{0}(\Omega)$ with
respect to the norm  $\lVert
u\rVert_{\gamma}=\left(\int_{\Omega}\lvert\nabla_{\gamma}u\rvert^{2}\,{\mathrm{d}z}\right)^{1/2}.$
. Since the embedding $\mathscr{H}_{\gamma}(B_{R}(0))\hookrightarrow
L^{q}(B_{R}(0))$ is compact, see [26, Proposition 3.2], it follows that
$D_{\gamma}(\mathbb{R}^{N})\hookrightarrow L^{q}(B_{R}(0))$ is a compact
embedding as well. Hence,

$u_{j}(x)\to 0\leavevmode\nobreak\ \hbox{a.e. in $\mathbb{R}^{N}$}$

for some subsequence. As $D_{\gamma}(\mathbb{R}^{N})\hookrightarrow
L^{2^{*}_{\gamma}}(\mathbb{R}^{N})$ is a continuous embedding, we have that
$\\{\lvert u_{j}\rvert^{q}\\}$ is a bounded sequence in
$L^{2^{*}_{\gamma}/q}(\mathbb{R}^{N})$. By the Brezis-Lieb Lemma, [10, Theorem
1], up to a subsequence if necessary

$\lvert u_{j}\rvert^{q}\rightharpoonup 0\leavevmode\nobreak\ \hbox{in
$L^{2^{*}_{\gamma}/q}(\mathbb{R}^{N})$},$

or equivalently,

$\int_{\mathbb{R}^{N}}\lvert u_{j}\rvert^{q}\varphi\,{\mathrm{d}z}\to
0,\leavevmode\nobreak\ \forall\varphi\in L^{p}(\mathbb{R}^{N}),$

where $p$ is the Hölder conjugate of $2^{*}_{\gamma}/q$. From the assumptions
on $g$ we know that $g\in L^{r}(\mathbb{R}^{N})$ for all $r\geq 1$. It follows
that

$\int_{\mathbb{R}^{N}}g(z)\lvert u_{j}\rvert^{q}\,{\mathrm{d}z}\to 0.$

This shows that $u_{j}\to 0$ in $L^{q}(\mathbb{R}^{N},\lvert g\rvert)$,
concluding the proof. ∎

##  4 Existence of solutions for (SP)

In the next lemma, depending on the values of $a$, we will establish the
mountain pass geometry for the functional $I_{a}$.

######  Lemma 4.1.

Let $N\geq 3$. Let $f$ satisfies _( f1)_, _( f2)_ and $g\in
L^{1}(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})$ be positive. Then the
following hold:

  1. (i)

There exists $r>0$ such that if $\rho\in(0,r)$ and $\lVert
u\rVert_{\gamma}=\rho$, then there are $\delta=\delta(\rho)>0$ and
$a_{1}=a_{1}(\rho)>0$ such that if $I_{a}(u)\geq\delta$ for all
$a\in(0,a_{1})$.

  2. (ii)

There exists $v\in D_{\gamma}(\mathbb{R}^{N})$ such that $\lVert
v\rVert_{\gamma}>\rho$ and $I_{a}(v)<0$, for all $a\in(0,a_{1})$.

###### Proof.

  1. (i)

Using (2), we estimate

$\displaystyle\int_{\mathbb{R}^{N}}g(z)F_{a}(u)\,{\mathrm{d}z}$
$\displaystyle\leq\int_{\mathbb{R}^{N}}g(z)\left(\varepsilon\lvert
u\rvert^{2}+C(f,\varepsilon)\lvert u\rvert^{\alpha}+a\lvert
u\rvert\right)\,{\mathrm{d}z}$
$\displaystyle=\varepsilon\int_{\mathbb{R}^{N}}g(z)\lvert
u\rvert^{2}\,{\mathrm{d}z}+C(f,\varepsilon)\int_{\mathbb{R}^{N}}g(z)\lvert
u\rvert^{\alpha}\,{\mathrm{d}z}+a\int_{\mathbb{R}^{N}}g(z)\lvert
u\rvert\,{\mathrm{d}z}$ $\displaystyle\leq\varepsilon C_{1}\lVert
u\rVert_{\gamma}^{2}+C_{2}\lVert u\rVert_{\gamma}^{\alpha}+aC_{3}\lVert
u\rVert_{\gamma},$

where $C_{1}$, $C_{2}$ and $C_{3}$ are the embedding constants of
$D_{\gamma}(\mathbb{R}^{N})\hookrightarrow L^{q}(\mathbb{R}^{N},\lvert
g\rvert)$, see Lemma 3.1. In particular, for $\lVert u\rVert_{\gamma}=\rho$,

$\displaystyle I_{a}(\rho)$ $\displaystyle\geq\frac{1}{2}\rho^{2}-\varepsilon
C_{1}\rho^{2}-C_{2}\rho^{\alpha}-aC_{3}\rho$
$\displaystyle=\rho^{2}\left(\frac{1}{2}-\varepsilon
C_{1}-C_{2}\rho^{\alpha-2}\right)-aC_{3}\rho.$ (6)

We choose $\varepsilon<(2C_{1})^{-1}$. Then we write $I_{a}(u)\geq
A(\rho)-aC_{3}\rho$, where
$A(\rho)=C\rho^{2}\left(1-\tilde{C}\rho^{\alpha-2}\right)$, with $C$,
$\tilde{C}$ independent of $a$. Let $r$ be the first non-trivial zero of $A$.
For $\rho<r$, we fix $a_{1}\in\left(0,\frac{A(\rho)}{C_{3}\rho}\right)$ and
$\delta=A(\rho)-a_{1}C_{3}\rho$. Thus, by (6) we get $I_{a}(u)\geq\delta$ for
every $a\in(0,a_{1})$.

  2. (ii)

Fix a function

$\varphi\in C^{\infty}_{0}(\mathbb{R}^{N})\setminus\\{0\\},\quad\hbox{with
$\varphi\geq 0$\leavevmode\nobreak\ and $\lVert\varphi\rVert_{\gamma}=1$.}$

Notice that for all $t>0$,

$\displaystyle
I_{a}(t\varphi)=\frac{1}{2}\int_{\Omega}\lvert\nabla_{\gamma}t\varphi\rvert^{2}\,{\mathrm{d}z}-\int_{\Omega}g(z)F_{a}(t\varphi)\,{\mathrm{d}z}$
$\displaystyle=\frac{t^{2}}{2}-\int_{\Omega}g(z)F(t\varphi)\,{\mathrm{d}z}+a\int_{\Omega}g(z)t\varphi\,{\mathrm{d}z}$

where $\Omega=\operatorname{supp}\varphi$. By (3),

$I_{a}(t\varphi)\leq\frac{t^{2}}{2}-M_{1}t^{\theta}\int_{\Omega}g(z)\lvert\varphi\rvert^{\theta}\,{\mathrm{d}z}+ta\lVert
g\rVert_{\infty}+M_{2}\lVert g\rVert_{1}.$

Since $\theta>2$ and $a\in(0,a_{1})$, we can fix $t_{0}>1$ large enough so
that $I_{a}(v)<0$, where $v=t_{0}\varphi\in D_{\gamma}(\mathbb{R}^{N})$.

∎

As a further step we prove the Palais-Smale condition of $I_{a}$. Recall that
from the estimate found in (4), for a fixed $a>0$,

$\theta F_{a}(t)\leq f_{a}(t)t+M\quad\hbox{for all $t\in\mathbb{R}$},$ (7)

for a suitable positive $M$.

######  Lemma 4.2.

The functional $I_{a}$ satisfies the Palais-Smale condition for all $a>0$.

###### Proof.

Let $\\{u_{j}\\}_{j}$ be a Palais-Smale sequence in
$D_{\gamma}(\mathbb{R}^{N})$. First of all we claim that

the sequence $\\{u_{j}\\}$ is bounded in $D_{\gamma}(\mathbb{R}^{N})$. (8)

Since $\\{u_{j}\\}_{j}$ is a Palais-Smale sequence, there exists
$j_{0}\in\mathbb{N}$ such that

$\left|\left\langle I^{\prime}_{a}(u_{j}),u_{j}\right\rangle\right|\leq\lVert
u_{j}\rVert_{\gamma}$

for $j\geq j_{0}$, and there exists $k>0$ such that for any $j\in\mathbb{N}$

$\lvert I_{a}(u_{j})\rvert\leq k.$

So

$\displaystyle-\lVert u_{j}\rVert_{\gamma}-\lVert
u_{j}\rVert_{\gamma}^{2}\leq-\int_{\mathbb{R}^{N}}g(z)f_{a}(u_{j})u_{j}\,{\mathrm{d}z}$
(9)

and

$\displaystyle\frac{1}{2}\lVert
u_{j}\rVert_{\gamma}^{2}-\int_{\mathbb{R}^{N}}g(z)F_{a}(u_{j})\,{\mathrm{d}z}\leq
k$ (10)

for all $j\geq j_{0}$. From (7) and (10),

$\frac{1}{2}\lVert
u_{j}\rVert_{\gamma}^{2}-\frac{1}{\theta}\int_{\mathbb{R}^{N}}g(z)f_{a}(u_{j})u_{j}\,{\mathrm{d}z}-\frac{1}{\theta}M\lVert
g\rVert_{1}\leq k$ (11)

for all $j\geq j_{0}$. Thereby, by (9) and (11),

$\frac{1}{2}\lVert u_{j}\rVert_{\gamma}^{2}-\frac{1}{\theta}\lVert
u_{j}\rVert_{\gamma}-\frac{1}{\theta}\lVert u_{j}\rVert_{\gamma}^{2}\leq
k+\frac{1}{\theta}M\lVert g\rVert_{1},$

or equivalently,

$\left(\frac{1}{2}-\frac{1}{\theta}\right)\lVert
u_{j}\rVert_{\gamma}^{2}-\frac{1}{\theta}\lVert u_{j}\rVert_{\gamma}\leq
k+\frac{1}{\theta}M\lVert g\rVert_{1},$

for $j$ large enough, so (8) is proved.

Up to a subsequence, still denoted by $\\{u_{j}\\}_{j}$, there exists $u\in
D_{\gamma}(\mathbb{R}^{N})$ such that

$u_{j}\rightharpoonup u\leavevmode\nobreak\ \hbox{ in
$D_{\gamma}(\mathbb{R}^{N})$}$

and

$u_{j}\to u\quad\hbox{a.e. in\leavevmode\nobreak\ $\mathbb{R}^{N}$}.$

From (2) we can estimate

$\lvert g(z)f_{a}(u_{j})\left(u_{j}-u\right)\rvert\leq Cg(z)\lvert
u_{j}-u\rvert\left(\lvert u_{j}\rvert+\lvert u_{j}\rvert^{\alpha-1}+a\right)$

for some constant $C>0$ independent of $a$. Now we we claim that

$\int_{\mathbb{R}^{N}}\left(g(z)\lvert u_{j}-u\rvert\left(\lvert
u_{j}\rvert+\lvert u_{j}\rvert^{\alpha-1}+a\right)\,\right){\mathrm{d}z}\to
0\leavevmode\nobreak\ \hbox{as \leavevmode\nobreak\ $j\to+\infty$}.$ (12)

In fact, we will only show the limit

$\int_{\mathbb{R}^{N}}\left(g(z)\lvert u_{j}-u\rvert\lvert
u_{j}\rvert^{\alpha-1}\,\right){\mathrm{d}z}\to 0\leavevmode\nobreak\ \hbox{as
\leavevmode\nobreak\ $j\to+\infty$}.$ (13)

because the limits involving the other terms follow with the same idea. The
sequence $\\{\lvert u_{j}\rvert^{\alpha-1}\\}_{j}$ is bounded in
$L^{\frac{2^{*}_{\gamma}}{\alpha-1}}(\mathbb{R}^{N})$, as (8) holds. Moreover,
applying Lemma 3.1 with $q=\frac{2^{*}_{\gamma}}{2^{*}_{\gamma}-(\alpha-1)}$,
we get

$u_{j}\to u\in
L^{\frac{2^{*}_{\gamma}}{2^{*}_{\gamma}-(\alpha-1)}}(\mathbb{R}^{N},\lvert
g\rvert).$ (14)

Then, by Hölder’s inequality with conjugate pair
$\left(\frac{2^{*}_{\gamma}}{\alpha-1},\frac{2^{*}_{\gamma}}{2^{*}_{\gamma}-\left(\alpha-1\right)}\right)$,

$\int_{\mathbb{R}^{N}}g(z)\lvert u_{j}-u\rvert\lvert
u_{j}\rvert^{\alpha-1}\,{\mathrm{d}z}\leq\lVert
g\left(u_{j}-u\right)\rVert_{{2^{*}_{\gamma}}/{2^{*}_{\gamma}-\left(\alpha-1\right)}}\lVert
u_{j}\rVert_{2^{*}_{\gamma}}^{\alpha-1}.$

Now, by (14) we get (13).

A consequence of (12) is the limit

$\int_{\mathbb{R}^{N}}g(z)f_{a}(u_{j})(u_{j}-u)\,{\mathrm{d}z}\to 0$

and so

$\int_{\mathbb{R}^{N}}\nabla_{\gamma}u_{j}\nabla_{\gamma}\left(u_{j}-u\right)\,{\mathrm{d}z}\to
0.$ (15)

The weak convergence $u_{j}\to u$ in $D_{\gamma}(\mathbb{R}^{N})$ yields

$\int_{\mathbb{R}^{N}}\nabla_{\gamma}u\nabla_{\gamma}\left(u_{j}-u\right)\,{\mathrm{d}z}\to
0.$ (16)

From (15) and (16),

$\int_{\mathbb{R}^{N}}\lvert\nabla_{\gamma}u_{j}-\nabla_{\gamma}u\rvert^{2}\,{\mathrm{d}z}\to
0,$

concluding the proof. ∎

Now we are in a position to apply the Mountain Pass Theorem [3, Theorem 2.1],
which ensures the existence and uniform boundedness of solutions to the (SP)
problem.

######  Theorem 4.3.

Let $a_{1}>0$ be given in Lemma 4.1. Then for each $a\in(0,a_{1})$, (SP) has a
solution $u_{a}\in D_{\gamma}(\mathbb{R}^{N})$. Moreover, there exists $C>0$
such that $\lVert u\rVert_{\gamma}\leq C$ for all $a\in(0,a_{1})$.

###### Proof.

Consider $a_{1}$, $\delta$ and $v$ as given in Lemma 4.1. For $a\in(0,a_{1})$,
using Lemma 4.1 and 4.2, the hypotheses of the mountain pass theorem [3,
Theorem 2.1] are verified. So there exists a non-trivial critical point
$u_{a}\in D_{\gamma}(\mathbb{R}^{N})$ of $I_{a}$ satisfying

$I_{a}(u_{a})=c_{a}=\inf_{\gamma\in\Gamma_{v}}\max_{t\in[0,1]}I_{a}(\gamma(t))\geq\delta\leavevmode\nobreak\
\hbox{and $I^{\prime}_{a}(u_{a})=0$},$ (17)

where $\Gamma_{v}=\\{\gamma\in
C\left([0,1],D_{\gamma}(\mathbb{R}^{N})\right)\mid\gamma(0)=0,\gamma(1)=v\\}$.
Thus, $u_{a}$ is a non-trivial solution of (SP). To prove the uniform
boundedness of $u_{a}$ in $D_{\gamma}(\mathbb{R}^{N})$, we first show that the
set $\\{I_{a}(u_{a})\mid a\in(0,a_{1})\\}$ is uniformly bounded. Considering
$t_{1}$ and $\varphi$ as in Lemma 4.1-(ii), Let us introduce a path
$\tilde{\gamma}\colon[0,1]\to D_{\gamma}(\mathbb{R}^{N})$ defined as
$\tilde{\gamma}(\sigma)=\sigma v$, where $v=t\varphi$ for some $t>t_{1}$. From
(17) we have that

$I_{a}(u_{a})\leq\max_{\sigma\in[0,1]}I_{a}(\tilde{\gamma}(\sigma))=\max_{\sigma\in[0,1]}I_{a}(\sigma
t\varphi).$ (18)

Now

$\displaystyle I_{a}(\sigma t\varphi)$
$\displaystyle\leq\frac{\sigma^{2}t^{2}}{2}\lVert\varphi\rVert_{\gamma}^{2}-M_{1}\sigma^{\theta}t^{\theta}\int_{\mathbb{R}^{N}}g(z)\left(\varphi(z)\right)^{\theta}\,{\mathrm{d}z}+M_{2}\int_{\mathbb{R}^{N}}g(z)\,{\mathrm{d}z}+a\sigma
t\int_{\mathbb{R}^{N}}g(z)\varphi(z)\,{\mathrm{d}z}$
$\displaystyle\leq\frac{t^{2}}{2}+M_{2}\lVert
g\rVert_{1}+a_{1}tC_{1}\lVert\varphi\rVert_{\gamma}.$ (19)

From (18) and (19), there exists $C=C(N_{\gamma},M_{2},g,a_{1})$ such that

$I_{a}(u_{a})\leq C,\leavevmode\nobreak\ \hbox{for all $a\in(0,a_{1})$.}$ (20)

Now we are going to prove that the solutions $u_{a}$ are uniformly bounded in
$D_{\gamma}(\mathbb{R}^{N})$ with respect to $a\in(0,a_{1})$. From (18) we
know that

$\lVert
u_{a}\rVert_{\gamma}^{2}-\int_{\mathbb{R}^{N}}g(z)f_{a}(z)u_{a}\,{\mathrm{d}z}=0.$
(21)

Also, by (20) we have

$\frac{1}{2}\lVert
u_{a}\rVert_{\gamma}^{2}-\int_{\mathbb{R}^{N}}g(z)F_{a}(u_{a})\,{\mathrm{d}z}\leq
C.$ (22)

Now first multiplying (21) by $\frac{1}{\theta}$ and then subtracting into
(22) gives the following

$\left(\frac{1}{2}-\frac{1}{\theta}\right)\lVert
u_{a}\rVert_{\gamma}^{2}+\int_{\mathbb{R}^{N}}g(z)\left(\frac{1}{\theta}f_{a}(u_{a})u_{a}-F_{a}(u_{a})\right)\,{\mathrm{d}z}\leq
C,$

and combining the above with (7) yields

$\left(\frac{1}{2}-\frac{1}{\theta}\right)\lVert
u_{a}\rVert_{\gamma}^{2}-\frac{1}{\theta}M_{3}\lVert g\rVert_{1}\leq C.$

So, there exists $C$ such that $\lVert u_{a}\rVert_{\gamma}\leq C$ for every
$a\in(0,a_{1})$. ∎

##  5 Properties of the solutions

Our next result establishes that the solution $u_{a}$ belongs to
$L^{\infty}(\mathbb{R}^{N})$ if the parameter $a$ is sufficiently small. Using
the iterative argument of [32, Lemma B.3] we prove that the solution $u_{a}\in
L^{q}(\mathbb{R}^{N})$, for every $q\in[2^{*}_{\gamma},\infty)$, provided $a$
is small enough. Moreover the argument in [30, Theorem 3.1] can be applied to
establish the boundedness of $u_{a}$.

######  Lemma 5.1.

There exists $a_{2}\in(0,a_{1})$ such that $u_{a}\in
L^{\infty}(\mathbb{R}^{N})\cap C(\mathbb{R}^{N})$ for all $a\in(0,a_{2})$.

###### Proof.

In order to prove the lemma, it is enough to show that for any sequence
$a_{j}\to 0$, the sequence of solutions $u_{j}=u_{a_{j}}$ possesses a
subsequence, still denoted by itself, which is bounded in
$L^{\infty}(\mathbb{R}^{N})$. By Lemma 4.3, the sequence $\\{u_{j}\\}$ is
bounded in $D_{\gamma}(\mathbb{R}^{N})$, then for some subsequence, there is
$u\in D_{\gamma}(\mathbb{R}^{N})$ such that

$u_{j}\rightharpoonup u\quad\hbox{in $D_{\gamma}(\mathbb{R}^{N})$}$

and

$u_{j}\to u\quad\hbox{a.e. in $\mathbb{R}^{N}$}.$

With the same approach explored in the proof of Lemma 4.2, we have that

$u_{j}\to u\quad\hbox{in $D_{\gamma}(\mathbb{R}^{N})$}.$

Consequently,

$\ u_{j}\to u\quad\hbox{in $L^{2^{*}_{\gamma}}(\mathbb{R}^{N})$}$ (23)

and for some subsequence, there is $h\in L^{2^{*}_{\gamma}}(\mathbb{R}^{N})$
such that

$\lvert u_{j}(z)\rvert\leq h(z),\quad\hbox{a.e. in $\mathbb{R}^{N}$ and
$\forall j\in\mathbb{N}$}.$

Setting the function

$V_{j}(z)=g(z)\frac{\left(1+\lvert
u_{j}\rvert^{\frac{N_{\gamma}+2}{N_{\gamma}-2}}\right)}{1+\lvert
u_{j}\rvert},$

it follows that

$\lvert V_{j}\rvert\leq g(1+\lvert u_{j}\rvert^{4/{N_{\gamma}-2}})\leq
g(1+\lvert h\rvert^{4/{N_{\gamma}-2}}),\quad\forall j\in\mathbb{N}.$ (24)

By the assumption on the function $g$, $V_{j}\in L^{p}(\mathbb{R}^{N})$, for
every $p\in[1,N_{\gamma}/2]$. Thereby, the Dominated Convergence Theorem with
(23) and (24) implies that $V_{j}\to V$ in $L^{p}(\mathbb{R}^{N})$, for some
$V\in L^{p}(\mathbb{R}^{N})$, $p\in[1,2^{*}_{\gamma}]$. So there exist a
constant $C>0$ and $j_{0}\in\mathbb{N}$ such that

$\lvert g(z)f_{a_{j}}(u_{j})\rvert\leq CV(z)\left(1+\lvert
u_{j}\rvert\right),\quad\forall j\geq j_{0}.$

In the rest of the proof we will assume $C=1$ to save notation.

Step 1. Given $q\in[2^{*}_{\gamma},\infty)$, there exists a constant $K_{q}$
such that $\lVert u_{j}\rVert_{q}\leq K_{q}$ for all $j\geq j_{0}$.

Let us prove that if $u_{j}\in L^{q}(\mathbb{R}^{N})$ for some $q\geq
2^{*}_{\gamma}$, then $u_{j}\in L^{\frac{2^{*}_{\gamma}}{2}q}$. Let $L\geq 0$
and $G(u_{j})=G_{L}(u_{j})=u_{j}\min\\{{u_{j}^{(q/2)-1},L}\\}$ and define

$F(u_{j})=F_{L}(u_{j})=\int_{0}^{u_{j}}\lvert
G^{\prime}(t)\rvert^{2}\,{\mathrm{d}t}\in D_{\gamma}(\mathbb{R}^{N}).$

First of all, observe that

$\int_{\mathbb{R}^{N}}\nabla_{\gamma}F(u_{j})\nabla_{\gamma}u_{j}\,{\mathrm{d}z}=\int_{\mathbb{R}^{N}}\lvert\nabla_{\gamma}G(u_{j})\rvert^{2}\,{\mathrm{d}z}\leavevmode\nobreak\
\hbox{for every $u_{j}\in D_{\gamma}(\mathbb{R}^{N})$}.$ (25)

and the following growth condition holds

$u_{j}F(u_{j})\leq C_{q}G(u_{j})^{2}\leq C_{q}u_{j}^{q},$

with a constant $C_{q}$ that can be explicitly given by
$C_{q}=\frac{q^{2}}{4(q-1)}$.

Using $F(u_{j})$ as a test function in (SP) we get

$\int_{\mathbb{R}^{N}}\nabla_{\gamma}F(u_{j})\nabla_{\gamma}u_{j}\,{\mathrm{d}z}\leq\int_{\mathbb{R}^{N}}V(z)(1+\lvert
u_{j}\rvert)u_{j}\min\\{u_{j}^{q-2},L^{2}\\}\,{\mathrm{d}z}\\\
\leq\int_{\mathbb{R}^{N}}V(z)(1+2\lvert
u_{j}\rvert^{2})\min\\{u_{j}^{q-2},L^{2}\\}\,{\mathrm{d}z}\\\
\leq\int_{\mathbb{R}^{N}}V(z)\,{\mathrm{d}z}+3\int_{\mathbb{R}^{N}}V(z)\lvert
u_{j}\rvert^{2}\min\\{u_{j}^{q-2},L^{2}\\}\,{\mathrm{d}z}.$ (26)

Moreover, if $q\in L^{q}(\mathbb{R}^{N})$, then for any $K\geq 1$ by (25) and
(26) there holds

$\int_{\mathbb{R}^{N}}\lvert\nabla_{\gamma}G(u_{j})\rvert^{2}\,{\mathrm{d}z}\leq
c+3\int_{\mathbb{R}^{N}}V(z)\lvert
u_{j}\rvert^{2}\min\\{u_{j}^{q-2},L^{2}\\}\,{\mathrm{d}z}\\\ \leq
c+3K\int_{\mathbb{R}^{N}}\lvert
u_{j}\rvert^{2}\min\\{u_{j}^{q-2},L^{2}\\}\,{\mathrm{d}z}+3\int_{\\{z\in\mathbb{R}^{N}\mid
V(z)\geq K\\}}\lvert
u_{j}\rvert^{2}\min\\{u_{j}^{q-2},L^{2}\\}\,{\mathrm{d}z}\\\ \leq
c(1+K)+c\left(\int_{\\{z\in\mathbb{R}^{N}\mid V(z)\geq
K\\}}V(z)^{N_{\gamma}/2}\,{\mathrm{d}z}\right)^{N_{\gamma}/2}\times\left(\int_{\mathbb{R}^{N}}\lvert
G(u_{j})\rvert^{\frac{2N_{\gamma}}{N_{\gamma}-2}}\,{\mathrm{d}z}\right)^{\frac{N_{\gamma}-2}{N_{\gamma}}}\\\
\leq
c(1+K)+c\varepsilon(K)\int_{\mathbb{R}^{N}}\lvert\nabla_{\gamma}G(u_{j})\rvert^{2}\,{\mathrm{d}z},$

with some constants $c$ depending on the $L^{q}(\mathbb{R}^{N})$ norm of
$u_{j}$ and

$\varepsilon(K)=\left(\int_{\\{z\in\mathbb{R}^{N}\mid V(z)\geq
K\\}}V(z)^{N_{\gamma}/2}\right)^{2/N_{\gamma}}\to 0\quad\hbox{as
$K\to\infty$}.$

Fix $K$ such that $\varepsilon(K)\leq 1/2$, for this choice of $K$ we may
conclude that

$\int_{\mathbb{R}^{N}}\lvert\nabla_{\gamma}G(u_{j})\rvert^{2}\,{\mathrm{d}z}$

remains uniformly bounded in $L$. Hence we may let $L\to\infty$ to derive that
$u_{j}\in L^{\frac{2^{*}_{\gamma}}{2}q}(\mathbb{R}^{N})$. We can conclude our
argument by iterating the process.

Step 2. There exists $C>0$ such that $\lVert u_{j}\rVert_{\infty}\leq C$ for
all $j\geq j_{0}$.

Since, by the previous Step, $u_{j}\in L^{q}(\mathbb{R}^{N})$ for all
$q\in[1,\infty)$ then also $V\in L^{q}(\mathbb{R}^{N})$ for every
$q\in[1,\infty)$. Let $t_{0}>N_{\gamma}/2$, we construct a sequence of
$L^{q_{k}}(\mathbb{R}^{N})$ norms of $u_{j}$, with $q_{k}\to\infty$, which are
uniformly bounded by the $L^{q_{0}}(\mathbb{R}^{N})$ norm of $u_{j}$, where
$q_{0}=t_{0}^{\prime}2^{*}_{\gamma}$ and $t_{0}^{\prime}$ is such that
$1/t_{0}+1/t_{0}^{\prime}=1$. As before, using again the test function
$F(u_{j})$, we estimate the right-hand side of (SP) as follows

$\int_{\mathbb{R}^{N}}V(z)(1+\lvert
u_{j}\rvert)F(u_{j})\,{\mathrm{d}z}\leq\lVert V\rVert_{t_{0}}\lVert
u_{j}F(u_{j})\rVert_{t_{0}^{\prime}}+C_{q}\lVert V\rVert_{\tilde{t}_{0}}\lVert
u_{j}\rVert_{qt_{0}^{\prime}}^{q-1}\\\ \leq C_{q}\lVert
u_{j}\rVert_{qt_{0}^{\prime}}^{q}\left(\lVert V\rVert_{t_{0}}+\lVert
V\rVert_{\tilde{t}_{0}}\lVert
u_{j}\rVert_{qt_{0}^{\prime}}^{-1}\right)=\tilde{C}C_{q}\lVert
V\rVert_{t_{0}}\lVert u_{j}\rVert_{qt_{0}^{\prime}}^{q}$ (27)

for some $\tilde{t}_{0}>N_{\gamma}/2$ and the constant
$\tilde{C}=\tilde{C}(q,V,u_{j})$. Hence, by (25) and (27), we obtain

$\int_{\mathbb{R}^{N}}\lvert\nabla_{\gamma}G(u)\rvert^{2}\,{\mathrm{d}z}\leq\tilde{C}C_{q}\lVert
V\rVert_{t_{0}}\lVert u_{j}\rVert_{qt_{0}^{\prime}}^{q}.$

By Sobolev inequality and letting $L\to\infty$, we get

$\lVert u_{j}\rVert_{\frac{2^{*}_{\gamma}}{2}q}^{q}\leq CC_{q}\lVert
V\rVert_{t_{0}}\lVert u_{j}\rVert_{qt_{0}^{\prime}}^{q}.$

Let $\delta=\frac{2^{*}_{\gamma}}{2t_{0}^{\prime}}>1$. With this notation, the
preceding estimate can be rewritten as

$\lVert u_{j}\rVert_{\delta
qt_{0}^{\prime}}\leq\left(CC_{q}\right)^{1/q}\lVert
V\rVert_{t_{0}}^{1/q}\lVert u_{j}\rVert_{qt_{0}^{\prime}}.$

Now, define $q_{0}=2^{*}_{\gamma}t_{0}^{\prime}$ and $q_{k}=\delta^{k}q_{0}$
for $k\geq 1$. Observing that $C_{q}\leq Cq$, we obtain

$\lVert
u_{j}\rVert_{q^{k}}\leq\left(\prod_{i=0}^{k-1}\left(Cq_{i}\right)^{1/q_{i}}\right)\lVert
V\rVert_{t_{0}}^{\sum_{i=0}^{k-1}\frac{1}{q_{i}}}\lVert u_{j}\rVert_{q_{0}}$

whose right-hand side is finite since

$\sum_{i=0}^{\infty}\frac{1}{q_{i}}=\frac{1}{q_{0}}\sum_{i=0}^{\infty}\frac{1}{\delta_{i}}<\infty\quad\text{and}\quad\sum_{i=0}^{\infty}\frac{\log
q_{i}}{q_{i}}<\infty.$

Therefore, letting $k\to\infty$, we can conclude.

Step 3. The continuity of the solutions $u_{j}$ comes from the non-homogeneous
Harnack inequality for $X$-elliptic operators established in [22, Theorem 5.5]
combined with some standard arguments, see for example [23, Corollary 4.18]
for further details. ∎

In the next two lemmas, we prove that the solution $\\{u_{a}\\}$ is uniformly
bounded from below over several spaces.

######  Lemma 5.2.

Let $a_{1}$ as in Theorem 4.3. Then there exists $C_{1}>0$ such that $\lVert
u_{a}\rVert_{\gamma}\geq C_{1}$ for all $a\in(0,a_{1})$.

###### Proof.

We notice that $F_{a}(t)\geq-a\lvert t\rvert$ for all $t\in\mathbb{R}$. For
$\delta$ as given in Theorem 4.3 and from (17) we write
$I_{a}(u_{a})\geq\delta$, for all $a\in(0,a_{1})$. Using the embedding of
$D_{\gamma}(\mathbb{R}^{N})$ proved in Lemma 3.1 we have

$\delta\leq I_{a}(u_{a})\leq\frac{1}{2}\lVert
u_{a}\rVert_{\gamma}^{2}+a\int_{\mathbb{R}^{N}}g(z)\lvert
u_{a}(z)\rvert\,{\mathrm{d}z}\leq\frac{1}{2}\lVert
u_{a}\rVert_{\gamma}^{2}+a_{1}C\lVert u_{a}\rVert_{\gamma}.$

So there exists $C_{1}=C_{1}(N,\gamma,g,a_{1},\delta)$ such that $\lVert
u_{a}\rVert_{\gamma}\leq C_{1}$ for all $a\in(0,a_{1})$. ∎

In what follows, we show an estimate from below to the norm
$L^{\infty}(\mathbb{R}^{N})$ of $u_{a}$ for $a$ small enough.

######  Lemma 5.3.

There exists $a_{3}\in(0,a_{2})$ and $\beta>0$ that does not depend on
$a\in(0,a_{3})$, such that $\lVert u_{a}\rVert_{\infty}\geq\beta$ for all
$a\in(0,a_{3})$.

###### Proof.

For the constant $\delta$ given in Theorem 4.3, from (17) we write
$I_{a}(u_{a})\geq\delta$ for all $a\in(0,a_{1})$. Further, we have (notice
again that $F_{a}(t)\geq-a\lvert t\rvert$ for all $t\in\mathbb{R}$),

$\frac{\lVert
u_{a}\rVert_{\gamma}^{2}}{2}=I_{a}(u_{a})+\int_{\mathbb{R}^{N}}g(z)F_{a}(u_{a})\,{\mathrm{d}z}\geq\delta-a\int_{\mathbb{R}^{N}}g(z)\lvert
u_{a}\rvert\,{\mathrm{d}z}.$

For every $a\in(0,a_{1})$, using the continuous embedding
$D_{\gamma}(\mathbb{R}^{N})\hookrightarrow L^{1}(\mathbb{R}^{N},\lvert
g\rvert)$, Lemma 3.1, with embedding constant $C_{2}=C_{2}(N,\gamma,g)$ and
the uniform boundeness of $\\{u_{a}\\}$ in $D_{\gamma}(\mathbb{R}^{N})$,
Theorem 4.3, we obtain

$\frac{\lVert u_{a}\rVert_{\gamma}^{2}}{2}\geq\delta-aC_{2}\lVert
u_{a}\rVert_{\gamma}\geq\delta-aC_{2}C=\delta-aC_{3},$

where $C_{3}=C_{2}C$. Now if we choose $a_{3}$ such that
$0<a_{3}<\min\left\\{\frac{\delta}{C_{3}},a_{2}\right\\}$, then

$\frac{\lVert u_{a}\rVert_{\gamma}^{2}}{2}\geq\delta_{0}=\delta-
a_{3}C_{3}>0,\quad\forall a\in(0,a_{3}).$ (28)

Since $u_{a}$ is a weak solution of (SP), then

$\int_{\mathbb{R}^{N}}\nabla_{\gamma}u_{a}\nabla_{\gamma}\varphi\,{\mathrm{d}z}=\int_{\mathbb{R}^{N}}g(z)f_{a}(u_{a})\varphi\,{\mathrm{d}z},\quad\forall\varphi\in
D_{\gamma}(\mathbb{R}^{N}).$

For $\varphi=u_{a}$ we have

$\int_{\mathbb{R}^{N}}\lvert\nabla_{\gamma}u_{a}\rvert^{2}\,{\mathrm{d}z}=\int_{\mathbb{R}^{N}}g(z)f_{a}(u_{a})u_{a}\,{\mathrm{d}z}$

So, by the estimation found in (28)

$\int_{\mathbb{R}^{N}}g(z)f_{a}(u_{a})u_{a}\,{\mathrm{d}z}\geq
2\delta_{0}>0,\quad\forall a\in(0,a_{3}).$

Thus, (2) gives

$2\delta_{0}\leq\int_{\mathbb{R}^{N}}g(z)\left(C\left(\lvert
u_{a}\rvert^{\alpha-1}+\lvert
u_{a}\rvert\right)+a\right)\,{\mathrm{d}z}\leq\left(C\left(\lVert
u_{a}\rVert_{\infty}^{\alpha-1}+\lVert
u_{a}\rVert_{\infty}\right)+a\right)\lVert g\rVert_{1}.$

This implies that $\lVert u_{a}\rVert_{\infty}\geq\beta$ for some $\beta>0$
for all $a\in(0,a_{3})$, taking $a_{3}$ smaller if necessary. ∎

##  6 Positive solutions

As in the previous Section, given a sequence $a_{j}\to 0$ as $j\to\infty$, let
us consider the solution $u_{j}=u_{a_{j}}$ for (SP). We denote by $u$ the
limit of $u_{j}$ as in Lemma 5.1.

Now consider the function $f_{0}$

$f_{0}(t)=\begin{cases}f(t),\quad&\hbox{if $t\geq 0$};\\\ 0,\quad&\hbox{if
$t<0$}.\end{cases}$

######  Proposition 6.1.

$u$ is a weak solution of

$-\Delta_{\gamma}u=g(z)f_{0}(u)\quad\hbox{in $\mathbb{R}^{N}$}.$ (29)

Moreover, $u\in L^{p}(\mathbb{R}^{N})\cap C(\mathbb{R}^{N})$ for every
$p\in[2^{*}_{\gamma},\infty]$.

###### Proof.

Since $a_{j}\to 0$, there exists $j_{1}\in\mathbb{N}$ such that $a_{j}\leq
a_{2}$ for all $j\geq j_{1}$ (with $a_{2}$ we denote the parameter $a_{2}$ as
in the statement of Lemma 5.1). Now, for every $\varphi\in
D_{\gamma}(\mathbb{R}^{N})$ with $\varphi\geq 0$ we show that

$\int_{\mathbb{R}^{N}}g(z)f_{j}(u_{j})\varphi(z)\,{\mathrm{d}z}\to\int_{\mathbb{R}^{N}}g(z)f_{0}(u)\varphi(z)\,{\mathrm{d}z},\quad\hbox{as
$j\to\infty$}.$

We split

$\lvert f_{j}(u_{j})-f_{0}(u)\rvert\leq\lvert
f_{j}(u_{j})-f_{0}(u_{j})\rvert+\lvert f_{0}(u_{j})-f_{0}(u)\rvert.$

By the continuity of $K_{0}^{\prime}$,

$\int_{\mathbb{R}^{N}}g(z)\lvert
f_{0}(u_{j})-f_{0}(u)\rvert\varphi(z)\,{\mathrm{d}z}\to 0,\quad\hbox{as
$j\to\infty$}.$

Further, $\lvert f_{j}(u_{j})-f_{0}(u_{j})\rvert\leq a_{j}$. Therefore,

$\int_{\mathbb{R}^{N}}g(z)\lvert
f_{j}(a_{j})-f_{0}(u)\rvert\varphi(z)\,{\mathrm{d}z}\\\ \leq
a_{j}\int_{\mathbb{R}^{N}}g(z)\varphi(z)\,{\mathrm{d}z}+\int_{\mathbb{R}^{N}}g(z)\lvert
f_{0}(u_{j})-f_{0}(u)\rvert\varphi(z)\,{\mathrm{d}z}\to
0\quad\hbox{as}\leavevmode\nobreak\ j\to\infty.$

From the weak formulation

$\int_{\mathbb{R}^{N}}\nabla_{\gamma}u_{j}(z)\nabla_{\gamma}\varphi(z)\,{\mathrm{d}z}=\int_{\mathbb{R}^{N}}g(z)f_{j}(u_{j})\varphi(z)\,{\mathrm{d}z},\forall\varphi\in
D_{\gamma}(\mathbb{R}^{N}),\quad\varphi\geq 0.$

Taking the limit as $j\to\infty$ gives

$\int_{\mathbb{R}^{N}}\nabla_{\gamma}u(z)\nabla_{\gamma}\varphi(z)\,{\mathrm{d}z}=\int_{\mathbb{R}^{N}}g(z)f(u)\varphi(z)\,{\mathrm{d}z},\forall\varphi\in
D_{\gamma}(\mathbb{R}^{N}),\quad\varphi\geq 0.$

Now, for any $\varphi\in D_{\gamma}(\mathbb{R}^{N})$, we write
$\varphi=\varphi^{+}-\varphi^{-}$. We see that the above identity holds for
both $\varphi^{+}$ and $\varphi^{-}$, so

$\int_{\mathbb{R}^{N}}\nabla_{\gamma}u(z)\nabla_{\gamma}\varphi(z)\,{\mathrm{d}z}=\int_{\mathbb{R}^{N}}g(z)f(u)\varphi(z)\,{\mathrm{d}z},\quad\forall\varphi\in
D_{\gamma}(\mathbb{R}^{N}),$ (30)

that is, $u$ is a weak solution of (29). With the same argument of Lemma 5.1,
we obtain $u\in L^{p}(\mathbb{R}^{N})$, $p\in[2^{*}_{\gamma},\infty]$ and $u$
is a continuous function in $\mathbb{R}^{N}$. ∎

######  Proposition 6.2.

The function $u$ is positive.

###### Proof.

Taking the test function $\varphi=u^{-}=\min\left\\{u,0\right\\}$ in (30) we
find that $\lVert u^{-}\rVert_{\gamma}=0$, so $u\geq 0$ a.e. in
$\mathbb{R}^{N}$.

Let $\Sigma=\\{0\\}\times\mathbb{R}^{\ell}$ be the degenerate set for the
Grushin operator. If $\Omega$ is any bounded domain contained in
$\mathbb{R}^{N}\setminus\Sigma$, then $\Delta_{\gamma}$ is a uniformly
elliptic operator on $\Omega$. Hence $u\in C^{2}(\Omega)$ and either $u\equiv
0$ on $\Omega$ or $u>0$ in $\Omega$. In the former case, it follows from [26,
Proposition 2.5] that $u\equiv 0$ on the connected component of
$\mathbb{R}^{N}\setminus\Sigma$ which contains $\Omega$. Therefore $u\equiv 0$
on $\mathbb{R}^{N}\setminus\Sigma$.

Now, it is clear that $\Sigma$ has empty interior, and it follows from [25,
Theorem 9, Chapter 1] that $\mathbb{R}^{N}\setminus\Sigma$ is dense in
$\mathbb{R}^{N}$. Recalling that $u$ is a continuous function, we conclude
that $u\equiv 0$ on $\Sigma$. To this stage the function $u$ might be zero in
(the interior of) some connected component of $\mathbb{R}^{N}\setminus\Sigma$
and strictly positive in (the interior of) some other component. However this
is again impossible, since [26, Proposition 2.5] applies again on any ball
which meets two such components across $\Sigma$. Hence either $u\equiv 0$ on
$\mathbb{R}^{N}$, or $u>0$ on $\mathbb{R}^{N}$. The former is not admissible
due to the Lemma 5.3. ∎

######  Remark 6.3.

When $\gamma$ is an even integer, a very elegant proof of the positivity of
$u$ can be devised. We follow an idea of [6]. When $\gamma$ is an integer, it
is known that $-\Delta_{\gamma}$ satisfies the Hörmander condition. Recalling
that $g>0$ and $f_{0}\geq 0$, we see that
$\int_{\mathbb{R}^{N}}\nabla_{\gamma}u\nabla_{\gamma}\varphi\,{\mathrm{d}z}\geq
0$ for every $\varphi\in C_{0}^{\infty}(\mathbb{R}^{N})$ such that
$\varphi\geq 0$. Hence
$\int_{\mathbb{R}^{N}}u(-\Delta_{\gamma}\varphi)\,{\mathrm{d}z}\geq 0$ for
every such $\varphi$. Recalling that $u$ is a continuous function, it follows
that $u$ is a viscosity subsolution of the operator $-\Delta_{\gamma}$. The
strong maximum principle proved in [4, Corollary 1.4] yields that either
$u\equiv 0$ or $u>0$ in $\mathbb{R}^{N}$. The first option, again, is not
admissible due to the Lemma 5.3.

Now we are going to show that $u_{j}\to u$ in
$L^{\infty}(\\{0\\}\times\mathbb{R}^{\ell})$. We essentially follow [1, 7],
although a few important differences will arise.

For $z=(x,y)\in\mathbb{R}^{N}=\mathbb{R}^{m}\times\mathbb{R}^{\ell}$, we
denote by

$d(z)=\left(\lvert x\rvert^{2\left(\gamma+1\right)}+(\gamma+1)^{2}\lvert
y\rvert^{2}\right)^{\frac{1}{2\left(\gamma+1\right)}}$ (31)

the homogeneous norm associated to $\Delta_{\gamma}$. Note that this norm is
homogeneous of degree $1$ with respect to the anisotropic dilatation

$\delta_{\lambda}(x,y)=(\lambda x,\lambda^{\gamma+1}y),\quad\lambda>0.$

and, for $\gamma=0$, it reduces to the usual Euclidean distance in
$\mathbb{R}^{N}$.

The function

$\Gamma(z)=\frac{C}{d(z)^{N_{\gamma}-2}}$

is a fundamental solution for the Grushin operator with singularity at $z=0$,
see [5] or [16, Appendix B].

The constant $C$ is a suitable positive constant that depends on $m,\ell$ and
$\gamma$. Precisely,

$C^{-1}=(N_{\gamma}-2)\int\limits_{\\{z\mid d(z)=1\\}}\frac{\lvert
x\rvert^{2\gamma}}{\left(\lvert
x\rvert^{2\left(1+2\gamma\right)}+\left(1+\gamma\right)^{2}\lvert
y\rvert^{2}\right)^{1/2}}\,\mathrm{d}S,$

where $S$ denotes the surface measure of the manifold $\left\\{z\mid
d(z)=1\right\\}$.

We recall the following definition.

######  Definition 6.4.

A function $u\colon\mathbb{R}^{N}\to\mathbb{R}$ is radially symmetric (radial
for short) if there exists a function $v\colon\mathbb{R}\to\mathbb{R}$ such
that

$u(z)=v(r)$

where $r=d(z)$, for all $z\in\mathbb{R}^{N}$.

For $R>0$ we set $B_{R}=\left\\{\xi\in\mathbb{R}^{N}\mid d(\xi)<R\right\\}$.

Let us consider a spherical shell
$\Omega=B_{R_{2}}\setminus\overline{B_{R_{1}}}$, with $0\leq
R_{1}<R_{2}\leq+\infty$. If $u$ is a radial function, by [15, Eq. (2.4)] we
get the formula

$\int_{\Omega}u(d(\xi))\,\mathrm{d}\xi=s_{N}\int_{R_{1}}^{R_{2}}\rho^{N_{\gamma}-1}u(\rho)\,\mathrm{d}\rho,$
(32)

where $s_{N}$ is an explicit constant that depends only on $N$.

######  Proposition 6.5.

$\lVert u_{j}-u\rVert_{L^{\infty}(\\{0\\}\times\mathbb{R}^{\ell})}\to 0$ as
$j\to\infty$.

###### Proof.

By [18, Sect. 3.1] and recalling that $u$ is a continuous function, we have
the representation formula

$u(z)=C\int_{\mathbb{R}^{N}}\frac{g(\xi)f_{0}(u(\xi))}{d(z-\xi)^{N_{\gamma}-2}}\,\mathrm{d}\xi\quad\hbox{for
a.e. $z\in\\{0\\}\times\mathbb{R}^{\ell}$}.$ (33)

Moreover, since $u_{j}$ is a solution of (SP), we also have

$u_{j}(z)=C\int_{\mathbb{R}^{N}}\frac{g(\xi)f_{j}(u_{j}(\xi))}{d(z-\xi)^{N_{\gamma}-2}}\,\mathrm{d}\xi\quad\hbox{for
a.e. $z\in\\{0\\}\times\mathbb{R}^{\ell}$}.$ (34)

Using (33) and (34) we estimate $\lvert u-u_{j}\rvert$ as follows: for every
$z\in\\{0\\}\times\mathbb{R}^{\ell}$,

$\lvert u(z)-u_{j}(z)\rvert\leq C\int_{\mathbb{R}^{N}}g(\xi)\frac{\lvert
f_{j}(u_{j}(\xi))-f_{0}(u(\xi))\rvert}{d(z-\xi)^{N_{\gamma}-2}}\,\mathrm{d}\xi\\\
\leq C\left(\int_{B_{1}}g(\xi)\frac{\lvert
f_{j}(u_{j}(\xi))-f_{0}(u(\xi))\rvert}{d(z-\xi)^{N_{\gamma}-2}}\,\mathrm{d}\xi+\int_{\mathbb{R}^{N}\setminus
B_{1}}g(\xi)\frac{\lvert
f_{j}(u_{j}(\xi))-f_{0}(u(\xi))\rvert}{d(z-\xi)^{N_{\gamma}-2}}\,\mathrm{d}\xi\right).$
(35)

Take $\delta>1$. Applying the Hölder’s inequality with the conjugate pairs
$(\delta,\tilde{\delta})$ we estimate the first integral of (35) as

$\int_{B_{1}}g(\xi)\frac{\left|f_{j}(u_{j}(\xi))-f_{0}(u(\xi))\right|}{d(z-\xi)^{N_{\gamma}-2}}\,\mathrm{d}\xi\leq\left(\int_{B_{1}}\frac{1}{d(z-\xi)^{(N_{\gamma}-2)\delta}}\,\mathrm{d}\xi\right)^{\frac{1}{\delta}}\times\\\
\left(\int_{B_{1}}g(\xi)^{\tilde{\delta}}\left|f_{j}(u_{j}(\xi))-f_{0}(u(\xi))\right|^{\tilde{\delta}}\,\mathrm{d}\xi\right)^{\frac{1}{\tilde{\delta}}}.$
(36)

We compute, using (32)

$\int_{B_{1}}\frac{1}{d(z-\xi)^{(N_{\gamma}-2)\delta}}\,\mathrm{d}\xi=s_{N}\int_{0}^{1}\frac{\rho^{N_{\gamma}-1}}{\rho^{(N_{\gamma}-2)\delta}}d\rho\leq
C(N_{\gamma}).$

Moreover, the second term in the inequality (36) converges to zero. In fact,
proceeding as in Lemma 5.1, $u_{j}\to u$ in
$L^{p},\,p\in[2^{*}_{\gamma},\infty)$. Then we can apply the Dominated
Convergence Theorem.

Next, the second integral of (36) has the following bound:

$\int_{\mathbb{R}^{N}\setminus B_{1}}g(\xi)\frac{\lvert
f_{j}(u_{j}(\xi))-f_{0}(u(\xi))\rvert}{d(z-\xi)^{N_{\gamma}-2}}\,\mathrm{d}\xi\leq\int_{\mathbb{R}^{N}\setminus
B_{1}}g(\xi)\lvert f_{j}(u_{j}(\xi))-f_{0}(u(\xi))\rvert\,\mathrm{d}\xi.$

Again, by the generalized dominated convergence theorem,

$\lim_{j\to+\infty}\int_{\mathbb{R}^{N}\setminus B_{1}}g(\xi)\frac{\lvert
f_{j}(u_{j}(\xi))-f_{0}(u(\xi))\rvert}{d(z-\xi)^{N_{\gamma}-2}}\,\mathrm{d}\xi=0.$

Therefore, (35) yields $u_{j}\to u$ in
$L^{\infty}(\\{0\\}\times\mathbb{R}^{\ell})$ as $j\to\infty$. ∎

The immediate corollary concludes the proof of the first statement of Theorem
1.1.

######  Corollary 6.6.

$u_{j}\geq 0$ a.e. in $\\{0\\}\times\mathbb{R}^{\ell}$ for $j$ large enough.

Let us now conclude the proof of Theorem 1.1.

######  Theorem 6.7.

If (f3) and (g1) are satisfied then $u_{j}>0$ a.e. in
$\\{0\\}\times\mathbb{R}^{\ell}$.

###### Proof.

For each $j\in\mathbb{N}$, since $f_{j}$ is locally Lipschitz (f3) and $0\leq
u_{j},u\leq C$, we have

$\lvert f_{j}(u_{j}(z))-f_{0}(u(z))\rvert\leq M\lvert u_{j}(z)-u(z)\rvert,$

for some $M>0$. For $z\in\\{0\\}\times\mathbb{R}^{\ell}\setminus\\{(0,0)\\}$,
using (33) and (34), we get

$\lvert u_{j}(z)-u(z)\rvert\leq
C\left(M\int_{\mathbb{R}^{N}}g(\xi)\frac{\lvert
u_{j}(\xi)-u(\xi)\rvert}{d(z-\xi)^{N_{\gamma}-2}}\,\mathrm{d}\xi+a_{j}\int_{\mathbb{R}^{N}}\frac{g(\xi)}{d(z-\xi)^{N_{\gamma}-2}}\,\mathrm{d}\xi\right).$

Since $g$ satisfies (g1), from the above inequality and the previous theorem
we get

$\lvert u_{j}(z)-u(z)\rvert\leq C\left(M\lVert
u_{j}-u\rVert_{\infty}+a_{n}\right)\frac{C(g)}{d(z)^{N_{\gamma}-2}}.$

Hence

$\sup_{z\in\mathbb{R}^{N}\setminus\\{0\\}}d(z)^{N_{\gamma}-2}\lvert
u_{j}(z)-u(z)\rvert\to 0\leavevmode\nobreak\ \hbox{as $j\to\infty$}.$ (37)

We claim that

$\lim_{d(z)\to+\infty}d(z)^{N_{\gamma}-2}u(z)>0.$ (38)

Using (33) it follows that

$\lim_{d(z)\to+\infty}d(z)^{N_{\gamma}-2}u(z)=C\lim_{d(z)\to+\infty}\int_{\mathbb{R}^{N}}\frac{g(\xi)f_{0}(u(\xi))d(z)^{N_{\gamma}-2}}{d(z-\xi)^{N_{\gamma}-2}}\,\mathrm{d}\xi\\\
\geq
C\int_{B_{R}}\frac{g(\xi)f_{0}(u(\xi))d(z)^{N_{\gamma}-2}}{d(z-\xi)^{N_{\gamma}-2}}\,\mathrm{d}\xi$
(39)

for any $R>0$. For any $R>0$ there exists $z\in\\{0\\}\times\mathbb{R}^{\ell}$
such that $d(z)>2R+1$. Hence

$d(z-\xi)^{N_{\gamma}-2}\geq\lvert d(z)-d(\xi)\rvert^{N_{\gamma}-2}\geq\lvert
d(z)-R\rvert^{N_{\gamma}-2}\geq
2^{2-N_{\gamma}}\left(1+d(z)\right)^{N_{\gamma}-2},$

for every $\xi\in B_{R}$. Using the above estimate, for $\xi\in B_{R}$ we get

$\frac{g(\xi)f_{0}(u(\xi))d(z)^{N_{\gamma}-2}}{d(z-\xi)^{N_{\gamma}-2}}\leq
2^{N_{\gamma}-2}g(\xi)f_{0}(u(\xi)).$

Furthermore,

$\frac{g(\xi)f_{0}(u(\xi))d(z)^{N_{\gamma}-2}}{d(z-\xi)^{N_{\gamma}-2}}\to
g(\xi)f_{0}(u(\xi))$

a.e. in $B_{R}$, as $d(z)\to\infty$. Therefore, the Dominated Convergence
Theorem yields

$\lim_{d(z)\to+\infty}\int_{B_{R}}\frac{g(\xi)f(u(\xi))d(z)^{N_{\gamma}-2}}{d(z-\xi)^{N_{\gamma}-2}}\,\mathrm{d}\xi=\int_{B_{R}}g(\xi)f_{0}(u(\xi))\,\mathrm{d}\xi.$

So, from (39) we conclude that

$\lim_{d(z)\to+\infty}d(z)^{N_{\gamma}-2}u(z)\geq
C\int_{B_{R}}g(\xi)f_{0}(u(\xi))\,\mathrm{d}\xi.$

Letting $R\to\infty$ and applying the Fatou’s lemma we get

$\lim_{d(z)\to\infty}d(z)^{N_{\gamma}-2}u(z)\geq
C\int_{\mathbb{R}^{N}}g(\xi)f_{0}(u(\xi))\,\mathrm{d}\xi.$

So (38) follows by (33) and Proposition 6.2.

Therefore, by (37) there exists $j_{2}\in\mathbb{N}$ and $R>>1$ such that for
$j\geq j_{2},\,u_{j}>0$ a.e. on $(\\{0\\}\times\mathbb{R}^{\ell})\setminus
B_{R}$. Moreover, since $u$ is continuous, there exists $\eta>0$ such that
$u>\eta$ on $\overline{B_{R}}$. Therefore, from Proposition 6.5, there exists
$j_{3}\in\mathbb{N}$ such that for $j\geq j_{3},\,u_{j}>0$ a.e. on
$\overline{B_{R}}\cap(\\{0\\}\times\mathbb{R}^{\ell})$. Thus, by choosing
$j_{4}=\max\left\\{j_{2},j_{3}\right\\}$, we see that for $j\geq
j_{4},\,u_{j}>0$ a.e. on $\\{0\\}\times\mathbb{R}^{\ell}$. This completes the
proof. ∎

######  Remark 6.8.

The validity of the previous argument relies heavily on the representation
formula (33), which is typically derived from the existence of a Green
function associated to the differential operator. For linear differential
operators with constant coefficients, the existence of a fundamental solution
allows one to construct (continuous) solutions via convolution. by exploiting
the invariance of the operator under translations. The Grushin operator
$\Delta_{\gamma}$ is a differential operator with variable coefficients, and
in particular it is invariant under translations only in the subspace
$\\{0\\}\times\mathbb{R}^{\ell}$ of $\mathbb{R}^{N}$. So, even if a
fundamental solution with singularity at a fixed point is available, it is not
trivial to use it in order to derive an explicit Green function. As we have
seen, it was proved in [18] that (33) holds true at points of the form $(0,y)$
at which the weak solution $u$ is continuous. It is interesting to observe
that the authors of [17] proved the equivalence of a representation formula
like (33) and of a Liouville-type result for the operator $\Delta_{\gamma}$,
but only for $C^{2}$ solutions. However a full regularity theory for
$\Delta_{\gamma}$ is not available, this approach remains unapplicable.

Furthermore, the existence of an _approximate_ Green function for
$\Delta_{\gamma}$ was proved in [29] and exploited in [2]. Unfortunately the
known properties of it are too weak for our purposes, and would lead to
replace (g1) with a completely implicit assumption.

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Paolo Malanchini, Dipartimento di Matematica e Applicazioni, Università degli
Studi di Milano Bicocca, via R. Cozzi 55, I-20125 Milano, Italy.

Giovanni Molica Bisci, Dipartimento di Scienze Pure e Applicate, Università di
Urbino Carlo Bo, Piazza della Repubblica 13, Urbino, Italy.

Simone Secchi, Dipartimento di Matematica e Applicazioni, Università degli
Studi di Milano Bicocca, via R. Cozzi 55, I-20125 Milano, Italy.