Existence of positive radial solutions for a problem involving the weighted Heisenberg $p(\cdot)$-Laplacian operator

1.   Introduction

Let $\Omega$ be the unit Korányi ball in the Heisenberg group $\mathbb{H}^n \; (n\geq1)$; the $p(\cdot)$-Laplacian problem

with Dirichlet boundary condition is studied. Assume

where $\alpha, \beta\in L^\infty(0, 1)$ such that $\alpha$ is a positive non-constant radially non-decreasing function and $\beta$ is a non-negative radially non-increasing function. $p, \theta, \vartheta \in C_+(\Omega)$ such that

where

for $s$ with

The weighted operator $\mathcal{L}$ is defined by

where $w$ is a Muckenhoupt weight function of class $A_s$. And, finally, $\mathcal{R}:\Omega\rightarrow [0, +\infty)$ belongs to $L^\infty(\Omega)$ such that ${\rm{ess}} \;{\rm{inf}}_{\Omega}\mathcal{R} > 0$.

The topicality of the theory of Sobolev spaces on Heisenberg groups is explained by numerous applications of it to the study of solutions for subelliptic differential equations, quasiconformal analysis, and many other related problems. The Heisenberg groups represent the best known and, in many respects, a model case of the Carnot-Carathéodory spaces.

Most papers in the literature on Muckenhoupt weighted problems focus their attention on the $p$-Laplacian operator (see ^{[1,3,4,6,13,14,15,18,19,20,21,25,26,28,29,30,31,32,35]} and the references therein). A sharp distinction between this note and the mentioned works is that this paper discusses on the existence of a suitable interval for embedding of weighted Heisenberg Sobolev spaces with variable exponents into the Lebesgue spaces and using that for study of the existence of solutions for a weighted Heisenberg $p(\cdot)$-Laplacian problem.

Here, we are going to prove that the problem $({\cal{P}})$, under the aforementioned assumptions, has at least one positive radial solution in $HW^{1, p(\cdot)}(\Omega, w)\cap L^{\theta(\cdot)}_\rho (\Omega)\cap L^{\vartheta(\cdot)}_\varrho (\Omega)$, where

which has the norm

and, similarly,

which has the norm

for $\rho, \varrho \in L^\infty(\Omega)$ as in (1.1), $\theta$ and $\vartheta$ are as above. We show that a radial weak solution to the problem ($\cal{P}$) is as follows.

Definition 1.1. We say that

is a non-trivial radial weak solution of ($\cal{P}$) if $u > 0$ in $\Omega$ is radial and the following equality is true:

in the weak sense; that is

for any $\phi \in C^\infty_0(\Omega)$.

The paper is arranged as follows. In Section 2 we present the notations and some facts related to the Heisenberg groups and Muckenhoupt weight functions. Additionally, we obtain a suitable interval for the embedding of weighted Heisenberg Sobolev spaces with variable exponents into the Lebesgue spaces. Plus that we bring some briefs from variational calculus and we introduce our main tool. Section 3 is devoted to the main result of the note and proof of the approach.

2.   Notations and auxiliary remarks

In this note, $\mathbb{H}^n \; (n\geq 1)$ is the Heisenberg Lie group which has $\mathbb{R}^{2n+1}$ as a background manifold and is endowed with the following noncommutative law of product:

where $x, x^{\prime}, y, y^{\prime} \in \mathbb {R}^n, \; t, t^{\prime} \in \mathbb{R}$ and $\langle \cdot | \cdot \rangle$ denotes the standard inner product in $\mathbb{R}^n$. We denote by $|\cdot|_{\mathbb{H}^n}$ Korányi norm with respect to the parabolic dilation $\delta_\lambda \xi = (\lambda x, \lambda y, \lambda^2t)$, i.e.,

for $z = (x, y) \in \mathbb{R}^{2n}$ and $\xi = (z, t) \in \mathbb{H}^n$.

Definition 2.1. (Radial Function) Let $\Omega\subset \mathbb{H}^n$ be a bounded open set. The function $u: \Omega\rightarrow \mathbb{R}$ is called a radial function if $u(x, y, t) = \phi({\bf r})$, where ${\bf r} = |(x, y, t)|_{\mathbb{H}^n}$ and $\phi:[0, +\infty)\rightarrow \mathbb{R}$.

A Korányi ball with the center $\xi_0$ and radius $\kappa$ is defined by

and it satisfies the following equalities:

where $|U|$ denotes the $(2n+1)$-dimensional Lebesgue measure of $U$ and $Q = 2n+2$ is the homogeneous dimension of $\mathbb{H}^n$. The Heisenberg gradient is given by

where

are vector fields that constitute a basis for the real Lie algebra of left-invariant vector fields on $\mathbb{H}^n$; More precisely, the family

satisfies the Hörmander's condition which means that it spans the whole tangent space $T\mathbb{R}^{2n+1}$. Let us recall that Hörmander's condition is a crucial condition for many problems consisting of hypoelliptic operators (see more details in ^[13] and the references therein).

For any horizontal vector field function $X = X(\xi), X = \{x_iX_i+x_i^\prime Y_i\}_{i = 1}^n$, of the class $C^1(\mathbb{H}^n, \mathbb{R}^{2n})$, we define the horizontal divergence of $X$ by

Definition 2.2. (Horizontal Curve) A piecewise smooth curve $y:[0, 1]\rightarrow \mathbb{H}^n$ is called a horizontal curve if $\dot{y}(t)$ belongs to the span of $\{X_i, Y_i\}_{i = 1}^n$ a.e. in $[0, 1]$. The horizontal length of $y$ is defined as follows

where

for each $X = \{x_iX_i+x_i^\prime Y_i\}_{i = 1}^n$ and $Y = \{y_iX_i+y_i^\prime Y_i\}_{i = 1}^n$.

The Carnot-Carathéodory distance of two points $\xi_1, \xi_2\in \mathbb{H}^n$ is defined by

Notice that according to the Chow-Rashevsky theorem ^[5,24], for any two arbitrary points $\xi_1, \xi_2\in \mathbb{H}^n$, there is a horizontal curve between them in $\mathbb{H}^n$; then, the above definition is well-defined. $d_{cc}$ is a left invariant metric on $\mathbb{H}^n$ and has a homogeneity of degree $1$ with respect to dilations $\delta_\lambda$, that is

for all $\xi_1, \xi_2\in \mathbb{H}^n$. In the case of the Heisenberg group, it is easy to check that the Lebesgue measure on $\mathbb{R}^{2n+1}$ is invariant under left translations. Thus, from here on, we denote by $d\xi$ the Haar measure on $\mathbb{H}^n$ that coincides with the $(2n +1)$-Lebesgue measure, this is because the Haar measures on Lie groups are unique up to constant multipliers.

As usual, for any measurable set $\Omega \subset \mathbb{H}^n\; (n\geq 1)$ and $m > 1$, we denote by $L^m(\Omega)$ the canonical Banach space, endowed with the norm

The first-order Heisenberg Sobolev space on $\Omega$ is defined as follows

endowed with the norm

Definition 2.3. (Poincaré-Sobolev Domain) An open set $\Omega$ of $\mathbb{H}^n$ is said to be a Poincaré-Sobolev domain, if there exists a bounded open set $U\subset\mathbb{H}^n$, with $\Omega\subset\bar{\Omega}\subset U$, which is a covering $\{B\}_{B\in \mathfrak{F}}$ of $\Omega$ by Carnot-Carathéodory balls $B$ and the numbers $N > 0, \alpha\geq1$ and $\nu \geq 1$ such that

$(i)$ $\sum_{B\in \mathfrak{F}} {\bf 1}_{(a+1)B}\leq N {\bf 1}_\Omega$ in $U$, where ${\bf 1}_D$ is the characteristic function of a Lebesgue measurable subset $D$.

$(ii)$ there exists a (central) ball $B_0\in \mathfrak{F}$ such that, for all $B\in \mathfrak{F}$ there is a finite chain $B_0, B_1, \cdots, B_{s(B)}$, with $B_i\cap B_{i+1}\neq \emptyset$ and

and moreover, $B\subset\nu B_i$ for $i = 0, 1, \cdots, s(B)$.

This definition is purely metric. There is a multiplicity of Poincaré-Sobolev domains in $\mathbb{H}^n$, as explained in details in ^[10]. The next result is a special case of [11,Theorem 1.3.1].

Theorem 2.1. $(i)$ Let $\Omega$ be a bounded Poincaré-Sobolev domain in $\mathbb{H}^n$, and let $1\leq m\leq Q$. Then, the embedding

is compact for all $\sigma$, where $m^* = \frac{mQ}{Q-m}$ is the critical Sobolev exponent related to $m$.

$(ii)$ The Carnot-Carathéodory balls are Poincaré-Sobolev domains.

Remark 2.1. Combining Theorem 2.1, with the fact that the Carnot-Carathéodory distance and the Korányi distance are equivalent on $\mathbb{H}^n$, we get that the following embedding is compact

when $1\leq m\leq Q$ and $\Omega$ is any Korányi ball centered at $\xi_0\in \mathbb{H}^n$ with a radius $R > 0$. Furthermore, there exists a $C_\sigma > 0$ such that

for all $u\in HW^{1, m}(\Omega)$.

From now on we denote by $\Omega$ the unit Korányi ball centered at the origin, and we set

for $q \in C_+(\bar{\Omega}) = \{g\in C(\bar{\Omega}): g^- > 1\}$. The generalized Lebesgue space $L^{q(\cdot)}(\Omega)$ is the collection of all measurable functions $u$ on $\Omega$ for which there exists a $\lambda > 0$ such that

and it has the norm

We know that for any $u\in L^{q(\cdot)}(\Omega)$ and $v \in L^{q' (\cdot)}(\Omega)$, i.e., the conjugate space of $L^{q (\cdot)}(\Omega)$, the Hölder type inequality

holds true. Following the authors of ^[23], for any $\kappa > 0$, we put

and

for $r\in C_+(\Omega)$. Then the well-known proposition $2.7$ of ^[12] will be rewritten as follows.

Proposition 2.1. For each $u\in L^{q(\cdot)}(\Omega)$ and $q\in C_+(\Omega)$, we have

The next lemma was established in ^[9].

Lemma 2.1. Assume that $q, r \in C_+(\bar{ \Omega})$. If $q(\xi) \leq r(\xi)$ for all $\xi \in \bar{\Omega}$, then $L^{r(\cdot)}(\Omega) \hookrightarrow L^{q(\cdot)}(\Omega).$

Remark 2.2. Let $q\in C_+(\Omega)$ with $q(\xi) < q^+ < m^*$ a.a. in $\Omega$. Thanks to Remark 2.1 and Lemma 2.1, we have the following compact embedding:

as $1\leq m\leq Q$. Thus, there exists a $C_{q(\cdot)} > 0$ such that

and for every bounded sequence $\{u_n\}$ in $HW^{1, m}(\Omega)$, up to the subsequence, $\{u_n\}$ converges to some $\bar{u}$ in $L^{q(\cdot)}(\Omega)$.

We continue by defining the Muckenhoupt weight functions rewritten on the Heisenberg groups.

Definition 2.4. (Muckenhoupt Weight) Let $w:\mathbb{H}^n\rightarrow (0, \infty)$ be a locally integrable function. Then, we say that $w$ belongs to the Muckenhoupt class $A_m$ if there exists a positive constant $c_{m, w}$ depending only on $m$ and $w$ such that, for all Korányi balls $B$ in $\mathbb{H}^n$,

A subclass of $A_m$: Let us define a subclass of $A_m$ by

Example 2.1. $w(\xi) = |\xi|^\alpha\in A_s\subset A_m$, for any $-\frac{Q}{s} < \alpha < \frac{Q}{s}$, provided $1 < m < Q$.

For $1 < m < Q$ and $w\in A_s$ with $s\in [\frac{1}{m-1}, \infty)\cap(\frac{Q}{m}, \infty)$, we set

Notice that by simple calculations one can show that $1\leq m_s < m < m^*_s$. Define

which has the norm

We define the weighted Heisenberg-Sobolev space $HW^{1, m}(\Omega, w)$ by

endowed with the norm

We need the following fact for embeddings.

Lemma 2.2. We have

Proof. Let $u\in HW^{1, m}(\Omega, w)$. Since $\frac{m}{m_s} > 1$, using the Hölder inequality with the exponents $\frac{m}{m_s}$ and $(\frac{m}{m_s})^\prime = s+1$, we obtain

which implies that

Replacing $u$ by $\nabla_{\mathbb{H}^n} u$, we gain

Adding (2.2) and (2.3), one has $\|u\|_{1, m_s}\leq |w^{-s}|_1^\frac{1}{sm}\|u\|_{1, m, w}$. Thus,

Remark 2.3. It is easy to see that, by standard embeddings in the Heisenberg-Sobolev spaces mentioned in Remark 2.1 and Lemma 2.2, one has

as $1\leq m\leq Q$; so, there exists a constant $k_\sigma$ such that

for each $u\in HW^{1, m}(\Omega, w)$.

Notice that embedding $HW^{1, m}(\Omega, w)\hookrightarrow L^\sigma (\Omega)$ is compact if $1\leq \sigma < m_s^*$.

Remark 2.4. Let $q\in C_+(\Omega)$ such that $q(\xi) < q^+ < m_s^*$ a.a. in $\Omega$. Thanks to Remark 2.2 and Lemma 2.2, for $1\leq m\leq Q$, we have the following compact embedding

Now, for $p\in C_+(\Omega)$, define the weighted Lebesgue space with a variable exponent as follows:

which has the norm

We denote the weighted Heisenberg-Sobolev space with a variable exponent by

equipped with the norm

Here-in-after, for $p\in C_+(\Omega)$ with $p(\xi)\leq p^+\leq Q$ a.a. in $\Omega$, we put

with the norm $\|u\|$.

Remark 2.5. Let $q\in C_+(\Omega)$ such that $q(\xi) < q^+ < \mathfrak{p}^*: = (p^-_s)^*$ a.a. in $\Omega$. Then,

Furthermore, we denote the Sobolev embedding constant of this compact embedding by $K_{q(\cdot)} > 0$, i.e.,

for each $u\in X$.

Proof. As a consequence of Lemma 2.1, for $p, q \in C_+(\bar{ \Omega})$, one has

if $q(\xi) \leq p(\xi)$ a.e. $\xi \in \Omega$. In a special case, we gain

On the other hand, from Remark 2.4, for $q\in C_+(\Omega)$ with $q(\xi) < q^+ < \mathfrak{p} = (p^-_s)^* \; \text{a.a. in}\; \Omega$, we have

So, the proof is completed.

Remark 2.6. For $u \in X$, there exist $\mu, M > 0$ such that

Proof. Since ${\rm{ess}}\;{\rm{inf}}_{\Omega}\mathcal{R} > 0$, there exists $0 < \delta < 1$ such that $\delta < \mathcal{R}(\xi)$ a.e. in $\Omega$. Using Proposition 2.1 and the hypothesis $\mathcal{R} \in L^{\infty}(\Omega)$, we gain

and

Bearing in mind the following elementary inequality due to J.A. Clarkson: for all $\gamma > 0$, there exists a $C_\gamma > 0$ such that

for all $a, b\in \mathbb{R}$. Then, we deduce

So the proof is complete; it is enough to put $\mu = \frac{\delta}{C_{\check{p}}}, M = 1+\|\mathcal{R}\|_\infty$.

We continue by providing some briefs from variational calculus; the interested reader can see more details in ^[22] and the references therein.

Let $V$ be a real Banach space and $V^*$ be its topological dual; and, also assume that the pairing between $V$ and $V^*$ denoted by $\langle \; , \; \rangle$.

Definition 2.5. (Subdifferential) Let $\Psi : V \rightarrow (-\infty, +\infty]$ be a proper (i.e. Dom $\Psi \ne \emptyset$), convex function. The subdifferential (generalized gradient) of $\Psi$ denoted by $\partial \Psi$, $\partial \Psi : V \rightarrow 2^{V^*}$, for $u\in Dom(\Psi) = \{ v\in V; \Psi(v) < \infty\}$, is defined as the following set-value operator

and $\partial \Psi (u) = \emptyset$ if $u\notin Dom(\Psi)$.

Notice that, if $\Psi$ is Gâteaux differentiable at $u$, which has a derivative that is denoted by $D \Psi (u)$, $\partial \Psi (u)$ is a singleton. In this case, $\partial \Psi (u) = \{D\Psi(u)\}$.

Definition 2.6. (Critical Point) Let $V$ be a real Banach space, $\Phi \in C^1 (V, \mathbb {R})$ and $\Psi : V \rightarrow (-\infty, +\infty]$ be a proper convex and lower semicontinuous function. Let $K\subset V$ be a weakly closed convex set. Define the function $\Psi_K :V\rightarrow (-\infty, +\infty]$ by

Consider the functional

a point $u \in V$ is called a critical point of $I$, if $D \Phi (u) \in \partial \Psi _K (u)$ or, equivalently, it satisfies the following inequality:

The following result has been proved in [2,Theorem 1.5.6].

Theorem 2.2. Let $V$ be a reflexive Banach space and $I: V\rightarrow \mathbb{R}$ be a continuous, convex and coercive functional. Then, $I$ has a global minimum point.

Notice that a global minimum point is a critical point.

Definition 2.7. ((PS) Condition) We say that $I$ mentioned in (2.5) satisfies the Palais-Smale compactness condition (in short, (PS) condition) if, for every sequence $\{ u_n \}$, the following states are satisfied:

● $I(u_n) \rightarrow c \in \mathbb {R}$;

● $\langle D\Phi (u_n), u_n-v \rangle +\Psi_K (v) - \Psi_K (u_n) \geq -\epsilon_n \| v-u_n\|$ for all $v\in V$ as $\epsilon_n \rightarrow 0$;

then, $\{u_n\}$ possesses a convergent subsequence.

The following mountain pass geometry (MPG) theorem was proved in ^[34].

Theorem 2.3. Suppose that $I: V \rightarrow (-\infty, + \infty]$ is of the form (2.5) and satisfies the (PS) condition and the following conditions:

(i) $I(0) = 0$;

(ii) there exists $e \in V$ such that $I(e) \leq 0$;

(iii) there exists a positive constant $\lambda$ such that $I(u) > 0$, if $\|u\| = \lambda$;

then, $I$ has a critical value $c \geq \lambda$ which is characterized by

where $\Gamma = \{ g \in C([0, 1], V): g(0) = 0\; , \; g(1) = e\}$.

Definition 2.8. (Pointwise Invariance Condition) Let $\Phi, \Psi:V\rightarrow \mathbb{R}$ be defined as in Definition 2.6 and $K$ be any subset of $V$. We say that the triple $(\Psi, \Phi, K)$ satisfies the pointwise invariance condition at a point $u\in V$ if there exist a convex Gâteaux-differentiable function $G: V\rightarrow \mathbb{R}$ and a point $v\in K$ such that

Here, we recall a variational principle established in ^[16] which we apply to prove our main approach.

Theorem 2.4. Let $V$ be a reflexive Banach space and $K$ be a convex and weakly closed subset of $V$. Let $\Psi :V \rightarrow \mathbb{R} \cup \{+\infty \}$ be a convex, lower semicontinuous function which is Gâteaux differentiable on $K$, and let $\Phi \in C^1(V, \mathbb{R})$. Assume that the following two assertions hold:

(i) The functional $I :V \rightarrow \mathbb{R} \cup \{+\infty \}$ defined by $I(w) = \Psi_K (w)- \Phi (w)$ has a critical point $u \in V$ in the sense of Definition 2.6,

(ii) The triple $(\Psi, \Phi, K)$ satisfies the pointwise invariance condition at the point u.

Then, $u \in K$ is a (weak) solution of the equation

Remark 2.7. Notice that if $\Psi$ is Gâteaux differentiable on $Dom \Psi$, $u$ is a critical point of $I(w) = \Psi (w)- \Phi (w)$ and there exists $v\in Dom\Psi$ such that

then $u$ is a solution of $D\Psi(u) = D \Phi (u)$, but it does not necessarily belongs to $K$.

The next is a fact mentioned in [17,problem 127,page 81] or in ^[23].

Theorem 2.5. Assume that $\{u_n\}$ is a sequence of monotonic (continuous or discontinuous) real functions on $[c, d]$ which converge pointwise to a continuous function $u: [c, d] \rightarrow \mathbb{R}$; then, the convergence is uniform.

Remark 2.8. Let $\Omega$ be a bounded open domain. Consider the closed convex set $K$ as follows:

Suppose that $\{u_n\}$ is a sequence in $K$ such that $u_n\to\bar{u}$ a.e. in $\Omega$. Then, regardless of a set of zero measures, $\{|u_n-\bar{u}|\}_{n\in\mathbb{N}}$ converge to zero uniformly.

Proof. Clearly $\bar{u}$ is a positive radial function; moreover, $u\in K$, since K is closed. If $\bar{u}$ is a continuous function, then Theorem 2.5 deduces $u_k\to\bar{u}$ uniformly. Otherwise, imagine that $E$ contains all of the discontinuous points of $\bar{u}$. According to [27,Theorem 4.30] every monotonic function is discontinuous at a countable set of points at most, so $E$ is at most countable with a Lebesgue measure of zero. Thus $\bar{u}$ is continuous on $\Omega\backslash E$ and the convergence of $\{|u_k-\bar{u}|\}_{k\in\mathbb{N}}$ to zero is uniform.

3.   Existence result

Here, we state the main result of this paper.

Theorem 3.1. Let $\Omega$ be the unit Korányi ball in the Heisenberg group $\mathbb{H}^n\; (n\geq1)$ and $p\in C_+(\Omega)$ with $p(\xi)\leq p^+\leq Q$ a.a. in $\Omega$. Let $\theta, \vartheta \in C_+(\Omega)$ such that

where

for $s$ with

Let $w$ be a Muckenhoupt weight function of the class $A_s$ and $\mathcal{R}:\Omega\rightarrow [0, +\infty)$ belong to $L^\infty(\Omega)$ such that ${\rm{ess}}\;{\rm{inf}}_{\Omega}\mathcal{R} > 0$. Assume $\rho, \varrho\in L^{\infty} (\Omega)$ satisfy the condition $(1.1)$. Then the Dirichlet problem ($\cal{P}$) admits at least one radially increasing (weak) solution.

Set

equipped with the norm

where

It is clear that $V$ is a reflexive Banach space. Now, consider the Euler-Lagrange energy functional corresponding to the problem ($\mathcal{P}$), i.e.,

as well as the closed convex set

To adapt Theorem 2.4 to our problem, we define $\psi, \varphi:V\rightarrow \mathbb{R}$ by

and

Notice that $\psi$ is a proper, convex, lower semicontinuous function and $D\varphi (u) = \rho(\xi)|u|^{\theta(\xi)-2}u$; therefore, $\varphi$ is a $C^1$- function on the space $V$. Let us introduce the functional $I : V\rightarrow (-\infty, +\infty]$ as follows:

where $\psi_K$ is defined as (2.4).

We should be aware that $I$ is indeed the Euler-Lagrange functional corresponding to our problem (denoted by $E(\cdot)$) as restricted to $K$, and it is clear that the critical points of $I$ are exactly the radially increasing weak solutions of ($\cal{P}$).

We prove Theorem 3.1 in two steps:

Step1. We show that $I$ has a critical point and, for this reason, we need the following lemma.

Lemma 3.1. Let $V = X_{rad} \cap L^{\theta(\cdot)}_\rho (\Omega)\cap L^{\vartheta(\cdot)}_\varrho (\Omega)$ and consider the functional $I: V \rightarrow \mathbb{R}$ by applying

as in $(3.1)$. Then, $I$ has a nontrivial critical point in $K$.

Proof. We apply the MPG theorem (Theorem 2.3) to prove this lemma.

First, we verify that $I$ satisfies the following MPG conditions:

It is clear that $I(0) = 0$. Take $e \in K$. From Remark 2.6, we have the following estimate

since $\theta^- > \vartheta^+$ and $\theta^- > p^+$, for $t$ sufficiently large, $I(te)$ is negative. We now prove Condition $(iii)$ of the MPG theorem. Take $u \in Dom(\psi)$ with $\|u\| = \lambda > 0$. Notice that from Lemmas 2.5 and 2.6, for $u\in K$, we have

Thus

provided $\lambda > 0$ is small enough as $2\leq \check{p} < \hat{\theta}$ and $C_1$ is a positive constant. If $u \notin Dom(\psi)$, clearly, $I(u) > 0$. Therefore, the MPG holds for the functional $I$.

Second, we verify the following (PS) condition:

Suppose that $\{u_n\}$ is a sequence in $K$ such that

and let, for all $v \in V$,

We show that $\{ u_n\}$ has a convergent subsequence in $V$. First notice that $u_n \in Dom(\psi)$; then,

Thus, for large values of $n$ we have

Now, consider the function $g(t) = t^{\vartheta^+} - \theta^- (t-1) -1$ on the interval $(1, +\infty)$ and set $\hat{t} = (\frac{\theta^-}{\vartheta^+})^\frac{1}{\vartheta^+-1}$. It is easy to see that for every $t\in (1, \hat{ t})$ we have $g(t) < 0$. We choose such a number $t$ for which we have $t > 1$ and $t^{\vartheta^+}-1 < \theta^- (t-1)$. In (3.2), set $v = tu_n$; then,

Furthermore,

Since $t^{\vartheta^+}-1 < \theta^-(t-1)$, we can take $\gamma > 0$ such that

Multiplying (3.4) by $\gamma$ and adding it to (3.3) we obtain

So, using Remark 2.6 for some suitable constant $C ^\prime > 0$, we have

Therefore, $\{u_n\}$ is a bounded sequence in the reflexive space $X$. Thanks to Remark 2.5, we gain that there exists $\bar{u} \in X$ such that, up to the subsequences, the following holds true

● $u_n \rightharpoonup \bar{u}$ in $X$;

● $u_n \rightarrow \bar{u}$ in $L^{q(\xi)}(\Omega), \; q\in C_+(\Omega) \; \; \text{and}\; \; q(\xi) < q^+ < \mathfrak{p}^*$;

● $u_n(\xi) \rightarrow \bar{u}(\xi)$ a.e in $\Omega$;

On the one hand, $\{u_n\} \subset K$; so, according to Remark 2.8, regardless of a set of measure of zeros, $\{|u_n-\bar{u}|\}_{n\in\mathbb{N}}$ converges to zero uniformly. Then,

On the other hand, Remark 2.6 gives

From the inequalities (3.6) and (3.7), we deduce that $\|u_n-\bar{u}\|\to 0$. As we mentioned before, by applying the standard embeddings

$u_n \rightarrow \bar{u}$ strongly in $V$, as desired.

Remark 3.1. Notice that for each $n\in \mathbb{N}$, $u_n\in K$ is radial, so $\bar{u}$ is radial. $K$ is a closed subset of $V$; then, $\bar{u} \in K$. Therefore, the MPG theorem guarantees that the existence of a critical point belongs to $K$, namely $\hat{u}$.

Step2. We show that for any $u\in K$, in a special case $\hat{u}$, the triple $(\psi_K, \varphi, Dom \psi)$ satisfies the pointwise invariance condition at $u$ when $G = 0$. To this end, we shall need following lemma.

Lemma 3.2. Let $\mathcal{R}\in L^\infty(\Omega)$ be a nonnegative real functional. Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be continuous and there exist constants $a, b > 0$ such that

where $\gamma\in (1, \mathfrak{p}^*)$; moreover,

Then for every $h\in L^\frac{sp^-}{s(p^–1)-1}(\Omega)$ the problem

where

admits at least one solution.

Proof. First notice that by integration one can see that there exist $a_1, b_1 > 0$ such that

and that $F(t)\leq 0$ for all $t\in \mathbb{R}$, where

Now, consider the following energy functional on $X$ which is corresponding to Problem (3.9):

Because of the growth condition (3.8) $J$ is well-defined on $X$. According to the Hölder inequality, one has $\int_\Omega hud\xi\leq |u|_{\mathfrak{p}}|h|_{\mathfrak{p}^\prime}$ where, by (2.2),

and by the assumption of the lemma $h\in L^{\mathfrak{p}^\prime}(\Omega), \; \mathfrak{p}^\prime = \frac{sp^-}{s(p^–1)-1}$. Then thanks to Remark 2.2 we have

This is because $p^- > 1$, $J$ is coercive. Thus according to Theorem 2.2, $J$ has a global minimum point, meaning that Problem (3.9) admits at least one solution.

Lemma 3.3. Let $u \in K$, $\mathcal{R}\in L^\infty(\Omega)$ and $\rho, \varrho\in L^{\infty} (\Omega)$ be defined as in (1.1). Let $p, \theta, \vartheta \in C_+(\Omega)$ with

Then, there exists $v \in Dom\psi$ such that

in the weak sense.

Proof. Let $u \in K$; so, $0 \leq u \in K \subset X$; also, set

Then, thanks to Lemma 3.2, it is enough to show that

But $u$ is a radial function (i.e., $u(\xi) = \phi({\bf r})$); so, by [7,equation (2.4)] we have $|\nabla_{\mathbb{H}^n}u| = \frac{{\bf r}}{r}|\phi^ \prime |$, where ${\bf r} = |\xi|_{\mathbb{H}^n} = |(z, t)|_{\mathbb{H}^n}$, $r = |z|$. Using the fundamental theorem of calculus and Hölder inequality, one has following estimate:

where $\omega_{Q-1}$ is the measure of the unit ball in $\mathbb{H}^n$ and $C, C'$ and $C_*$ are positive constants. A computation shows that $u(\xi)^{\theta-1}\in L^{\mathfrak{p}^\prime}(\Omega)$. So, the proof is complete.

4.   Conclusions

In this paper, first, we looked for a suitable interval embedding of weighted Heisenberg-Sobolev spaces with variable exponents into the Lebesgue spaces in a step by step manner. In Remark 2.3, using Lemma 2.2, we found the following embedding

as $1\leq m\leq Q$. Thanks to Remark 2.2 and Lemma 2.2, for $1\leq m\leq Q$, we generalized the result as follows:

Finally, in Remark 2.5, we proved that if $q\in C_+(\Omega)$ such that $q(\xi) < q^+ < \mathfrak{p}^*: = (p^-_s)^*$ a.a. in $\Omega$, then

Employing the result and MPG theorem, we proved that $I = \psi_K - \varphi$ has a critical point in $K$, namely $\hat{u}$, which is radial but may not necessarily be a solution of (${\cal{P}}$). In Lemma 3.3, we showed that, for any $u\in K$, particularly for $\hat{u}$, there exists $v \in Dom \psi$ satisfying the equation $D\psi (v) = D \varphi (u)$. Indeed, we showed that the triple $(\psi, \varphi, Dom \psi)$ satisfies the point wise invariance condition at any $u\in K$, especially at $\hat{u}$, given $G = 0$. Therefore, Theorem 2.4, Remark 2.7 and the maximum principle for the $p(\cdot)$-Laplacian operator ensure the trueness of Theorem 3.1.

Acknowledgments

M. A. Ragusa was partially supported by Piano della Ricerca 2016–2018-Linea di intervento 2: "Metodi variazionali ed equazioni differenziali".

Conflict of interest

All authors declare no conflicts of interest regarding this study.