1 Introduction

We consider the following Leray–Lions type problem with Steklov boundary conditions

$$\begin{aligned} \left\{ \begin{array}{lc} -div(a(x,\nabla u))+\mathcal {R}(x)|u|^{p(x)-2}u=f(x,u)-g(x,u)&{} x\in \Omega ,\\ u>0 &{} x\in \Omega ,\\ a(x,u)\frac{\partial u}{\partial \nu }= h(x) &{} x\in \partial \Omega , \end{array} \right. \end{aligned}$$
(1.1)

where \(\Omega\) is a bounded region in \(\mathbb {R}^N\), \(N>1\), with smooth boundary \(\partial \Omega\), \(\nu (x)=(\nu _1(x),\cdots ,\nu _N(x))\) is the outward unit normal to the smooth surface \(\partial \Omega\) at x. We assume that

$$\begin{aligned} a:\overline{\Omega }\times \mathbb {R}\rightarrow \mathbb {R} \end{aligned}$$

satisfies the following conditions:

  1. (a1)

    a is a Carathéodory function such that \(a(x, 0)=0\), for a.e. \(x\in \overline{\Omega }\).

  2. (a2)

    There exist positive functions \(\alpha ,\beta \in \big (L^\infty (\Omega ), |\cdot |_\infty \big )\) such that

    $$\begin{aligned} |a(x, t)|\le \alpha (x)+\beta (x)|t|^{p(x)-1}, \end{aligned}$$

    for a.e. \(x\in \Omega\) and all \(t\in \mathbb {R}\) where \(p\in C_+(\Omega )\).

  3. (a3)

    For all \(s, t\in \mathbb {R}\) with \(t\ne s\) the inequality

    $$\begin{aligned} (a(x, t)-a(x, s))(t-s)>0 \end{aligned}$$

    holds, for a.e. \(x\in \Omega\). For which antiderivative

    $$\begin{aligned} A:\overline{\Omega }\times \mathbb {R}\rightarrow \mathbb {R} \end{aligned}$$

    defined by

    $$\begin{aligned} A(x, t):=\int _{0}^{t}a(x, s)ds, \end{aligned}$$

    holds in the following condition

  4. (a4)

    There exists \(\underline{c}\ge 1\) such that

    $$\begin{aligned} \underline{c}|t|^{p(x)}\le \min \{a(x, t)t, p(x) A(x, t)\}, \end{aligned}$$

    for a.e. \(x\in \Omega\) and all \(t\in \mathbb {R}\).

Also, we suppose that \(\mathcal {R} \in L^{\infty } (\Omega )\) is a radial function with \({{\,\mathrm{ess\inf }\,}}_{\Omega } \mathcal {R}>0\) and the Carathéodory function

$$\begin{aligned} f:\overline{\Omega }\times \mathbb {R}\rightarrow \mathbb {R} \end{aligned}$$

is continuously differentiable and odd with respect to its second variable satisfying the hypotheses \((f_1)-(f_3)\):

\((f_1)\):

\(\partial _t f(x,0)=0\) and there exists \(q\in C(\Omega )\) with \(p(x)\le q(x)\le p^*(x)\) a.e. in \(\Omega\) such that

$$\begin{aligned} \lim {{\,\mathrm{sup\ {}}\,}}_{t\rightarrow +\infty } \frac{\partial _t f (x,t)}{t^{q(x)-2}}<+\infty ; \end{aligned}$$

where \(\partial _t f (x, t)\) denotes the partial derivative of f at the point (x, t) with respect to the second variable.

\((f_2)\):

There exists \(\mu \in C(\Omega )\) with \(p(x)\le \mu (x)\le p^*(x)\) a.e. in \(\Omega\) such that

$$\begin{aligned} f(x,t)t \ge \mu (x) F(x,t), \end{aligned}$$

for all \(t\in \mathbb {R}\), where \(= \int _0^t f(x,s)ds\);

\((f_3)\):

For every \(t\in (0,+\infty )\) there results \(\partial _t f(x,t)t>f(x,t)\);

and the Carathéodory function

$$\begin{aligned} g:\overline{\Omega }\times \mathbb {R}\rightarrow \mathbb {R} \end{aligned}$$

is continuously differentiable and odd with respect to its second variable satisfying the following assertions:

\((g_1)\):

\(\partial _t g(x,0)=0\) and there exists \(\theta \in C(\Omega )\) with \(p(x)\le \theta (x)\le \mu (x)\) a.e. in \(\Omega\) such that

$$\begin{aligned} \lim {{\,\mathrm{sup\ {}}\,}}_{t\rightarrow \infty } \frac{\partial _t g (x,t)}{t^{\theta (x)-2}}<+\infty ; \end{aligned}$$
\((g_2)\):

the inequalities \(g(x,t)\ge 0\) and

$$\begin{aligned} \partial _t g(x,t)t\le (\theta (x)-1)g(x,t) \end{aligned}$$

hold for every \(t\in (0,+\infty )\).

Finally, we assume that \(h\in L^{p'(\cdot )}(\Omega )\) is a positive continuous function such that \(|h|_{p'(\cdot )}\) is small enough.

Faria et al. [4] have studied the existence of positive solutions for the following nonlinear elliptic problems under Dirichlet boundary condition

$$\begin{aligned} \left\{ \begin{array}{ll} -\sum _{i=1}^N\frac{\partial }{\partial x_i} a_i(x,u(x),\nabla u(x))=g(x,u(x),\nabla u(x)) &{} \quad x\in \Omega , \\ u=0 &{} \quad x\in \partial \Omega . \end{array} \right. \end{aligned}$$

Their approach relies on the method of sub-supersolution and nonlinear regularity theory (see [5, 16, 17] for sub-supersolution methods). Hai Ha et al. [6] have proved the existence of infinitely many solutions for a generalized \(p(\cdot )\)-Laplace equation involving Leray–Lions operators

$$\begin{aligned} \left\{ \begin{array}{lc} -div(a(x,\nabla u))= f(x,u) &{} \quad x\in \Omega , \\ u=0 &{} \quad x\in \partial \Omega , \end{array} \right. \end{aligned}$$

where \(\Omega\) is a bounded domain in \(\mathbb {R}^N\) with a Lipchitz boundary \(\partial \Omega\); \(a:\Omega \times \mathbb {R}^N\rightarrow \mathbb {R}^N\) and \(f:\Omega \times \mathbb {R}\rightarrow \mathbb {R}\) are Carathéodory functions with suitable growth conditions. Firstly, under a \(p(\cdot )\)-sublinear condition for nonlinear term, they obtained a sequence of solutions approaching 0 by showing a priori bound for solutions. Secondly, for a \(p(\cdot )\)-superlinear condition, they produced a sequence of solutions whose Sobolev norms diverge to infinity when the nonlinear term satisfies a couple of generalized Ambrosetti-Rabinowitz type condition.

Recently, Musbah et al. [11] study the existence of multiple solutions for the following fourth-order problem

$$\begin{aligned} \left\{ \begin{array}{lcl} \Delta (a(x, \Delta u))=\lambda f(x, u)&{}\quad \text {in}&{} \Omega ,\\ u={\Delta u}=0 &{}\quad \text {on}&{} \partial \Omega , \end{array} \right. \end{aligned}$$
(1.2)

involving Leray–Lions type operator via variational methods, where \(\Omega\) is a bounded domain in \({\mathbb {R}^{N}}(N\ge 2)\) with a smooth boundary \(\partial \Omega\), \(\lambda >0\) is a parameter, f is a Carathéodory function, \(p\in C(\overline{\Omega })\) satisfies the inequality

$$\begin{aligned} \inf _{x\in {\Omega }}p(x)>\frac{N}{2}\quad \text {for all}\,\ x\in \Omega , \end{aligned}$$

and \(\Delta (a(x, \Delta u))\) is Leray–Lions operator of the fourth-order, where a satisfies a set of conditions (see [9, 14] for variational method).

The purpose of this paper is to establish the existence of at least one positive radial increasing weak solution of the problem (1.1) in the first order Sobolev space with variable exponent. We point out the authors have proved the existence of solutions to the problems in some special cases of f and g for \(a(x,t)=|t|^{p(x)-2}t\) on the Heisenberg groups (see [15, 19,20,21,22,23,24] for more details).

2 Initial definitions and auxiliary remarks

In this section, first of all we present a brief survey of notions and results of Lebesgue and Sobolev spaces with variable exponents which we shall need later. The interested reader is refereed to [3] for a fuller treatment of the subject.

Let \(\Omega\) be a bounded simply connected domain with smooth boundary. We set

$$\begin{aligned} p^- =\inf _{x \in \Omega } {p(x)}\quad \& \quad p^+= \sup _{x \in \Omega } {p(x)}, \end{aligned}$$

where \(p \in C_+(\overline{\Omega }) =\{g\in C(\overline{\Omega }): g^->1\}\). The generalized Lebesgue space \(L^{p(\cdot )}(\Omega )\) is the collection of all measurable functions u on \(\Omega\) for which \(\int _\Omega |u(x)|^{p(x)}dx < +\infty\) and has the norm

$$\begin{aligned} |u|_{p(\cdot )}= inf\{\lambda >0 : \int _\Omega |\frac{u(x)}{\lambda }|^{p(x)}dx\le 1\}. \end{aligned}$$

For any \(u\in L^{p(\cdot )}(\Omega )\) and \(v \in L^{p' (\cdot )}(\Omega )\), where \(L^{p' (\cdot )}(\Omega )\) is the conjugate space of \(L^{p (\cdot )}(\Omega )\), the Hölder type inequality

$$\begin{aligned} |\int _\Omega uv dx| \le \left(\frac{1}{p^-}+\frac{1}{p'^-}\right) |u|_{p(\cdot )} |v|_{p'(\cdot )}, \end{aligned}$$

holds true. The following theorem is in [8, Theorem 2.8].

Theorem 2.1

Assume that \(\Omega \subset \mathbb {R}^N\) is a bounded domain and \(p, q \in C_+(\overline{ \Omega })\). Then

$$\begin{aligned} L^{p(\cdot )}(\Omega ) \hookrightarrow L^{q(\cdot )}(\Omega ) \end{aligned}$$

if and only if \(q(x) \le p(x)\) a.e. \(x \in \Omega\).

Following the authors of paper [15], for any \(\iota >0\), we put

$$\begin{aligned} \iota ^{\check{r}}:= \left\{ \begin{array}{ll} \iota ^{r^+}\quad &{} \quad \iota <1 , \\ \iota ^{r^-}\; &{} \quad \iota \ge 1; \end{array}\right. \end{aligned}$$

and

$$\begin{aligned} \iota ^{\hat{r}}:= \left\{ \begin{array}{ll} \iota ^{r^-}\quad &{} \quad \iota <1 , \\ \iota ^{r^+}\; &{} \quad \iota \ge 1; \end{array}\right. \end{aligned}$$

for \(r\in C_+(\Omega )\). Then the well-known proposition [7, Proposition 2.7] will be rewritten as follows.

Proposition 2.1

For each \(u\in L^{p(\cdot )}(\Omega )\), we have

$$\begin{aligned} |u|_{p(\cdot )}^{\check{p}}\le \int _\Omega |u(x)|^{p(x)}dx \le |u|_{p(\cdot )}^{\hat{p}}. \end{aligned}$$

Normally, the first order Sobolev space associated with \(L^{p(\cdot )}(\Omega )\) is defined as follows

$$\begin{aligned} W^{1,p(\cdot )}(\Omega ) :=\{u: \Omega \rightarrow \mathbb {R}:\; u , |\nabla u| \in L^{p(\cdot )}(\Omega ) \}, \end{aligned}$$

endowed with the norm

$$\begin{aligned} \Vert u\Vert := |\nabla u|_{p(\cdot )}+|u|_{p(\cdot )}, \end{aligned}$$

where \(\nabla u=(\frac{\partial u}{\partial x_1}(x), \cdots , \frac{\partial u}{\partial x_N}(x))\) is the gradient of u at \(x=(x_1, \cdots ,x_N)\) and as usual \(|\nabla u|= \big ( \sum _{i=1}^N |\frac{\partial u}{\partial x_i}|^2 \big )^\frac{1}{2}\). The next is Sobolev embedding established in [3, Theorem 8.2.4].

Proposition 2.2

Assume that \(\Omega\) is a bounded and smooth domain in \(\mathbb {R}^N\), \(N>1\), and \(p, q\in C_+(\overline{\Omega })\). If \(q(x)<p^*(x)\) for any \(x\in \overline{\Omega }\), the embedding

$$\begin{aligned} W^{1,p(\cdot )}(\Omega )\hookrightarrow L^{q(\cdot )}(\Omega ) \end{aligned}$$

is compact and continuous, where

$$\begin{aligned} p^*(x)= \left\{ \begin{array}{ll} \frac{Np(x)}{N-p(x)}\; &{} \quad p(x) < N, \\ +\infty \; &{}\quad p(x)\ge N; \end{array} \right. \end{aligned}$$

moreover, there exists \(\kappa _q>0\) such that

$$\begin{aligned} |u|_{q(\cdot )}\le \kappa _q\Vert u\Vert . \end{aligned}$$

Remark 2.1

Suppose that conditions (a1)–(a4) hold, then we have

  • A(x, t) defined as above is a \(C^{1}\)-Carathéodory function.

  • Thanks to [2, Proposition 1.5.10], from (a3), the functional A is strictly convex.

  • There exists a constant \(\bar{c}\ge \max \{ |\alpha |_\infty , \frac{|\beta |_\infty }{p^+}\}\) such that

    $$\begin{aligned} \frac{\underline{c}}{p^+}|t|^{p(x)}\le |A(x, t)|\le \bar{c}(|t|+|t|^{p(x)}), \end{aligned}$$

    for a.e. \(x\in \Omega\) and all \(t\in \mathbb {R}\). So, one has

    $$\begin{aligned} |\int _\Omega A(x,\nabla u) dx|&\le \int _\Omega |A(x,\nabla u) |dx\nonumber \\&\le \bar{c}\int _\Omega (|\nabla u|+|\nabla u|^{p(x)})dx \nonumber \\&\le \bar{C} (\Vert u\Vert +\Vert u\Vert ^{\hat{p}}), \end{aligned}$$
    (2.1)

    where \(\bar{C}= \bar{c}(\kappa _1+1)\).

Following remark consists of some properties of f(t) and \(F(x,t):=\int _0^tf(x,s)ds\).

Remark 2.2

We have

(1):

\(\lim _{t \rightarrow 0} \frac{F(x,t)}{t^2} =0\).

(2):

There exist positive constants \(M_1, M_2\) such that

$$\begin{aligned} |\partial _t f(x,t)|\le M_1 |t|^ {q(x)-2}, \end{aligned}$$

for \(|t|>M_2\) and a.e. \(x\in \Omega\).

(3):

There exist positive constants \(M_3, M_4\) such that

$$\begin{aligned} |F(x,t)| + |f(x,t)t|\le M_3 |t|^{q(x)}, \end{aligned}$$

for \(|t|>M_4\) and a.e. \(x\in \Omega\).

(4):

For every \(\epsilon >0\), there exists \(C_\epsilon >0\) such that

$$\begin{aligned} |F(x,t)| + |f(x,t)t|\le \epsilon t^2+C_\epsilon |t|^{q(x)}, \end{aligned}$$

for all \(t \in \mathbb {R}\) and a.e. \(x\in \Omega\).

(5):

There exists a positive constant D such that for every \(t\ge 1\) and \(x\in \Omega\) one has

$$\begin{aligned} f(x,t)t \ge Dt^{\mu (x)} \quad \& \quad F(x,t)\ge Dt^{\mu (x)}. \end{aligned}$$

Proof

These results follow at once from assumptions \((f_1)--(f_3)\) and the definitions of f and F.

In the next remarks, we mention to some properties of g(t) and its antiderivative

$$\begin{aligned} G(x,t):= \left\{ \begin{array}{ll} \int _0^tg(x,s)ds\qquad &{} \quad t>0 , \\ 0 &{} \quad t\le 0; \end{array}\right. \end{aligned}$$

Remark 2.3

We have

(1):

\(\lim _{t \rightarrow 0} \frac{G(x,t)}{t^2} =0\).

(2):

There exist positive constants \(M_5, M_6\) such that

$$\begin{aligned} |\partial _t g(x,t)|\le M_5|t|^{\theta (x)-2}, \end{aligned}$$

for \(|t|>M_6\) and a.e \(x\in \Omega\).

(3):

There exist positive constants \(D_1\) and \(D_2\) such that

$$\begin{aligned} |G(x,t)| + |g(x,t)t|\le D_1 |t|^2+D_2|t|^{\theta (x)}, \end{aligned}$$

for all \(t\in \mathbb {R}\) and a.e \(x\in \Omega\).

Proof

These results follow directly from hypotheses \((g_1)--(g_2)\) and the definitions of g and G.

Remark 2.4

For \(u \in W^{1,p(\cdot )}(\Omega )\), there exist \(\lambda ,\Lambda >0\) such that

$$\begin{aligned} \lambda \Vert u\Vert ^{\check{p}}\le \int _{\Omega }\big (A(x,\nabla u)+\frac{1}{p(x)}\mathcal {R}(x)\vert u\vert ^{p(x)}\big )dx \le \Lambda (\Vert u\Vert +\Vert u\Vert ^{\hat{p}}). \end{aligned}$$

Proof

Since \({{\,\mathrm{ess\inf }\,}}_{\Omega }\mathcal {R}>0\), there exists \(0<\delta <1\) such that \(\delta <\mathcal {R}(x)\) a.e. in \(\Omega\). Using Proposition 2.1, Hölder inequality and hypothesis \(\mathcal {R} \in L^{\infty }(\Omega )\), we gain

$$\begin{aligned} \frac{\delta }{p^+}|u|_{p(\cdot )}^{\check{p}}\le \int _\Omega \frac{1}{p(x)} \mathcal {R}(x)|u(x)|^{p(x)}dx \le |\mathcal {R}|_\infty |u|_{p(\cdot )}^{\hat{p}}\le \frac{1}{p^-}|\mathcal {R}|_\infty \Vert u\Vert ^{\hat{p}}, \end{aligned}$$

and from hypotheses (a2) and relation (2.1), one has

$$\begin{aligned} \frac{\delta }{p^+} |\nabla u|_{p(\cdot )}^{\check{p}}&\le \frac{\delta }{p^+} \;\underline{c}\;|\nabla u|_{p(\cdot )}^{\check{p}}\\ {}&\le \frac{\underline{c}}{p^+} |\nabla u|_{p(x)}^{\check{p}}\\ {}&\le \int _\Omega |A(x, \nabla u)|dx\\ {}&\le \bar{C}(\Vert u\Vert +\Vert u\Vert ^{\hat{p}}). \end{aligned}$$

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

$$\begin{aligned} |s+t|^\gamma \le |s+t|^\gamma + |s-t|^\gamma \le C_\gamma (|s|^\gamma +|t|^\gamma ), \end{aligned}$$

for all \(s,t\in \mathbb {R}\). Then we deduce

$$\begin{aligned} \frac{\delta }{p^+C_{\check{p}}} \Vert u\Vert ^{\check{p}}\le \int _{\Omega }(\vert \nabla u\vert ^{p(x)}+\frac{1}{p(x)}\mathcal {R}(x)\vert u\vert ^{p(x)})dx\le (\bar{C}+\frac{1}{p^-}|\mathcal {R}|_\infty )(\Vert u\Vert +\Vert u\Vert ^{\hat{p}}). \end{aligned}$$

So, the proof is complete; It is enough to put \(\lambda =\frac{\delta }{p^+C_{\check{p}}}, \Lambda =\bar{C}+\frac{1}{p^-}|\mathcal {R}|_\infty\).

The next two theorems have been proved in [2].

Theorem 2.2

[2, Theorem 1.5.6] Let X be a reflexive Banach space and \(I: X\rightarrow \mathbb {R}\) be a continuous, convex and coercive functional. Then I has a global minimum point.

Theorem 2.3

[2, Theorem 1.5.8] Let X be a reflexive Banach space and \(I: X\rightarrow \mathbb {R}\) be strictly convex. Then I has at most one minimum point in X.

We continue by a short review of the main definitions and results concerning the variational calculus. A complete discussion of this subject can be found for example in [13].

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

Definition 2.1

(Subdifferential) Let \(\Psi : X \rightarrow (-\infty ,+\infty ]\) be a proper convex function and \(2^{X^*}\) be the set of all subsets of \(X^*\). The subdifferential of \(\Psi\) denoted by \(\partial \Psi\), \(\partial \Psi : X \rightarrow 2^{X^*}\), is defined to be following set-value operator

$$\begin{aligned} \partial \Psi (u): =\{ u^* \in X^* : \Psi (v) \ge \Psi (u)+\langle u^* , v-u \rangle \ \text {for all}\ v \in X \}, \end{aligned}$$

for \(u\in Dom(\Psi )=\{ v\in X:\Psi (v)<\infty \}\), and \(\partial \Psi (u) = \emptyset\) if \(u\notin Dom(\Psi )\).

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

Let X be a reflexive Banach space and \(K\subset X\) be a convex closed set. Let \(\Phi \in C^1 (X, \mathbb {R})\) and \(\Psi : X \rightarrow ( -\infty , +\infty ]\) be a proper (i.e. Dom \(\Psi \ne \emptyset\)), convex and lower semicontinuous function. Define the function

$$\begin{aligned} \Psi _K :X\rightarrow (-\infty , +\infty ] \end{aligned}$$
$$\begin{aligned} \Psi _K (u):= \left\{ \begin{array}{ll} \Psi (u) \qquad &{} u\in K, \\ +\infty &{} u\notin K, \end{array} \right. \end{aligned}$$
(2.2)

and consider the functional

$$\begin{aligned} I_K:=\Psi _K- \Phi . \end{aligned}$$
(2.3)

Definition 2.2

(Critical Point) We say that the point \(u \in X\) is a critical point of \(I_K\), if \(D \Phi (u) \in \partial \Psi _K (u)\) or equivalently, it satisfies the following inequality

$$\begin{aligned} \langle D \Phi (u), u-v \rangle +\Psi _K (v) - \Psi _K (u) \ge 0, \quad \text {for all} \; v \in X . \end{aligned}$$
(2.4)

Notice that a global minimum point is a critical point.

Definition 2.3

(Point-Wise Invariance Condition) The triple \((\Psi , \Phi , K)\) satisfies the point-wise invariance condition at a point \(u\in X\) if there exist a convex Gâteaux differentiable function \(\mathcal {G}: X\rightarrow \mathbb {R}\) and a point \(v\in K\) such that

$$\begin{aligned} D\Psi (v) + D\mathcal {G}(v) = D\Phi (u) + D\mathcal {G}(u). \end{aligned}$$

Now, we bring a variational principle verified in [10] which is the main tool of this paper.

Theorem 2.4

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

  1. (i)

    The functional \(I_K :X \rightarrow (-\infty ,+\infty ]\) defined by

    $$\begin{aligned} I_K (w)=\Psi _K (w)- \Phi (w), \end{aligned}$$

    where \(\Psi _K\) is defined as (2.2), has a critical point \(u \in X\) as in Definition 2.2;

  2. (ii)

    the triple \((\Psi _K, \Phi , K)\) satisfies the point-wise invariance condition at the point u;

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

$$\begin{aligned} D\Psi (u)= D\Phi (u). \end{aligned}$$

Here, we present another proof of this theorem.

Proof

Let \(u\in Dom(\Psi _K)\) be a critical point of \(I_K\), so we have

$$\begin{aligned} \Psi _K (w) - \Psi _K (u) \ge \langle D\Phi (u)\;,\;w-u \rangle ,\quad \text {for all} \quad w \in X. \end{aligned}$$
(2.5)

Since \(I_K(u)\) is a finite number, so \(u\in K\). By assumptions of the theorem, there exists \(v \in K\) satisfying the linear equation

$$\begin{aligned} D\Psi _K (v) + D\mathcal {G}(v) = D\Phi (u) + D\mathcal {G}(u) . \end{aligned}$$
(2.6)

Substituting \(w=v\) in inequality (2.5) we gain

$$\begin{aligned} \Psi _K(v)-\Psi _K(u) \ge \langle D\Phi (u),v-u \rangle . \end{aligned}$$

As a consequence of (2.6) we obtain

$$\begin{aligned} \Psi _K(v)-\Psi _K(u) \ge \langle D\Psi _K(v),v-u \rangle +\langle D\mathcal {G}(v)-D\mathcal {G}(u), v-u\rangle . \end{aligned}$$

On the other hand, \(\Psi _K\) is Gâteaux differentiable at \(v\in K\), so

$$\begin{aligned} D\Psi _K(v) \in \partial \Psi _K(v), \end{aligned}$$

which means

$$\begin{aligned} \Psi _K (u) \ge \Psi _K (v)+\langle D\Psi _K(v) , u-v \rangle . \end{aligned}$$

Thus

$$\begin{aligned} \langle D\mathcal {G}(v)-D\mathcal {G}(u), v-u\rangle \le 0. \end{aligned}$$

But \(\mathcal {G}\) is convex, so we have

$$\begin{aligned} \langle D\mathcal {G}(v)-D\mathcal {G}(u), v-u\rangle \ge 0. \end{aligned}$$

Thereby gaining that \(u=v\). It then follows that \(D\Psi (u) =D \varphi (u)\) as claimed.

Remark 2.5

Notice that if \(\Psi _K\) is Gâteaux differentiable at each point of \(Dom \Psi\), u is a critical point of \(I_K(w)=\Psi _K (w)- \Phi (w)\) and there exists \(v\in Dom\Psi\) such that

$$\begin{aligned} D\Psi _K (v) + D\mathcal {G}(v) = D\Phi (u) + D\mathcal {G}(u), \end{aligned}$$

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

The Mountain Pass Theorem (MPT) is an existence theorem from the calculus of variations and is as follows [1].

Theorem 2.5

Let \((X,\Vert .\Vert _X)\) be a reflexive Banach space. Suppose that the functional \(I: X \rightarrow (-\infty , + \infty ]\) satisfies (PS) compactness condition and also the following assertions

  1. (i)

    \(I(0)=0\);

  2. (ii)

    there exists \(e \in X\) such that \(I(e) \le 0\);

  3. (iii)

    there exists positive constant \(\rho\) such that \(I(u) > 0\), if \(\Vert u\Vert _X= \rho\);

Then I has a critical value \(c \ge \rho\) which is characterized by

$$\begin{aligned} c= \mathop {\mathrm {inf \; sup}}\limits _{\eta \in \Gamma \ t \in [0,1]} {I(\eta (t))}, \end{aligned}$$

where \(\Gamma =\{ \eta \in C([0,1], X): \eta (0)=0\; ,\; \eta (1)=e\}\).

Here, we recall the (PS) compactness condition.

Definition 2.4

((PS) Compactness Condition) We say that \(I\in C(X,\mathbb {R})\) satisfies the Palais-Smale (PS) compactness condition if any sequence \(\{ u_k \}\subset X\) such that

  • \(\{I(u_k)\}\) is bounded;

  • \(I^\prime (u_k)\rightarrow 0\) in X;

has a convergent subsequence in X.

The next is a fact [12, problem 127, P. 81] established in [15].

Proposition 2.3

Assume that \(u_k:[a,b] \rightarrow \mathbb {R}, \; k \ge 1\) is a monotone increasing sequence of (not necessarily continuous) functions which converge pointwise to a continuous function \(u: [a,b] \rightarrow \mathbb {R}\) then the convergence is uniform.

Remark 2.6

Let \(\Omega\) be a bounded open domain. Consider the closed convex set K as follows

$$\begin{aligned} K=\{u:\Omega \rightarrow \mathbb {R}: u \ge 0,\; u\; \text {is increasing radial function} \}. \end{aligned}$$

Suppose that \(\{u_k\}\) is a sequence in K such that \(u_k\rightarrow \bar{u}\) a.e. in \(\Omega\). Then, regardless of a set of measure zero, \(\{|u_k-\bar{u}|\}_{k\in \mathbb {N}}\) converge to zero uniformly.

Proof

Clearly \(\bar{u}\) is a positive radial function. On the other hand, since K is closed, we have \(K=\overline{K}\) [18, Theorem 2.27] and so \(\bar{u}\in K\). If \(\bar{u}\) is a continuous function, then Theorem 2.3 deduces \(u_k\rightarrow \bar{u}\) uniformly. Otherwise, imagine that E contains all the discontinuous points of \(\bar{u}\). According to [18, Theorem 4.30 ] every monotonic function is discontinuous at a countable set of points at most, so E is at most countable with Lebesgue measure zero. Thus \(\bar{u}\) is continuous on \(\Omega \backslash E\) and convergence of \(\{|u_k-\bar{u}|\}_{k\in \mathbb {N}}\) to zero is uniform.

Now, we are ready to sketch the proof of main result.

3 Existence of radially increasing solutions

We begin this section by definition of the radial increasing weak solution of the problem (1.1).

Definition 3.1

(Weak Solution) We say that \(u\in W^{1,p(\cdot )}(\Omega )\) is a radial increasing weak solution of problem (1.1) if it is radially increasing function with \(u>0\) in \(\Omega\) such that

$$\begin{aligned} a(x,u).\frac{\partial u}{\partial \nu }= h(x)\;\; \text {on}\; \partial \Omega , \end{aligned}$$

and one has

$$\begin{aligned}&\int _\Omega a(x,\nabla u)w dx+\int _\Omega \mathcal {R}(x)|u|^{p(x)-2}uwdx\\&\quad +\int _\Omega g(x,u)wdx-\int _\Omega f(x,u)wdx=\int _{\partial \Omega } h(x)wd\sigma , \end{aligned}$$

for all \(w\in W^{1,p(\cdot )}(\Omega )\).

We set

$$\begin{aligned} X:=W_{rad}^{1,p(\cdot )}(\Omega )=\{u:\ u \in W^{1,p(\cdot )}(\Omega )\ \text {and} \ u \;\text {is radial} \}, \end{aligned}$$

with respect to the norm \(\Vert \cdot \Vert\) and we apply Theorem 2.4. We consider the Euler-Lagrange energy functional corresponding to the problem (1.1); i.e.,

$$\begin{aligned} E(u)&:=\int _\Omega \big (A(x,\nabla u)+\frac{1}{p(x)}\mathcal {R}(x)|u|^{p(x)}\big ) dx \\&\qquad +\int _\Omega G(x,u)dx-\int _\Omega F(x,u)dx-\int _{\partial \Omega }h(x)ud\sigma . \end{aligned}$$

Conforming Theorem 2.4 with our problem, we define \(\psi ,\varphi :X\rightarrow \mathbb {R}\) by

$$\begin{aligned} \psi (u):= \int _\Omega \big (A(x,\nabla u)+\frac{1}{p(x)}\mathcal {R}(x)|u|^{p(x)} \big )dx, \end{aligned}$$

and

$$\begin{aligned} \varphi (u):=\int _\Omega F(x,u)dx-\int _\Omega G(x,u)dx+\int _{\partial \Omega }h(x)ud\sigma . \end{aligned}$$

over the closed set

$$\begin{aligned} K=\{ u\in X: u\ge 0,\; u \; \text {is increasing radial function}\}. \end{aligned}$$
(3.1)

Notice that \(\psi\) is a proper, from Remark 2.1, it is convex, lower semicontinuous and thanks to Remark 2.4, we have

$$\begin{aligned} \lambda \Vert u\Vert ^{\check{p}}\le \psi (u) \le \Lambda (\Vert u\Vert +\Vert u\Vert ^{\hat{p}}). \end{aligned}$$

And

$$\begin{aligned} D\varphi (u)=f(x,u)-g(x,u)+ h(x)\in C(X,\mathbb {R}), \end{aligned}$$

so, \(\varphi\) is a \(C^1\)- function on the space X.

Let us introduce the functional \(I_K:X\rightarrow (-\infty ,+\infty ]\) defined by

$$\begin{aligned} I_K(u)= \psi _K(u) - \varphi (u), \end{aligned}$$
(3.2)

where \(\psi _K\) is defined as (2.2). We prove our claim, which is the existence of at least one nontrivial radial increasing weak solution to the problem (1.1), in two steps.

Step1. We show that \(I_K: X \rightarrow \mathbb {R}\), defined as in (3.2), has a nontrivial critical point in K. To this end, we use the MPT (Theorem 2.5).

Firstly, We verify (PS) compactness condition.

Proof

Let \(\{u_k\}\) is a sequence in K such that \(I(u_k)<+\infty\) and \(I^\prime (u_k)\rightarrow 0\). Then one has the following estimate

$$\begin{aligned} \langle I'(u_k) , u_k \rangle&=\int _\Omega a(x, \nabla u_k)u_ndx+\int _\Omega \mathcal {R}(x)|u_k|^{p(x)}dx+\int _\Omega g(x,u_k)u_kdx\\&\quad -\int _\Omega f(x,u_k)u_kdx-\int _{\partial \Omega } h(x)u_kd\sigma \\&\le |\alpha |_\infty \Vert u_k\Vert +|\beta |_\infty \Vert u_k\Vert ^{\hat{p}}+ |\mathcal {R}|_\infty \Vert u_k\Vert ^{\hat{p}}+ D_1\Vert u_k\Vert ^2\\&\quad +D_2\Vert u_k\Vert ^{\hat{\theta }}-D\Vert u_k\Vert ^{\check{\mu }}. \end{aligned}$$

Then for large values of k we have

$$\begin{aligned} D\Vert u_k\Vert ^{\check{\mu }} \le |\alpha |_\infty \Vert u_k\Vert +(|\beta |_\infty +|\mathcal {R}|_\infty )\Vert u_k\Vert ^{\hat{p}}+ D_1 \Vert u_k\Vert ^2+D_2\Vert u_k\Vert ^{\hat{\theta }}. \end{aligned}$$

On the other hand, according to our hypothesis, \(2\le p(x)\le \theta (x)\le \mu (x)\) a.e. on \(\Omega\), then \(\{u_k\}\) is a bounded sequence in the reflexive Banach space X, so the standard results in Sobolev space imply that there exists \(\bar{u} \in W^{1,p(\cdot )}(\Omega )\) such that, up to subsequences,

  • \(u_k \rightharpoonup \bar{u}\)   in X;

  • \(u_k \rightarrow \bar{u}\)   in \(L^{s(\cdot )}(\Omega )\), where \(1 \le s(x)<p^*(x)\) a.e in \(\Omega\);

  • \(u_k(x) \rightarrow {\bar{u}}(x)\) a.e. in \(\Omega\);

Then thanks to Remark 2.6, not only \(\bar{u} \in K\), but also \(u_k \rightarrow \bar{u}\) (strongly) in X as desired.

Secondly, we verify that \(I_K\) satisfies in MPT conditions.

Proof

It is clear that \(I_K(0)=0\). Take \(e \in K\) then it follows that

$$\begin{aligned} I_K(te)\le \Lambda (t\Vert e\Vert + t^{\hat{p}}\Vert e\Vert ^{\hat{p}})+D_1 t^2\Vert u\Vert ^2+D_2 t^{\hat{\theta }}\Vert e\Vert ^{\hat{\theta }}-Dt^{\check{\mu }} \Vert u\Vert ^{\check{\mu }}. \end{aligned}$$

Now, since \(\mu (x)>\theta (x)>p(x)>1\) a.e in \(\Omega\), for t large enough I(te) is negative. We now prove condition (iii) of MPT. Take \(u \in Dom(\psi )\) with \(\Vert u\Vert =\rho >0\). With respect to part (4) of Remark 2.2, Hölder inequality, and standard embeddings, one has

$$\begin{aligned} \varphi (u)&=\int _\Omega F(x,u)dx-\int _\Omega G(x,u)dx+\int _{\partial \Omega } h(x)ud\sigma \\ {}&\le \epsilon \int _\Omega |u|^2\,dx+C_\epsilon \int _\Omega |u|^{q(x)}dx+2|h|_{p'(\cdot )}|u|_{p(\cdot )}\\ {}&\le \kappa _1\epsilon \Vert u\Vert ^2+\kappa _q C_\epsilon \Vert u\Vert ^{\hat{q}}+2|h|_{p'(\cdot )}\Vert u\Vert ^{\hat{p}}. \end{aligned}$$

We put \(\epsilon =\frac{\kappa _q}{\kappa _1}\rho ^{\hat{q}-2}\) and so we gain

$$\begin{aligned} I_K(u)&\ge \lambda \rho ^{\check{p}}-\kappa _q(1+C_\epsilon )\rho ^{\hat{q}}-2|h|_{p'(\cdot )}\rho ^{\hat{p}} \nonumber \\&>\big (\lambda -2|h|_{p'(\cdot )}\big )\rho ^{\hat{p}}-\kappa _q(1+C_\epsilon )\rho ^{\hat{q}} >0, \end{aligned}$$
(3.3)

provided \(\rho\) is small enough and

$$\begin{aligned} |h|_{p'(\cdot )}<\frac{\lambda }{2}=\frac{\frac{\delta }{2}}{p^+C_{\check{p}}}, \end{aligned}$$

as \(p(x)<q(x)\) a.e. in \(\Omega\). If \(u \notin Dom(\psi )\), then clearly \(I_K(u)>0\). Therefore, all of states of MPT (Theorem 2.5) are held for the functional \(I_K\).

By the above argument and MPT, \(I_K\) has a critical value

$$\begin{aligned} c= \mathop {\mathrm {inf \; sup}}\limits _{\eta \in \Gamma \ t \in [0,1]} {I(\eta (t))}, \end{aligned}$$

where \(\Gamma =\{ \eta \in C([0,1],X): \eta (0)=0\; ,\; \eta (1)=e\}\). In the light of the above discussion, it seems reasonable to certify that \(c>0\). Then \(I_K\) has a nontrivial critical point belongs to K.

Step 2. We show that the triple \((\psi _K, \varphi , K)\) satisfies the point-wise invariance condition at u when \(\mathcal {G}=0\). To show this assertion, we need following lemma.

Lemma 3.1

Let \(f, g, \mathcal {R}\) be as before and \(\mathcal {H}\in L^{{p^*}^\prime (\cdot )}(\Omega )\) where \({p^*}^\prime (\cdot )=\frac{p^*(\cdot )}{p^*(\cdot )-1}\). Then the problem

$$\begin{aligned} \left\{ \begin{array}{lc} -div(a(x,\nabla u))+\mathcal {R} (x) |u|^{p(x)-2}u= \mathcal {H}(x)\quad &{} \quad x \in \Omega , \\ a(x,u).\frac{\partial u}{\partial \nu }= h(x) &{} \quad x\in \partial \Omega , \end{array} \right. \end{aligned}$$
(3.4)

admits a unique weak solution.

Proof

Consider the following energy functional corresponding to the problem (3.4) on \(W^{1,p(\cdot )} (\Omega )\) :

$$\begin{aligned} J(u) = \int _\Omega \big (A(x,\nabla u)+ \frac{\mathcal {R}(x)}{p(x)}|u|^{p(x)}\big )dx - \int _\Omega \mathcal {H}udx-\int _{\partial \Omega } h(x)ud\sigma . \end{aligned}$$

J is differentiable on \(W^{1,p(\cdot )}(\Omega )\) with

$$\begin{aligned} \langle J'(u) , v \rangle =\int _\Omega a(x,\nabla u)\nabla v+\mathcal {R}(x) |u|^{p(x)-2}uv dx-\int _\Omega \mathcal {H}vdx - \int _{\partial \Omega } hvdx. \end{aligned}$$

So, one has

$$\begin{aligned} \langle J'(u)-J'(v) ,u- v \rangle&=\int _\Omega (a(x,\nabla u)-a(x,\nabla v))(\nabla u-\nabla v)dx\\&\quad +\int _\Omega \mathcal {R}(x) (|u|^{p(x)-2}u-|v|^{p(x)-2}v)(u-v) dx, \end{aligned}$$

then using assumption (a3), J is strictly convex. In addition, by standard embeddings in Sobolev spaces one has

$$\begin{aligned} J(u)\ge \lambda \Vert u\Vert ^{\check{p}} - |\mathcal {H}|_{{{p^*}^\prime (\cdot )}}|u|_{p^*(\cdot )}-|h|_{p^\prime (\cdot )}|u|_{p(\cdot )}\ge \lambda \Vert u\Vert ^{\check{p}} -C\Vert u\Vert , \end{aligned}$$

so that J is coercive. Applying Theorems 2.2 and 2.3, J has a unique global minimum point which means problem (3.4) admits a unique solution.

Lemma 3.2

Let \(u \in Dom (\psi )\). Then there exists \(v \in Dom (\psi )\) such that

$$\begin{aligned} \left\{ \begin{array}{lc} -div(a(x,\nabla v))+\mathcal {R}(x)|v|^{p(x)-2}v= f(x,u)-g(x,u)\quad &{} \quad x\in \Omega , \\ v>0 &{} \quad x\in \Omega ,\\ a(x,v).\frac{\partial v}{\partial \nu }= h(x) &{} \quad x\in \partial \Omega , \end{array} \right. \end{aligned}$$
(3.5)

Proof

Let \(u \in Dom (\psi )\). According to Lemma 3.1, it is enough to show that

$$\begin{aligned} \mathcal {H}(x)=f(x,u(x))-g(x,u(x))\in L^{{p^*}^\prime (\cdot )}(\Omega ). \end{aligned}$$

From Lemma 2.2, we have

$$\begin{aligned} \int _\Omega |f(x,u(x))|^{{p^*}^\prime (x)}dx\le M_3\int _\Omega |u(x)|^{{p^*}^\prime (x)(q(x)-1)}dx<+\infty , \end{aligned}$$

since \(q(x)-1\le \frac{p^*(x)}{{p^*}^\prime (x)}=p^*(x)-1\) a.e. in \(\Omega\). From Lemma 2.3 and Minkowski’s inequality we conclude

$$\begin{aligned} \int _\Omega |g(x,u(x))|^{{p^*}^\prime (x)}dx\le D^\prime _1\int _\Omega |u(x)|^{{p^*}^\prime (x)}dx+D^\prime _2\int _\Omega |u(x)|^{{p^*}^\prime (x)(\theta (x)-1)}dx<+\infty , \end{aligned}$$

since \(\theta (x)-1<\mu (x)-1\le \frac{p^*(x)}{{p^*}^\prime (x)}=p^*(x)-1\). Then \(f,g\in L^{{p^*}^\prime (\cdot )}(\Omega )\). On the other hand, \(L^{{p^*}^\prime (\cdot )}(\Omega )\) is closed under subtraction, thus the result would be achieved.

Remark 3.1

Employing MPT (Theorem 2.5), we proved \(I_K =\psi _K - \varphi\) has a nontrivial critical point, namely \(u\in K\) not necessarily solution of problem (1.1). By Lemma 3.2, we displayed there exists a unique \(v \in Dom\psi\) satisfying the equation \(D\psi (v)=D \varphi (u)\). Then Theorem 2.4 implies that problem (1.1) has at least one nontrivial increasing radial weak solution.