Existence result for a Neumann boundary value problem governed by a class of p ( x ) -Laplacian-like equation

Abstract

In this article, we consider a Neumann boundary value problem driven by $p (x)$ -Laplacian-like operator with a reaction term depending also on the gradient (convection) and on three real parameters, originated from a capillary phenomena, of the following form: $\begin{matrix} \{\begin{matrix} - Δ_{p (x)}^{l} u + δ | u |^{ζ (x) - 2} u = μ g (x, u) + λ f (x, u, \nabla u) & in Ω, \\ \frac{\partial u}{\partial η} = 0 & on \partial Ω, \end{matrix} \end{matrix}$ where $Δ_{p (x)}^{l} u$ is the $p (x)$ -Laplacian-like operator, Ω is a smooth bounded domain in $R^{N}$ , δ, μ and λ are three real parameters, $p (x), ζ (x) \in C_{+} (\overline{Ω})$ , η is the outer unit normal to $\partial Ω$ and g, f are Carathéodory functions. Under suitable nonstandard growth conditions on g and f and using the topological degree for a class of demicontinuous operator of generalized $(S_{+})$ type and the theory of variable exponent Sobolev spaces, we establish the existence of weak solution for the above problem.

Keywords

Neumann boundary value problem capillarity phenomena weak solution topological degree methods variable exponent Sobolev spaces

1. Introduction and motivation

Partial differential equations with nonlinearities and nonconstant exponents has been received considerable attention in recent years. Perhaps the impulse for this comes from the new search field that reflects a new type of physical phenomenon is a class of nonlinear problems with variable exponents. Modeling with classic Lebesgue and Sobolev spaces has been demonstrated to be limited for a number of materials with inhomogeneities. In the subject of fluid mechanics, for example, great emphasis has been paid to the study of electrorological fluids, which have the ability to modify their mechanical properties when exposed to an electric field. Rajagopal and M. Ruzicka recently developed a very interesting model for these fluids in [21] (see also [23]), taking into account the delicate interaction between the electric field $E (x)$ and the moving liquid. This type of problem’s energy is provided by $\int_{Ω} | \nabla u |^{p (x)} d x$ . This type of energy can also be found in elasticity problems [29]. Other applications relate to image processing [3,8], elasticity [28], the flow in porous media [5,12], and problems in the calculus of variations involving variational integrals with nonstandard growth [4,16,28].

Let Ω be a smooth bounded domain in $R^{N}$ ( $N ⩾ 2$ ), with a Lipschitz boundary denoted by $\partial Ω$ . In this paper, We consider the following Neumann boundary value problem involving the $p (x)$ -Laplacian-like operator with a reaction term depending also on the gradient (convection) and on three real parameters $\begin{matrix} (1.1) & \{\begin{matrix} - Δ_{p (x)}^{l} u + δ | u |^{ζ (x) - 2} u = μ g (x, u) + λ f (x, u, \nabla u) & in Ω, \\ \frac{\partial u}{\partial η} = 0 & on \partial Ω, \end{matrix} \end{matrix}$ where $\begin{matrix} Δ_{p (x)}^{l} u : = div (| \nabla u |^{p (x) - 2} \nabla u + \frac{| \nabla u |^{2 p (x) - 2} \nabla u}{\sqrt{1 + | \nabla u |^{2 p (x)}}}) \end{matrix}$ is the $p (x)$ -Laplacian-like operator, $p (x), ζ (x) \in C_{+} (\overline{Ω})$ with $p (x)$ is log-Hölder continuous function, δ, μ and λ are three real parameters, η is the outer unit normal to $\partial Ω$ , $g : Ω \times R \to R$ and $f : Ω \times R \times R^{N} \to R$ are Carathéodory functions that satisfy the assumption of growth. The expression $f (x, u, \nabla u)$ is often referred to as a convection term.

The motivation for this research originated from the application of similar problems in physics to model the behavior of electrorheological fluids (see [21,23]), specifically the phenomenon of capillarity, which depends on solid-liquid interfacial characteristics as surface tension, contact angle, and solid surface geometry. Recently problems like (1.1) has received more and more attention, such as [6,13,24,30].

In [17], Obersnel et al. established the existence and multiplicity of positive solutions of the problem $\begin{matrix} (1.2) & \{\begin{matrix} - div (\frac{\nabla u}{\sqrt{1 + | \nabla u |^{2}}}) = λ f (x, u) & in Ω, \\ u = 0 & on \partial Ω, \end{matrix} \end{matrix}$ where $λ > 0$ and $f : Ω \times R \to R$ is a Carathéodory function. Their discussion is based on variational and critical point theory.

Note that if $μ = δ = 0$ , $λ > 0$ and f independent of $\nabla u$ with Dirichlet boundary condition, then the problem (1.1) has a nontrivial solutions [22].

Vetro [25] established the existence of weak solutions to the problem (1.1) where $ζ (x) = p (x)$ , $δ = 1$ , $μ = 0$ , $λ > 0$ and f independent of $\nabla u$ with Dirichlet boundary condition.

In this paper, we will generalize these works, by proving the existence of weak solution for (1.1) by using another approach based on the topological degree theory.

The rest of the paper is structured as follows. In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces. In Section 3, we shall present some classes of operator of generalized $(S_{+})$ type and topological Berkovits degrees. At last, we introduce our assumptions, technical lemmas, and we present and prove the main result of the paper in Section 4.

2. Preliminaries

In the analysis of problem (1.1), we will use the theory of the generalized Lebesgue-Sobolev spaces $L^{p (x)} (Ω)$ and $W^{1, p (x)} (Ω)$ . In this section, we recall only basic background facts that will be used later (see [10,15,18–20,27] for more details).

Let Ω be a smooth bounded domain in $R^{N}$ ( $N ⩾ 2$ ), with a Lipschitz boundary denoted by $\partial Ω$ . Set $\begin{matrix} C_{+} (\overline{Ω}) = {p : p \in C (\overline{Ω}) such that p (x) > 1 for any x \in \overline{Ω}} . \end{matrix}$ For each $p \in C_{+} (\overline{Ω})$ , we define $\begin{matrix} p^{+} : = max {p (x), x \in \overline{Ω}} and p^{-} : = min {p (x), x \in \overline{Ω}} . \end{matrix}$ For every $p \in C_{+} (\overline{Ω})$ , we define $\begin{matrix} L^{p (x)} (Ω) = {u : Ω \to R is measurable such that \int_{Ω} {| u (x) |}^{p (x)} d x < + \infty}, \end{matrix}$ equipped with the Luxemburg norm $\begin{matrix} | u |_{p (x)} = inf {λ > 0 : ρ_{p (x)} (\frac{u}{λ}) ⩽ 1}, \end{matrix}$ where $\begin{matrix} ρ_{p (x)} (u) = \int_{Ω} {| u (x) |}^{p (x)} d x, \forall u \in L^{p (x)} (Ω) . \end{matrix}$

Proposition 2.1 ([10]).

Let $(u_{n})$ and $u \in L^{p (x)} (Ω)$ , then $\begin{array}{l} (2.1) & | u |_{p (x)} < 1 (resp. = 1; > 1) \Leftrightarrow ρ_{p (x)} (u) < 1 (resp. = 1; > 1), \\ (2.2) & | u |_{p (x)} > 1 \Rightarrow | u |_{p (x)}^{p^{-}} ⩽ ρ_{p (x)} (u) ⩽ | u |_{p (x)}^{p^{+}}, \\ (2.3) & | u |_{p (x)} < 1 \Rightarrow | u |_{p (x)}^{p^{+}} ⩽ ρ_{p (x)} (u) ⩽ | u |_{p (x)}^{p^{-}}, \\ (2.4) & lim_{n \to \infty} | u_{n} - u |_{p (x)} = 0 \Leftrightarrow lim_{n \to \infty} ρ_{p (x)} (u_{n} - u) = 0 . \end{array}$

Remark 2.1.
According to (2.2) and (2.3), we have $\begin{array}{l} (2.5) & | u |_{p (x)} ⩽ ρ_{p (x)} (u) + 1, \\ (2.6) & ρ_{p (x)} (u) ⩽ | u |_{p (x)}^{p^{-}} + | u |_{p (x)}^{p^{+}} . \end{array}$
Proposition 2.2 ([15, Theorem 2.5 and Corollary 2.7]).

The space $(L^{p (x)} (Ω), | u |_{p (x)})$ is a separable and reflexive Banach space.

Proposition 2.3 ([15, Theorem 2.1]).

The conjugate space of $L^{p (x)} (Ω)$ is $L^{p^{'} (x)} (Ω)$ where $\frac{1}{p (x)} + \frac{1}{p^{'} (x)} = 1$ for all $x \in Ω$ . For any $u \in L^{p (x)} (Ω)$ and $v \in L^{p^{'} (x)} (Ω)$ , we have the following Hölder-type inequality $\begin{matrix} (2.7) & | \int_{Ω} u v d x | ⩽ (\frac{1}{p -} + \frac{1}{p^{' -}}) | u |_{p (x)} | v |_{p^{'} (x)} ⩽ 2 | u |_{p (x)} | v |_{p^{'} (x)} . \end{matrix}$

Remark 2.2.
If $p_{1}, p_{2} \in C_{+} (\overline{Ω})$ with $p_{1} (x) ⩽ p_{2} (x)$ for any $x \in \overline{Ω}$ , then we have $L^{p_{2} (x)} (Ω) ↪ L^{p_{1} (x)} (Ω)$ .

Now, let $p \in C_{+} (\overline{Ω})$ and we define $W^{1, p (x)} (Ω)$ as $\begin{matrix} W^{1, p (x)} (Ω) = {u \in L^{p (x)} (Ω) such that | \nabla u | \in L^{p (x)} (Ω)}, \end{matrix}$ equipped with the norm $\begin{matrix} | u |_{1, p (x)} = | u |_{p (x)} + | \nabla u |_{p (x)} . \end{matrix}$ Furthermore, we have the compact embedding $W^{1, p (x)} (Ω) ↪ L^{p (x)} (Ω)$ (see [15]). Remark 2.3.
Note that for all $u \in W^{1, p (x)} (Ω)$ , we have $\begin{matrix} | u |_{p (x)} ⩽ | u |_{1, p (x)} and | \nabla u |_{p (x)} ⩽ | u |_{1, p (x)} . \end{matrix}$
Proposition 2.4 ([10,15]).

The space $(W^{1, p (x)} (Ω), | \cdot |_{1, p (x)})$ is a separable and reflexive Banach space.

Remark 2.4.
The dual space of $W^{1, p (x)} (Ω)$ denoted $W^{- 1, p^{'} (x)} (Ω)$ , is equipped with the norm $\begin{matrix} | u |_{- 1, p^{'} (x)} = inf {| u_{0} |_{p^{'} (x)} + \sum_{i = 1}^{N} | u_{i} |_{p^{'} (x)}}, \end{matrix}$ where the infinimum is taken on all possible decompositions $u = u_{0} - div F$ with $u_{0} \in L^{p^{'} (x)} (Ω)$ and $F = (u_{1}, \dots, u_{N}) \in {(L^{p^{'} (x)} (Ω))}^{N}$ .

3. A review on some classes of mappings and topological degree theory

Now, we give some results and properties from the theory of topological degree. The readers can find more information about the history of this theory in [1,2,7,9,14].

In what follows, let X be a real separable reflexive Banach space and $X^{*}$ be its dual space with dual pairing $⟨ \cdot, \cdot ⟩$ and given a nonempty subset Ω of X. Strong (weak) convergence is represented by the symbol $\to (⇀)$ .

Definition 3.1.
Let Y be real Banach space. A operator $F : Ω \subset X \to Y$ is said to be
bounded, if it takes any bounded set into a bounded set.

demicontinuous, if for any sequence $(u_{n}) \subset Ω$ , $u_{n} \to u$ implies $F (u_{n}) ⇀ F (u)$ .

compact, if it is continuous and the image of any bounded set is relatively compact.

Definition 3.2.
A mapping $F : Ω \subset X \to X^{}$ is said to be
of type $(S_{+})$ , if for any sequence $(u_{n}) \subset Ω$ with $u_{n} ⇀ u$ and ${lim sup}_{n \to \infty} ⟨ F u_{n}, u_{n} - u ⟩ ⩽ 0$ , we have $u_{n} \to u$ .

quasimonotone, if for any sequence $(u_{n}) \subset Ω$ with $u_{n} ⇀ u$ , we have ${lim sup}_{n \to \infty} ⟨ F u_{n}, u_{n} - u ⟩ ⩾ 0$ .

Definition 3.3.
Let T: $Ω_{1} \subset X \to X^{}$ be a bounded operator such that $Ω \subset Ω_{1}$ . For any operator F: $Ω \subset X \to X$ , we say that
F of type ${(S_{+})}_{T}$ , if for any sequence $(u_{n}) \subset Ω$ with $u_{n} ⇀ u$ , $y_{n} : = T u_{n} ⇀ y$ and ${lim sup}_{n \to \infty} ⟨ F u_{n}, y_{n} - y ⟩ ⩽ 0$ , we have $u_{n} \to u$ .

F has the property ${(Q M)}_{T}$ , if for any sequence $(u_{n}) \subset Ω$ with $u_{n} ⇀ u$ , $y_{n} : = T u_{n} ⇀ y$ , we have ${lim sup}_{n \to \infty} ⟨ F u_{n}, y - y_{n} ⟩ ⩾ 0$ .

In the sequel, we consider the following classes of operators: $\begin{array}{l} F_{1} (Ω) : = {F : Ω \to X^{*} : F is bounded, demicontinuous and of type (S_{+})}, \\ F_{T, B} (Ω) : = {F : Ω \to X : F is bounded, demicontinuous and of type {(S_{+})}_{T}}, \\ F_{T} (Ω) : = {F : Ω \to X : F is demicontinuous and of type {(S_{+})}_{T}}, \end{array}$ for any $Ω \subset D (F)$ , where $D (F)$ denotes the domain of F, and any $T \in F_{1} (Ω)$ .

Now, let $O$ be the collection of all bounded open sets in X and we define $\begin{matrix} F (X) : = {F \in F_{T} (\overline{E}) : E \in O, T \in F_{1} (\overline{E})}, \end{matrix}$ where, $T \in F_{1} (\overline{E})$ is called an essential inner map to F.
Lemma 3.1 ([14, Lemma 2.3]).

Let $T \in F_{1} (\overline{E})$ be continuous and $S : D (S) \subset X^{*} \to X$ be demicontinuous such that $T (\overline{E}) \subset D (S)$ , where E is a bounded open set in a real reflexive Banach space X. Then the following statements are true:

If S is quasimonotone, then $I + S \circ T \in F_{T} (\overline{E})$ , where I denotes the identity operator.

If S is of type $(S_{+})$ , then $S \circ T \in F_{T} (\overline{E})$ .

Definition 3.4.
Suppose that E is bounded open subset of a real reflexive Banach space X, $T \in F_{1} (\overline{E})$ is continuous and $F, S \in F_{T} (\overline{E})$ . The affine homotopy $H$ : $[0, 1] \times \overline{E} \to X$ defined by $\begin{matrix} H (t, u) : = (1 - t) F u + t S u, for all (t, u) \in [0, 1] \times \overline{E} \end{matrix}$ is called an admissible affine homotopy with the common continuous essential inner map T.
Remark 3.1 ([14, Lemma 2.5]).

The above affine homotopy is of type ${(S_{+})}_{T}$ .

Next, as in [14] we give the topological degree for the type $F (X)$ .

Theorem 3.1.
Let $M = {(F, E, h) : E \in O, T \in F_{1} (\overline{E}), F \in F_{T, B} (\overline{E}), h \notin F (\partial E)}$ . Then, there exists a unique degree function $d : M ⟶ Z$ that satisfies the following properties:
(Normalization) For any $h \in E$ , we have $\begin{matrix} d (I, E, h) = 1 . \end{matrix}$

(Additivity) Let $F \in F_{T, B} (\overline{E})$ . If $E_{1}$ and $E_{2}$ are two disjoint open subsets of E such that $h \notin F (\overline{E} ∖ (E_{1} \cup E_{2}))$ , then we have $\begin{matrix} d (F, E, h) = d (F, E_{1}, h) + d (F, E_{2}, h) . \end{matrix}$

(Homotopy invariance) If $H$ : $[0, 1] \times \overline{E} \to X$ is a bounded admissible affine homotopy with a common continuous essential inner map and h: $[0, 1] \to X$ is a continuous path in X such that $h (t) \notin H (t, \partial E)$ for all $t \in [0, 1]$ , then $\begin{matrix} d (H (t, \cdot), E, h (t)) = C for all t \in [0, 1] . \end{matrix}$

(Existence) If $d (F, E, h) \neq 0$ , then the equation $F u = h$ has a solution in E.

Definition 3.5 ([14, Definition 3.3]).

The above degree is defined as follows: $\begin{matrix} d (F, E, h) : = d_{B} (F |_{{\overline{E}}_{0}}, E_{0}, h), \end{matrix}$ where $d_{B}$ is the Berkovits degree [7] and $E_{0}$ is any open subset of E with $F^{- 1} (h) \subset E_{0}$ and F is bounded on ${\overline{E}}_{0}$ .

4. Assumptions and main results

In this section, we will discuss the existence of weak solution of (1.1).

We assume that $Ω \subset R^{N}$ ( $N ⩾ 2$ ) is a bounded domain with a Lipschitz boundary $\partial Ω$ , $p \in C_{+} (\overline{Ω})$ is a log-Hölder continuous function, $δ, μ, λ \in R$ , $ζ \in C_{+} (\overline{Ω})$ with $2 ⩽ ζ^{-} ⩽ ζ (x) ⩽ ζ^{+} < p^{-}$ , $g : Ω \times R \to R$ and $f : Ω \times R \times R^{N} \to R$ are functions such that:

f is a Carathéodory function.

$\exists C_{1} > 0$ and $a \in L^{p^{'} (x)} (Ω)$ such that $\begin{matrix} | f (x, y, z) | ⩽ C_{1} (a (x) + | y |^{q (x) - 1} + | z |^{q (x) - 1}) . \end{matrix}$

g is a Carathéodory function.

$\exists C_{2} > 0$ and $b \in L^{p^{'} (x)} (Ω)$ such that $\begin{matrix} | g (x, y) | ⩽ C_{2} (b (x) + | y |^{s (x) - 1}), \end{matrix}$ for a.e. $x \in Ω$ and all $(y, z) \in R \times R^{N}$ , where $q, s \in C_{+} (\overline{Ω})$ with $2 ⩽ q^{-} ⩽ q (x) ⩽ q^{+} < p^{-}$ and $2 ⩽ s^{-} ⩽ s (x) ⩽ s^{+} < p^{-}$ .

Remark 4.1.

Note that $\int_{Ω} (| \nabla u |^{p (x) - 2} \nabla u + \frac{| \nabla u |^{2 p (x) - 2} \nabla u}{\sqrt{1 + | \nabla u |^{2 p (x)}}}) \nabla ϑ d x$ is well defined (see [22]).

$δ | u |^{ζ (x) - 2} u \in L^{p^{'} (x)} (Ω)$ , $μ g (x, u) \in L^{p^{'} (x)} (Ω)$ and $λ f (x, u, \nabla u) \in L^{p^{'} (x)} (Ω)$ under $u \in W^{1, p (x)} (Ω)$ , the assumptions $(A_{2})$ and $(A_{4})$ and the given hypotheses about the exponents p, ζ, q and s because: $a \in L^{p^{'} (x)} (Ω)$ , $b \in L^{p^{'} (x)} (Ω)$ , $r (x) = (q (x) - 1) p^{'} (x) \in C_{+} (\overline{Ω})$ with $r (x) < p (x)$ , $β (x) = (ζ (x) - 1) p^{'} (x) \in C_{+} (\overline{Ω})$ with $β (x) < p (x)$ and $κ (x) = (s (x) - 1) p^{'} (x) \in C_{+} (\overline{Ω})$ with $κ (x) < p (x)$ .

Then, by Remark 2.2 we can conclude that $\begin{matrix} L^{p (x)} ↪ L^{r (x)}, L^{p (x)} ↪ L^{β (x)} and L^{p (x)} ↪ L^{κ (x)} . \end{matrix}$ Hence, since $ϑ \in L^{p (x)} (Ω)$ , we have $\begin{matrix} (- δ | u |^{ζ (x) - 2} u + μ g (x, u) + λ f (x, u, \nabla u)) ϑ \in L^{1} (Ω) . \end{matrix}$ This implies that, the integral $\begin{matrix} \int_{Ω} (- δ | u |^{ζ (x) - 2} u + μ g (x, u) + λ f (x, u, \nabla u)) ϑ d x \end{matrix}$ exists.

Then, we shall use the definition of weak solution for problem (1.1) in the following sense:

Definition 4.1.

We say that a function $u \in W^{1, p (x)} (Ω)$ is a weak solution of (1.1), if for any $ϑ \in W^{1, p (x)} (Ω)$ , it satisfies the following: $\begin{matrix} \int_{Ω} (| \nabla u |^{p (x) - 2} \nabla u + \frac{| \nabla u |^{2 p (x) - 2} \nabla u}{\sqrt{1 + | \nabla u |^{2 p (x)}}}) \nabla ϑ d x = \int_{Ω} (- δ | u |^{ζ (x) - 2} u + μ g (x, u) + λ f (x, u, \nabla u)) ϑ d x . \end{matrix}$

Before giving our main result we first give two results that will be used later.

Lemma 4.1.

The operator $G : W^{1, p (x)} (Ω) \to W^{- 1, p^{'} (x)} (Ω)$ defined by $\begin{matrix} ⟨ G u, ϑ ⟩ = \int_{Ω} (| \nabla u |^{p (x) - 2} \nabla u + \frac{| \nabla u |^{2 p (x) - 2} \nabla u}{\sqrt{1 + | \nabla u |^{2 p (x)}}}) \nabla ϑ d x, \end{matrix}$ is continuous, bounded, strictly monotone and is of type $(S_{+})$ .

Proof.

Let us consider the following functional: $\begin{matrix} J (u) : = \int_{Ω} \frac{1}{p (x)} (| \nabla u |^{p (x)} + \sqrt{1 + | \nabla u |^{2 p (x)}}) d x . \end{matrix}$ From [22], it is obvious that the derivative operator of the functional $J$ in the weak sense at the point $u \in W^{1, p (x)} (Ω)$ is the functional $G (u) : = J^{'} (u) \in W^{- 1, p^{'} (x)} (Ω)$ given by $\begin{matrix} ⟨ G u, ϑ ⟩ = \int_{Ω} (| \nabla u |^{p (x) - 2} \nabla u + \frac{| \nabla u |^{2 p (x) - 2} \nabla u}{\sqrt{1 + | \nabla u |^{2 p (x)}}}) \nabla ϑ d x, \end{matrix}$ for all $u, ϑ \in W^{1, p (x)} (Ω)$ where $⟨ \cdot, \cdot ⟩$ means the duality pairing between $W^{- 1, p^{'} (x)} (Ω)$ and $W^{1, p (x)} (Ω)$ .

Hence, by using the similar argument as in the Proposition 3.1 of [22], we conclude that $G$ is continuous, bounded, strictly monotone and is of type $(S_{+})$ . □

Lemma 4.2.

If the assumptions ( $A_{1}$ )–( $A_{4}$ ) hold, then the operator $\begin{matrix} A : & W^{1, p (x)} (Ω) \to W^{- 1, p^{'} (x)} (Ω) \\ ⟨ A u, ϑ ⟩ = - \int_{Ω} (- δ | u |^{ζ (x) - 2} u + μ g (x, u) + λ f (x, u, \nabla u)) ϑ d x, \end{matrix}$ is compact.

Proof.

To prove this lemma, we follow four steps.

First step: Let Υ: $W^{1, p (x)} (Ω) \to L^{p^{'} (x)} (Ω)$ be an operator defined by $\begin{matrix} Υ u (x) : = - μ g (x, u) . \end{matrix}$ we will prove that the operator Υ is bounded and continuous.

Let $u \in W^{1, p (x)} (Ω)$ , bearing $(A_{4})$ in mind and using (2.5) and (2.6), we have $\begin{matrix} | Υ u |_{p^{'} (x)} & ⩽ ρ_{p^{'} (x)} (Υ u) + 1 \\ = \int_{Ω} {| μ g (x, u (x)) |}^{p^{'} (x)} d x + 1 \\ = \int_{Ω} | μ |^{p^{'} (x)} {| g (x, u (x)) |}^{p^{'} (x)} d x + 1 \\ ⩽ (| μ |^{p^{' -}} + | μ |^{p^{' +}}) \int_{Ω} {| C_{2} (b (x) + | u |^{s (x) - 1}) |}^{p^{'} (x)} d x + 1 \\ ⩽ C (| μ |^{p^{' -}} + | μ |^{p^{' +}}) \int_{Ω} ({| b (x) |}^{p^{'} (x)} + | u |^{κ (x)}) d x + 1 \\ ⩽ C (| μ |^{p^{' -}} + | μ |^{p^{' +}}) (ρ_{p^{'} (x)} (b) + ρ_{κ (x)} (u)) + 1 \\ ⩽ C (| b |_{p (x)}^{p^{' +}} + | u |_{κ (x)}^{κ^{+}} + | u |_{κ (x)}^{κ^{-}}) + 1 . \end{matrix}$ Next, from Remark 2.3 and $L^{p (x)} ↪ L^{κ (x)}$ , we deduce that $\begin{matrix} | Υ u |_{p^{'} (x)} ⩽ C (| b |_{p (x)}^{p^{' +}} + | u |_{1, p (x)}^{κ^{+}} + | u |_{1, p (x)}^{κ^{-}}) + 1, \end{matrix}$ that means Υ is bounded on $W^{1, p (x)} (Ω)$ .

Now, we show that the operator Υ is continuous. To this purpose let $u_{n} \to u$ in $W^{1, p (x)} (Ω)$ . We need to show that $Υ u_{n} \to Υ u$ in $L^{p^{'} (x)} (Ω)$ . We will apply the Lebesgue’s theorem.

We have $u_{n} \to u$ in $W^{1, p (x)} (Ω)$ , then $u_{n} \to u$ in $L^{p (x)} (Ω)$ . So there exist a subsequence $(u_{k})$ of $(u_{n})$ and ϕ in $L^{p (x)} (Ω)$ such that $\begin{matrix} (4.1) & u_{k} (x) \to u (x) and | u_{k} (x) | ⩽ ϕ (x) . \end{matrix}$ Then, by $(A_{2})$ and (4.1), we get $\begin{matrix} | g (x, u_{k} (x)) | ⩽ C_{2} (b (x) + {| ϕ (x) |}^{s (x) - 1}) . \end{matrix}$ On the other side, thanks to $(A_{3})$ and (4.1), we obtain $\begin{matrix} g (x, u_{k} (x)) \to g (x, u (x)) a.e. x \in Ω as k ⟶ \infty . \end{matrix}$ Seeing that $\begin{matrix} b + | ϕ |^{s (x) - 1} \in L^{p^{'} (x)} (Ω) and ρ_{p^{'} (x)} (Υ u_{k} - Υ u) = \int_{Ω} {| g (x, u_{k} (x)) - g (x, u (x)) |}^{p^{'} (x)} d x, \end{matrix}$ then, by (2.4) and the Lebesgue’s theorem, we get $\begin{matrix} Υ u_{k} \to Υ u in L^{p^{'} (x)} (Ω), \end{matrix}$ thus $\begin{matrix} Υ u_{n} \to Υ u in L^{p^{'} (x)} (Ω), \end{matrix}$ that is, Υ is continuous.

Second step: We define the operator Ψ: $W^{1, p (x)} (Ω) \to L^{p^{'} (x)} (Ω)$ by $\begin{matrix} Ψ u (x) : = δ {| u (x) |}^{ζ (x) - 2} u (x) . \end{matrix}$ We will prove that Ψ is bounded and continuous.

It is obvious that Ψ is continuous. We then demonstrate that Ψ is bounded.

Let $u \in W^{1, p (x)} (Ω)$ and using (2.5) and (2.6), we get $\begin{matrix} | Ψ u |_{p^{'} (x)} & ⩽ ρ_{p^{'} (x)} (Ψ u) + 1 \\ = \int_{Ω} {| δ | u |^{ζ (x) - 2} u |}^{p^{'} (x)} d x + 1 \\ = \int_{Ω} | δ |^{p^{'} (x)} | u |^{(ζ (x) - 1) p^{'} (x)} d x + 1 \\ ⩽ (| δ |^{p^{' -}} + | δ |^{p^{' +}}) \int_{Ω} | u |^{β (x)} d x + 1 \\ = (| δ |^{p^{' -}} + | δ |^{p^{' +}}) ρ_{β (x)} (u) + 1 \\ ⩽ (| δ |^{p^{' -}} + | δ |^{p^{' +}}) (| u |_{β (x)}^{β^{-}} + | u |_{β (x)}^{β^{+}}) + 1 . \end{matrix}$ Hence, we deduce from $L^{p (x)} ↪ L^{β (x)}$ and Remark 2.3 that $\begin{matrix} | Ψ u |_{p^{'} (x)} ⩽ C (| u |_{1, p (x)}^{β^{-}} + | u |_{1, p (x)}^{β^{+}}) + 1, \end{matrix}$ so Ψ is bounded on $W^{1, p (x)} (Ω)$ .

Third step: Let us define the operator Φ: $W^{1, p (x)} (Ω) \to L^{p^{'} (x)} (Ω)$ by $\begin{matrix} Φ u (x) : = - λ f (x, u (x), \nabla u (x)) . \end{matrix}$ We will show that Φ is bounded and continuous.

Let $u \in W^{1, p (x)} (Ω)$ . From $(A_{2})$ , (2.5) and (2.6), we get $\begin{matrix} | Φ u |_{p^{'} (x)} & ⩽ ρ_{p^{'} (x)} (Φ u) + 1 \\ = \int_{Ω} {| λ f (x, u (x), \nabla u (x)) |}^{p^{'} (x)} d x + 1 \\ = \int_{Ω} | λ |^{p^{'} (x)} {| f (x, u (x), \nabla u (x)) |}^{p^{'} (x)} d x + 1 \\ ⩽ (| λ |^{p^{' -}} + | λ |^{p^{' +}}) \int_{Ω} {| C_{1} (a (x) + | u |^{q (x) - 1} + | \nabla u |^{q (x) - 1}) |}^{p^{'} (x)} d x + 1 \\ ⩽ C (| λ |^{p^{' -}} + | λ |^{p^{' +}}) \int_{Ω} ({| a (x) |}^{p^{'} (x)} + | u |^{r (x)} + | \nabla u |^{r (x)}) d x + 1 \\ ⩽ C (| λ |^{p^{' -}} + | λ |^{p^{' +}}) (ρ_{p^{'} (x)} (a) + ρ_{r (x)} (u) + ρ_{r (x)} (\nabla u)) + 1 \\ ⩽ C (| a |_{p (x)}^{p^{' +}} + | u |_{r (x)}^{r^{+}} + | u |_{r (x)}^{r^{-}} + | \nabla u |_{r (x)}^{r^{+}} + | \nabla u |_{r (x)}^{r^{-}}) + 1 . \end{matrix}$ Bearing in mind that $L^{p (x)} ↪ L^{r (x)}$ and Remark 2.3, we have then $\begin{matrix} | Φ u |_{p^{'} (x)} ⩽ C (| a |_{p (x)}^{p^{' +}} + | u |_{1, p (x)}^{r^{+}} + | u |_{1, p (x)}^{r^{-}}) + 1, \end{matrix}$ and consequently Φ is bounded on $W^{1, p (x)} (Ω)$ .

It remains to show that Φ is continuous. Let $u_{n} \to u$ in $W^{1, p (x)} (Ω)$ , we need to show that $Φ u_{n} \to Φ u$ in $L^{p^{'} (x)} (Ω)$ . We will apply the Lebesgue’s theorem.

We have $u_{n} \to u$ in $W^{1, p (x)} (Ω)$ , then $u_{n} \to u$ in $L^{p (x)} (Ω)$ and $\nabla u_{n} \to \nabla u$ in ${(L^{p (x)} (Ω))}^{N}$ . Hence, there exist a subsequence $(u_{k})$ and ϕ in $L^{p (x)} (Ω)$ and ψ in ${(L^{p (x)} (Ω))}^{N}$ such that $\begin{array}{l} (4.2) & u_{k} (x) \to u (x) and \nabla u_{k} (x) \to \nabla u (x), \\ (4.3) & | u_{k} (x) | ⩽ ϕ (x) and | \nabla u_{k} (x) | ⩽ | ψ (x) | . \end{array}$ Hence, thanks to $(A_{1})$ and (4.2), we get, as $k ⟶ \infty$ $\begin{matrix} f (x, u_{k} (x), \nabla u_{k} (x)) \to f (x, u (x), \nabla u (x)) a.e. x \in Ω . \end{matrix}$ On the other hand, from $(A_{2})$ and (4.3), we can deduce the estimate $\begin{matrix} | f (x, u_{k} (x), \nabla u_{k} (x)) | ⩽ C_{1} (a (x) + {| ϕ (x) |}^{q (x) - 1} + {| ψ (x) |}^{q (x) - 1}) . \end{matrix}$ Seeing that $\begin{matrix} a + | ϕ |^{q (x) - 1} + {| ψ (x) |}^{q (x) - 1} \in L^{p^{'} (x)} (Ω), \end{matrix}$ and taking into account the equality $\begin{matrix} ρ_{p^{'} (x)} (Φ u_{k} - Φ u) = \int_{Ω} {| f (x, u_{k} (x), \nabla u_{k} (x)) - f (x, u (x), \nabla u (x)) |}^{p^{'} (x)} d x, \end{matrix}$ then, we conclude from (2.4) and the Lebesgue’s theorem that $\begin{matrix} Φ u_{k} \to Φ u in L^{p^{'} (x)} (Ω), \end{matrix}$ and consequently $\begin{matrix} Φ u_{n} \to Φ u in L^{p^{'} (x)} (Ω), \end{matrix}$ and then Φ is continuous.

Fourth step: Let $I^{*}$ : $L^{p^{'} (x)} (Ω) \to W^{- 1, p^{'} (x)} (Ω)$ be the adjoint operator of the operator $I : W^{1, p (x)} (Ω) \to L^{p (x)} (Ω)$ . We then define $\begin{array}{l} I^{*} \circ Υ : W^{1, p (x)} (Ω) \to W^{- 1, p^{'} (x)} (Ω), \\ I^{*} \circ Ψ : W^{1, p (x)} (Ω) \to W^{- 1, p^{'} (x)} (Ω), \end{array}$ and $\begin{matrix} I^{*} \circ Φ : W^{1, p (x)} (Ω) \to W^{- 1, p^{'} (x)} (Ω) . \end{matrix}$ On the other hand, if we take into account that I is compact, then $I^{*}$ is compact. Hence, $I^{*} \circ Υ$ , $I^{*} \circ Ψ$ and $I^{*} \circ Φ$ are compact, which means that $A = I^{*} \circ Υ + I^{*} \circ Ψ + I^{*} \circ Φ$ is compact. □

Now, we formulate our main result as follows.

Theorem 4.1.

If hypotheses ( $A_{1}$ )–( $A_{4}$ ) hold, then for each $δ, μ, λ \in R$ the problem ( 1.1 ) possesses at least one weak solution u in $W^{1, p (x)} (Ω)$ .

Proof.

We will reduce the problem (1.1) to a new one governed by a Hammerstein equation, and we will apply the theory of topological degree introduced in Section 3.

For all $u, ϑ \in W^{1, p (x)} (Ω)$ , we define the operators $G$ and $A$ , as defined in Lemmas 4.1 and 4.2 respectively, by $\begin{array}{l} ⟨ G u, ϑ ⟩ = \int_{Ω} (| \nabla u |^{p (x) - 2} \nabla u + \frac{| \nabla u |^{2 p (x) - 2} \nabla u}{\sqrt{1 + | \nabla u |^{2 p (x)}}}) \nabla ϑ d x, \\ ⟨ A u, ϑ ⟩ = - \int_{Ω} (- δ | u |^{ζ (x) - 2} u + μ g (x, u) + λ f (x, u, \nabla u)) ϑ d x . \end{array}$ Consequently, the problem (1.1) is equivalent to the equation $\begin{matrix} (4.4) & G u = - A u, u \in W^{1, p (x)} (Ω) . \end{matrix}$ Taking into account that, by Lemma 4.1, the operator $G$ is a continuous, bounded, strictly monotone and of type $(S_{+})$ , then, by [26, Theorem 26 A], the inverse operator $\begin{matrix} N : = G^{- 1} : W^{- 1, p^{'} (x)} (Ω) \to W^{1, p (x)} (Ω), \end{matrix}$ is also bounded, continuous, strictly monotone and of type $(S_{+})$ .

On another side, according to Lemma 4.2, we have that the operator $A$ is bounded, continuous and quasimonotone.

Consequently, following Zeidler’s terminology [26], the equation (4.4) is equivalent to the following abstract Hammerstein equation $\begin{matrix} (4.5) & u = N ϑ and ϑ + A \circ N ϑ = 0, u \in W^{1, p (x)} (Ω) and ϑ \in W^{- 1, p^{'} (x)} (Ω) . \end{matrix}$ Seeing that (4.4) is equivalent to (4.5), then to solve (4.4) it is thus enough to solve (4.5). In order to solve (4.5), we will apply the Berkovits topological degree introduced in Section 3.

First, let us set $\begin{matrix} R : = {ϑ \in W^{- 1, p^{'} (x)} (Ω) : \exists t \in [0, 1] such that ϑ + t A \circ N ϑ = 0} . \end{matrix}$ Now, we show that $R$ is bounded in $\in W^{- 1, p^{'} (x)} (Ω)$ .

Let us put $u : = N ϑ$ for all $ϑ \in R$ . Taking into account that $| N ϑ |_{1, p (x)} = | \nabla u |_{p (x)}$ , then we have the following two cases:

First case: If $| \nabla u |_{p (x)} ⩽ 1$ .

Then $| N ϑ |_{1, p (x)} ⩽ 1$ , that means ${N ϑ : ϑ \in R}$ is bounded.

Second case: If $| \nabla u |_{p (x)} > 1$ .

Then, we deduce from (2.2), $(A_{2})$ and $(A_{4})$ , the inequalities (2.7) and (2.6) and the Young’s inequality that $\begin{matrix} | N ϑ |_{1, p (x)}^{p^{-}} & = | \nabla u |_{p (x)}^{p -} \\ ⩽ ρ_{p (x)} (\nabla u) \\ ⩽ ⟨ G u, u ⟩ \\ = ⟨ ϑ, N ϑ ⟩ \\ = - t ⟨ A \circ N ϑ, N ϑ ⟩ \\ = t \int_{Ω} (- δ | u |^{ζ (x) - 2} u + μ g (x, u) + λ f (x, u, \nabla u)) u d x \\ ⩽ t max (| δ |, C_{2} | μ |, C_{1} | λ |) (\int_{Ω} | u |^{ζ (x)} d x + \int_{Ω} | b (x) u (x) | d x + \int_{Ω} {| u (x) |}^{s (x)} d x \\ + \int_{Ω} | a (x) u (x) | d x + \int_{Ω} {| u (x) |}^{q (x)} d x + \int_{Ω} | \nabla u |^{q (x) - 1} | u | d x) \\ = t max (| δ |, C_{2} | μ |, C_{1} | λ |) (ρ_{ζ (x)} (u) + \int_{Ω} | b (x) u (x) | d x + \int_{Ω} | a (x) u (x) | d x \\ + ρ_{s (x)} (u) + ρ_{q (x)} (u) + \int_{Ω} | \nabla u |^{q (x) - 1} | u | d x) \\ ⩽ C (| u |_{ζ (x)}^{ζ^{-}} + | u |_{ζ (x)}^{ζ^{+}} + | b |_{p^{'} (x)} | u |_{p (x)} + | a |_{p^{'} (x)} | u |_{p (x)} + | u |_{s (x)}^{s^{+}} + | u |_{s (x)}^{s^{-}} \\ + | u |_{q (x)}^{q^{+}} + | u |_{q (x)}^{q^{-}} + \frac{1}{q^{' -}} ρ_{q (x)} (\nabla u) + \frac{1}{q -} ρ_{q (x)} (u)) \\ ⩽ C (| u |_{ζ (x)}^{ζ^{-}} + | u |_{ζ (x)}^{ζ^{+}} + | u |_{p (x)} + | u |_{s (x)}^{s^{+}} + | u |_{s (x)}^{s^{-}} + | u |_{q (x)}^{q^{+}} + | u |_{q (x)}^{q^{-}} + | \nabla u |_{q (x)}^{q^{+}}), \end{matrix}$ then, according to $L^{p (x)} ↪ L^{ζ (x)}$ , $L^{p (x)} ↪ L^{s (x)}$ and $L^{p (x)} ↪ L^{q (x)}$ , we get $\begin{matrix} | N ϑ |_{1, p (x)}^{p^{-}} ⩽ C (| N ϑ |_{1, p (x)}^{ζ^{+}} + | N ϑ |_{1, p (x)} + | N ϑ |_{1, p (x)}^{s^{+}} + | N ϑ |_{1, p (x)}^{q^{+}}), \end{matrix}$ what implies ${N ϑ : ϑ \in R}$ is bounded.

On the other hand, we have that the operator is $A$ is bounded, then $A \circ N ϑ$ is bounded. Thus, thanks to (4.5), we have that $R$ is bounded in $W^{- 1, p^{'} (x)} (Ω)$ .

However, $\exists r > 0$ such that $\begin{matrix} | ϑ |_{- 1, p^{'} (x)} < r for all ϑ \in R, \end{matrix}$ which leads to $\begin{matrix} ϑ + t A \circ N ϑ \neq 0, ϑ \in \partial R_{r} (0) and t \in [0, 1], \end{matrix}$ where $R_{r} (0)$ is the ball of center 0 and radius r in $W^{- 1, p^{'} (x)} (Ω)$ .

Moreover, by Lemma 3.1, we conclude that $\begin{matrix} I + A \circ N \in F_{N} (\overline{R_{r} (0)}) and I = G \circ N \in F_{N} (\overline{R_{r} (0)}) . \end{matrix}$ On another side, taking into account that I, $A$ and $N$ are bounded, then $I + A \circ N$ is bounded. Hence, we infer that $\begin{matrix} I + A \circ N \in F_{N, B} (\overline{R_{r} (0)}) and I = G \circ N \in F_{N, B} (\overline{R_{r} (0)}) . \end{matrix}$ Next, we define the homotopy $\begin{matrix} H : & [0, 1] \times \overline{R_{r} (0)} \to W^{- 1, p^{'} (x)} (Ω) \\ (t, ϑ) \mapsto H (t, ϑ) : = ϑ + t A \circ N ϑ . \end{matrix}$ Hence, thanks to the properties of the degree d seen in Theorem 3.1, we obtain $\begin{matrix} d (I + A \circ N, R_{r} (0), 0) = d (I, R_{r} (0), 0) = 1 \neq 0, \end{matrix}$ what implies that there exists $ϑ \in R_{r} (0)$ which verifies $\begin{matrix} ϑ + A \circ N ϑ = 0 . \end{matrix}$ And finally, we deduce that $u = N ϑ$ is a weak solution of (1.1). The proof is done. □

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Existence result for a Neumann boundary value problem governed by a class of p ( x ) -Laplacian-like equation

Abstract

Keywords

1. Introduction and motivation

2. Preliminaries

Proposition 2.1 ([10]).

Remark 2.1. According to (2.2) and (2.3), we have (2.5) | u | p ( x ) ⩽ ρ p ( x ) ( u ) + 1 , (2.6) ρ p ( x ) ( u ) ⩽ | u | p ( x ) p − + | u | p ( x ) p + . Proposition 2.2 ([15, Theorem 2.5 and Corollary 2.7]).

Proposition 2.3 ([15, Theorem 2.1]).

4. Assumptions and main results

References

Remark 2.1.
According to (2.2) and (2.3), we have $\begin{array}{l} (2.5) & | u |_{p (x)} ⩽ ρ_{p (x)} (u) + 1, \\ (2.6) & ρ_{p (x)} (u) ⩽ | u |_{p (x)}^{p^{-}} + | u |_{p (x)}^{p^{+}} . \end{array}$
Proposition 2.2 ([15, Theorem 2.5 and Corollary 2.7]).