Existence of radial weak solutions to Steklov problem involving Leray–Lions type operator

We make use of variational methods to prove the existence of at least one positive radial increasing weak solution to a Leray–Lions type problem under Steklov boundary conditions.


Introduction
We consider the following Leray-Lions type problem with Steklov boundary conditions where Ω is a bounded region in ℝ N , N > 1 , with smooth boundary Ω , (x) = ( 1 (x), ⋯ , N (x)) is the outward unit normal to the smooth surface Ω at x. We assume that x ∈ Ω, a ∶ Ω × ℝ → ℝ satisfies the following conditions: (a1) a is a Carathéodory function such that a(x, 0) = 0 , for a.e. x ∈ Ω.
(a2) There exist positive functions , ∈ L ∞ (Ω), | ⋅ | ∞ such that for a.e. x ∈ Ω and all t ∈ ℝ where p ∈ C + (Ω). (a3) For all s, t ∈ ℝ with t ≠ s the inequality holds, for a.e. x ∈ Ω . For which antiderivative defined by holds in the following condition (a4) There exists c ≥ 1 such that for a.e. x ∈ Ω and all t ∈ ℝ. Also, we suppose that R ∈ L ∞ (Ω) is a radial function with ess inf Ω R > 0 and the Carathéodory function is continuously differentiable and odd with respect to its second variable satisfying the hypotheses (f 1 ) − (f 3 ) : (f 1 ) t f (x, 0) = 0 and there exists q ∈ C(Ω) with p(x) ≤ q(x) ≤ p * (x) a.e. in Ω such that where t f (x, t) denotes the partial derivative of f at the point (x, t) with respect to the second variable.
a(x, s)ds, and the Carathéodory function is continuously differentiable and odd with respect to its second variable satisfying the following assertions: in Ω such that (g 2 ) the inequalities g(x, t) ≥ 0 and hold for every t ∈ (0, +∞). Finally, we assume that h ∈ L p � (⋅) (Ω) is a positive continuous function such that |h| p � (⋅) is small enough. Faria et al. [4] have studied the existence of positive solutions for the following nonlinear elliptic problems under Dirichlet boundary condition 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(⋅)-Laplace equation involving Leray-Lions operators where Ω is a bounded domain in ℝ N with a Lipchitz boundary Ω ; a ∶ Ω × ℝ N → ℝ N and f ∶ Ω × ℝ → ℝ are Carathéodory functions with suitable growth conditions. Firstly, under a p(⋅)-sublinear condition for nonlinear term, they obtained a sequence of solutions approaching 0 by showing a priori bound for solutions. Secondly, for a p(⋅)-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 involving Leray-Lions type operator via variational methods, where Ω is a bounded domain in ℝ N (N ≥ 2) with a smooth boundary Ω , > 0 is a parameter, f is a Carathéodory function, p ∈ C(Ω) satisfies the inequality and Δ(a(x, Δu)) 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).

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 Ω be a bounded simply connected domain with smooth boundary. We set is the collection of all measurable functions u on Ω for which ∫ Ω |u(x)| p(x) dx < +∞ and has the norm Following the authors of paper [15], for any > 0 , we put and for r ∈ C + (Ω) . Then the well-known proposition [7, Proposition 2.7] will be rewritten as follows.
Normally, the first order Sobolev space associated with L p(⋅) (Ω) is defined as follows endowed with the norm

Proposition 2.2 Assume that Ω is a bounded and smooth domain in
for a.e. x ∈ Ω and all t ∈ ℝ . So, one has where C =c( 1 + 1).
Following remark consists of some properties of f(t) and  (4) For every > 0 , there exists C > 0 such that for all t ∈ ℝ and a.e. x ∈ Ω. In the next remarks, we mention to some properties of g(t) and its antiderivative ,

Remark 2.3
We have (3) There exist positive constants D 1 and D 2 such that for all t ∈ ℝ and a.e x ∈ Ω.
Proof These results follow directly from hypotheses (g 1 ) − −(g 2 ) and the definitions of g and G.
Using Proposition 2.1, Hölder inequality and hypothesis R ∈ L ∞ (Ω) , we gain and from hypotheses (a2) and relation (2.1), one has Bearing in mind the following elementary inequality due to J.A. Clarkson: for all > 0 , there exists C > 0 such that for all s, t ∈ ℝ . Then we deduce So, the proof is complete; It is enough to put The next two theorems have been proved in [2]. 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 ⟨ , ⟩. Definition 2.1 (Subdifferential) Let Ψ ∶ X → (−∞, +∞] be a proper convex function and 2 X * be the set of all subsets of X * . The subdifferential of Ψ denoted by Ψ , Ψ ∶ X → 2 X * , is defined to be following set-value operator for u ∈ Dom(Ψ) = {v ∈ X ∶ Ψ(v) < ∞} , and Ψ(u) = � if u ∉ Dom(Ψ).
Notice that if Ψ is Gâteaux differentiable at u, which its derivative is denoted by DΨ(u) , then Ψ(u) is a singleton. In this case, Ψ(u) = {DΨ(u)}.

Definition 2.2 (Critical Point) We say that the point u ∈ X is a critical point of I K , if DΦ(u) ∈ Ψ K (u) or equivalently, it satisfies the following inequality
Notice that a global minimum point is a critical point.

Definition 2.3 (Point-Wise Invariance
Condition) The triple (Ψ, Φ, K) satisfies the point-wise invariance condition at a point u ∈ X if there exist a convex Gâteaux differentiable function G ∶ X → ℝ and a point v ∈ K such that 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 Ψ ∶ X → (−∞, +∞] be a convex, lower semicontinuous function which is Gâteaux differentiable on K, and let Φ ∈ C 1 (X, ℝ) . Assume that the following two assertions hold: where Ψ K is defined as (2.2), has a critical point u ∈ X as in Definition 2.2; (ii) the triple (Ψ K , Φ, K) satisfies the point-wise invariance condition at the point u; Then u ∈ K is a solution of the equation Here, we present another proof of this theorem.
Proof Let u ∈ Dom(Ψ K ) be a critical point of I K , so we have Since I K (u) is a finite number, so u ∈ K . By assumptions of the theorem, there exists v ∈ K satisfying the linear equation Substituting w = v in inequality (2.5) we gain As a consequence of (2.6) we obtain On the other hand, Ψ K is Gâteaux differentiable at v ∈ K , so which means Thus But G is convex, so we have Thereby gaining that u = v . It then follows that DΨ(u) = D (u) as claimed. Here, we recall the (PS) compactness condition.

Definition 2.4 ((PS) Compactness Condition)
We say that I ∈ C(X, ℝ) satisfies the Palais-Smale (PS) compactness condition if any sequence {u k } ⊂ X such that 1] I( (t)), has a convergent subsequence in X.

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 ∈ W 1,p(⋅) (Ω) is a radial increasing weak solution of problem (1.1) if it is radially increasing function with u > 0 in Ω such that and one has for all w ∈ W 1,p(⋅) (Ω).
We where 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.
and u is radial}, Step1. We show that I K ∶ X → ℝ , defined as in (3.2), has a nontrivial critical point in K. To this end, we use the MPT (Theorem 2.5).
Proof Let {u k } is a sequence in K such that I(u k ) < +∞ and I � (u k ) → 0 . Then one has the following estimate Then thanks to Remark 2.6, not only ū ∈ K , but also u k →ū (strongly) in X as desired.
Secondly, we verify that I K satisfies in MPT conditions.
By the above argument and MPT, I K has a critical value where Γ = { ∈ C([0, 1], X) ∶ (0) = 0 , (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.