Infinitely many positive solutions for a double phase problem

This paper is concerned with the existence of infinitely many positive solutions to a class of double phase problem. By variational methods and the theory of the Musielak–Orlicz–Sobolev space, we establish the existence of infinitely many positive solutions whose W01,H(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$W_{0}^{1,H}(\varOmega )$\end{document}-norms and L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{\infty }$\end{document}-norms tend to zero under suitable hypotheses about nonlinearity.


Introduction and main results
The study of differential equations and variational problems with double phase operator is a new and interesting topic. Such interest is widely justified by many physical examples, such as elasticity, strongly anisotropic materials and Lavrentiev's phenomenon (e.g., see Refs. [1][2][3][4]). More precisely, their research is related to the following energy functional: The aim of this paper is to obtain infinitely many distinct positive solutions for the following double phase problem: ⎧ ⎨ ⎩ -div(|∇u| p-2 ∇u + a(x)|∇u| q-2 ∇u) = f (x, u), in Ω, where Ω is a smooth bounded domain in R N (N ≥ 2), 1 < p < q < N , and f : Ω × R → R satisfy Carathéodory condition and there exists t 0 > 0 such that In the past decade, many authors considered the existence and multiplicity of solutions of (P). For example, Liu and Dai [14] got one sign-changing ground state solution for problem (P) using the Nehari manifold method. Additionally, Liu and Dai in [15] also obtained the existence of at least three ground state solutions of (P) by using the strong maximum principle. In a recent paper [12], Ge and Chen obtained the same result as in [14] for problem (P) under more general assumptions on f . In [16], by using the fountain and dual theorem with Cerami condition, we obtained some existence of infinitely many solutions for the above problem under some weaker assumptions on f . The aim of the present paper is to establish the existence of infinitely many distinct positive solutions for problem (P) under suitable oscillatory assumptions on the nonlinear term f at zero.
In order to state the main result of this paper, let us introduce the following assumptions for problem (P): (h 1 ) There are two sequences {a k } ∞ k=1 , {b k } ∞ k=1 such that 0 < a k < b k , lim k→+∞ b k = 0, and for almost all x ∈ Ω and k ∈ N .
We are now in the position to state our main results.
The rest of this paper is organized as follows. In Sect. 2, we present some necessary preliminary knowledge on space W 1,H 0 (Ω). In Sect. 3, we establish the variational framework associated with problem (P), and we complete the proofs of Theorem 1.1.

Preliminaries
In order to discuss problem (P), we need some facts on space W 1,H 0 (Ω) which are called Musielak-Orlicz-Sobolev space. For this reason, we will recall some properties involving the Musielak-Orlicz spaces, which can be found in [16,[21][22][23] and the references therein.

It is clear that H ∈ N(Ω) is a locally integrable and
endowed with the Luxemburg norm and it is equipped with the norm is an equivalent norm on W 1,H 0 (Ω). Furthermore, we have the following embedding theorem.
for any u ∈ W 1,H 0 (Ω). From now on, we denote by E the space W 1,H 0 (Ω). In order to discuss the problem (P), we need to define a functional in E: We know that J ∈ C 1 (E, R) and double phase operator -div(|∇u| p-2 ∇u + a(x)|∇u| q-2 ∇u) is the derivative operator of J in the weak sense. We denote L = J : E → E * , then Here E * denotes the dual space of E and ·, · denotes the pairing between E and E * . Then we have the following result.

Proposition 2.2 ([14, Proposition 3.1]) If L is as above, then
(1) L : E → E * is a continuous, bounded and strictly monotone operator; (2) L : E → E * is a mapping of type (S) + , i.e., if u n u in E and lim sup n→+∞ L(u n ) -L(u), u nu ≤ 0, implies u n → u in E; (3) L : E → E * is a homeomorphism.

Variational setting and the proof of Theorem 1.1
To prove our Theorem 1.1, we recall the variational setting corresponding to the problem (P).
We observe that problem (P) has a variational structure, and as a matter of fact, its solutions can be searched as critical points of the energy functional ϕ : E → R defined as follows: where and Ψ (u) = Ω F(x, u(x)) dx. Thus, in [14], it is shown that Φ(u) is a Gâteaux differentiable functional in E whose derivative is given by for all v ∈ E. Finally, Φ(u) is weakly lower semi-continuous and coercive. Moreover, standard arguments show that Ψ is a well defined and continuously Gâteaux differentiable functional whose Gâteaux derivative Definition 3. 1 We say that u ∈ E is a weak solution of (P) if Next, we will prove Theorem 1.1 by virtue of some idea due to Kristaly, Morosanu and Tersian [24], where the infinitely many homoclinic solutions for a p-Laplace equation was obtained. Firstly, by our assumptions on f , there exist d 0 > 0 and t 0 > 0 such that |f (x, t)| ≤ d 0 , for every t ∈ [0, t 0 ] and a.e. x ∈ Ω. Without loss of generality, we suppose that, for every Thus, we have f (x, t) ≤ d 0 , ∀t ∈ R and a.e. x ∈ Ω.
Now, we consider the following problem: Hence, the weak solutions of (3.4) are the critical points of the functional where F(x, u) = u 0 f (x, s) ds. By (3.3), it is easy to see that ϕ is well defined, weakly sequentially lower semi-continuous and Gâteaux differentiable in E. For every fixed k ∈ N , consider the set S k = {u ∈ E : u(x) = 0 and 0 ≤ u(x) ≤ b k a.e. x ∈ Ω}.

Lemma 3.2 For every k ∈ N , the functional ϕ is bounded from below on S k and its infimum m k on S k is attained at u k ∈ S k .
Proof For every k ∈ N , we obtain, for any every u ∈ S k , This means that ϕ is bounded from below on S k . Moreover, it is clear that S k is convex and closed, thus weakly closed in E. Let m k = inf u∈S k ϕ(u), and {u n } be a sequence in S k such that m k ≤ ϕ(u n ) ≤ m k + 1 n for all n ∈ N . Then, if u n ≤ 1, we have done it, otherwise, we have From this, we deduce that {u n } is bounded in E. So, up to a subsequence, {u n } weakly converges to some u k ∈ S k . At this point, we obtain ϕ(u k ) = m k in view of the weakly sequentially lower semi-continuity of ϕ. This completes the proof of the lemma. Proof Let T = {x ∈ Ω : a k < u k (x) ≤ b k } and assume that meas(T) > 0. Define the function h(t) = min{t + , a k } and v k = h(u k ), where t + = max{0, t}. It is obvious that h is continuous in (3.7) Moreover, we have On the other hand, by v ∈ S k , we have Using (2.1) and (3.9) in (3.8), we obtain where τ = p(q) when uv ≥ 1(≤ 1).
Since h is continuous, there exists δ > 0 such that, for every u ∈ E with uu k < δ, uv ≤ 1 qC 0 c θ+1 , which implies that u k is a local minimum of ϕ.  Proof In view of (h 2 ), one easily deduces c k ∈ S k . Hence, (3.10) Now we will prove that lim k→+∞ m k = 0. As a result of Lemma 3.2, for every k ∈ N and u k ∈ S k , we obtain Since lim k→+∞ b k = 0, we have lim k→+∞ m k ≥ 0. Note that m k < 0, hence lim k→+∞ m k = 0.