Existence of solutions for a boundary problem involving p(x)-biharmonic operator

Abdel Rachid El Amrouss,Anass Ourraoui

doi:10.5269/bspm.v31i1.15148

Abdel Rachid El Amrouss, Anass Ourraoui

Open Access

https://doi.org/10.5269/bspm.v31i1.15148

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Abstract

In this paper, we establish the existence of at least three solutions to a boundary problem involving the p(x)-biharmonic operator. Our technical approach is based on theorem obtained by B. Ricceri's variational principale and local mountain pass theorem without (Palais.Smale) condition.

Highlights

Many authors consider the existence of multiple nontrivial solutions for some fourth order problems [11,16]
El Amrouss interested to the spectrum of a fourth order elliptic equation with variable exponent. They proved the existence of infinitely many eigenvalue sequences and supΛ = +∞, where Λ is the set of all eigenvalues
(2) the mapping L′ : X → X′ is a strictly monotone, bounded homeomorphism and is of type S+, namely, un ⇀ u and lim supn→∞ L′(un)(un − u) ≤ 0 implies that un → u, where → and ⇀ denote the strong and weak convergence respectively

Summary

Variable exponent space and Sobolev Spaces

R ∈ C+(Ω) such that r(x) ≤ p∗k(x) for all x ∈ Ω, there is a continuous and compact embedding. 3) By the above remark and proposition 2.2 there is a continuous and compact embedding of W 2,p(x)(Ω) ∩ W01,p(x)(Ω) into Lr(x)(Ω), where r(x) < p∗(x) for all x ∈ Ω. (2) the mapping L′ : X → X′ is a strictly monotone, bounded homeomorphism and is of type S+ , namely, un ⇀ u and lim supn→∞ L′(un)(un − u) ≤ 0 implies that un → u, where → and ⇀ denote the strong and weak convergence respectively. Thanks to a Minty-Browder [15], L′ is surjective and admits an inverse mapping It suffices to show the continuity of (L′)−1. By the coercivity of L′, one deducts that the sequence (un) is bounded in the reflexive space X. Using Holder’s inequality and the continuous embedding of X into Lq(x)(Ω), we obtain

Ricceri’s variational principle

Proof of the main result