Eigenvalues of the $p(x)-$biharmonic operator with indefinite weight] {Eigenvalues of the $p(x)-$biharmonic operator with indefinite weight under Neumann boundary conditions

Abstract

In this paper we will study the existence of solutions for the nonhomogeneous elliptic equation with variable exponent ∆ 2 p(x) u = λV (x)|u| q(x)−2 u, in a smooth bounded domain,under Neumann boundary conditions, where λ is a positive real number, p, q : Ω → R, are continuous functions, and V is an indefinite weight function. Considering different situations concerning the growth rates involved in the above quoted problem, we will prove the existence of a continuous family of eigenvalues.

Existence of solutions for a class of p(x)-biharmonic problems without (A-R) type conditions

“…By Lemmas 2.3, 2.4 and 2.5, the functional J satisfies the conditions of the classical mountain pass theorem due to Ambrosettiand Rabinowitz [2]. Thus, we obtain a nontrivial weak solution of problem (1). If, further, f is odd, then J is even.…”

“…El Amrouss first studied the spectrum of a fourth order elliptic equation with variable exponent. After that, many authors studied the existence of solutions for problems of this type, see for examples [1,3,11,12,13,16]. In [3], A. El Amrouss et al used the mountain pass theorem to study the existence of nontrivial solutions.…”

“…In [12,16], the authors studied the multiplicity of solutions for a class of Navier boundary value problems involving the p(x)-biharmonic operator. We also refer the readers to recent papers [1,11,13], in which the authors study the existence of eigenvalues of the p(x)-biharmonic operator. Motivated by the ideas introduced in [7] and some properties of the p(x)-biharmonic operator in [3,4,16], we study the existence and multiplicity of nontrivial solutions for a class of p(x)-biharmonic problems without (A-R) type conditions.…”

Existence of two non-zero weak solutions for a $p(\cdot)$-biharmonic problem with Navier boundary conditions

“…• elliptic problems involving p. /-biharmonic operator with different boundary conditions (see for instance [2,3,10,13,15,16,18,20,22,23,25,27,29,32,37,38,40,41,43]).…”