EXISTENCE OF THREE SOLUTIONS FOR A NAVIER BOUNDARY VALUE PROBLEM INVOLVING THE p(x)-BIHARMONIC

Abstract

Abstract. The existence of at least three weak solutions is established for a class of quasilinear elliptic equations involving the p(x)-biharmonic operators with Navier boundary value conditions. The technical approach is mainly based on a three critical points theorem due to Ricceri [11].

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

“…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.…”

“…For this purpose, the authors need the Ambrosetti-Rabinowitz (A-R) type condition (see [2]) to prove the energy functional satisfies the Palais-Smale (PS) condition. 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.…”

“…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. Our result here is different from one introduced in the previous paper [15].…”

On a nonlinear differential equation involving the p(x)-triharmonic operator

“…Partial differential equations involving the operators with variable exponents growth conditions have been the object of increasing amount of attention in recent years. For background and recent results, we refer the reader to [1,2,3,4,5,6,7,8,9,10]. These equations are interesting in applications and raise many problems such as the model of the motion of electroheological fluids, mathematical modeling of stationary thermo-rheological viscous flows of non-Newtonian fluids and other phenomena related to image processing, elasticity and the flow in porous media [11,12,13].…”

On ap(⋅)-biharmonic problem with no-flux boundary condition

“…The line of investigation was continued by several authors, see [1,2,4,21,34,30,31,47]. Notice that all these studies focus on problems with the Navier boundary condition (1) and only one of them, [21], also considers the Neumann type boundary condition ∂u ∂ν = ∂ ∂ν (|∆u| p(x)−2 ∆u) = 0 on ∂Ω.…”