Infinitely many solutions for a nonlinear Navier boundary systems involving $(p(x),q(x))$-biharmonic

Abstract

In this article, we study the following (p(x), q(x))-biharmonic type systemWe prove the existence of infinitely many solutions of the problem by applying a general variational principle due to B. Ricceri and the theory of the variable exponent Sobolev spaces.

Multiple Solutions for a Nonlocal Elliptic Problem Involving p x

“…It well known that the variable exponent case possess more complicated properties than the constant exponent case, and some methods used in the (p, q)-biharmonic case cannot be applied to the (p(x), q(x))-biharmonic case. erefore, Allaoui et al [9] have made a great contribution to such problems, and they continued to extend (p, q)-biharmonic operator in [8] to (p(x), q(x))-biharmonic case, on the basis of Ricceri's variational principle [10] and the basic theory of Sobolev space, and the following system is solved:…”

Multiple Solutions for Nonlinear Navier Boundary Systems Involving(p1(x),…,pnMiao1 2016Discrete Dynamics in Nature and Society3010View full textAdd to dashboardCite

“…For the fixed ∈ Λ, the other step is to show that the functional has no global minimum. Arguing as in [15],…”

“…For the fixed ∈ Λ, the other step is to show that the functional has not a local minimum at zero. Arguing as in [15], since 1/ < ( ∑ =1 ( + ) 1/ + )̂1 0 , we can consider 8 Discrete Dynamics in Nature and Society positive real sequences { , } =1 and > 0 such that √∑ =1 2 , → 0 as → +∞ and…”

“…In [12], when the nonlinearity has the subcritical growth and via variational methods [13], the authors obtained the existence of at least one, two, or three weak solutions for a class of differential equations with ( )-Laplacian whenever the parameter belongs to a precise positive interval. Recently, the ( )-biharmonic problems have attracted more and more attention; we refer the reader to [11,[14][15][16][17][18][19][20][21]. In [16], El Amrouss and Ourraoui studied the ( )-biharmonic equation with Navier and Neumann boundary condition; the technical approach is based on Ricceri's variational principle and local mountain pass theorem, without Palais-Smale condition.…”

“…In [20], the authors established the existence of at least three solutions for elliptic systems involving the ( ( ), ( ))-biharmonic operator. In [15], Allaoui et al considered the existence of infinitely many solutions for the ( ( ), ( ))-biharmonic problem by a general Ricceri's variational principle. However, there are rare results on ( 1 ( ), .…”