On a bi‐nonlocal p(x)‐Kirchhoff equation via Krasnoselskii's genus

Abstract

We study existence and multiplicity of solutions to the following bi‐nonlocal p(x)‐Kirchhoff equation via Krasnoselskii's genus, on the Sobolev space with variable exponent, MathClass-bin−M()MathClass-op∫Ω1p(x)MathClass-rel|MathClass-rel∇uMathClass-rel|p(x)Δp(x)uMathClass-rel=f(xMathClass-punc,u)[]MathClass-op∫ΩF(xMathClass-punc,u)r1emnbsp1emnbsp1emnbspin1emnbsp1emnbspΩMathClass-punc,1emnbsp1emnbsp1emnbspuMathClass-rel=01emnbsp1emnbspon1emnbsp1emnbsp∂ΩMathClass-punc, where Ω is a bounded smooth domain of I RN, 1 < p(x) < N, M and f are continuous functions, f is an odd function, F(xMathClass-punc,u)MathClass-rel=MathClass-op∫0uf(xMathClass-punc,ξ)dξ and r > 0 is a real parameter. Copyright © 2014 John Wiley & Sons, Ltd.

Existence results for a class of nonlocal problems involving p(x)-Laplacian

“…After the work by Lions [19], various equations of Kirchhoff-type have been studied extensively, see [2,5]. The study of Kirchhoff-type equations has already been extended to the case involving the p-Laplacian (for details, see [3,4,10,11]) and p(x)-Laplacian (see [9,12]). Motivated by the above papers and the results in [8,20], we consider (1.1) to study the existence of weak solutions.…”

EXISTENCE OF WEAK SOLUTIONS FOR $p(x)$-KIRCHHOFF-TYPE EQUATION

“…In recent years, elliptic problems involving p−Kirchhoff and p (x) −Kirchhoff type operators have been studied in many papers, we refer to [2,5,7,8,9,10,18,23,25].…”

Multiple solutions for a class of p(x)-Kirchhoff type problems with Neumann boundary conditions

“…Moreover, problem (1.1) is related to the stationary version of the Kirchhoff equation which is presented by Kirchhoff in 1883, see [16] for details. One of the important hypotheses in these papers is that the Kirchhoff function M is non-degenerate, i.e., M.t / m 0 > 0 for all t 2 R C 0 WD OE0; C1/: Motivated by the ideas introduced in [8,10,12,22,23], the goal of this paper is to study the multiplicity of solutions for problem (1.1) in the case when the Kirchhoff function M.t/ may be degenerate at zero. One of the important hypotheses in these papers is that the Kirchhoff function M is non-degenerate, i.e., M.t / m 0 > 0 for all t 2 R C 0 WD OE0; C1/: Motivated by the ideas introduced in [8,10,12,22,23], the goal of this paper is to study the multiplicity of solutions for problem (1.1) in the case when the Kirchhoff function M.t/ may be degenerate at zero.…”