Existence and multiplicity of solutions for nonlocal p(x)-Laplacian equations with nonlinear Neumann boundary conditions

Abstract

In this article, we study the nonlocal p(x)-Laplacian problem of the following form ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

Multiple solutions for a class of bi-nonlocal problems with nonlinear Neumann boundary conditions

“…By the sequentially weakly continuity of the functional L 2 combining with the continuity of the function M 2 , we get I ′ (u n ) → I ′ (u) in X * . Similarly, since the embedding X ֒→ L p (∂Ω) is compact, we can see that the functional ψ ′ is sequentially weaklystrongly continuous (see [12]).…”

“…In [9], Correa et al considered problem (1.4) in the case when b(t) = t r , r > 0 is a constant and proved the existence of infinitely many solutions for (1.4) by using Krasnoselskii's genus. In [12], Guo et al developed the results of Fan [11] for the p(x)-Kirchhoff type problem with Neumann nonlinear boundary condition. Some further results on Kirchhoff type problems with Neumann nonlinear boundary condition can be found in [14,16,20,21], in which the authors studied the existence and multiplicity of solutions for the problem by using the Nehari manifold and fibering maps, Ekeland variational principle or the variational principles due to Bonanno et al [3,4].…”

Existence and Multiplicity of Solutions for a Biharmonic Equation with p(x)-Kirchhoff Type

“…As we all know, the Ambrosetti-Rabinowitz [14] (it is written as (AR)) condition can not only ensure that f(x, s) is superlinear with respect to the variable s at infinity but also ensure the boundedness of the Palais-Smale (it is written as (PS)) sequence. erefore, (AR) condition is essential for the study of many boundary value problems, see [15][16][17][18]. In particular, when a � 1 and b � 0 in problem ( 1) and (AR) condition holds, the existence of solution of problem ( 1) is discussed in [16].…”

“…In particular, when a � 1 and b � 0 in problem ( 1) and (AR) condition holds, the existence of solution of problem ( 1) is discussed in [16]. In [17], Guo and Zhao made reasonable presupposition on the nonlinear terms M and f, respectively, and obtained the existence number of solutions for p(x)-Laplace equation in the whole space by applying the basic theory of weighted function and variable exponent space. In [18], Chung, respectively, applied the minimum principle or critical point theory to discuss the problem when the perturbation term f is sublinear or superlinear at infinity.…”

Sobre una ecuación elíptica del tipo p(x)-Kirchhoff con términos fuente no local

“…Para los problemas del tipo p(x)-Laplace con condiciones no lineales en la frontera, puede verse Guo y Zhao [4] y sus referencias.…”