On an elliptic system of p(x)-Kirchhoff-type under neumann boundary condition

In the present paper, by using the direct variational method and the Ekeland variational principle, we study the existence of solutions for an elliptic system of p(x)-Kirchhoff-type under Neumann boundary condition and show the existence of a weak solution

Nontrivial Solution for a Nonlocal Elliptic Transmission Problem in Variable Exponent Sobolev Spaces

In this paper, by means of adequate variational techniques and the theory of the variable exponent Sobolev spaces, we show the existence of nontrivial solution for a transmission problem given by a system of two nonlinear elliptic equations of p x -Kirchhoff type with nonstandard growth condition

Multiple small solutions for p(x)p(x)-Schrödinger equations with local sublinear nonlinearities via genus theory

In this paper, we deal with the following $p(x)$-Schrödinger problem: \begin{equation*} \begin{cases} -\text{div}(|\nabla u|^{p(x)-2}\nabla u)+V(x)\left\vert u\right\vert ^{p(x)-2}u=f(x,u) & \hbox{in $\mathbb{R}^{N}$ ;} \\ u\in W^{1,p(x)}(\mathbb{R}^{N}), & \hbox{} \end{cases} \end{equation*} where the nonlinearity is sublinear. We present the existence of infinitely many solutions for the problem. The main tool used here is a variational method and Krasnoselskii's genus theory combined with the theory of variable exponent Sobolev spaces. We also establish a Bartsch–Wang type compact embedding theorem for the variable exponent spaces