The fractional p(.,.)-Neumann boundary conditions for the nonlocal p(.,.)-Laplacian operator

Abstract

In this paper, we deal with the fractional $p(.,.)$-Laplacian problem with nonlocal Neumann boundary conditions $\{\begin{array}{ll}(-\mathrm{\Delta }{)}_{p(.,.)}^{s}u+|u{|}^{\overline{p}(x)-2}u=\lambda V_1(x)|u{|}^{q(x)-2}u,& \text{in}\mathrm{\Omega },\\ {\mathcal{N}}_{s,p(.,.)}u=\mu V_2(x)|u{|}^{r(x)-2}u,& \text{on}{\mathbb{R}}^N\setminus \mathrm{\Omega },\end{array}$here $\lambda ,\mu >0$ are parameters and $V_1,V_2$ are two nonnegative weighted functions. The domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$ $(N\ge 1)$ is smooth and bounded, $\overline{p}(x)=p(x,x),q,r$ are continuous bounded functions. Applying the Ekeland principle and the variational method, under appropriate assumptions, we show that the above problem has nontrivial weak solutions.

Keywords:

• Fractional $p(..)$-Laplacian
• $p(..)$-Neumann boundary condition
• Ekeland variational principle

2010 AMS Mathematics subject Classifications:

• 35D05
• 35D30
• 35J58
• 35J60
• 35J65

Disclosure statement

No potential conflict of interest was reported by the author(s).