Existence of a weak solution for the fractional $p$-Laplacian equations with discontinuous nonlinearities via the Berkovits-Tienari degree theory

Abstract

We are concerned with the following nonlinear elliptic equations of the fractional $p$-Laplace type: \begin{equation*} \begin{cases} (-\Delta)_p^su \in \lambda[\underline{f}(x,u(x)), \overline{f}(x,u(x))] &\textmd{in } \Omega, u= 0 &\text{on } \mathbb{R}^N\setminus\Omega, \end{cases} \end{equation*} where $(-\Delta)_p^s$ is the fractional $p$-Laplacian operator, $\lambda$ is a parameter, $0< s< 1< p< +\infty$, $sp< N$, and the measurable functions $\underline{f}$, $ \overline{f}$ are induced by a possibly discontinuous at the second variable function $f\colon \Omega\times\mathbb R \to \mathbb R$. By using the Berkovits-Tienari degree theory for upper semicontinuous set-valued operators of type (S$_+)$ in reflexive Banach spaces, we show that our problem with the discontinuous nonlinearity $f$ possesses at least one nontrivial weak solution. In addition, we show the existence of two nontrivial weak solutions in which one has negative energy and another has positive energy.