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

Yun-Ho Kim

DOI: http://dx.doi.org/10.12775/TMNA.2017.064

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.

Keywords


Fractional p-Laplacian; weak solution; critical point; degree theory

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