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On discontinuous elliptic problems involving the fractional $p$-Laplacian in $\mathbb R^{N}$

김연호|박기섭|김인현
대한수학회 (The Korean Mathematical Society)|2018.11|대한수학회보 (Bulletin of the KMS)|제55권 제6호|pp. 1869-1889

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참고문헌

키워드

critical pointweak solutiondegree theoryfractional $p$-Laplacian
주제분류수학

초록

We are concerned with the following fractional $p$-Laplacian inclusion: \begin{equation*} (-\Delta)_p^su+V(x)|u|^{p-2}u\in \lambda[\underline{f}(x,u(x)),\, \overline{f}(x,u(x))] \quad \textmd{in} \ \ \mathbb R^{N}, \end{equation*} where $(-\Delta)_p^s$ is the fractional $p$-Laplacian operator, $0<s<1<p<+\infty$, $sp<N$, and $f:\mathbb R^{N}\times\mathbb R \to \mathbb R$ is measurable with respect to each variable separately. We show that our problem with the discontinuous nonlinearity $f$ admits at least one or two nontrivial weak solutions. In order to do this, the main tool is the Berkovits-Tienari degree theory for weakly upper semicontinuous set-valued operators. In addition, our main assertions continue to hold when $(-\Delta)_p^su$ is replaced by any non-local integro-differential operator.

Abstract

We are concerned with the following fractional $p$-Laplacian inclusion: \begin{equation*} (-\Delta)_p^su+V(x)|u|^{p-2}u\in \lambda[\underline{f}(x,u(x)),\, \overline{f}(x,u(x))] \quad \textmd{in} \ \ \mathbb R^{N}, \end{equation*} where $(-\Delta)_p^s$ is the fractional $p$-Laplacian operator, $0<s<1<p<+\infty$, $sp<N$, and $f:\mathbb R^{N}\times\mathbb R \to \mathbb R$ is measurable with respect to each variable separately. We show that our problem with the discontinuous nonlinearity $f$ admits at least one or two nontrivial weak solutions. In order to do this, the main tool is the Berkovits-Tienari degree theory for weakly upper semicontinuous set-valued operators. In addition, our main assertions continue to hold when $(-\Delta)_p^su$ is replaced by any non-local integro-differential operator.

참고문헌 (41)

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