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Existence of weak solutions to a class of Schr\"{o}dinger type equations involving the fractional $p$-Laplacian in $\mathbb R^{N}$

김연호|이종락|김재명
대한수학회 (The Korean Mathematical Society)|2019.11|대한수학회지 (Journal of the KMS)|제56권 제6호|pp. 1529-1560|DOI 10.4134/JKMS.j180785

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키워드

critical point theoryvariational methodsfractional $p$-Laplacian
주제분류수학

초록

We are concerned with the following elliptic equations: \begin{equation*} (-\Delta)_p^su+V(x)|u|^{p-2}u=\lambda g(x,u) \quad \textmd{in} \quad \mathbb R^{N}, \end{equation*} where $(-\Delta)^s_p$ is the fractional $p$-Laplacian operator with $0<s<1<p<+\infty$, $sp<N$, the potential function $V:\mathbb R^{N}\to(0,\infty)$ is a continuous potential function, and $g:\mathbb R^{N}\times\mathbb R \to \mathbb R$ satisfies a Carath\'eodory condition. We show the existence of at least one weak solution for the problem above without the Ambrosetti and Rabinowitz condition. Moreover, we give a positive interval of the parameter $\lambda$ for which the problem admits at least one nontrivial weak solution when the nonlinearity $g$ has the subcritical growth condition.

Abstract

We are concerned with the following elliptic equations: \begin{equation*} (-\Delta)_p^su+V(x)|u|^{p-2}u=\lambda g(x,u) \quad \textmd{in} \quad \mathbb R^{N}, \end{equation*} where $(-\Delta)^s_p$ is the fractional $p$-Laplacian operator with $0<s<1<p<+\infty$, $sp<N$, the potential function $V:\mathbb R^{N}\to(0,\infty)$ is a continuous potential function, and $g:\mathbb R^{N}\times\mathbb R \to \mathbb R$ satisfies a Carath\'eodory condition. We show the existence of at least one weak solution for the problem above without the Ambrosetti and Rabinowitz condition. Moreover, we give a positive interval of the parameter $\lambda$ for which the problem admits at least one nontrivial weak solution when the nonlinearity $g$ has the subcritical growth condition.

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