Rocky Mountain Journal of Mathematics

On nonlocal fractional Laplacian problems with oscillating potentials

Vincenzo Ambrosio, Luigi D'Onofrio, and Giovanni Molica Bisci

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In this paper, we deal with the following fractional nonlocal $p$-Laplacian problem: \begin{equation} \begin{cases}(-\Delta )^{s}_{p}u= \lambda \beta (x) u^q + f(u) &\mbox {in } \Omega ,\\ u\geq 0,\ u\not \equiv 0 &\mbox {in } \Omega ,\\ u=0 &\mbox {in } \mathbb{R} ^{N}\setminus \Omega , \end{cases} \end{equation} where $\Omega \subset \mathbb{R} ^{N}$ is a bounded domain with a smooth boun\-dary of $\mathbb{R} ^N$, $s\in (0,1)$, $p\in (1, \infty )$, $N> s p$, $\lambda $ is a real parameter, $\beta \in L^\infty (\Omega )$ is allowed to be indefinite in sign, $q>0$ and $f:[0,+\infty )\to \mathbb{R} $ is a continuous function oscillating near the origin or at infinity. By using variational and topological methods, we obtain the existence of infinitely many solutions for the problem under consideration. The main results obtained here represent some new interesting phenomena in the nonlocal setting.

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Rocky Mountain J. Math., Volume 48, Number 5 (2018), 1399-1436.

First available in Project Euclid: 19 October 2018

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Zentralblatt MATH identifier

Primary: 35J20: Variational methods for second-order elliptic equations 35J62: Quasilinear elliptic equations 35J92: Quasilinear elliptic equations with p-Laplacian
Secondary: 35J15: Second-order elliptic equations 47J30: Variational methods [See also 58Exx]

$p$-fractional Laplacian operator infinitely many solutions variational methods


Ambrosio, Vincenzo; D'Onofrio, Luigi; Bisci, Giovanni Molica. On nonlocal fractional Laplacian problems with oscillating potentials. Rocky Mountain J. Math. 48 (2018), no. 5, 1399--1436. doi:10.1216/RMJ-2018-48-5-1399.

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