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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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 19 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1539936029

Digital Object Identifier
doi:10.1216/RMJ-2018-48-5-1399

Mathematical Reviews number (MathSciNet)
MR3866552

Zentralblatt MATH identifier
06958785

Subjects
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]

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

Citation

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. https://projecteuclid.org/euclid.rmjm/1539936029


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