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2018 On nonlocal fractional Laplacian problems with oscillating potentials
Vincenzo Ambrosio, Luigi D'Onofrio, Giovanni Molica Bisci
Rocky Mountain J. Math. 48(5): 1399-1436 (2018). DOI: 10.1216/RMJ-2018-48-5-1399

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.

Citation

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Vincenzo Ambrosio. Luigi D'Onofrio. Giovanni Molica Bisci. "On nonlocal fractional Laplacian problems with oscillating potentials." Rocky Mountain J. Math. 48 (5) 1399 - 1436, 2018. https://doi.org/10.1216/RMJ-2018-48-5-1399

Information

Published: 2018
First available in Project Euclid: 19 October 2018

zbMATH: 06958785
MathSciNet: MR3866552
Digital Object Identifier: 10.1216/RMJ-2018-48-5-1399

Subjects:
Primary: 35J20, 35J62, 35J92
Secondary: 35J15, 47J30

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

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Vol.48 • No. 5 • 2018
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