Multiplicity of nontrivial solutions for a class of fractional elliptic equations

Tarek Kenzizi

doi:10.1216/jie.2021.33.315

Abstract

We are concerned with the following fractional Laplacian equation

$\matrix{ {{{\left( { - {\rm{\Delta }}} \right)}^s}u = a\left( x \right)\left| {u\left( x \right){|^{q - 2}}u\left( x \right) + \lambda b\left( x \right)} \right|u\left( x \right){|^{p - 2}}u\left( x \right)\quad {\rm{\;in\;}}\left( { - 1,1} \right),} \cr {\quad u > 0\quad {\rm{\;in\;}}\left( { - 1,1} \right),\quad u = 0\quad {\rm{\;in\;}} \mathbb{R} \setminus \left( { - 1,1} \right),} \cr }$

where $s \in \left( {0,1} \right)$ , $\lambda > 0$ . Using variational methods, we show existence and multiplicity of positive solutions.

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Tarek Kenzizi. "Multiplicity of nontrivial solutions for a class of fractional elliptic equations." J. Integral Equations Applications 33 (3) 315 - 325, Fall 2021. https://doi.org/10.1216/jie.2021.33.315

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Received: 24 March 2020; Revised: 12 November 2020; Accepted: 2 January 2021; Published: Fall 2021

First available in Project Euclid: 17 February 2022

MathSciNet: MR4383255

zbMATH: 1502.34033

Digital Object Identifier: 10.1216/jie.2021.33.315

Subjects:

Primary: 34B15

Secondary: 35R20 , 37C25

Keywords: fractional Laplacian operator , multiple solutions , Nehari manifold

Abstract

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