Nonlinear equations involving the square root of the Laplacian

Vincenzo Ambrosio; Giovanni Molica Bisci; Dušan Repovš

doi:10.3934/dcdss.2019011

Article Contents

2019, Volume 12, Issue 2: 151-170. Doi: 10.3934/dcdss.2019011

This issue Previous Article Perturbation effects for the minimal surface equation with multiple variable exponents Next Article Singular solutions of a nonlinear equation in a punctured domain of

$\mathbb{R}^{2}$

Nonlinear equations involving the square root of the Laplacian

1.
Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino 'Carlo Bo' Piazza della Repubblica, 13, 61029 Urbino, Pesaro e Urbino, Italy
2.
Dipartimento PAU, Università degli Studi 'Mediterranea' di Reggio Calabria, Salita Melissari - Feo di Vito, 89100 Reggio Calabria, Italy
3.
Faculty of Education, and Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia

* Corresponding author: Giovanni Molica Bisci
* Corresponding author: Giovanni Molica Bisci

Received: April 30, 2017

Revised: November 30, 2017

Published: April 2019

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Abstract

In this paper we discuss the existence and non-existence of weak solutions to parametric fractional equations involving the square root of the Laplacian $A_{1/2}$ in a smooth bounded domain $Ω\subset \mathbb{R}^{n}$ ($n≥2$) and with zero Dirichlet boundary conditions. Namely, our simple model is the following equation

$\left\{ \begin{align} &{{A}_{1/2}}u = \lambda f(u) \\ &u = 0 \\ \end{align} \right.\begin{array}{*{35}{l}} {}&\text{in}\ \Omega \\ {}&\text{on }\partial \Omega . \\\end{array}$

The existence of at least two non-trivial $L^{∞}$-bounded weak solutions is established for large value of the parameter $λ$, requiring that the nonlinear term $f$ is continuous, superlinear at zero and sublinear at infinity. Our approach is based on variational arguments and a suitable variant of the Caffarelli-Silvestre extension method.

Keywords:

Mathematics Subject Classification: Primary: 49J35, 35A15, 35S15; Secondary: 47G20, 45G05.

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