In this paper we consider the following critical nonlocal problem, where s∈(0, 1), Ω is an open bounded subset of Rn, n>2s, with continuous boundary, λ is a positive real parameter, 2*:=2n/(n-2s) is the fractional critical Sobolev exponent, while LK is the nonlocal integrodifferential operator, whose model is given by the fractional Laplacian -(-δ)s.Along the paper, we prove a multiplicity and bifurcation result for this problem, using a classical theorem in critical points theory. Precisely, we show that in a suitable left neighborhood of any eigenvalue of -LK (with Dirichlet boundary data) the number of nontrivial solutions for the problem under consideration is at least twice the multiplicity of the eigenvalue. Hence, we extend the result got by Cerami, Fortunato and Struwe in [14] for classical elliptic equations, to the case of nonlocal fractional operators.

Fiscella, A., Molica Bisci, G., Servadei, R. (2016). Bifurcation and multiplicity results for critical nonlocal fractional Laplacian problems. BULLETIN DES SCIENCES MATHEMATIQUES, 140(1), 14-35 [http://dx.doi.org/10.1016/j.bulsci.2015.10.001].

Bifurcation and multiplicity results for critical nonlocal fractional Laplacian problems

Fiscella A;
2016

Abstract

In this paper we consider the following critical nonlocal problem, where s∈(0, 1), Ω is an open bounded subset of Rn, n>2s, with continuous boundary, λ is a positive real parameter, 2*:=2n/(n-2s) is the fractional critical Sobolev exponent, while LK is the nonlocal integrodifferential operator, whose model is given by the fractional Laplacian -(-δ)s.Along the paper, we prove a multiplicity and bifurcation result for this problem, using a classical theorem in critical points theory. Precisely, we show that in a suitable left neighborhood of any eigenvalue of -LK (with Dirichlet boundary data) the number of nontrivial solutions for the problem under consideration is at least twice the multiplicity of the eigenvalue. Hence, we extend the result got by Cerami, Fortunato and Struwe in [14] for classical elliptic equations, to the case of nonlocal fractional operators.
Articolo in rivista - Articolo scientifico
fractional Laplacian, critical nonlinearities, best fractional critical Sobolev constant, variational techniques, integrodifferential operators.
English
14
35
22
Fiscella, A., Molica Bisci, G., Servadei, R. (2016). Bifurcation and multiplicity results for critical nonlocal fractional Laplacian problems. BULLETIN DES SCIENCES MATHEMATIQUES, 140(1), 14-35 [http://dx.doi.org/10.1016/j.bulsci.2015.10.001].
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/338138
Citazioni
  • Scopus 38
  • ???jsp.display-item.citation.isi??? 38
Social impact