On multiple solutions for nonlocal fractional problems via\n \∇-theorems

2015/10/29 by Giovanni Molica Bisci, Bisci, Giovanni Molica, Dimitri Mugnai +3 · 1 citation
Mathematics · #35S15 #45G05 #47G20 #5J20 #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1510.08701

openalex publication_date 2015/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The aim of this paper is to prove multiplicity of solutions for nonlocal\nfractional equations modeled by
left
\(-
Delta)^sν-
lambda u=f(x,u) · amp;
mbox in
Omega

u=0 · amp;
mbox in \n
mathbbRⁿ
setminus
Omega
,, \
right. where s\∈ (0,1) is\nfixed, (-\Δ)^s is the fractional Laplace operator, \λ is a real\nparameter, \Ω\⊂ \ℝⁿ, n>2s, is an open bounded set with\ncontinuous boundary and nonlinearity f satisfies natural superlinear and\nsubcritical growth assumptions. Precisely, along the paper we prove the\nexistence of at least three non-trivial solutions for this problem in a\nsuitable left neighborhood of any eigenvalue of (-\Δ)^s. At this purpose\nwe employ a variational theorem of mixed type (one of the so-called\n\∇-theorems).\n

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