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An Ambrosetti–Prodi type result for fractional spectral problems
Author(s) -
Ambrosio Vincenzo
Publication year - 2020
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800416
Subject(s) - mathematics , bounded function , eigenfunction , lipschitz continuity , type (biology) , domain (mathematical analysis) , nonlinear system , lipschitz domain , fractional laplacian , mathematical analysis , harmonic function , laplace operator , pure mathematics , boundary (topology) , fractional calculus , eigenvalues and eigenvectors , ecology , biology , physics , quantum mechanics
We consider the following class of fractional parametric problems( − Δ Dir ) s u = f ( x , u ) + t φ 1 + hin Ω ,u = 0on ∂ Ω ,where Ω ⊂ R Nis a smooth bounded domain, s ∈ ( 0 , 1 ) , N > 2 s ,( − Δ Dir ) s is the fractional Dirichlet Laplacian, f : Ω ¯ × R → R is a locally Lipschitz nonlinearity having linear or superlinear growth and satisfying Ambrosetti–Prodi type assumptions, t ∈ R , φ 1 is the first eigenfunction of the Laplacian with homogenous boundary conditions, and h : Ω → R is a bounded function. Using variational methods, we prove that there exists at 0 ∈ R such that the above problem admits at least two distinct solutions for any t ≤ t 0 . We also discuss the existence of solutions for a fractional periodic Ambrosetti–Prodi type problem.