An Ambrosetti–Prodi type result for fractional spectral problems

Vincenzo Ambrosio

Mathematische Nachrichten, 2020, vol. 293, issue 3, 412-429

Abstract: We consider the following class of fractional parametric problems (−ΔDir)su=f(x,u)+tφ1+hinΩ,u=0on∂Ω,where Ω⊂RN is a smooth bounded domain, s∈(0,1), N>2s, (−ΔDir)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, φ1 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 a t0∈R such that the above problem admits at least two distinct solutions for any t≤t0. We also discuss the existence of solutions for a fractional periodic Ambrosetti–Prodi type problem.

Date: 2020
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