Multiplicity results for non-local operators of elliptic type
Year: 2025
Type: article
Abstract: In this paper, we study a class of problems proposed by Servadei and Valdinoci (2013); namely, (1) − L K u ( x ) − λ u ( x ) = f ( x , u ) , for x ∈ Ω ; u = 0 in R N ∖ Ω , where Ω ⊂ R N is an open bounded set with Lipschitz boundary, λ ∈ R , f ∈ C 1 ( Ω ¯ × R , R ) , with f ( x , 0 ) = 0 for x ∈ Ω , and L K is a non-local integrodifferential operator with homogeneous Dirichlet boundary condition. By computing the critical groups of the associated energy functional for problem (1) at the origin and at infinity, respectively, we prove that problem (1) has three nontrivial solutions for the case λ < λ 1 and two nontrivial solutions for the case λ ⩾ λ 1 , where λ 1 is the first eigenvalue of the operator − L K . Finally, assuming that the nonlinearity f is odd in the second variable, we prove the existence of an unbounded sequence of weak solutions of problem (1) for the case λ ⩾ λ 1 . We use variational methods and infinite-dimensional Morse theory to obtain the results.
Source: Results in Applied Mathematics
Authors: Emer Lopera, Leandro Recôva, Adolfo Rumbos
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $1,770
OpenAlex ID: W4413289713