A multiplicity result via Ljusternick-Schnirelmann category and morse theory for a fractional schrödinger equation in Rⁿ

Original · EN

In this work we study the following class of problems in Rⁿ, N>2s ε²ˢ (-Δ)ˢu + V(z)u=f(u), u(z) > 0 where 0<s<1, (-Δ)ˢ is the fractional Laplacian, ε is a positive parameter, the potential V:Rⁿ %is a continuous functions and the nonlinearity f:R → R satisfy suitable assumptions; in particular it is assumed that V achieves its positive minimum on some set M. By using variational methods we prove existence, multiplicity and concentration of maxima of positive solutions when ε→ 0⁺. In particular the multiplicity result is obtained by means of the Ljusternick-Schnirelmann and Morse theory, by exploiting the "topological complexity" of the set M.