A multiplicity result via Ljusternick-Schnirelmann category and Morse theory for a fractional Schrödinger equation in $${\mathbb R^{N}}$$RN

By Giovany M. Figueiredo, Gaetano Siciliano. 2015

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In this work we study the following class of problems in $${\mathbb R^{N}, N > 2s}$$RN,N>2s$$\varepsilon^{2s}(-\Delta)^{s}u + V(z)u = f(u), \,\,\,u(z) > 0$$ε2s(-Δ)su+V(z)u=f(u),u(z)>0where $${0 < s < 1}$$0<s<1, $${(-\Delta)^{s}}$$(-Δ)s is the fractional Laplacian, $${\varepsilon}$$ε is a positive parameter, the potential $${V : \mathbb{R}^N \to \mathbb{R}}$$V:RN→R and the nonlinearity $${f : \mathbb R \to \mathbb R}$$f:R→R satisfy suitable assumptions; in particular it is assumed that $${V}$$V achieves its positive minimum on some set $${M.}$$M. By using variational methods we prove existence and multiplicity of positive solutions when $${\varepsilon \to 0^{+}}$$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 $${M}$$M.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7bfe3e30438dce777fc4f78ddcf7a7c399785271