Ground state solutions for a fractional Schrödinger equation with critical growth

Article type: Research Article

Authors: Ambrosio, Vincenzo^{a; *} | Figueiredo, Giovany M.^b

Affiliations: [a] Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino ‘Carlo Bo’, Piazza della Repubblica, 13, 61029 Urbino, Pesaro e Urbino, Italy. E-mail: vincenzo.ambrosio@uniurb.it | [b] Departamento de Matemática, Universidade de Brasília, 70.910-900, Brasília – DF, Brazil. E-mail: giovany@unb.br

Correspondence: [*] Corresponding author. E-mail: vincenzo.ambrosio@uniurb.it.

Abstract: In this paper we investigate the existence of nontrivial ground state solutions for the following fractional scalar field equation (−Δ)su+V(x)u=f(u)in RN, where s∈(0,1), N>2s, (−Δ)s is the fractional Laplacian, V:RN→R is a bounded potential satisfying suitable assumptions, and f∈C1,β(R,R) has critical growth. We first analyze the case V constant, and then we develop a Jeanjean–Tanaka argument [Indiana Univ. Math. J. 54 (2005), 443–464] to deal with the non autonomous case. As far as we know, all results presented here are new.

Keywords: Fractional Laplacian, monotonicity trick, critical exponent, compactness lemma

DOI: 10.3233/ASY-171438

Journal: Asymptotic Analysis, vol. 105, no. 3-4, pp. 159-191, 2017