Ground state solutions for quasilinear scalar field equations arising in nonlinear optics

Abstract

In this paper, we study a class of quasilinear elliptic equations which appears in nonlinear optics. By using the mountain pass theorem together with a technique of adding one dimension of space (Hirata et al. in Topol Methods Nonlinear Anal 35:253–276, 2010; Jeanjean in Nonlinear Anal Theory Methods Appl 28:1633–1659, 1997), we prove the existence of a non-trivial weak solution for general nonlinear terms of Berestycki–Lions’ type. The existence of a radial ground state solution and a ground state solution is also established under stronger assumptions on the quasilinear term.