Quasilinear Schrödinger equations with singular and vanishing potentials involving exponential critical growth in R2. (July 2020)

• Record Type: Journal Article
• Authors: Severo, Uberlandio B.; de Carvalho, Gilson M.
• Is Part Of: Nonlinear analysis. Volume 196(2020)
• Publication Date: 2020-07
• Subjects: 35J20 -- 35J25 -- 35J60 -- 35Q60; Schrödinger equations -- Singular potentials -- Trudinger–Moser inequality -- Ground state solution; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2020.111800
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study the existence and nonexistence of solution for the following class of quasilinear Schrödinger equations: − Δ u + V ( | x | ) u − [ Δ ( u 2 ) ] u = Q ( | x | ) h ( u ), x ∈ R 2, u ( x ) → 0 as | x | → ∞, where V and Q are potentials that can be singular at the origin, unbounded or vanishing at infinity and the nonlinearity h ( s ) is allowed to satisfy the exponential critical growth with respect to the Trudinger–Moser inequality. By combining variational methods in a suitable weighted Orlicz space with a version of the Trudinger–Moser inequality for this space, we obtain the existence of a nonnegative ground state solution. For this, we have used some regularity results and we need to explore a symmetric criticality type argument. Moreover, under some conditions on V, Q and h ( s ), we prove that this problem does not have positive radial solution. Schrödinger equations of this type have been studied as models of several physical phenomena.