Ground states of degenerate quasilinear Schrödinger equation with vanishing potentials. (December 2019)

• Record Type: Journal Article
• Authors: Miyagaki, Olimpio H.; Santana, Cláudia R.; Vieira, Rônei S.
• Is Part Of: Nonlinear analysis. Volume 189(2019)
• Publication Date: 2019-12
• Subjects: 35J10 -- 35J70 -- 35J92; Degenerate elliptic equations -- p-Laplacian type equation -- Schrödinger operator; 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.2019.111587
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper we study the existence of nontrivial ground state solutions for the following class of p -Laplacian type equation − div a ( x, ∇ u ) + V ( x ) | x | − α p ∗ | u | p − 2 u = K ( x ) | x | − α p ∗ f ( u ) in R N, where 1 < p < N, N ≥ 3, − ∞ < α < N − p p, α ≤ e ≤ α + 1, d = 1 + α − e, p ∗ ≔ p ∗ ( α, e ) = N p N − d p (critical Hardy–Sobolev exponent); f has a quasicritical growth; V and K are nonnegative potentials; the function a satisfies | a ( x, ∇ u ) | ≤ c 0 | x | − α p h 0 ( x ) | ∇ u | p − 1 + c 0 ( 1 + | x | − α p ) h 1 ( x ) | ∇ u | p − 1 for any ξ ∈ R N, a.e. x ∈ R N, for any two positive functions h 1 ∈ L l o c ∞ ( R N ), h 0 ∈ L α ¯ p p − 1 ( R N ), with α ¯ = α p p ∗ .