Infinitely many solutions for a class of perturbed degenerate elliptic equations involving the Grushin operator. Issue 12 (1st December 2020)

• Record Type: Journal Article
• Authors: Luyen, Duong Trong; Tri, Nguyen Minh
• Is Part Of: Complex variables and elliptic equations. Volume 65:Issue 12(2020)
• Page Start: 2135
• Page End: 2150
• Publication Date: 2020-12-01
• Subjects: Semilinear degenerate elliptic equations -- boundary value problems -- critical points -- perturbation methods -- multiple solutions -- Grushin operator; Primary: 35J60 -- Secondary: 35B33 -- 35J25 -- 35J70; Functions of complex variables -- Periodicals; Differential equations, Elliptic -- Periodicals; 515.905
• DOI: 10.1080/17476933.2020.1730824
• Languages: English
• ISSNs: 1747-6933

Abstract

Abstract : In this paper, we study the multiplicity of weak solutions to the boundary value problem − G α u = f ( x, y, u ) + g ( x, y, u ) in Ω, u = 0 on ∂ Ω, where Ω is a bounded domain with smooth boundary in R N ( N ≥ 2 ), α ∈ N, f ( x, y, ξ ) is odd in ξ and g ( x, y, ξ ) is a perturbation term. Under some growth conditions on f and g, we show that there are infinitely many weak solutions to the problem. Here we do not require that f satisfies the Ambrosetti-Rabinowitz (AR) condition. The conditions on f and g are relatively weak and our result is new even in the case α = 0, i.e. for the classical Laplace equation with the Dirichlet boundary condition.