Existence of infinitely many radial nodal solutions for a Dirichlet problem involving mean curvature operator in Minkowski space

Man Xu,Ruyun Ma

doi:10.14232/ejqtde.2020.1.27

Abstract

In this paper, we show the existence of infinitely many radial nodal solutions for the following Dirichlet problem involving mean curvature operator in Minkowski space −div � ∇y 1−|∇y| 2 = λh(y) + g(|x|, y) in B, y = 0 on ∂B, where B = {x ∈ RN : |x| < 1} is the unit ball in RN, N ≥ 1, λ ≥ 0 is a parameter, h ∈ C(R) and g ∈ C(R+ × R). By bifurcation and topological methods, we prove the problem possesses infinitely many component of radial solutions branching off at λ = 0 from the trivial solution, each component being characterized by nodal properties.

Highlights

The purpose of this paper is to deal with radial nodal solutions for the following 0-Dirichlet problem with mean curvature operator in the Minkowski space ∇y 1 − |∇y|2 (1.1)y = 0 on ∂B, where B = {x ∈ RN : |x| < 1} is the unit ball in RN, N ≥ 1, λ ≥ 0 is a parameter, h(y) |y|q−2y, 1 < q < 2 near y = 0 and g is of higher order with respect to h at y = 0
By bifurcation and topological methods, we prove the problem possesses infinitely many component of radial solutions branching off at λ = 0 from the trivial solution, each component being characterized by nodal properties
Y = 0 on ∂B, where B = {x ∈ RN : |x| < 1} is the unit ball in RN, N ≥ 1, λ ≥ 0 is a parameter, h(y) |y|q−2y, 1 < q < 2 near y = 0 and g is of higher order with respect to h at y = 0

Summary

Introduction

Few results on the existence of radial nodal solutions [15], even positive solutions, have been established for problem with mean curvature operator on general domain. We will show an existence result of infinitely many radial nodal solutions for Dirichlet problem (1.1) by bifurcation and topological methods. Let (λ, u) be a solution of (1.3), it follows from |u (r)| < 1 that u ∞ < 1 This leads to the bifurcation diagrams mainly depend on the behavior of h = h(s) and g = g(r, s) near s = 0. Theorem 1.1 improves some well-known existence results of positive solutions [5] and radial nodal solutions [15] for related problems.

Superior limit and component

Topological degree jumping result

Findings

Proof of the main result