Nodal Solutions for Problems with Mean Curvature Operator in Minkowski Space with Nonlinearity Jumping Only at the Origin

Wenguo Shen

doi:10.1155/2020/9801931

Abstract

In this paper, we establish a unilateral global bifurcation result for half-linear perturbation problems with mean curvature operator in Minkowski space. As applications of the abovementioned result, we shall prove the existence of nodal solutions for the following problem −div∇v/1−∇v2=αxv++βxv−+λaxfv, in BR0,vx=0, on ∂BR0, where λ ≠ 0 is a parameter, R is a positive constant, and BR0=x∈ℝN:x<R is the standard open ball in the Euclidean space ℝNN≥1 which is centered at the origin and has radius R. a(|x|) ∈ C[0, R] is positive, v+ = max{v, 0}, v− = −min{v, 0}, α(|x|), β(|x|) ∈ C[0, R]; f∈Cℝ,ℝ, s f (s) > 0 for s ≠ 0, and f0 ∈ [0, ∞], where f0 = lim|s|⟶0 f(s)/s. We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results.

Highlights

We first consider the following problem with mean curvature operator in Minkowski space:⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩−v(dxiv)⎛⎜⎜⎝ 􏽱0, 1 −∇ |v ∇ v | 2 ⎞⎟⎟⎠ λa(|x|)v + F(|x|, v, λ), in BR(0), on zBR(0), (1)where λ ≠ 0 is a parameter, R is a positive constant, and BR(0) 􏼈x ∈ RN: |x| < R􏼉 is the standard open ball in the Euclidean space RN(N ≥ 1), which is centered at the origin and has radius R
Where λ ≠ 0 is a parameter, R is a positive constant, and BR(0) 􏼈x ∈ RN: |x| < R􏼉 is the standard open ball in the Euclidean space RN(N ≥ 1), which is centered at the origin and has radius R
Using the same method to prove ([16], eorem 2) with obvious changes, we may get the following result about Lemma 2

Summary

Introduction

We first consider the following problem with mean curvature operator in Minkowski space:⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩−v(dxiv)⎛⎜⎜⎝ 􏽱0, 1 −∇ |v ∇ v | 2 ⎞⎟⎟⎠ λa(|x|)v + F(|x|, v, λ), in BR(0), on zBR(0), (1)where λ ≠ 0 is a parameter, R is a positive constant, and BR(0) 􏼈x ∈ RN: |x| < R􏼉 is the standard open ball in the Euclidean space RN(N ≥ 1), which is centered at the origin and has radius R. Using the same method to prove ([16, eorem 1]) with obvious changes, we may get the following global bifurcation result about Lemma 1. Using the same method to prove ([16], eorem 2) with obvious changes, we may get the following result about Lemma 2.

Results

Conclusion