Bifurcation of solutions for the problems involving jumping non-linearities with mean curvature operator on general domain(一般区域上含跳跃项平均曲率算子方程解的全局分歧)

Abstract

In this paper, we firstly establish an global bifurcation result for the problems involving jumping non-linearities with mean curvature operator in Minkowski space. As applications of the above results, we study the existence of solutions for the following problem-div(∇u/1-∇u2)=α(x)u++β(x)u-+λa(x)f(u), x∈Ω,u=0, x∈∂Ω,whereis a real parameter,Ω is a general C2 bounded domain in RN with a smooth boundary ∂Ω and N≥1,a∈C(Ω¯,(0,∞)),u+=maxu,0,u-=min-u,0, α,β∈C(Ω¯); f∈C(R,R), sf(s)>0 for s≠0,and f0∈[0,∞],where f0=lims→0f(s)/s.(建立了一般区域上含跳跃项平均曲率算子方程解的全局分歧定理，研究了问题-div(∇u/1-∇u2)=α(x)u++β(x)u-+λa(x)f(u), x∈Ω,u=0, x∈∂Ω解的存在性，其中λ≠0为实参数，Ω为在RN中有界且在其边界上光滑的C2区域，N≥1，a∈C(Ω¯,(0,∞))，u+=maxu,0，u-=min-u,0，α,β∈C(Ω¯)；f∈C(R,R)，对于s≠0，满足sf(s)>0；f0∈[0,∞]，其中f0=lims→0f(s)/s。)

Keywords

• mean curvature operator(平均曲率算子)
• the problems involving jumping non-linearities(含跳跃非线性项问题)
• global bifurcation of solutions(解的全局分歧)