一般有界区域上高维变权p-Laplacian问题保号解的存在性

Existence of One-sign Solutions to the High-Dimensional Sign-Changing Weight p-Laplacian on General Domain

研究一般有界区域上高维变权p-Laplacian方程-div(φ_(p)▽(u))=γm(x)f(u),u(x)=0,x∈∂Ω保号解的存在性.其中Ω是R^(N)上的一个有界且在其边界上光滑的区域,m(x)∈C(Ω),γ是一个参数,f∈C(ℝ,ℝ),对于s≠0满足sf(s)>0.当满足f_(0)■(0,∞)或f_(∞)(0,∞)(其中f_(0)=lim|s|→0 f(s)/φ_(p)(s),f_(∞)=lim|s|→∞f(s)/φ_(p)(s)),且γ≠0属于一定区间时,本文研究上述高维p-Laplacian方程保号解的存在性.我们用全局分歧技巧和连通序列集取极限的方法获得主要结果.

In this paper,we shall study the existence of one-sign solutions for thep-Laplacian problem:-div(φ_(p)▽(u))=γm(x)f(u),u(x)=0,x∈∂Ω,whereΩis a bounded domain in ℝ^(N) with a smooth boundary∂Ω,and m(x)∈C(Ω)is a sign changing function,γis a parameter,f∈C(ℝ,ℝ),sf(s)>0 for s≠0.Based on the bifurcation result of Dai et al.[9,Theorem 5.1],we give the intervals for the parameterγ≠0 which ensure the existence of one-sign solutions for the above high-dimensional p-Laplacian problems if f_(0)■(0,∞)or f_(∞)(0,∞),where f_(0)=lim|s|→0 f(s)/φ_(p)(s),f_(∞)=lim|s|→∞f(s)/φ_(p)(s).We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results.