一类带权函数的拟线性椭圆方程

该文利用变分方法讨论了方程 -△_p u=λa(x)(u⁺)^p-1-μa(x)(u^-)^p-1+f(x, u), u∈W₀^1,p(\Omega)在(λ, μ)\not\in ∑_p和(λ, μ) ∈ ∑_p 两种情况下的可解性, 其中\Omega是 R^N(N≥3)中的有界光滑区域, ∑_p为方程 -△_p u=α a(x)(u⁺)^p-1-βa(x)(u^-)^p-1, u∈ W₀^1,p(\Omega)的Fucik谱, 权重函数a(x)∈ L^r(\Omega) (r≥ N/p)$且a(x)>0 a.e.于\Omega, f满足一定的条件.

A Class of Quasilinear Elliptic Equations with Weights

In this paper, by using the variational methods, the authors consider the solvability of the equation -△_p u=λa(x)(u⁺)^p-1-μa(x)(u^-)^p-1+f(x, u), u∈W₀^1,p(\Omega) in the cases of (λ, μ)\not\in ∑_p and (λ, μ) ∈ ∑_p, where \Omega is a bounded domain in R^N(N≥3) with smooth boundary $\partial\Omega$, ∑_p is the Fucik spectrum of the equation -△_p u=α a(x)(u⁺)^p-1-βa(x)(u^-)^p-1, u∈ W₀^1,p(\Omega), the weight function a(x) ∈ L^r(\Omega)(r≥ N/p) with a(x)> 0 a.e. in \Omega, f satisfies some conditions to be specified.