Existence of solutions of degenerate semilinear elliptic boundary value problems (original) (raw)

Existence of solutions for quasilinear degenerate elliptic equations

2001

In this paper, we study the existence of solutions for quasilinear degenerate elliptic equations of the form A(u) + g(x, u, ∇u) = h, where A is a Leray-Lions operator from W 1,p 0 (Ω, w) to its dual. On the nonlinear term g(x, s, ξ), we assume growth conditions on ξ, not on s, and a sign condition on s. * Mathematics Subject Classifications: 35J15, 35J20, 35J70.

Boundary singularities for weak solutions of semilinear elliptic problems

Journal of Functional Analysis, 2007

Let Ω be a bounded domain in R N , N ≥ 2, with smooth boundary ∂Ω. We construct positive weak solutions of the problem ∆u + u p = 0 in Ω, which vanish in suitable trace sense on ∂Ω, but which are singular at prescribed single points if p is equal or slightly above N+1 N−1 . Similar constructions are carried out for solutions which are singular on any given embedded submanifold of ∂Ω of dimension 0 ≤ k ≤ N − 2, if p equals or it is slightly above N−k+1 N−k−1 , and even on countable families of these objects, dense on a given closed set. The role of this exponent, first discovered by Brezis and Turner [1] for boundary regularity when p < N+1 N−1 , parallels that of p = N N−2 for interior singularities.

Existence of Weak Non-Negative Solutions for a Class of Nonuniformly Boundary Value Problem

Bulletin of the Korean Mathematical Society, 2012

The goal of this paper is to study the existence of non-trivial non-negative weak solution for the nonlinear elliptic equation: −div(h(x)∇u) = f (x, u) in Ω with Dirichlet boundary condition in a bounded domain Ω ⊂ R N , N ≥ 3, where h(x) ∈ L 1 loc (Ω), f (x, s) has asymptotically linear behavior. The solutions will be obtained in a subspace of the space H 1 0 (Ω) and the proofs rely essentially on a variation of the mountain pass theorem in [12].