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Existence and nonexistence of solutions for a singular elliptic problem with a nonlinear boundary condition

Zonghu XiuCaisheng Chen — 2013

Annales Polonici Mathematici

We consider the existence and nonexistence of solutions for the following singular quasi-linear elliptic problem with concave and convex nonlinearities: ⎧ - d i v ( | x | - a p | u | p - 2 u ) + h ( x ) | u | p - 2 u = g ( x ) | u | r - 2 u , x ∈ Ω, ⎨ ⎩ | x | - a p | u | p - 2 u / ν = λ f ( x ) | u | q - 2 u , x ∈ ∂Ω, where Ω is an exterior domain in N , that is, Ω = N D , where D is a bounded domain in N with smooth boundary ∂D(=∂Ω), and 0 ∈ Ω. Here λ > 0, 0 ≤ a < (N-p)/p, 1 < p< N, ∂/∂ν is the outward normal derivative on ∂Ω. By the variational method, we prove the existence of multiple solutions. By the test function method,...

Positive solution for a quasilinear equation with critical growth in N

Lin ChenCaisheng ChenZonghu Xiu — 2016

Annales Polonici Mathematici

We study the existence of positive solutions of the quasilinear problem ⎧ - Δ N u + V ( x ) | u | N - 2 u = f ( u , | u | N - 2 u ) , x N , ⎨ ⎩ u(x) > 0, x N , where Δ N u = d i v ( | u | N - 2 u ) is the N-Laplacian operator, V : N is a continuous potential, f : × N is a continuous function. The main result follows from an iterative method based on Mountain Pass techniques.

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