Abstract
We study the existence and non-existence of positive solutions for $(p,q)$-Laplacian Steklov problem with two parameters. The main result of our research is the construction of a continuous curve in plane, which becomes a threshold between the existence and non-existence of positive solutions.
Highlights
Under the zero Dirichlet boundary condition in Ω, the authors obtained in [15] reasonably complete description of the subsets of (α, β) plane which correspond to the existence/nonexistence of positive solutions to the following problem:
Our goal in this paper is to provide a complete description of 2-dimensional sets in the (α, β) plane, which correspond to the existence and non-existence of positive solutions for (Pα,β) by generalizing and complementing the research [21,22], and seems more natural, due to the structure of the equation
MinW 1,p(Ω) E[α0,βu] < 0 holds, and (Pα,β) has a positive solution belonging to intC1(Ω)+
Summary
We give some preliminary results and definitions which well be used . A function u ∈ W 1,p(Ω) is called a super-solution of (Pα,β) if the following holds. Let us show the following elementary result which implies the existence of global minimizer (cf [13], Theorem 1.1 ). It well known that the properties of E[αu,,βu] started in lemma 2.6, imply the existence of a global minimizer u of E[αu,,βu] (cf [13], Theorem 1.1 ), which becomes a solution of (Pα,β). Assume that β > λ1(q) and u ∈ intC1(Ω)+ is a positive super-solution of (Pα,β). MinW 1,p(Ω) E[α0,,βu] < 0 holds, and (Pα,β) has a positive solution belonging to intC1(Ω)+. Let β > λ1(q) and u ∈ intC1(Ω)+ be a positive super-solution of (Pα,β). There exists ρ > 0 such that for any differentiable functions u > 0 and φ ≥ 0 in Ω it holds (|∇u|p−2 + |∇u|q−2)|∇u|∇u φp up−1 + uq−1
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