On a positive solution for $(p,q)$-Laplace equation with Nonlinear

Abstract

In the presentp aper, we study the existence and non-existence results of a positive solution for the Steklov eigenvalue problem driven by nonhomogeneous operator $(p,q)$-Laplacian with indefinite weights. We also prove that in the case where $\mu>0$ and with $1<q<p<\infty$ the results are completely different from those for the usua lSteklov eigenvalue problem involving the $p$-Laplacian with indefinite weight, which is retrieved when $\mu=0$. Precisely, we show that when $\mu>0$ there exists an interval of principal eigenvalues for our Steklov eigenvalue problem.

On a positive solutions for $(p,q)$-Laplacian Steklov problem with two parameters

“…First, we state the result of non-existence which generalize Theorem 2.1 from [21] for the problem (P α,β , m p , m q ) with non-negative weights.…”

“…In this section, we give some preliminary results and definitions which well be used in the following sections. First, we give three results from [21,22], where they were proved using the variational methods.…”

“…Theorem 2.2. ( [21], Theorem 3.1) One supposes that m p ∈ M p , m q ∈ M q and λ 1 (p, m p ) = λ 1 (q, m q ). If min{λ 1 (p, m p ), λ 1 (q, m q )} < λ < max{λ 1 (p, m p ), λ 1 (q, m q )}, then the problem (P λ,λ,mp,mq ) has at least one positive solution.…”

“…where m r ∈ L ∞ (Ω), m r ≡ 0 and m r ≥ 0 a.e in Ω for r = p, q. In [21,22] the authors of the present article studied the existence and non-existence results of a positive solution for the Steklov eigenvalue problem P (λ,λ) ( α = β = λ).…”

“…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 (P α,β ) by generalizing and complementing the research [21,22], and seems more natural, due to the structure of the equation. We restrict ourselves to the case where m p and m q are constants, to save transparency and simplicity of presentation.…”