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.
Highlights
Where △ru = div (|∇u|r−2∇u) indicate the r-Laplacian, has attracted considerable attention
Our purpose in this article is to extend the results of the classical eigenvalue problem involving the (p, q)-Laplacian and generalize some results knouwn in the classical p-Laplacian Steklov problems
Letting μ → 0+, our problem (Pλ,μ) turns into the (p−1)-homogeneous problem known as the usual weighted eigenvalue Steklov problem for the p-Laplacian with indefinite weight mp: (Pλ,mp )
Summary
We show non-existence results for positive solutions of the eigenvalue problem (Pλ,μ) formulated as Theorem 2.5. We study the non-resonant case (Theorem 3.1) which prove that when μ > 0 there exists an interval of positive eigenvalues for the problem (Pλ,μ). This section concerns the Rayleigh quotient and non-existence results for our eigenvalue Steklov problem (Pλ,μ).
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