In this paper we study the regularity of the solutions to the problem ∆pu = |u| p−2 u in the bounded smooth domain Ω ⊂ R N , with |∇u| p−2 ∂u ∂ν = λV (x)|u| p−2 u+h as a nonlinear boundary condition, where ∂Ω is C 2,β with β ∈]0, 1[, and V is a weight in L s (∂Ω) and h ∈ L s (∂Ω) for some s ≥ 1. We prove that all solutions are in L ∞ (∂Ω) L ∞ (Ω), and using the D.Debenedetto's theorem of regularity in [1] we conclude that those solutions are in C 1,α Ω for some α ∈]0, 1[.
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.
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