Regularity of the solutions to a nonlinear boundary problem with indefinite weight

Abstract

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[.

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

“…thus u − = 0. We can check that u ∈ C 1,α (Ω) for some α ∈ (0, 1) (see [1]). Then the maximum principle of Vasquez [16] can be applied to ensure positiveness of u.…”

“…Passing to the limit in (3.9), we see that v is a non-negative and non-trivial solution of problem (P λ,mp ) (note v ≥ 0 and v W 1,p (Ω) = 1). The eigenfunction v is C 1,α (Ω) for some α ∈ (0, 1) (see [1]). According to maximum principle of Vasquez, we have v > 0 in W 1,p (Ω).…”

Multiple solutions for p-Laplacian eigenproblem with nonlinear boundary conditions

“…In particular, for w = v − , we obtain |∇v Since the regularity of solutions at this problem are C 1,α (Ω), (see [3]). We can apply Harnack inequality, we have v > 0 in Ω, so λ 1 (h 1 (x), h 2 (x)) = 1.…”

A non resonance under and between the two first eigenvalues in a nonlinear boundary problem

“…Supposing that u 1 ≥ 0 in ∂Ω, one show that u 1 > 0 on ∂Ω. Indeed, if there exists x ∈ ∂Ω such that u 1 (x) = 0, by the regularity proven in [2], u 1 ∈ C 1,α Ω and by the maximum principle of Vazquez…”

“…In [2], one has proved that, in the case g (x, u) = λV (x) |u| p−2 u + h with V satisfies the same last conditions and h ∈ L s (∂Ω), the solutions are in C 1,α Ω for some α in ]0, 1[. Now we will study the case…”