José C. Sabina de Lis

On the perturbation of eigenvalues for the p-Laplacian

Abstract In this Note the differentiability with respect to the domain of the first Dirichlet eigenvalue of the minus p -Laplacian is shown for the first time. An explicit formula for the first variation is obtained. The proof, based on the variational formulation and C 1,α -type estimates, even simplifies the corresponding to the linear case p=2 . An application to the profile behaviour of positive solutions to a class of logistic problems is given.

Existence, Asymptotic Behavior and Uniqueness for Large Solutions to Δu = eq(x)u

Abstract In this paper we consider existence, asymptotic behavior near the boundary and uniqueness for solutions to Δu = eq(x)u in a bounded smooth domain Ω with the boundary condition u(x) → + ∞ as dist(x, ∂Ω) → 0. The exponent q(x) is assumed to be a Hölder continuous function which is either positive on ∂Ω or is positive in a neighborhood of ∂Ω maybe vanishing on ∂Ω. When dealing with nonnegative exponents q we are allowing nonempty interior regions Ω0 ⊂ Ω where q vanishes. Changing sign exponents q will be also considered.

Layer Profiles of Solutions to Elliptic Problems under Parameter-Dependent Boundary Conditions

We consider the unique positive solution to the equation ∆u = u in Ω, where r > 1 and Ω is a smooth bounded domain of R , under one of the boundary conditions u = λ, ∂u ∂ν = λ, ∂u ∂ν = λu or ∂u ∂ν = λu− u on ∂Ω, q > 1. The main interest is determining the exact layer behavior of this solution near ∂Ω in terms of the parameter λ as λ → ∞. Our analysis is completed with the study of the same type of problems involving the p-Laplacian operator.

A variable exponent diffusion problem of concave-convex nature

We deal with the problem

Existence and uniqueness of positive solutions to elliptic problems with sublinear mixed boundary conditions

\begin{cases} -\Delta u = \lambda u^{q(x)} & \text{if } x\in \Omega,\\ u = 0 & \text{if } x\in \p\Omega, \end{cases}

Large solutions for the laplacian with a power nonlinearity given by a variable exponent

where

A bifurcation problem governed by the boundary condition II

\Omega\subset \R^N

Large solutions to the p-Laplacian for large p

is a bounded smooth domain,