J. Sabina de Lis

Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up

In this paper we prove uniqueness of positive solutions to logistic singular problems −∆u = λ(x)u − a(x)up, u|∂Ω = +∞, p > 1, a > 0 in Ω, where the main feature is the fact that a|∂Ω = 0. More importantly, we provide exact asymptotic estimates describing, in the form of a two-term expansion, the blow-up rate for the solutions near ∂Ω. This expansion involves both the distance function d(x) = dist(x, ∂Ω) and the mean curvature H of ∂Ω.

The behaviour of stationary solutions to some nonlinear diffusion problems

where the parameter λ > 0 becomes large, i.e., when λ → +∞. The operator ∆p stands for the p-Laplacian, i. e. the operator ∆p := div (|∇u|p−2∇u), and the number p > 1. It will be assumed that Ω ⊂ RN is a class C2+β bounded domain with 0 0, with 0 0 and k ∈ N (generically k = 1). For later use, we will set σ = h(0+) = limu→0+ f(u) up−1 (for simplicity we shall only consider the case σ < +∞) meanwhile we will term F = F (u) as the zero valued at u = 0 primitive of f(u), i. e. F (u) = ∫ u 0 f(s) ds. In the present work the attention will be focused on positive solutions u ∈ W 1,p 0 (Ω) ∩ L∞(Ω) to (1.1), where the concept of solution is understood in the weak sense. Namely, u ∈ W 1,p 0 (Ω) and ∫

Eigenvalue analysis for the p -Laplacian convective perturbation

Abstract In this paper a detailed analysis of the eigenvalue problem under convection −(|u′| p−2 u′)′−c|u′| p−2 u′=λ|u| p−2 u, 0 u(0)=u(1)=0 (′= d / d x) is performed. The analysis is based on a complete study of the phase space of the family of equations (|u′| p−2 u′)′+a|u′| p−2 u′+b|u| p−2 u=0, a,b constants.

Multiplicity of solutions to a degenerate diffusion problem

Abstract In this paper it is shown that the Dirichlet problem −Δ p u=λf(u), u ∂B =0 in a ball B⊂ R N , loses the property of uniqueness of positive solutions u under the sole condition max B u→ u 0 as λ→+∞, with u 0 >0 certain prefixed zero of f, provided p>k+1, k being the order of u 0 , what is in contrast with the so-called “nondegenerate case” p⩽k+1 where such hypothesis implies uniqueness. This also proves that a slightly stronger convergence condition for uniqueness introduced by the authors in a previous work cannot be relaxed.