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Semilinear elliptic problems in unbounded domains

Aleksandra Orpel (2006)

Applicationes Mathematicae

We investigate the existence of positive solutions and their continuous dependence on functional parameters for a semilinear Dirichlet problem. We discuss the case when the domain is unbounded and the nonlinearity is smooth and convex on a certain interval only.

Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation

Jacques Giacomoni, Ian Schindler, Peter Takáč (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We investigate the following quasilinear and singular problem, t o 2 . 7 c m - Δ p u = λ u δ + u q in Ω ; u | Ω = 0 , u > 0 in Ω , t o 2 . 7 c m (P) where Ω is an open bounded domain with smooth boundary, 1 < p < , p - 1 < q p * - 1 , λ > 0 , and 0 < δ < 1 . As usual, p * = N p N - p if 1 < p < N , p * ( p , ) is arbitrarily large if p = N , and p * = if p > N . We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in W 0 1 , p ( Ω ) . While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle and a regularity result for solutions...

Solutions to a class of singular quasilinear elliptic equations

Lin Wei, Zuodong Yang (2010)

Annales Polonici Mathematici

We study the existence of positive solutions to ⎧ d i v ( | u | p - 2 u ) + q ( x ) u - γ = 0 on Ω, ⎨ ⎩ u = 0 on ∂Ω, where Ω is N or an unbounded domain, q(x) is locally Hölder continuous on Ω and p > 1, γ > -(p-1).

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