Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity

Abstract

Abstract. We study the existence, nonexistence and multiplicity of positive solutions for the family of problems − u = f λ (x, u), u ∈ H 1 0 ( ), where is a bounded domain in R N , N ≥ 3 and λ > 0 is a parameter. The results include the well-known nonlinearities of the Ambrosetti-Brezis-Cerami type in a more general form, namely λa(x)u q + b(x)u p , where 0 ≤ q < 1 < p ≤ 2 * − 1. The coefficient a(x) is assumed to be nonnegative but b(x) is allowed to change sign, even in the critical case. The notions of local superlinearity and local sublinearity introduced in [9] are essential in this more general framework. The techniques used in the proofs are lower and upper solutions and variational methods.

Superlinear Elliptic Problems With Sign Changing Coefficients

“…In the case of a critical superlinearity, that is, when p = 2 * = 2N N −2 , the result in [1] still holds, while the case of a variable coefficient b(x) was considered in [4], finding again two positive solutions but under the condition b ∈ L ∞ and b ≥ 0; the latter requirement is important because it implies a monotone dependence on λ of the nonlinearity, which is crucial for the application of the argument of sub and supersolutions.…”

MULTIPLICIDADE DE SOLUÇÕES PARA UMA CLASSE DE PROBLEMAS QUASELINEARES CRÍTICOS EM R<SUP>N</SUP> ENVOLVENDO FUNÇÃO PESO COM MUDANÇA DE SINAL

“…For more general results in bounded domains see e.g. the papers by Ambrosetti et al (1996); Birindelli e Demengel (2004); Pacella et al (1997) de Figueiredo et al (2006); Silva e Xavier (2003); Azore ro et al (2000) and their references.…”

On a Liouville-type equation with sign-changing weight

“…More recently, in [8] and [9], more general nonlinearities were considered which include nonlinearities of the form f (x, u) = λa(x)u q + b(x)u p , where the coefficients a(x), b(x) are allowed to change sign.…”