JournalsjemsVol. 8, No. 2pp. 269–288

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

  • Djairo Guedes de Figueiredo

    IMECC - UNICAMP, Campinas, Brazil
  • Jean-Pierre Gossez

    Université Libre de Bruxelles, Belgium
  • Pedro Ubilla

    Universidad de Santiago de Chile, Chile
Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity cover
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Abstract

In this paper we study the existence, nonexistence and multiplicity of positive solutions for the family of problems Δu=fλ(x,u)-\Delta u = f_\lambda (x,u), uH01(Ω)u \in H^1_0(\Omega), where Ω\Omega is a bounded domain in RN\mathbb{R}^N, N3N\geq 3 and λ>0\lambda>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)uq+b(x)up\lambda a (x)u^q + b(x) u^p , where 0q<1<p210 \leq q<1<p\leq 2^{\ast}-1. The coefficient a(x) is assumed nonnegative but b(x) is allowed to change sign, even in the critical case. The notions of local superlinearity and local sublinearity introduced in \cite{dF-G-U} are essential in this more general framework. The techniques used in the proofs are lower and upper solutions and variational methods.

Cite this article

Djairo Guedes de Figueiredo, Jean-Pierre Gossez, Pedro Ubilla, Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity. J. Eur. Math. Soc. 8 (2006), no. 2, pp. 269–288

DOI 10.4171/JEMS/52