Existence of entire solutions for a class of quasilinear elliptic equations

  • Giuseppina Autuori
  • Patrizia PucciEmail author


The paper deals with the existence of entire solutions for a quasilinear equation \({(\mathcal E)_\lambda}\) in \({\mathbb{R}^N}\) , depending on a real parameter λ, which involves a general elliptic operator in divergence form A and two main nonlinearities. The competing nonlinear terms combine each other, being the first subcritical and the latter supercritical. We prove the existence of a critical value λ* > 0 with the property that \({(\mathcal E)_\lambda}\) admits nontrivial non-negative entire solutions if and only if λ ≥ λ*. Furthermore, when \({\lambda > \overline{\lambda} \ge \lambda^*}\) , the existence of a second independent nontrivial non-negative entire solution of \({(\mathcal{E})_\lambda}\) is proved under a further natural assumption on A.

Mathematics Subject Classification (2010)

Primary 35J62 35J70 Secondary 35J20 


Quasilinear elliptic equations Variational methods Divergence type operators Weighted Sobolev spaces Ekeland’sprinciple 


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© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità degli Studi di PerugiaPerugiaItaly

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