High perturbations of quasilinear problems with double criticality

Claudianor O. Alves, Prashanta Garain, Vicenţiu D. Rădulescu

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


This paper is concerned with the qualitative analysis of solutions to the following class of quasilinear problems {-ΔΦu=f(x,u)inΩ,u=0on∂Ω,where ΔΦu=div(φ(x,|∇u|)∇u) and Φ(x,t)=∫0|t|φ(x,s)sds is a generalized N-function. We assume that Ω ⊂ RN is a smooth bounded domain that contains two open regions Ω N, Ω p with Ω ¯ N∩ Ω ¯ p= ∅. The features of this paper are that - Δ Φu behaves like - Δ Nu on Ω N and - Δ pu on Ω p, and that the growth of f: Ω × R→ R is like that of eα|t|NN-1 on Ω N and as |t|p∗-2t on Ω p when |t| is large enough. The main result establishes the existence of solutions in a suitable Musielak–Sobolev space in the case of high perturbations with respect to the values of a positive parameter.

Original languageEnglish
Pages (from-to)1875-1895
Number of pages21
JournalMathematische Zeitschrift
Issue number3-4
StatePublished - 1 Dec 2021
Externally publishedYes


  • Musielak–Sobolev space
  • Quasilinear problems
  • Variational methods

ASJC Scopus subject areas

  • Mathematics (all)


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