Groundstate asymptotics for a class of singularly perturbed p-Laplacian problems in \({{\mathbb {R}}}^N\)

  • Wedad Albalawi
  • Carlo MercuriEmail author
  • Vitaly Moroz


We study the asymptotic behaviour of positive groundstate solutions to the quasilinear elliptic equation
where \(1<p<N \), \(p<q<l<+\infty \) and \(\varepsilon > 0 \) is a small parameter. For \({\varepsilon }\rightarrow 0\), we give a characterization of asymptotic regimes as a function of the parameters q, l and N. In particular, we show that the behaviour of the groundstates is sensitive to whether q is less than, equal to, or greater than the critical Sobolev exponent \(p^{*} :=\frac{pN}{N-p}\).


Groundstates Liouville-type theorems Quasilinear equations Singular perturbation 

Mathematics Subject Classification

35J92 (35B33 · 35B53 · 35B38) 



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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.College of SciencesPrincess Nourah Bint Abdulrahman UniversityRiyadhKingdom of Saudi Arabia
  2. 2.Department of Mathematics, Computational FoundrySwansea UniversitySwanseaUK

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