• Nontrivial solution of a quasilinear elliptic equation with critical growth in ℝn

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      https://www.ias.ac.in/article/fulltext/pmsc/105/04/0425-0444

    • Keywords

       

      Elliptic equation; critical growth; Palais-Smale condition; concentration compactness; mountain pass lemma

    • Abstract

       

      Suppose Δnu = div (¦ ∇u ¦n-2∇u) denotes then-Laplacian. We prove the existence of a nontrivial solution for the problem$$\left\{ \begin{gathered} - \Delta _n u + \left| u \right|^{n - 2} u = \int {(x,u)u^{n - 2} in \mathbb{R}^n } \hfill \\ u \in W^{1,n} (\mathbb{R}^n ) \hfill \\ \end{gathered} \right.$$ wheref(x, t) =o(t) ast → 0 and ¦f(x, t)¦ ≤C exp(αn¦t¦n/(n-1)) for some constantC > 0 and for allx∈ℝ;t∈ℝ with αn =nωn1/(n-1), ωn = surface measure ofSn-1.

    • Author Affiliations

       

      Ratikanta Panda1

      1. T.I.F.R. Centre, I.I.Sc. Campus, P.O. Box No. 1234, Bangalore - 560012, India
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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