Continuum of solutions for an elliptic problem with critical growth in the gradient

David Arcoya, Colette De Coster, Louis Jeanjean, Kazunaga Tanaka

Research output: Contribution to journalArticle

21 Citations (Scopus)

Abstract

We consider the boundary value problem(Pλ)u∈H01(Ω)∩L∞(Ω):-δu=λc(x)u+μ(x)|∇u|2+h(x), where Ω⊂RN, N≥3 is a bounded domain with smooth boundary. It is assumed that c{greater-than but not equal to}0, c, h belong to Lp(Ω) for some p>N/2 and that μ∈L(Ω). We explicitly describe a condition which guarantees the existence of a unique solution of (Pλ) when λ<0 and we show that these solutions belong to a continuum. The behaviour of the continuum depends in an essential way on the existence of a solution of (P0). It crosses the axis λ=0 if (P0) has a solution, otherwise it bifurcates from infinity at the left of the axis λ=0. Assuming that (P0) has a solution and strengthening our assumptions to μ(x)≥μ1>0 and h{greater-than but not equal to}0, we show that the continuum bifurcates from infinity on the right of the axis λ=0 and this implies, in particular, the existence of two solutions for any λ>0 sufficiently small.

Original languageEnglish
Pages (from-to)2298-2335
Number of pages38
JournalJournal of Functional Analysis
Volume268
Issue number8
DOIs
Publication statusPublished - 2015 Apr 15

Keywords

  • Continuum of solutions
  • Elliptic equations
  • Quadratic growth in the gradient
  • Topological degree

ASJC Scopus subject areas

  • Analysis

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