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

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

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    17 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

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    Keywords

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

    ASJC Scopus subject areas

    • Analysis

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