A priori estimates and bifurcation of solutions for an elliptic equation with semidefinite critical growth in the gradient. (July 2015)

• Record Type: Journal Article
• Authors: Souplet, Philippe
• Is Part Of: Nonlinear analysis. Volume 121(2015)
• Page Start: 412
• Page End: 423
• Publication Date: 2015-07
• Subjects: Elliptic equation -- Critical growth in the gradient -- A priori estimates -- Existence -- Multiplicity -- Bifurcation -- Lδp spaces -- Weighted Sobolev inequalities -- Hardy inequalities; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.02.005
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study nonnegative solutions of the boundary value problem ( P λ ) − Δ u = λ c ( x ) u + μ ( x ) ∣ ∇ u ∣ 2 + h ( x ), u ∈ H 0 1 ( Ω ) ∩ L ∞ ( Ω ), where Ω is a smooth bounded domain of R n, μ, c ∈ L ∞ ( Ω ), h ∈ L r ( Ω ) for some r > n / 2 and μ, c, h ≩ 0 . Our main motivation is to study the "semidefinite" case. Namely, unlike in previous work on the subject, we do not assume μ to be uniformly positive in Ω, nor even positive everywhere. In space dimensions up to n = 5, we establish uniform a priori estimates for weak solutions of ( P λ ) when λ > 0 is bounded away from 0 . This is proved under the assumption that the supports of μ and c intersect, a condition that we show to be actually necessary, and in some cases we further assume that μ is uniformly positive on the support of c and/or some other conditions. As a consequence of our a priori estimates, assuming that ( P 0 ) has a solution, we deduce the existence of a continuum C of solutions, such that the projection of C onto the λ -axis is an interval of the form [ 0, a ] for some a > 0 and that the continuum C bifurcates from infinity to the right of the axis λ = 0 . In particular, for each λ > 0 small enough, problem ( P λ ) has at least two distinct solutions.