TY - JOUR

T1 - Continuum of solutions for an elliptic problem with critical growth in the gradient

AU - Arcoya, David

AU - De Coster, Colette

AU - Jeanjean, Louis

AU - Tanaka, Kazunaga

N1 - Publisher Copyright:
© 2015 Elsevier Inc.
Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

PY - 2015/4/15

Y1 - 2015/4/15

N2 - 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.

AB - 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.

KW - Continuum of solutions

KW - Elliptic equations

KW - Quadratic growth in the gradient

KW - Topological degree

UR - http://www.scopus.com/inward/record.url?scp=84924156116&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84924156116&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2015.01.014

DO - 10.1016/j.jfa.2015.01.014

M3 - Article

AN - SCOPUS:84924156116

VL - 268

SP - 2298

EP - 2335

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 8

ER -