2013/04/10 by David Arcoya, Arcoya, David, Colette De Coster +5 · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · doi:10.48550/arxiv.1304.3066
openalex publication_date 2013/04/10 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
We consider the boundary value problem - Δu = λc(x)u+ μ(x) |∇ u|2 + h(x), u ∈ H10(Ω) ∩ L∞(Ω) \eqno(Pλ) where Ω⊂ \RN, N ≥ 3 is a bounded domain with smooth boundary. It is assumed that c\gneqq 0, c,h belong to Lp(Ω) for some p > N/2 and that μ∈ L∞(Ω). We explicit 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 if bifurcates from infinity at the left of the axis λ=0. Assuming that (P0) has a solution and strenghtening our assumptions to μ(x)≥ μ1>0 and h\gneqq 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.