A priori bounds and multiplicity of solutions for an indefinite elliptic problem with critical growth in the gradient

Abstract

Let Ω⊂RN, N≥2, be a smooth bounded domain. We consider a boundary value problem of the form−Δu=cλ(x)u+μ(x)|∇u|2+h(x),u∈H01(Ω)∩L∞(Ω) where cλ depends on a parameter λ∈R, the coefficients cλ and h belong to Lq(Ω) with q>N/2 and μ∈L∞(Ω). Under suitable assumptions, but without imposing a sign condition on any of these coefficients, we obtain an a priori upper bound on the solutions. Our proof relies on a new boundary weak Harnack inequality. This inequality, which is of independent interest, is established in the general framework of the p-Laplacian. With this a priori bound at hand, we show the existence and multiplicity of solutions.