Uniqueness and multiplicity results for N-Laplace equation with critical and singular nonlinearity in a ball

Abstract

Let B₁ be the unit open ball with center at the origin in R^N, N≥2. We consider the following quasilinear problem depending on a real parameter λ>0:

−Δ_Nu=λ f(u), u>0 in Ω,

u=0 on ∂Ω,   (P_λ)

where f(t) is a nonlinearity that grows like e^tN/N−1 as t→∞ and behaves like t^α, for some α∈(−∞, 0), as t→0⁺. More precisely, we require f to satisfy assumptions (A1) and (A2) listed in Section 1. For such a general nonlinearity we show that if λ>0 is small enough, (P_λ) admits at least one weak solution (in the sense of distributions). We further study the question of uniqueness and multiplicity of solutions to (P_λ) when Ω=B₁ under additional structural conditions on f (see assumptions (A3)–(A8) in Section 2). Using shooting methods and asymptotic analysis of ODEs, under the additional assumptions (A3)–(A5), we prove uniqueness of solution to (P_λ) for all λ>0 small whereas under (A6), (A7) or (A8), we show multiplicity of solutions to (P_λ) for all λ>0 in a maximal interval. These results clearly show that the borderline between uniqueness and multiplicity is given by the growth condition lim inf _t→∞h(t)te^εt1/(N−1)=∞ ∀ε>0.