Uniqueness And Multiplicity Results For Tv-laplace Equation With Critical And Singular Nonlinearity In A Ball

摘要

Let B_1 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:rn{-△_Nu = λf(u), u > 0 u = 0}in Ω on δΩ (P_λ)rnwhere f(t) is a nonlinearity that grows like e~(t~(N/N-1)) as t → ∞ and behaves like t~α, for some α ∈ (- ∞, 0), as t → 0~+.rnMore 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 = B_1 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 betweenrnuniqueness and multiplicity is given by the growth condition lim inf_(t→∞)h(t)te~(εt~(1/(N-1))= ∞ anyε > 0.