Affiliations: L.M.A., Université de Pau et des Pays de l'Adour, B.P. 576, 64012 Pau cedex, France. E-mail: [email protected] | T.I.F.R. Center, I.I.Sc. Campus, P.B. No. 1234, Bangalore 560012, India. E-mail: [email protected] | Department of Mathematics, Indian Institute of Technology (Delhi), Hauz Khaz, New Delhi-16, India. E-mail: [email protected]
Abstract: Let B1 be the unit open ball with center at the origin in RN, N≥2. We consider the following quasilinear problem depending on a real parameter λ>0: −ΔNu=λ f(u), u>0 in Ω, u=0 on ∂Ω, (Pλ) where f(t) is a nonlinearity that grows like etN/N−1 as t→∞ and behaves like tα, 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, (Pλ) admits at least one weak solution (in the sense of distributions). We further study the question of uniqueness and multiplicity of solutions to (Pλ) 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 (Pλ) for all λ>0 small whereas under (A6), (A7) or (A8), we show multiplicity of solutions to (Pλ) 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)teεt1/(N−1)=∞ ∀ε>0.
