Multiple Solutions Of Two Classes Nonlinear Elliptic Problems

In this paper we study the existence and multiplicity of solutions of two classes nonlinear elliptic problems. LetΩ(?) RN is a bounded open set with smooth boundary. The first is about a quasilinear elliptic problem with a singular term N≥1, N< p<+∞,γ> 0 is a constant, andλ> 0 is a parameter. Three weak solutions of the problem (P1) are obtained by a three critical points theorem of B. Ricceri. Our methods are mainly based on the super-sub solutions and the cutoff method. The second problem is the existence and multiplicity of solutions of the following nonlinear elliptic problem (?) LetA:Ωx RN→R, A= A(x,ξ) be a continuous function inΩ×RN with continuous derivative with respect toξ, a(x,·)=(?)A(x,·)= A'. Two weak solutions are obtained by mountian pass lemma provided the nonlinear term via a (p - 1)-superlinear growth at infinity. Three weak solutions are obtained by a three critical points theorem provided the nonlinear term via a (p - 1)-sublinear growth at infinity.