【作者】 赵琳；

【导师】 赵培浩；

【作者基本信息】 兰州大学 ， 基础数学， 2010， 硕士

【摘要】 本文研究两类非线性椭圆问题的可解性和多解性.设Ω(?)N(N≥1)是光滑有界区域.首先讨论下面带有奇异项的拟线性椭圆问题的多解性,N<p<+∞，γ>0是常数,λ是非负的参数.我们运用Ricceri三解定理证明了问题(P1)存在三个不同的正解,主要的方法是上下解以及截断的技巧.其次讨论了如下的非线性椭圆问题的可解性和多解性其中A：Ω×RN→R,A=A(χ，ξ)在Ω×RN关于ξ有连续的偏导数,α(χ,.)=(?)A(χ,·)=A’.当f在无穷远处是(p-1)次超线性时,我们应用山路引理得到两个解；当f在无穷远处是(p-1)次次线性时,由Ricceri三解定理得到三个不同的解.更多还原

【Abstract】 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.更多还原