几类带奇异位势的非线性椭圆型边值问题的多解性研究

发布时间：2018-02-26 10:29

  本文关键词： 椭圆型边值问题 奇异位势 变分方法 多重正解的存在性　出处：《福建师范大学》2015年博士论文　论文类型：学位论文

【摘要】：非线性椭圆型边值问题正解的存在性、多解性及其它相关性质的研究具有十分重要的理论和现实意义.本文研究了三类带奇异位势的非线性椭圆型边值问题,主要工作如下：1.研究了一类带反平方位势和凹凸非线性的椭圆型边值问题：首先,利用Ekeland变分原理,在Nehari流形上构造适合的极小化问题,得到了保证问题(1P)至少有两个正解的充分条件.其次,作为证明多解性的另一收获,得到了问题(1P)取p=1+ε时的解当ε→ 0+时的爆破行为.这两个结果补充和推广了Sun [92, p.752,定理1.1和定理1.3]的结论.最后,结合多解性结果,并进一步利用上下解方法,研究了问题(1P)当h,W三1时的极值问题,得到了极值μ*的一致估计.2.研究了一类带Hardy项和奇异非线性的椭圆型边值问题：与问题(1P)相比,问题(2P)唯一不同的是,方程右端的h(x)u-q在点u=0奇异(当u→0时h(x)u-q→∞),因而问题(2P)对应的能量泛函不可微,这使得在利用变分方法研究解的存在性、对出现的h(x)u-q相关项进行讨论时,需要更多的分析技巧(应用两次Fatou引理).对问题(2P),我们得到了类似于问题(1P)的三个结果.当λ=0时,前两个结果即Sun和Li[94,p.2637-2638,定理1和推论2]的结论.3.研究了一类具有临界非线性和描述奇异性的椭圆问题：利用变分方法,通过构建适合的极小化问题,得到了保证问题(3P)存在多重正解的充分条件.所得结果补充并完善了Chen [28, p.141,定理1.1]的结论.
[Abstract]:The existence, multiplicity and other related properties of positive solutions for nonlinear elliptic boundary value problems are of great theoretical and practical significance. In this paper, three classes of nonlinear elliptic boundary value problems with singular potential are studied. The main work is as follows: 1. A class of elliptic boundary value problems with inverse square potential and concave convex nonlinearity are studied. Firstly, using the Ekeland variational principle, a suitable minimization problem is constructed on the Nehari manifold. In this paper, we obtain sufficient conditions to guarantee that the problem has at least two positive solutions. Secondly, as another gain to prove the multiplicity of solutions, we obtain the solution of the problem (1 P) when 蔚 is taken as p1 蔚. 鈫扵he blasting behavior at 0:00. These two results complement and generalize the conclusions of Sun [92, p. 752, Theorem 1.1 and Theorem 1.3]. Finally, combining with the results of multiple solutions and further using the method of upper and lower solutions, we study the extreme value problem of HW 3#time1#. In this paper, we obtain a uniform estimate of extreme value 渭 * .2.We study a class of elliptic boundary value problems with Hardy term and singular nonlinearity: the only difference between problem 2P and problem 1 P is that the right end of the equation is singular at the point UU 0 (if u). 鈫，

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