本文關(guān)鍵詞： 橢圓型邊值問題 奇異位勢 變分方法 多重正解的存在性　出處：《福建師范大學(xué)》2015年博士論文　論文類型：學(xué)位論文

【摘要】：非線性橢圓型邊值問題正解的存在性、多解性及其它相關(guān)性質(zhì)的研究具有十分重要的理論和現(xiàn)實意義.本文研究了三類帶奇異位勢的非線性橢圓型邊值問題,主要工作如下：1.研究了一類帶反平方位勢和凹凸非線性的橢圓型邊值問題：首先,利用Ekeland變分原理,在Nehari流形上構(gòu)造適合的極小化問題,得到了保證問題(1P)至少有兩個正解的充分條件.其次,作為證明多解性的另一收獲,得到了問題(1P)取p=1+ε時的解當(dāng)ε→ 0+時的爆破行為.這兩個結(jié)果補(bǔ)充和推廣了Sun [92, p.752,定理1.1和定理1.3]的結(jié)論.最后,結(jié)合多解性結(jié)果,并進(jìn)一步利用上下解方法,研究了問題(1P)當(dāng)h,W三1時的極值問題,得到了極值μ*的一致估計.2.研究了一類帶Hardy項和奇異非線性的橢圓型邊值問題：與問題(1P)相比,問題(2P)唯一不同的是,方程右端的h(x)u-q在點u=0奇異(當(dāng)u→0時h(x)u-q→∞),因而問題(2P)對應(yīng)的能量泛函不可微,這使得在利用變分方法研究解的存在性、對出現(xiàn)的h(x)u-q相關(guān)項進(jìn)行討論時,需要更多的分析技巧(應(yīng)用兩次Fatou引理).對問題(2P),我們得到了類似于問題(1P)的三個結(jié)果.當(dāng)λ=0時,前兩個結(jié)果即Sun和Li[94,p.2637-2638,定理1和推論2]的結(jié)論.3.研究了一類具有臨界非線性和描述奇異性的橢圓問題：利用變分方法,通過構(gòu)建適合的極小化問題,得到了保證問題(3P)存在多重正解的充分條件.所得結(jié)果補(bǔ)充并完善了Chen [28, p.141,定理1.1]的結(jié)論.
[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，

本文編號：1537626