Kirchhoff型椭圆方程正解和变号解的研究
发布时间：2018-08-10 18:58
【摘要】：本文运用约束变分法和一些分析技巧研究了 Kirchhoff型椭圆方程基态解的存在性和正解的多重性;变号解、基态变号解的存在性及渐近行为.首先,研究了如下有界区域上带临界指数的Kirchhoff型问题其中Ω(?)R3是一个具有光滑边界的有界区域,b0,λλ1，λ1是算子-△在Dirichlet边界条件下的第一特征值.由2*= 6为Sobolev临界指数可知,非线性项|u|4u达到临界增长.利用Nehari流形方法和Brezis-Lieb引理得到了当λ位于λ1的一个小的右邻域时正解的存在性和多重性.其次,研究了如下R3上具有零质量且带临界指数的Kirchhoff型问题其中b是一个正常数,为特征值问题-△u = λf(x)u,u ∈D1,2(R3)的第一特征值,D1,2(R3)= {u ∈ L2*(R3):%絬 ∈L2(R3)},f为非零非负函数.利用Nehari流形方法和集中紧性引理获得了两个正解,其中一个是基态解.再次,研究如下RN上带不定非线性项的Kirchhoff型问题其中 a0,b0,N≤3,λaλ1,λ1 为特征值问题-△u + u = λf(x)u,u ∈H1(RN)的第一特征值,权函数f非负非零,g变号.利用Weierstrass定理和Nehari流形方法得到bb0及bb0(b00)两种情况下,基态解的存在性及正解的多重性.之后,研究了下面的RN上带不定非线性项的Kirchhoff型方程其中a,b均为正常数,N≤3,1r2,4s2*=2N/N-2,λ0,权函数.f非负非零,g变号.利用Nehari流形方法获得了基态解的存在性和正解的多重性.最后,研究了如下Kirchhoff型问题变号解的存在性及不存在性其中Ω(?)R3是一个具有光滑边界的有界区域,a,b0,λaλ1.利用变号的Nehari流形方法和一些分析技巧,不仅证明了 0bA(A0)时,基态变号解的存在性、能量特征及b(?)0时基态变号解的收敛性,而且证明了 b ≥ A时变号解的不存在性.
[Abstract]:In this paper, the existence of ground state solutions and the multiplicity of positive solutions for Kirchhoff elliptic equations are studied by means of constrained variational method and some analytical techniques. Firstly, we study the Kirchhoff type problem with critical exponents on the bounded domains, where 惟 (?) R3 is a bounded region with smooth boundary, 位 1, 位 1 is the first eigenvalue of operator-under Dirichlet boundary condition. According to the Sobolev critical exponent of 2U = 6, the nonlinear term u 4u reaches critical growth. By using Nehari manifold method and Brezis-Lieb Lemma, we obtain the existence and multiplicity of positive solutions when 位 is located in a small right neighborhood of 位 1. Secondly, we study the Kirchhoff type problem with zero mass and critical exponent on the following R3, where b is a normal number and is the first eigenvalue of the eigenvalue problem-u = 位 f (x) UU 鈭，
本文编号：2175899
[Abstract]:In this paper, the existence of ground state solutions and the multiplicity of positive solutions for Kirchhoff elliptic equations are studied by means of constrained variational method and some analytical techniques. Firstly, we study the Kirchhoff type problem with critical exponents on the bounded domains, where 惟 (?) R3 is a bounded region with smooth boundary, 位 1, 位 1 is the first eigenvalue of operator-under Dirichlet boundary condition. According to the Sobolev critical exponent of 2U = 6, the nonlinear term u 4u reaches critical growth. By using Nehari manifold method and Brezis-Lieb Lemma, we obtain the existence and multiplicity of positive solutions when 位 is located in a small right neighborhood of 位 1. Secondly, we study the Kirchhoff type problem with zero mass and critical exponent on the following R3, where b is a normal number and is the first eigenvalue of the eigenvalue problem-u = 位 f (x) UU 鈭，
本文编号：2175899
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