几类具非局部项的椭圆型方程解的存在性研究

发布时间：2018-03-01 13:26

本文关键词： Kirchhoff方程 Schrodinger-Poisson方程 p-Laplacian K-irchhoff方程临界点 Nehari流形变分法　出处：《中央民族大学》2017年硕士论文　论文类型：学位论文

【摘要】：本文运用临界点理论研究几类非局部椭圆型偏微分方程,分别讨论了它们基态解、正解及变号解的存在性.在本文的假设条件下,非线性项f仅要求是连续的,从而Nehari流形不必是C1的.在第二章中我们研究一类带有周期位势Kirchhoff方程非平凡解的存在性,其中f满足更一般的超四次增长条件但不必是C1类函数.当f在无穷远处是次临界增长或临界增长情况下,应用Szulkin和Weth的广义Nehari流形方法我们证明了基态解的存在性.最后,当位势V(x)是常数时,我们证明了自治Kirchhoff型方程基态解的存在性.在第三章中我们研究一类具临界指数增长的Schrodinger-Poisson方程正解的存在性,其中f满足次临界增长条件,位势V在无穷远处衰减于零.利用变分法,我们证明了方程至少存在一个正解.在第四章中我们研究一类带有衰减位势的p-Laplacian Kirchhoff方程变号解的存在性,其中f不是C1类函数,利用Nehari流形方法和极小化原理我们证明了方程存在变号解.进一步,当f是奇函数时,我们证明了方程存在无穷多个非平凡解.
[Abstract]:In this paper, the critical point theory is used to study some kinds of nonlocal elliptic partial differential equations, and the existence of ground state solutions, positive solutions and sign changing solutions are discussed respectively. Under the assumption in this paper, the nonlinear term f is only required to be continuous. In chapter 2, we study the existence of nontrivial solutions for a class of Kirchhoff equations with periodic potential. When f is a subcritical growth or critical growth at infinity, the existence of ground state solutions is proved by using the generalized Nehari manifold method of Szulkin and Weth. When the potential is constant, we prove the existence of ground state solutions for autonomous Kirchhoff type equations. In Chapter 3, we study the existence of positive solutions for a class of Schrodinger-Poisson equations with critical exponential growth, where f satisfies the subcritical growth conditions. The potential V attenuates to zero at infinity. By using the variational method, we prove that there is at least one positive solution to the equation. In Chapter 4th, we study the existence of the signed solutions for a class of p-Laplacian Kirchhoff equations with attenuated potential, where f is not a C1 class function. Using the Nehari manifold method and the minimization principle, we prove that the equation has a variable sign solution. Furthermore, when f is an odd function, we prove that there are infinitely many nontrivial solutions to the equation.
【学位授予单位】：中央民族大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.25

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