【作者】 陈志辉；

【导师】 沈尧天；

【作者基本信息】 中国科学技术大学 ， 应用数学， 2006， 博士

【摘要】 本文主要讨论Hardy(型)不等式以及含临界位势的椭圆型方程多重解的存在性，全文共七章。 第一章，建立了R~4中相应的Rellich不等式，证明了常数是最佳的，由此确定了临界位势。随后，利用临界点理论证明了含临界位势的非线性椭圆型双调和方程多重解的存在性，其中非线性项为次临界增长。 第二章，建立了含一般权的一维Hardy不等式，并证明常数是最佳的，利用变量代换的方法得到含一般权的L~2-恒等式，以此式进行递推便得到含任意有限个余项的恒等式。进而，建立了含任意有限个余项的一维Hardy不等式。随后，利用对称重排的方法，得到了含任意有限个余项的N维L~p-Hardy不等式，其中权为幂函数。最后，研究了N=p时，含临界位势的p-Laplace方程第一特征值问题，给出了第一特征值的渐进性态。 第三章，给出了使Hardy不等式成立的(左右两端)权函数之间的一个关系式，得到了含一般权函数的N维L~pHardy不等式，其中常数是最佳的。类似于第二章，通过变量代换得到了含一般权的N维L~2-恒等式，以此恒等式出发便得到含无穷多个余项的L~2-Hardy不等式，其中常数和权函数都是最佳的。此外，建立了含任意多个余项的Hardy-Poincaré不等式。最后，研究了含一般权及余项的L~p-Hardy不等式，但没能找到最佳常数。 第四章，建立了第三章中定义的新空间中的嵌入定理及嵌入紧性，并利用临界点理论证明了一类含(一般径向)临界位势和临界参数的非线性退化椭圆型方程多重解的存在性。 第五章，在迹非零的函数空间中建立了相应的Hardy不等式，此时要求Hardy不等式包含边界项积分，用类似于前两章的方法在迹非零空间中建立了含一般权及余项的Hardy不等式以及Hardy-Poincaré不等式。定义了一个新的Hilbert空间，证明了一类半线性椭圆型方程Neumann边值问题的可解性。 第六章，建立了含到边界距离的Hardy-Poincaré不等式，由此定义一个新的Hilbert空间，并得到该空间嵌入L~p空间的紧性结果。在新空间中，证明了半线性椭圆型方程Dirichlet边值问题多重解的存在性，其中非线性项为次临界增长。 第七章，简单研究了含临界位势p-Laplace方程特征值问题，N＞p，通过直接定义的方法得到了特征值序列的存在性，并证明第一特征值的单重性及第一特征函数不变号。更多还原

【Abstract】 This doctoral dissertation is devoted to studying the Hardy(-type) inequality and the existence of many solutions to elliptic PDEs with critical potential. It is made up of seven chapters.In Chapter 1, it is obtained the Rellich inequality in R~4 with the best constant, and determined the corresponding critical potential. Then by using the critical point theory, prove the many solutions to a class of nonlinear biharmonic equation with critical potential when the nonlinear term has subcritical growth.In Chapter 2, prove a one-dimensional Hardy-type inequality with general weights, where the constant is optimal. By changing variable, establish a L~2-identity with general weight, and then a L~2-identity and an improved Hary-type inequality with a finite number of remainder terms. Using the symmetrization rearrangement method, obtain an improved N-dimensional Hardy-type inequality with remainder terms. At last, study the first eigenvalue problem for p-Laplace equation with critical potential when N — p, and give the asymptotic behavior of the first eigenvalues.In Chapter 3, give the relation between the weights in Hardy’s inequality and prove a N-dimensional Hardy-type inequality with general weights, where the constant is optimal. Using the analogous methods, get an improved Hardy-type inequality with a series of remainder terms, where both the constants and weights are the best possible. In addition, prove a Hary-Poincare inequality with general weights and remainder terms. At last, study the L~p-Hardy inequality with general weights and remainder terms, however, the constants are not optimal.In Chapter 4, establish the compactness of embedding of new spaces defined in Chapter 3, and obtain the many solutions to a class of degenerate elliptic equation including critical potential and critical parameter by the critical point theory.In Chapter 5, establish the L~2-Hardy inequality with general weights in spaces with trace being nonzero. In this case, the inequality must include some boundary integrals. Analogous to before, obtain a Hardy-type inequality and Hardy-Poincare-type inequality with general weights and remainder terms. Define a new Hilbert space, and where prove the solvability of a class of semilinear elliptic PDEs with Neumann boundary condition.In Chapter 6, prove a Hardy-Poincare-type inequality include the distance from boundary. It then follows that a new Hilbert space is embedded into L~p spaces. In this new space, obtain the many solutions to a seminliear elliptic PDEs with critical potential, where the nonlinear term has subcritical growth.In Chapter 7, devote to study the eigenvalue problem for p-Laplace equation with critical potential, N > p. By a direct method, prove the existence of an sequence of eigenvalues.更多还原