Hardy's Inequalities And Some Elliptic Equations With Critical Potentials

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.