Abstract: | Consider the existence of nontrivial solutions of homogeneous Dirichlet problem for a nonlinear elliptic equation with the
critical potential in ℝ2. By establishing a weighted inequality with the best constant, determine the critical potential in ℝ2, and study the eigenvalues of Laplace equation with the critical potential. By the Pohozaev identity of a solution with a
singular point and the Cauchy-Kovalevskaya theorem, obtain the nonexistence result of solutions with singular points to the
nonlinear elliptic equation. Moreover, for the same problem, the existence results of multiple solutions are proved by the
mountain pass theorem. |