Pohozaev identities and their applications to nonlinear elliptic equations

Pohozaev identity plays a very important role in proving the existence and nonexistence results for the nonlinear elliptic partial differential equations. In this review, we aim to introduce Pohozaev identity and its applications in some classical elliptic partial differential equations. First, we introduce Pohozaev identities for several typical elliptic equations on bounded domains or unbounded domains, and the necessary conditions for the existence of solutions, by which we get some existence and nonexistence results of the solutions. Next, we will show a local Pohozaev identity and its application in proving the compactness of the sequences of approximate solutions, by which we obtain infinitely many solutions of some typical nonlinear elliptic equations. At last, we also introduce some local uniqueness results of peak solutions for some nonlinear elliptic partial differential equations, which can also be obtained by using the local Pohozaev identity. Using this type of uniqueness result, we can prove that the peak solutions can keep the symmetry that the equations have.