Nontrivial solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations

In this paper we study boundary value problems for semilinear equations involving strongly degenerate elliptic differential operators. Via a Pohozaev’s type identity we show that if the nonlinear term grows faster than some power function then the boundary value problem has no nontrivial solution. Otherwise when the nonlinear term grows slower than the same power function, by establishing embedding theorems for weighted Sobolev spaces associated with the strongly degenerate elliptic equations, then applying the theory of critical values in Banach spaces, we prove that the problem has a nontrivial solution, or even infinite number of solutions provided that the nonlinear term is an odd function.