Existence of solutions to modified nonlinear Schrödinger equations on non-compact Riemannian manifolds

We study the existence of nontrivial solutions to a class of modified nonlinear Schrödinger equation on a complete non-compact N-dimensional (N≥3) Riemannian manifold with asymptotically non-negative Ricci curvature. Using the critical point theory, in a general Sobolev space instead of an Orlicz one, we prove the existence of a nontrivial non-negative solution to a class of modified Schrödinger equation with coercive potential. We also estimate the decay rate of the solution and show that the solution is exponentially decay.