Abstract:

A study is conducted on the following nonlinear Schrödinger equation:

where N ≥ 3, u:R^N→R is a positive function, f(x, t) tends to p(x) and q(x)∈L∞(R^N), respectively, as t→0 and t→+∞. Assuming some simple assumptions on f(x, t), V(x), but not requiring any symmetry property on them, it is proved that problem (P) has a positive solution and a ground state.

Key words: nonlinear Schrödinger equation, mountain pass theorem, asymptotically linear, ground states