Remarks on a Class of Nonlinear Schrödinger Equations with Potential Vanishing at Infinity

Abstract EN

We study the following nonlinear Schrödinger equation −Δu+V(x)u=K(x)f(u), x∈ℝN, u∈H1(ℝN), where the potential V(x) vanishes at infinity.

Working in weighted Sobolev space, we obtain the ground states of problem (?) under a Nahari type condition.

Furthermore, if V(x),K(x) are radically symmetric with respect to x∈ℝN, it is shown that problem (?) has a positive solution with some more general growth conditions of the nonlinearity.

Particularly, if f(u)=up, then the growth restriction σ≤p≤N+2/N-2 in Ambrosetti et al.

(2005) can be relaxed to σ~≤p≤N+2/N-2, where σ~<σ if 0<β<α<2.