The compactness of minimizing sequences for a nonlinear Schrödinger system with potentials

In this paper, we consider the following minimizing problem with two constraints: [Formula: see text] where [Formula: see text] and [Formula: see text] is defined by [Formula: see text] [Formula: see text] Here [Formula: see text], [Formula: see text] and [Formula: see text] [Formula: see text] are given functions. For [Formula: see text], we consider two cases: (i) both of [Formula: see text] and [Formula: see text] are bounded, (ii) one of [Formula: see text] and [Formula: see text] is bounded. Under some assumptions on [Formula: see text] and [Formula: see text], we discuss the compactness of any minimizing sequence.

Ground State Solution for an Autonomous Nonlinear Schrödinger System

In this paper, we study the following autonomous nonlinear Schrödinger system (discussed in the paper), where λ , μ , and ν are positive parameters; 2 ∗ = 2 N / N − 2 is the critical Sobolev exponent; and f satisfies general subcritical growth conditions. With the help of the Pohožaev manifold, a ground state solution is obtained.