Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations with potentials vanishing at infinity

Abstract

For a singularly perturbed nonlinear elliptic equation ε 2 ∆u − V (x)u + u p = 0, x ∈ R N , there are solutions concentrating around critical points of V with different energy scales with respect to small ε > 0. We combine the solutions when lim inf |x|→∞ |x| 2 V (x) > 0.

Multi-bump standing waves with critical frequency for nonlinear Schrödinger equations

“…Since the pioneering work [21], there haven been many further papers for the case inf x∈R n V (x) > E (refer to [1,2,4,5,[14][15][16][17][18][19][20][21]24,25,27,28,[31][32][33][34][37][38][39] and references therein). When inf x∈R n V (x) > 0, we see via a transformation v(x) ≡ u(εx) that the following equations with constant c > 0 serve as limiting equations of (1.3)…”