Multi-bump solutions and multi-tower solutions for equations onRN

Abstract

Let > 0 be a small parameter. In this paper, we study existence of multiple multi-bump positive solutions for the semilinear Schrödinger equationand lim |x|→∞ a(x) = 0. We also study existence of multiple multi-tower positive solutions for the prescribed scalar curvature equationwhere N 3, K ∈ C([0, ∞)), K(r) > 0 for r > 0, lim r→0 K(r) = 0, and lim r→∞ K(r) = 0.

Existence of multi-bump solutions for a semilinear Schrödinger–Poisson system

“…Here we compare our results of theorem 1.1 and corollary 1.2 with theorem 1.1 in [28]. On the one hand, in [28], the condition (A) implies b ∈ L ∞ (R 3 ), however our condition (R 2 ) can allow b to be an unbounded function. Moreover, we give the exact minima distance between the two-bumps, i.e.…”

“…Under the condition that 0 < a ∈ L 2 (R 3 ) and b ∈ L 6 5−p (R 3 ), they obtained 1-bump solution for (SP ε ). More recently, in [28], Lin and Liu considered the following nonlinear Schrödinger equation:…”

“…γ = min i =j |p i − p i | = β ln 1 ε for β ∈ (0, 1). On the other hand, we consider a problem with a non-local term here, and extend the results of [28] . Generally speaking, the non-local term does not affect the results as in [28], however, because of the presence of a non-local term in our problem, we must prove some new estimates throughout the whole paper, which brings some difficulties.…”

“…Proof. As in [28,29], we use a contradiction argument to prove this lemma. Precisely, assume that N (ε) = ∅ as ε m → 0.…”

Existence of multi-bump solutions for a nonlinear Kirchhoff equation

“…where a, b > 0, 2 < p < 6, and q(x) ∈ C R 3 , R + satisfies some suitable conditions. By using the Lyapunov-Schmidt reduction method, he extended the results in [17] to the Kirchhoff problem.…”