Multiplicity and concentration of positive solutions for the Schrödinger–Poisson equations

Abstract

This paper is concerned with the multiplicity and concentration of positive solutions for the nonlinear Schrödinger-Poisson equationswhere ε > 0 is a parameter, V : R 3 → R is a continuous function and f : R → R is a C 1 function having subcritical growth. The proof of the main result is based on minimax theorems and the Ljusternik-Schnirelmann theory. (2000). 35J60 · 35J25. Mathematics Subject Classification

Existence of multiple positive solutions for Schrödinger–Poisson systems with critical growth

“…In addition, in the present paper, the conditions on f are weaker than the previous papers [2,12,22,24,31,34]. Indeed, in the paper [34], the author shows that (1.3) is weaker than [2,12,22,24,31]. So here we only need to check that our conditions are weaker than the conditions in (1.3).…”

“…(ii) We treat the critical case and more general nonlinearity f for (SP ) ε . That is, the nonlinearity is allowed to be critical growth and the conditions on f are weaker than the previous works [2,24,31,34].…”

“…Later, by applying a standard Lyapunov-Schmidt reduction methods, Ruiz and Vaira [24] proved the existence of multi-bump solutions of (1.1), whose bumps concentrate around a local minimum of the potential a(x) when h(u) = u p with 3 < p < 5. Finally, we should point out the latest paper [34], in which the author considered the system (1.1), where g is independent of x and a C 1 and subcritical function such that g(s) s 3 is increasing on (0, ∞), 0 < μG(s) = μ s 0 g(t)dt ≤ sg(s), μ > 4, g (s)s 2 − 3g(s)s ≥ Cs σ , σ ∈ (4, 6), C > 0, and g(s) = o(s 3 ) as s → 0.…”

“…Now let us summarize some properties of φ u . From the papers [10,22,33,34], one infers that the following lemma holds.…”

“…One can refer to the papers [9,13,34] for the proof of the conclusions (1) and (3). Here we follow the idea of Lemma 2.1 of [13] to give the proof of the conclusion (2).…”