Abstract
In this paper, we study a class of the Kirchhoff-Schrödinger-Poisson system. By using the quantitative deformation lemma and degree theory, the existence result of the least energy sign-changing solution u0 is obtained. Meanwhile, the energy doubling property is proved, that is, we prove that the energy of any sign-changing solution is strictly larger than twice that of the least energy. Moreover, we also get the convergence properties of u0 as the parameters b↘0 and λ↘0.
Highlights
Introduction and the Main ResultsIn this paper, the following Kirchhoff-Schrödinger-Poisson system is considered: 8 ð >>>>< − a + b j∇uj2dx ΩΔu + λφx = gðuÞ, x∈Ω >>>>:−Δφ = u2, u = φ = 0, ð1Þ x ∈ Ω, x ∈ ∂Ω, where Ω ⊂ R3 is a bounded domain with a smooth boundary ∂Ω, a, b, λ ∈ R+ = ð0,+∞Þ, and g ∈ CðR, RÞ satisfies some basic assumptions
Motivated by the above papers, we study the problem
The main results of this paper are described as follows
Summary
The following Kirchhoff-Schrödinger-Poisson system is considered:. >>>>:. Assume that ðg1Þ − ðg4Þ hold, problem (1) possesses a least-energy sign-changing solution u0 ∈ M such that Fðu0Þ = inf M F > 0, which changes sign only once. For any sequence fbng with bn↘0 as n ⟶ ∞, there exists a subsequence of fubn g, still denoted by fubn g, such that ubn ⟶ ub0 in H10ðΩÞ, where ub0 ∈ Mb0 is a sign-changing solution of problem (2) with Fb0 ðub0 Þ = inf Mb0 Fb0 > 0: Theorem 4. For any sequence fλng with λn↘0 as n ⟶ ∞, there exists a subsequence of fuλn g, still denoted by fuλn g, such that uλn ⟶ uλ0 in H10ðΩÞ, where uλ0 ∈ Mλ0 is a sign-changing solution of (3) with Fλ0 ðuλ0 Þ = inf Mλ0 Fλ0 > 0: The rest of the paper is organized as follows.
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