In this paper, we investigate the existence of least energy sign-changing solutions for the Kirchhoff-type problem [Formula: see text], where a, b > 0 are parameters, [Formula: see text], and [Formula: see text]. Under weaker assumptions on V and f, by using variational methods with the aid of a new version of global compactness lemma, we prove that this problem has a least energy sign-changing solution with exactly two nodal domains, and its energy is strictly larger than twice that of least energy solutions.

This paper is concerned with the existence of ground state solutions for the Schrödinger–Poisson system −Δ u + V( x) u + ϕu = | u|4 u + λ| u| p−2 u in [Formula: see text] and −Δ ϕ = u2 in [Formula: see text], where λ > 0 and p ∈ [4, 6). Here, [Formula: see text], V( x) = V1( x) for x1 > 0, and V( x) = V2( x) for x1 < 0, where V1, V2 are periodic functions in each coordinate direction. In this paper, we give a splitting lemma corresponding to the nonperiodic potential and, then, prove the existence of ground state solutions for any λ > 0 when p ∈ (4, 6). Moreover, when p = 4, the above system possesses a ground state solution for λ > 0 sufficiently large. It is worth underlining that the technique employed in this paper is also valid for the Sobolev subcritical problem studied by Cheng and Wang [Discrete Contin. Dyn. Syst., Ser. B 27, 6295 (2022)].