Ground state solutions for an indefinite Kirchhoff type problem with steep potential well

High energy solutions of modified quasilinear fourth-order elliptic equation

“…It was pointed out in [1][2][3][4] that (1.1) models several physical and biological systems where u describes a process which relies on the mean of itself such as the population density. For more mathematical and physical background on Kirchhoff-type problems, we refer the reader to [1,[5][6][7][8] and the references therein. It is well known that fourth-order elliptic equation has been widely studied since Lazer and Mckenna [9] first proposed to study periodic oscillations and traveling waves in a suspension bridge.…”

Existence and multiplicity of solutions for a p-Kirchhoff equation on R N ${\mathbb {R}}^{N}$

“…Sun and Wu [4] investigated the existence and non-existence of nontrivial solutions with the following assumption: V (x) ≥ 0 and there exists c > 0 such that meas{x ∈ R N : V (x) < c} is nonempty and has finite measure. Wu [5] proved that problem (1.2) has a nontrivial solution and a sequence of high energy solutions where V (x) is continuous and satisfies inf V (x) ≥ a 1 > 0 and for each M > 0, meas{x ∈ R N : V (x) ≤ M} < +∞.…”

“…However, p-Kirchhoff problem in the following form: 4) or p-Kirchhoff problem like (1.1) seems to be considered by few researchers as far as we know. Alves et al [14] and Corrêa and Figueiredo [15] established the existence of a positive solution for problem (1.4) by the mountain pass lemma, where M is assumed to satisfy the following conditions:…”

Existence and Concentration Results for Kirchhoff-Type Schrö Dinger Systems With Steep Potential Well

“…Subsequently, Liu and He [12] proved the existence of infinitely many high energy solutions for (1.3) when f is a subcritical nonlinearity which does't need to satisfy the usual Ambrosetti-Rabinowitz-type growth conditions. More recently, by using variational methods, Sun and Wu [16] obtained the existence and concentration of ground state solutions for (1.3) when V (x) was replaced by λV (x), where λ is a positive parameter. We would also mention the recent papers [19,20] where the existence of high energy solutions for Kirchhoff-type Schrödinger systems was established.…”