Existence of ground state solutions for a super-biquadratic Kirchhoff-type equation with steep potential well

Existence and concentration of positive solutions for a class of Kirchhoff-type equations with steep potential well

“…The existence and the nonexistence of nontrivial solutions were obtained by using variational methods. When N = 3 and f (x, u) = |u| p−2 u with 4 < p < 6, the problem (1.6) was studied by Du et al [6], they proved the existence and asymptotic behavior of ground state solutions. Very recently, Zhang and Du [22] studied (1.6) when N = 3 and f (x, u) = |u| p−2 u in the case 2 < p < 4.…”

On the Existence of Solutions for a Class of Schrödinger–Kirchhoff-Type Equations with Sign-Changing Potential

“…Hence, the corresponding results in [9] have been obtained by using the variational techniques in a standard way. In [11][12][13], the authors considered Kirchhoff type problem (3) with a steep potential well. Precisely, the potential function satisfies the following conditions besides (V 2 ):…”

“…By using this conditions, Sun and Wu [11] considered (3) in the case where the nonlinearity f(x, s) is asymptotically k-linear (k � 1, 2, 4) with respect to s at infinity. Du et al [12] studied (3) when f(x, u) behaves like |u| p− 2 u with 4 < p < 6 and proved the existence and asymptotic behavior of ground state solutions. Zhang and Du [13] investigated the existence and asymptotic behavior of positive solutions for (3) by combining the truncation technique and the parameter-dependent compactness lemma for b small and λ large in the case where f(x, u) behave like |u| p− 2 u with 2 < p < 4.…”

Existence and Asymptotic Behavior of Ground State Solutions to Kirchhoff-Type Equations of General Convolution Nonlinearity with a Steep Potential Well

“…Additionally, the authors also explored the asymptotic behavior of nontrivial solutions for (7). Subsequently, with the help of the variational framework developed by [20], Du et al [21] studied (7) when N = 3 and g(x, u) behaved similar to |u| p−2 u with 4 < p < 6, and subsequently proved the existence and asymptotic behavior of ground state solutions. In [22], the authors obtained the existence of nontrivial solutions for the case of N = 3 and g(x, u) = |u| p−2 u with 4 ≤ p < 6.…”