Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in R3

In this paper we study the existence and concentration behaviors of positive solutions to the Kirchhoff type equations 

$$- \varepsilon^2 M \left(\varepsilon^{2-N}\!\!\int_{\mathbf{R}^N}|\nabla u|^2\,\mathrm{d} x \right) \Delta u \!+\! V(x) u \!=\! f(u) \quad{\rm in}\ \mathbf{R}^N, \quad u \!\in\! H^1(\mathbf{R}^N), \ N \!\geqq\!1,$$

where M and V are continuous functions. Under suitable conditions on M and general conditions on f, we construct a family of positive solutions ${(u_\varepsilon)_{\varepsilon \in (0,\tilde{\varepsilon}]}}$ which concentrates at a local minimum of V after extracting a subsequence (ek).

Existence and Concentration Result for the Kirchhoff Type Equations with General Nonlinearities

Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains☆

Ground state sign-changing solutions for Kirchhoff type problems in bounded domains ☆

This paper is dedicated to studying the following Kirchhoff-type problem 0.1
 $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\left( a+b\int _{\mathbb {R}^3}|\nabla u|^2\mathrm {d}x\right) \triangle u+V(x)u=f(u), &{} x\in \mathbb {R}^3; \\ u\in H^1(\mathbb {R}^3), \end{array} \right. \end{aligned}$$
 where $$a>0,\,b\ge 0$$
 are two constants, V(x) is differentiable and $$f\in \mathcal {C}(\mathbb {R}, \mathbb {R})$$
 . By introducing some new tricks, we prove that the above problem admits a ground state solution of Nehari–Pohozaev type and a least energy solution under some mild assumptions on V and f. Our results generalize and improve the ones in Guo (J Differ Equ 259:2884–2902, 2015) and Li and Ye (J Differ Equ 257:566–600, 2014) and some other related literature.

Ground state solutions of Nehari–Pohozaev type for Kirchhoff-type problems with general potentials

Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3

On the concentration phenomenon of L2-subcritical constrained minimizers for a class of Kirchhoff equations with potentials

In this paper, we study the existence of global L2$$ {L}^2 $$ ‐constrained minimizers related to the following Kirchhoff type equation: −a+b∫ℝN|∇u|2Δu−f(u)=λu,x∈ℝN,λ∈ℝ,$$ -\left(a+b{\int}_{{\mathbb{R}}^N}{\left|\nabla u\right|}^2\right)\Delta u-f(u)=\lambda u,x\in {\mathbb{R}}^N,\lambda \in \mathbb{R}, $$ where N≤3,a,b>0$$ N\le 3,a,b>0 $$ are constants, f(u)$$ f(u) $$ is a general L2$$ {L}^2 $$ ‐subcritical nonlinearity. By using the concentration compactness principle, we prove the sharp existence and nonexistence of global L2$$ {L}^2 $$ ‐constraint minimizers.

Hongyu Ye

Papers

Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3

On the concentration phenomenon of L2-subcritical constrained minimizers for a class of Kirchhoff equations with potentials

The existence and nonexistence of global L2‐constrained minimizers for Kirchhoff equations with L2‐subcritical general nonlinearity