https://scholarworks.wm.edu/cgi/viewcontent.cgi?article=2523&context=aspubs

Existence of a positive solution to Kirchhoff type problems without compactness conditions

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

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

In the present paper, the following Schrodinger–Kirchhoff-type problem: (1.1) − ( a + b ∫ R N | ∇ u | 2 d x ) Δ u + V ( x ) u = f ( x , u ) , in R N is studied and four new existence results for nontrivial solutions and a sequence of high energy solutions for problem (1.1) are obtained by using a symmetric Mountain Pass Theorem.

Existence of nontrivial solutions and high energy solutions for Schrödinger–Kirchhoff-type equations in RN

In this paper, we study the following quasi-linear elliptic equations of the form − ∑ i , j = 1 N D j ( a i , j ( x , u ) D i u ) + 1 2 ∑ i , j = 1 N D s a i , j ( x , u ) D i u D j u + V ( x ) u = h ( x , u ) , x ∈ R N , where h ∈ C ( R N × R , R ) , N ≥ 3 and V ∈ C ( R N , R ) . Under appropriate assumptions on V ( x ) and h ( x , u ) , some existence results for positive solutions, negative solutions and a sequence of high energy solutions are obtained via the perturbation method.

Existence of positive solutions, negative solutions and high energy solutions for quasi-linear elliptic equations on RN

In this paper, we study the existence and multiplicity of positive solutions for the fractional Schrodinger equations e 2 α ( − Δ ) α u + V ( x ) u = ∣ u ∣ 2 α ∗ − 2 u + σ g ( x , u ) , x ∈ R N , where e and σ are positive parameters, 0 α 1 , ( − Δ ) α denotes the fractional Laplacian of order α , N > 2 α and 2 α ∗ = 2 N N − 2 α is the fractional critical exponent, V ∈ C ( R N , R + ) , g ∈ C ( R N × R , R ) . By the variational methods we study the existence and multiplicity of positive solutions. Our main results improve corresponding results in Shang and Zhang (2014).

Xian Wu

Papers

Existence of nontrivial solutions and high energy solutions for Schrödinger–Kirchhoff-type equations in RN

Existence of positive solutions, negative solutions and high energy solutions for quasi-linear elliptic equations on RN

Existence and multiplicity of positive solutions for fractional Schrödinger equations with critical growth