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1. CMB Online first

He, Yubo; Qin, Dongdong; Tang, Xianhua
Ground state and multiple solutions for Kirchhoff type equations with critical exponent
In this paper, we consider the following critical Kirchhoff type equation: \begin{align*} \left\{ \begin{array}{lll} - \left(a+b\int_{\Omega}|\nabla u|^2 \right)\Delta u=Q(x)|u|^4u + \lambda |u|^{q-1}u,~~\mbox{in}~~\Omega, \\ u=0,\quad \text{on}\quad \partial \Omega, \end{array} \right. \end{align*} By using variational methods that are constrained to the Nehari manifold, we prove that the above equation has a ground state solution for the case when $3\lt q\lt 5$. The relation between the number of maxima of $Q$ and the number of positive solutions for the problem is also investigated.

Keywords:Kirchhoff type equation, variational methods, critical exponent, Nehari manifold, ground state
Categories:35J20, 35J60, 35J25

2. CMB 2017 (vol 60 pp. 422)

Tang, Xianhua
New Super-quadratic Conditions for Asymptotically Periodic Schrödinger Equations
This paper is dedicated to studying the semilinear Schrödinger equation $$ \left\{ \begin{array}{ll} -\triangle u+V(x)u=f(x, u), \ \ \ \ x\in {\mathbf{R}}^{N}, \\ u\in H^{1}({\mathbf{R}}^{N}), \end{array} \right. $$ where $f$ is a superlinear, subcritical nonlinearity. It focuses on the case where $V(x)=V_0(x)+V_1(x)$, $V_0\in C(\mathbf{R}^N)$, $V_0(x)$ is 1-periodic in each of $x_1, x_2, \ldots, x_N$ and $\sup[\sigma(-\triangle +V_0)\cap (-\infty, 0)]\lt 0\lt \inf[\sigma(-\triangle +V_0)\cap (0, \infty)]$, $V_1\in C(\mathbf{R}^N)$ and $\lim_{|x|\to\infty}V_1(x)=0$. A new super-quadratic condition is obtained, which is weaker than some well known results.

Keywords:Schrödinger equation, superlinear, asymptotically periodic, ground state solutions of Nehari-Pankov type
Categories:35J20, 35J60

3. CMB 2015 (vol 58 pp. 651)

Tang, Xianhua
Ground State Solutions of Nehari-Pankov Type for a Superlinear Hamiltonian Elliptic System on ${\mathbb{R}}^{N}$
This paper is concerned with the following elliptic system of Hamiltonian type \[ \left\{ \begin{array}{ll} -\triangle u+V(x)u=W_{v}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N}, \\ -\triangle v+V(x)v=W_{u}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N}, \\ u, v\in H^{1}({\mathbb{R}}^{N}), \end{array} \right. \] where the potential $V$ is periodic and $0$ lies in a gap of the spectrum of $-\Delta+V$, $W(x, s, t)$ is periodic in $x$ and superlinear in $s$ and $t$ at infinity. We develop a direct approach to find ground state solutions of Nehari-Pankov type for the above system. Especially, our method is applicable for the case when \[ W(x, u, v)=\sum_{i=1}^{k}\int_{0}^{|\alpha_iu+\beta_iv|}g_i(x, t)t\mathrm{d}t +\sum_{j=1}^{l}\int_{0}^{\sqrt{u^2+2b_juv+a_jv^2}}h_j(x, t)t\mathrm{d}t, \] where $\alpha_i, \beta_i, a_j, b_j\in \mathbb{R}$ with $\alpha_i^2+\beta_i^2\ne 0$ and $a_j\gt b_j^2$, $g_i(x, t)$ and $h_j(x, t)$ are nondecreasing in $t\in \mathbb{R}^{+}$ for every $x\in \mathbb{R}^N$ and $g_i(x, 0)=h_j(x, 0)=0$.

Keywords:Hamiltonian elliptic system, superlinear, ground state solutions of Nehari-Pankov type, strongly indefinite functionals
Categories:35J50, 35J55

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