CPAA
Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four
Rui-Qi Liu Chun-Lei Tang Jia-Feng Liao Xing-Ping Wu
Communications on Pure & Applied Analysis 2016, 15(5): 1841-1856 doi: 10.3934/cpaa.2016006
In this article, we study the existence and multiplicity of positive solutions for the Kirchhoff type problem with singular and critical nonlinearities \begin{eqnarray} \begin{cases} -\left(a+b\displaystyle\int_\Omega|\nabla u|^2dx\right)\Delta u=\mu u^{3}+\frac{\lambda}{|x|^{\beta}u^{\gamma}}, &\rm \mathrm{in}\ \ \Omega, \\ u>0, &\rm \mathrm{in}\ \ \Omega, \\ u=0, &\rm \mathrm{on}\ \ \partial\Omega, \end{cases} \end{eqnarray} where $\Omega\subset \mathbb{R}^{4}$ is a bounded smooth domain and $a, b>0$, $\lambda>0$. For all $\mu>0$ and $\gamma\in(0,1)$, $0\leq\beta < 3$, we obtain one positive solution. Particularly, we prove that this problem has at least two positive solutions for all $\mu> bS^{2}$ and $\gamma\in(0,\frac{1}{2})$, $2+2\gamma < \beta < 3$.
keywords: positive solution critical exponent perturbation method. singularity Kirchhoff type problem
CPAA
Four positive solutions of a quasilinear elliptic equation in $ R^N$
Fang-Fang Liao Chun-Lei Tang
Communications on Pure & Applied Analysis 2013, 12(6): 2577-2600 doi: 10.3934/cpaa.2013.12.2577
This paper deals with the existence of multiple positive solutions of a quasilinear elliptic equation \begin{eqnarray} -\Delta_p u+u^{p-1} = a(x)u^{q-1}+\lambda h(x) u^{r-1}, \text{in} R^N; \\ u\geq 0, \text{ a.e. }x \in R^N;\\ u \in W^{1,p}(R^N), \end{eqnarray} where $1 < p \leq 2$, $N>p$ and $1 < r < p$ $< q < p^* ( = \frac{pN}{N-p})$. A Nehari manifold is defined by a $C^1-$functional $I$ and is decomposed into two parts. Our work is to find four positive solutions of Eq. (1) when parameter $\lambda$ is sufficiently small.
keywords: positive solutions. Nehari manifold $p-$Laplacian Palais-Smale decomposition lemma
DCDS-B
Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition
Li-Li Wan Chun-Lei Tang
Discrete & Continuous Dynamical Systems - B 2011, 15(1): 255-271 doi: 10.3934/dcdsb.2011.15.255
The existence and multiplicity of homoclinic orbits for a class of the second order Hamiltonian systems $\ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0, \ \forall t \in \mathbb{R}$, are obtained via the concentration-compactness principle and the fountain theorem respectively, where $W(t, x)$ is superquadratic and need not satisfy the (AR) condition with respect to the second variable $ x\in\mathbb{R}^{N}$.
keywords: concentration-compactness principle fountain theorem. Homoclinic orbits second order Hamiltonian systems
DCDS
Resonance problems for Kirchhoff type equations
Jijiang Sun Chun-Lei Tang
Discrete & Continuous Dynamical Systems - A 2013, 33(5): 2139-2154 doi: 10.3934/dcds.2013.33.2139
The existence of weak solutions is obtained for some Kirchhoff type equations with Dirichlet boundary conditions which are resonant at an arbitrary eigenvalue under a Landesman-Lazer type condition by the minimax methods.
keywords: Resonance critical point Landesman-Lazer type condition Kirchhoff type problem link.
CPAA
The existence and nonexistence results of ground state nodal solutions for a Kirchhoff type problem
Xiao-Jing Zhong Chun-Lei Tang
Communications on Pure & Applied Analysis 2017, 16(2): 611-628 doi: 10.3934/cpaa.2017030
In this paper, we investigate the existence and nonexistence of ground state nodal solutions to a class of Kirchhoff type problems
$ -\left( a+b\int_{\Omega }{|}\nabla u{{|}.{2}}dx \right)\vartriangle u=\lambda u+|u{{|}.{2}}u,\ \ u\in H_{0}.{1}(\Omega ), $
where $a, b>0$, $\lambda < a\lambda_1$, $\lambda_1$ is the principal eigenvalue of $(-\triangle, H_0.{1}(\Omega))$. With the help of the Nehari manifold, we obtain that there is $\Lambda>0$ such that the Kirchhoff type problem possesses at least one ground state nodal solution $u_b$ for all $0 < b < \Lambda$ and $\lambda < a\lambda_1$ and prove that its energy is strictly larger than twice that of ground state solutions. Moreover, we give a convergence property of $u_b$ as $b\searrow 0$. Besides, we firstly establish the nonexistence result of nodal solutions for all $b\geq\Lambda$. This paper can be regarded as the extension and complementary work of W. Shuai (2015)[21], X.H. Tang and B.T. Cheng (2016)[22].
keywords: Kirchhoff type problem nonlocal term Nehari manifold nodal solution ground state
CPAA
Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent
Miao-Miao Li Chun-Lei Tang
Communications on Pure & Applied Analysis 2017, 16(5): 1587-1602 doi: 10.3934/cpaa.2017076
In this paper, we study the existence of multiple positive solutions of the following Schrödinger-Poisson system with critical exponent
$\begin{equation*}\begin{cases}-Δ u-l(x)φ u=λ h(x)|u|^{q-2}u+|u|^{4}u,\ \text{in}\ \mathbb{R}^{3}, \\-Δφ=l(x)u^{2},\ \text{in}\ \mathbb{R}^{3},\end{cases}\end{equation*}$
where
$1 < q < 2 $
and
$λ>0 $
. Under some appropriate conditions on
$ l$
and
$h $
, we show that there exists
$λ^{*}>0 $
such that the above problem has at least two positive solutions for each
$λ∈(0,λ^{*}) $
by using the Mountain Pass Theorem and Ekeland's Variational Principle.
keywords: Schrödinger-Poisson system Mountain Pass Theorem Ekeland's variational principle critical exponent concentration compactness principle
CPAA
Multiple positive solutions for Kirchhoff type problems involving concave-convex nonlinearities
Jia-Feng Liao Yang Pu Xiao-Feng Ke Chun-Lei Tang
Communications on Pure & Applied Analysis 2017, 16(6): 2157-2175 doi: 10.3934/cpaa.2017107

In this paper, we are interested in looking for multiple solutions for a class of Kirchhoff type problems with concave-convex nonlinearities. Under the combined effect of coefficient functions of concave-convex nonlinearities, by the Nehari method, we obtain two solutions, and one of them is a ground state solution. Under some stronger conditions, we point that the two solutions are positive solutions by the strong maximum principle.

keywords: Kirchhoff type problems concave-convex nonlinearities ground state solution Nehari method
CPAA
Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term
Gui-Dong Li Chun-Lei Tang
Communications on Pure & Applied Analysis 2019, 18(1): 285-300 doi: 10.3934/cpaa.2019015
In this paper, we investigate the following a class of Choquard equation
$\begin{equation*} -Δ u+u = (I_α*F(u))f(u) \ \ \ \ \ \ {\rm in} \ \mathbb{R}^N,\end{equation*}$
where
$N≥ 3,~α∈ (0,N),~I_α$
is the Riesz potential and
$F(s) = ∈t_{0}^{s}f(t)dt$
. If
$f$
satisfies almost necessary the upper critical growth conditions in the spirit of Berestycki and Lions, we obtain the existence of positive radial ground state solution by using the Pohožaev manifold and the compactness lemma of Strauss.
keywords: Choquard equation Pohožaev manifold compactness lemma of strauss positive ground state solution critical growth
CPAA
Positive solution for the Kirchhoff-type equations involving general subcritical growth
Jiu Liu Jia-Feng Liao Chun-Lei Tang
Communications on Pure & Applied Analysis 2016, 15(2): 445-455 doi: 10.3934/cpaa.2016.15.445
In this paper, the existence of a positive solution for the Kirchhoff-type equations in $\mathbb{R}^N$ is proved by using cut-off and monotonicity tricks, which unify and sharply improve the results of Li et al. [Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations 253 (2012) 2285--2294]. Our result cover the case where the nonlinearity satisfies asymptotically linear and superlinear at infinity.
keywords: Kirchhoff-type equation cut-off and monotonicity tricks Pohozaev equality.
CPAA
Existence and nonuniqueness of homoclinic solutions for second-order Hamiltonian systems with mixed nonlinearities
Dong-Lun Wu Chun-Lei Tang Xing-Ping Wu
Communications on Pure & Applied Analysis 2016, 15(1): 57-72 doi: 10.3934/cpaa.2016.15.57
In this paper, we study the existence of homoclinic solutions to the following second-order Hamiltonian systems \begin{eqnarray} \ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,\quad \forall t\in R, \end{eqnarray} where $L(t)$ is a symmetric and positive definite matrix for all $t\in R$. The nonlinear potential $W$ is a combination of superlinear and sublinear terms. By different conditions on the superlinear and sublinear terms, we obtain existence and nonuniqueness of nontrivial homoclinic solutions to above systems.
keywords: indefinite signs. Multiple homoclinic solutions second-order Hamiltonian systems mixed nonlinearities variational methods

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