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
Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent
Qi-Lin Xie Xing-Ping Wu Chun-Lei Tang
Communications on Pure & Applied Analysis 2013, 12(6): 2773-2786 doi: 10.3934/cpaa.2013.12.2773
In the present paper, the existence and multiplicity of solutions for Kirchhoff type problem involving critical exponent with Dirichlet boundary value conditions are obtained via the variational method.
keywords: Dirichlet problem Kirchhoff type problem Brezis-Lieb Lemma. critical exponent
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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