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
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 nonuniqueness of homoclinic solutions for second-order Hamiltonian systems with mixed nonlinearities
Dong-Lun Wu Chun-Lei Tang Xing-Ping Wu
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
CPAA
Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent
Qi-Lin Xie Xing-Ping Wu Chun-Lei Tang
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

Year of publication

Related Authors

Related Keywords

[Back to Top]