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Che, Guofeng,Chen, Haibo Korean Mathematical Society 2017 대한수학회보 Vol.54 No.3
This paper is concerned with the following fourth-order elliptic equations $${\Delta}^2u-{\Delta}u+V(x)u-{\frac{k}{2}}{\Delta}(u^2)u=f(x,u),\text{ in }{\mathbb{R}}^N$$, where $N{\leq}6$, ${\kappa}{\geq}0$. Under some appropriate assumptions on V(x) and f(x, u), we prove the existence of infinitely many negative-energy solutions for the above system via the genus properties in critical point theory. Some recent results from the literature are extended.
Infinitely many solutions for a class of modified nonlinear fourth-order elliptic equations on Rn
Guofeng Che,Haibo Chen 대한수학회 2017 대한수학회보 Vol.54 No.3
This paper is concerned with the following fourth-order elliptic equations $$ \triangle^{2}u-\Delta u+V(x)u-\frac{\kappa}{2}\Delta(u^{2})u=f(x,u),\rm \mbox{ \ \ }in~\mathbb{R}^{N}, $$ where $N\leq6$, $\kappa\geq0$. Under some appropriate assumptions on $V(x)$ and $f(x, u)$, we prove the existence of infinitely many negative-energy solutions for the above system via the genus properties in critical point theory. Some recent results from the literature are extended.
EXISTENCE AND MULTIPLICITY OF NONTRIVIAL SOLUTIONS FOR KLEIN-GORDON-MAXWELL SYSTEM WITH A PARAMETER
Guofeng Che,Haibo Chen 대한수학회 2017 대한수학회지 Vol.54 No.3
This paper is concerned with the following Klein-Gordon-Maxwell system: $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u+\lambda V(x)u-(2\omega+\phi)\phi u=f(x,u), \mbox{ \ \ }x\in\mathbb{R}^{3},\\ \Delta\phi=(\omega+\phi)u^{2},\mbox{ \ \ }x\in\mathbb{R}^{3}, \end{array} \right. $$ where $\omega>0$ is a constant and $\lambda$ is the parameter. Under some suitable assumptions on $V(x)$ and $f(x, u)$, we establish the existence and multiplicity of nontrivial solutions of the above system via variational methods. Our conditions weaken the Ambrosetti Rabinowitz type condition.
EXISTENCE AND MULTIPLICITY OF NONTRIVIAL SOLUTIONS FOR KLEIN-GORDON-MAXWELL SYSTEM WITH A PARAMETER
Che, Guofeng,Chen, Haibo Korean Mathematical Society 2017 대한수학회지 Vol.54 No.3
This paper is concerned with the following Klein-Gordon-Maxwell system: $$\{-{\Delta}u+{\lambda}V(x)u-(2{\omega}+{\phi}){\phi}u=f(x,u),\;x{\in}\mathbb{R}^3,\\{\Delta}{\phi}=({\omega}+{\phi})u^2,\;x{\in}\mathbb{R}^3$$ where ${\omega}$ > 0 is a constant and ${\lambda}$ is the parameter. Under some suitable assumptions on V (x) and f(x, u), we establish the existence and multiplicity of nontrivial solutions of the above system via variational methods. Our conditions weaken the Ambrosetti Rabinowitz type condition.
Guofeng Che,Haibo Chen 대한수학회 2020 대한수학회지 Vol.57 No.6
This paper is concerned with the following Kirchhoff-Schr\"{o}d\-inger-Poisson system $$ \scalebox{0.84}{$\displaystyle\left\{\!\! \begin{array}{ll} \displaystyle -\big(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\mathrm{d}x\big)\Delta u + V(x)u+\mu\phi u=\lambda f(x)|u|^{p-2}u+g(x)|u|^{q-2}u, &\mbox{ in }\mathbb{R}^{3},\\ -\Delta \phi= \mu |u|^{2}, &\mbox{ in }\mathbb{R}^{3},\\ \end{array} \right.$} $$ where $a>0,~b,~\mu\geq0$, $p\in(1,2)$, $q\in[4,6)$ and $\lambda>0$ is a parameter. Under some suitable assumptions on $V(x)$, $f(x)$ and $g(x)$, we prove that the above system has at least two different nontrivial solutions via the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory. Some recent results from the literature are improved and extended.