• Korean Journal of Mathematics
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• 제16권1호
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• pp.41-49
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• 2008

We show the existence of a negative solution for the system of the following nonlinear wave equations with critical growth, under Dirichlet boundary condition and periodic condition $$u_{tt}-u_{xx}=au+b{\upsilon}+\frac{2{\alpha}}{{\alpha}+{\beta}}u_+^{\alpha-1}{\upsilon}_+^{\beta}+s{\phi}_{00}+f,\\{\upsilon}_{tt}-{\upsilon}_{xx}=cu+d{\upsilon}+\frac{2{\alpha}}{{\alpha}+{\beta}}u_+^{\alpha}{\upsilon}_+^{{\beta}-1}+t{\phi}_{00}+g,$$ where ${\alpha},{\beta}>1$ are real constants, $u_+={\max}\{u,0\},\;s,\;t{\in}R,\;{\phi}_{00}$ is the eigenfunction corresponding to the positive eigenvalue ${\lambda}_{00}$ of the wave operator and f, g are ${\pi}$-periodic, even in x and t and bounded functions.

• 호남수학학술지
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• 제30권1호
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• pp.197-203
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• 2008

We show the existence of the unique solution of the following system of the nonlinear wave equations with Dirichlet boundary conditions and periodic conditions under some conditions $U_{tt}-U_{xx}+av^+=s{\phi}_{00}+f$ in $(-{\frac{\pi}{2},{\frac{\pi}{2}}){\times}R$, ${\upsilon}_{tt}-{\upsilon}_{xx}+bu^+=t{\phi}_{00}+g$ in $(-{\frac{\pi}{2},{\frac{\pi}{2}}){\times}R$, where $u^+$ = max{u, 0}, s, t ${\in}$ R, ${\phi}_{00}$ is the eigenfunction corresponding to the positive eigenvalue ${\lambda}_{00}$ of the wave operator. We first show that the system has a positive solution or a negative solution depending on the sand t, and then prove the uniqueness theorem by the contraction mapping principle on the Banach space.

• Korean Journal of Mathematics
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• 제19권3호
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• pp.331-342
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• 2011

We consider the multiplicity of the solutions for a class of a system of critical growth beam equations with periodic condition on t and Dirichlet boundary condition $$\{u_{tt}+u_{xxxx}=av+\frac{2{\alpha}}{{\alpha}+{\beta}}u_{+}^{{\alpha}-1}v_{+}^{\beta}+s{\phi}_{00}\;\;in\;(-\frac{\pi}{2},\;\frac{\pi}{2}){\times}R,\\u_{tt}+v_{xxxx}=bu+\frac{2{\alpha}}{{\alpha}+{\beta}}u_{+}^{\alpha}v_{+}^{{\beta}-1}+t{\phi}_{00}\;\;in\;(-\frac{\pi}{2},\;\frac{\pi}{2}){\times}R,$$ where ${\alpha}$, ${\beta}$ > 1 are real constants, $u_+=max\{u,0\}$, ${\phi}_{00}$ is the eigenfunction corresponding to the positive eigenvalue ${\lambda}_00=1$ of the eigenvalue problem $u_{tt}+u_{xxxx}={\lambda}_{mn}u$. We show that the system has a positive solution under suitable conditions on the matrix $A=\(\array{0&a\\b&0}\)$, s > 0, t > 0, and next show that the system has another solution for the same conditions on A by the linking arguments.