Nonlinear Differential Equations,
Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 325-333.

A condition on the potential for the existence of doubly periodic solutions of a semi-linear fourth-order partial differential equation

Chen Chang

We study the existence of solutions to the fourth order semi-linear equation
$\Delta ^2u=g(u)+h(x)$.
We show that there is a positive constant $C_*$, such that if $g(\xi )\xi \geq 0$ for $|\xi |\geq \xi _0$ and $\limsup _{|\xi |\to \infty } 2G(\xi )/\xi^2<C_*$, then for all $h\in L^2(Q)$ with $\int _Q h dx=0$, the above equation has a weak solution in $H^2_{2\pi}$.

Published October 31, 2000.
Math Subject Classifications: 35J30, 35B10.
Key Words: periodic solutions, elliptic fourth-order PDE.

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Chen Chang
Division of Mathematics and Statistics
the University of Texas at San Antonio
San Antonio, TX 78249 USA
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