Existence of solution for a class of biharmonic equations

• Main Authors: Najib Tsouli, Omar Chakrone, Omar Darhouche, Mostafa Rahmani
• Format: Article
• Language: English
• Published: Sociedade Brasileira de Matemática 2014-01-01
• Series: Boletim da Sociedade Paranaense de Matemática
• Subjects: Biharmonic equation; fourth elliptic equation; nonresonance
• Online Access: http://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/16178

In this paper, We prove the solvability of the biharmonic problem $$\begin{cases}\Delta^{2}u=f(x,u)+h ~~~~in~~\Omega, &\hbox{}\\ u=\Delta u=0 ~~~~~~on ~~\partial\Omega,\\\end{cases}$$ for a given function $h\in L^2(\Omega)$, if the limits at infinity of the quotients $f(x,s)/s$ and $2F(x,s)/s$ for a.e.$x\in\Omega$ lie between two consecutive eigenvalues of the biharmonic operator $\Delta^2$, where $F(x,s)$ denotes the primitive $F(x,s)=\int_{0}^{s}{f(x,t)dt}$.