Electron. J. Diff. Eqns., Vol. 2006(2006), No. 121, pp. 1-10.

A minimax inequality for a class of functionals and applications to the existence of solutions for two-point boundary-value problems

Ghasem Alizadeh Afrouzi, Shapour Heidarkhani

Abstract:
In this paper, we establish an equivalent statement to minimax inequality for a special class of functionals. As an application, we prove the existence of three solutions to the Dirichlet problem
$$\displaylines{ -u''(x)+m(x)u(x) =\lambda f(x,u(x)),\quad x\in (a,b),\cr u(a)=u(b)=0, }$$
where $\lambda$ greater than 0, $f:[a,b]\times \mathbb{R}\to \mathbb{R}$ is a continuous function which changes sign on $[a,b]\times \mathbb{R}$ and $m(x)\in C([a,b])$ is a positive function.

Submitted August 22, 2006. Published October 2, 2006.
Math Subject Classifications: 35J65.
Key Words: Minimax inequality; critical point; three solutions; multiplicity results; Dirichlet problem.

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Ghasem Alizadeh Afrouzi
Department of Mathematics
Faculty of Basic Sciences
Mazandaran University, Babolsar, Iran
email: afrouzi@umz.ac.ir
Shapour Heidarkhani
Department of Mathematics
Faculty of Basic Sciences
Mazandaran University, Babolsar, Iran
email: s.heidarkhani@umz.ac.ir

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