On the Eigenvalue of $p(x)$-Laplace Equation

The main purpose of this paper is to show that there exists a positive number $λ_{1}$, the first eigenvalue, such that some $p(x)$-Laplace equation admits a solution if $λ=λ_{1}$ and that $λ_{1}$ is simple, i.e., with respect to \textit{the first eigenvalue} solutions, which are not equal to zero a. e., of the $p(x)$-Laplace equation forms an one dimensional subset. Furthermore, by developing Moser method we obtained some results concerning Hölder continuity and bounded properties of the solutions. Our works are done in the setting of the Generalized-Sobolev Space. There are many perfect results about $p$-Laplace equations, but about $p(x)$-Laplace equation there are few results. The main reason is that a lot of methods which are very useful in dealing with $p$-Laplace equations are no longer valid for $p(x)$-Laplace equations. In this paper, many results are obtained by imposing some conditions on $p(x)$. Stimulated by the development of the study of elastic mechanics, interest in variational problems and differential equations has grown in recent decades, while Laplace equations with nonstandard growth conditions share a part. The equation discussed in this paper is derived from the elastic mechanics.

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