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

The main purpose of this paper is to show that there exists a positive number ${\lambda }_1$, the first eigenvalue, such that some $p(x)$-Laplace equation admits a solution if $\lambda ={\lambda }_1$ and that ${\lambda }_1$ is simple, i.e., with respect to $\mathit{\text{the}}\mathit{\text{first}}\mathit{\text{eigenvalue}}$ 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 H"{o}lder 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 The main reason is that a lot of methods which are very useful in dealing with $p$-Laplace equations are no longer valid for $p(x)$-Laplace equations.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.