Existence Of Positive Solutions For A Class Of Singular P(x)-Laplacian Elliptic Equations

In this paper, we consider the existence and multiplicity of positive solutions for a class of singular quasilinear p(x)-Laplace equations in a bounded domainΩ(?)Rⁿ. Applying the theory of variable exponent Sobolev space and p(x)-Laplace equation, we solve critical points of the corresponding functional to obtain solutions of the equation. In this equation, as there is a singular term au^-r(x) when u=0 and the conditions of the nonlinear term f are weak, the corresponding functional is not well defined in the space W₀^1,p(x) (Ω), Combing with sub-supersolution method and truncation we can get the existence of a positive solution of the equation in this paper, then Applying Mountain Pass Lemma we may conclude the existence of anther positive solution. The main difficulty in this paper is the inhomogeneity of p(x)-Laplace operator, e.g. we can't use the method similar to the constant case when p(x) = p to obtain the existence of the weak sub-solution u, which is the basis of the proofs of the main results, i.e. Theorem 1.1, Theorem 1.2 and Theorem 1.3, in this paper. Because of the weak conditions of f, solutions in the sense of distribution may not be the weak ones of the equation, thus, we can only have the solutions of the equation in the sense of distribution in Theorem 1.1 and Theorem 1.2. However, the fact that f also satisfies usual subcritical growth condition may imply the equation have two weak ordered positive solutions in theorem 1.3.