Hardy Type Inequality In Weighted Variable Exponent Spaces And Its Applications

The main goal of this graduate thesis for the master's degree is to investigate the Hardy type inequality in the weighted variable exponent Sobolev space. Meanwhile, we show an application of the derived inequality to the p(x)-Laplace type equations.In the thesis, the conditions for the one dimensional Hardy type inequality to hold are given first, and under such conditions the validity of Hardy type inequality is given too. Based on this result, the general N dimensional case is discussed. The relationships between the weight functions are stated, and the needed property of the weight function is investigated too. Meanwhile, some properties of the weighted variable exponent Sobolev spaces are given. An imbedding theorem in unbounded domain is proved. Finally, the existence of solutions of p(x) -Laplace type equations is proved, and it is an immediate application of the Hardy type inequality proved in this thesis.