Removable Singularities Of Solutions For Several Kinds Of Elliptic Equations Involving Variable Exponent

The removable singularities of solutions for partial differential equations come from many practical problems in physics and geometry. Therefore, domestic and overseas scholars pay high attention to the problems of singularities for solutions. Recently, most problems on the singularities of solutions are studied based on the classical Lebesgue and Sobolev spaces, but, in recent years, there are many problems with nonstandard growth, for instance, the electrorheological fluid, non-elastic mechanics, image processing models and so on. In the study of these problems, the theory of the constant exponent Lebesgue and Sobolev spaces are no longer applicable. Hence, it is necessary to study the singu-larities of solutions for the partial differential equations based on the theory of variable exponent Lebesgue and Sobolev spaces.In this thesis, we study the removable singularities of solutions for several kinds of elliptic equations based on the theory of variable exponent Lebesgue and Sobolev spaces.Firstly, the removability of isolated singular points for a class of nonlinear elliptic equations involving variable exponent is studied. The behavior of solutions near the iso-lated singular points is obtained by a new iterative technique and the L∞ estimate is given by constructing a logarithmic type test function based on the property of solution. Then, the sufficiency conditions for removability of isolated singular points for this equation are obtained and we prove that the singular points are removable under some conditions.Secondly, based on the theory of variable exponent Lebesgue and Sobolev spaces, the removability of 0 singular point for a class of divergence form elliptic equations with power degeneration in absorption term is studied. The behavior of solutions in a neigh-borhood of 0 singular point is obtained by constructing suitable test function and using Moser’s iteration method. Then, on this basis, the local boundedness of the solutions near 0 singular point is given by taking a logarithmic type test function and we prove that the singular point 0 is removable under some certain conditions.Thirdly, the removability of capacity zero singular sets for a class of nonlinear el-liptic equations with p(x)-growth is studied. We study the properties of solutions in a neighborhood of every point in a compact set. The Caccioppoli estimate is obtained by constructing suitable test functions. On this basis, the bounedness of the solutions in a neighborhood of a compact singular set is given. Then, we get the conditions for the removability of capacity zero singular sets.Last, the removability of compact singular sets for Holder continuous solutions of general elliptic equations involving variable exponent is studied, the sufficiency condition-s for removability of compact singular sets are given by the theory of obstacle problems and negative Radon measure. The existence of solutions to the obstacle problems is ob-tained by using the method of monotone operators. Then we prove Harnack estimates and the continuity of solutions for obstacle problems when the obstacle itself is continuous. On these basis, we get that the compact singular sets are removable for Holder continuous solutions when the certain Hausdorff measure of the compact sets equal zero by using the properties of the negative Radon measure.