Results On Existence And Regularity Of Solutions For Some Quasilinear Equations

In this paper, we consider the existence, regularity and decay of weak solutionsfor the following p&q-Laplace equations on R^N,where 1 < q < p < N and N is the dimension of the space.This equation comes from physics and related sciences, such as biophysics, plasmaphysics, and chemical reaction design. We will consider some properties about the weaksolutions of the equation from the view of PDE. We divided this paper into three parts.In Chapter II, we consider the existence of a positive solution for equation (2) onRN. We prove that when f(x,u) = g(x,u) - m|u|^p-2u - n|u|^q-2u, where m,n > 0 andg(x,u) satisfies some kind of asymptotically linear about u(see condition (C2)), thereexists a positive solution for equation (2). A type di-culty for this problem in RN isthe lack of compactness of the Sobolev imbedding. On the other hand, since f(x,u)satisfies condition (C2), the (AR)-condition does not hold. We follow the frame work ofG.B.Li&H.S.Zhou [46], by Vanishing, Nonvanishing and variable scaling we prove theboundedness of the corresponding (PS)-sequences, then get the existence of a positivesolution.In Chapter III, we prove the C^1,α regularity of the weak solution of equation (2).As the classical results in [18][40][62] could not be applied to the p&q-Laplace typeequation, careful analysis is needed. We follow the frame work of E.DiBenedetto[18]and P.Tolksdorf [62] to prove the weak solutions of equation (2) are of C1,a if f(x,u)and u are bounded with the help of some results in [40]. And by Morse iteration,we prove that the solutions are of C1,a even if f(x,u) is of critical growth. By usingkinds of test functions to cut o- some parts of u, we successfully compare p-Laplacewith q-Laplace in di-erent places to overcome the di-culties induced by both p andq-Laplacian equation.In Chapter IV, we prove that weak solutions of equation (2) decay exponentiallyat infinity, where f(x,u) = g(x,u) - m|u|^p-2u - n|u|^q-2u. To overcome the di-culties caused by both p and q-Laplace, we use a couple of test functions to get a couple ofinequalities, then use them to get the boundedness of the gradient of weak solutionof equation (2). Then we follow the frame work of G.B.Li&S.S.Yan [44] to get theexponentially decay of solutions of equation (2).Our results are new to our knowledge, they are the generalization of the results of[18][40][44][46][62].