The Properties Of Positive Solution For A Class Of Quasilinear Elliptic Equations

The study of Laplace elliptic equations with nonstandard growth conditions is a very active research branch in differential equations. The study has extensive theoretical and practical importance.p ( x )- Laplace operators and p ( x )- Laplace equations are the generalizations of the corresponding p - Laplace operators and p - Laplace equations. Many researches indicate that problems with p ( x )- growth conditions and problems with standard p - growth conditions have many essential differences. There are many results for the study of elliptic equations with p ( x )- growth conditions and regularity of the corresponding variational problems, but the study of existence, positive solutions and decay does not have many results. That is because of the lack of many required framework and theories. The study of p ( x )- Laplace operators and p ( x )- Laplace equations is on the horizon. There are many important problems to be solved. In this paper we study the asymptotic behavior of the solutions for the elliptic equations. First, we turn to the quasilinear elliptic system whereΩ=N .Here we study the asymptotic behavior of positive radial solutions of p ( x )- Laplace equations near infinity. We obtain lim ( ) lim ( ) 0r→∞u r = r→∞v r= and we obtain the estimate of the positive solutions. Because of the nonhomogeneity of Laplace operator and the absent of many usual methods, our results have a distance from that of p ( x )-p - Laplace operators.