Existence of solutions for a quasilinear elliptic system with local nonlinearity on DOUBLE-STRUCK CAPITAL RN

In this paper, we investigate the existence of solutions for a class of quasilinear elliptic system. By developing the Moser iteration technique, we obtain that system has a nontrivial solution (u(lambda), v(lambda)) with ||(u(lambda), v(lambda))||(infinity) <= 2 for every lambda large enough when the nonlinear term F satisfies some growth conditions only in a circle with center 0 and radius 4, and the families of solutions {(u(lambda), v(lambda))} satisfy that ||(u(lambda), v(lambda)) || -> 0 as lambda -> infinity. Moreover, because the interaction of u and v in elliptic system causes that the estimate of ||u||(infinity) cannot vary with ||u||, the conclusion for the elliptic system is weaker than the corresponding result for the quasilinear elliptic equation, which is given in the end as a comparison.