Existence of solutions for a quasilinear elliptic system with local nonlinearity on ℝN

Abstract

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 (uλ, vλ) with ‖(uλ, vλ)‖∞ ≤ 2 for every λ 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 {(uλ, vλ)} satisfy that ‖(uλ, vλ) ‖ → 0 as λ → ∞. Moreover, because the interaction of u and v in elliptic system causes that the estimate of ‖u‖∞ 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.