Solvability of nonlinear problem for some second-order nonstrongly elliptic system

Abstract

In this work, we investigate the second-order nonlinear elliptic system defined on an unbounded domain and with variable nonconstant coefficients. We establish the existence of the solution of the nonlinear problem for a second-order nonstrongly elliptic system by using a coercive estimate, which is obtained for the linear case. The nonlinear elliptic system is transformed into an equivalent fixed point problem for a suitable nonlinear operator in a convex subset of a Banach function space. Using the topological fixed-point approach and embedding theorems, the conditions for solvability of the semiperiodical nonlinear problem in a weighted Sobolev space are obtained.