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.

Keywords:

• Nonlinear elliptic equations
• singular elliptic equations
• weighted Sobolev space
• second-order elliptic systems
• boundary value problems for second-order elliptic systems

AMS Subject Classifications:

• 35J47
• 35J50
• 35J57
• 35J60
• 35J75

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author is supported by the project No. AP08855402 “Boundary value problems for systems of differential equations on geometric graphs and their application in the calculations of compounds of elastic thin rods” and the second author is supported by the projects No. AP08856281 “Nonlinear elliptic equations with unbounded coefficients” and No. AP05131649 “Elliptic equations with drift: regularity and approximate properties of solutions” from the Ministry of Science and Education of the Republic of Kazakhstan.