On a Class of Chemically Reacting Systems Involving Nonlocal Operators

G.A. Afrouzi, A. Firouzjai, S. Shakeri, S. Talebi

Abstract

In this article, using the sub and supersolutions method, we study the existence of a positive solution for a class of Kirchhoff type systems with singular weights.The concepts of sub- and super-solution were introduced by Nagumo [M. Nagumo, \"Uber die differentialgleichung $y'' = f (x, y, y')$, Proceedings of the Physico-Mathematical Society of Japan 19 (1937) 861--866] in 1937 who proved, using also the shooting method, the existence of at least one solution for a class of nonlinear Sturm-Liouville problems. In fact, the premises of the sub- and super-solution method can be traced back to Picard. He applied, in the early 1880s, the method of successive approximations to argue the existence of solutions for nonlinear elliptic equations that are suitable perturbations of uniquely solvable linear problems. This is the starting point of the use of sub- and super-solutions in connection with monotone methods. Picard's techniques were applied later by Poincar\'e [H. Poincar\'e,  Les fonctions fuchsiennes et l'\'equation $\Delta u = e^u$, J. Math. Pures Appl. 4 (1898) 137--230] in connection with problems arising in astrophysics. We refer to [V. R\u adulescu, Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations: Monotonicity, Analytic, and Variational Methods, Contemporary Mathematics and Its Applications, Vol. 6, Hindawi Publishing Corporation, New York, 2008].