Notions of sublinearity and superlinearity for nonvariational elliptic systems
This paper deals with \(n\times n\) elliptic systems of partial differential equations of the form \[ \left\{\begin{matrix} -\Delta \mathbf{u} =\text\textbf{f}(\text\textbf{u})\,\, \text{in } \Omega,\\ \text\textbf{u}=\text\textbf{0}\,\, \text{on } \partial \Omega.\end{matrix}\right. \] Here \(\Omega\subset {\mathbb R}^N\) is a bounded smooth domain, \(\mathbf{u }=(u_1,u_2,\cdots,u_n)\) is a vector function, and \(\mathbf{f}(\text\textbf{u})=(f_1(\text\textbf{u}),\cdots,f_n(\text\textbf{u}))\) is Hölder continuous on \({\mathbb R}^n_+=[0,\infty)^n\) such that \(f_i(\mathbf{0})=0\) for \(i=1,\cdots, n\). The author introduces the notions of sublinearity and superlinearity for such systems and provides conditions for the existence of positive solutions. These generalize earlier results on particular \(2\times 2\) systems by authors [\textit{Y. Liu}, On a class of semilinear elliptic systems, Proc. R. Soc. Edinb. 124 A, 609--620 (1994; Zbl 0806.35038)] and \textit{C. Alves} and \textit{D. G. Figueiredo} [''Nonvariational elliptic systems,'' Discrete Contin. Dyn. Syst. 8, No. 2, 289--302 (2002; Zbl 1162.35356)].
scientific article; zbMATH DE number 2177518