Positive solutions of inhomogeneous elliptic equations with indefinite data (Q1883455)

The existence of positive solutions to the following problem is studied: \[ -\Delta = | u| ^{p-1}u + \lambda f(x), \quad x \in \Omega,\qquad u=0 \text{ on } \partial\Omega, \] where \(\Omega\) is a smooth bounded domain and \(f \in C^1(\bar\Omega)\backslash\{0\}\). The cases \(p \in (1,(n+2)/(n-2))\) and \(p \in (0,1)\) are considered. Under suitable assumptions on the classification of \(f\), it is shown that this problem has exactly one or two positive solutions. Results rely on a multiplicity result due to \textit{Q. Dai} and \textit{Y. Gu}, [Proc. R. Soc. Edindb. 133, No. 2, 297--306 (2003; Zbl 1035.35041)], variational methods, and the method of sub and super-solutions.