Positive Solutions of Indefinite Semipositone Elliptic Problems

We are concerned with the parametrized family of problems

$$\begin{aligned} \left\{ \begin{aligned} \begin{array}{ll} {\mathcal {L}} u=\lambda a(x)(f(u)-l),\ \ \ \ \ &{}x\in \Omega ,\\ u=0, \ \ \ \ {} &{}x\in \partial \Omega ,\\ \end{array} \end{aligned} \right. \end{aligned}$$(P)

where \(\Omega \) is a bounded domain of \({\mathbb {R}}^N~(N\ge 3)\) with regular boundary \(\partial \Omega ,~{\mathcal {L}}\) is a general second-order uniformly elliptic operator, \(\lambda ,~l>0\), \(a:{\overline{\Omega }}\rightarrow {\mathbb {R}}\) is a continuous function which may change sign, \(f:{\mathbb {R}}^+\rightarrow {\mathbb {R}}\) is subcritical and superlinear at infinity. Under some suitable conditions, we obtain there exists \(\lambda _0 > 0\) such that (P) has positive solutions for all \(0 < \lambda \le \lambda _0 \) by topological degree argument and a priori estimates. In doing so, we require f to be of regular variation at infinity.