Existence of positive solutions for a semipositone \(p\)-Laplacian problem (Q2832273)

The authors prove existence of positive weak solutions to the Dirichlet problem for the \(p\)-Laplacian NEWLINE\[NEWLINE \begin{cases} -\Delta_p u=\lambda f(u) & \text{in}\;\Omega,\\ u=0 & \text{on}\;\partial\Omega, \end{cases} NEWLINE\]NEWLINE on a smooth bounded domain \(\Omega \subset\mathbb{R}^N,\) \(N>2,\) where \(p>2.\) The function \(f: \mathbb{R}\to \mathbb{R}\) is assumed to be differentiable, \(f(0)<0\) (the semipositone case) and such that there exist \(q\in (p-1, Np/(N-p)-1),\) \(A>0,\) \(B>0\) to have NEWLINE\[NEWLINE \begin{cases} A(u^q-1)\leq f(u)\leq B(u^q+1) & \text{for}\;u>0,\\ f(u)=0 & \text{for}\;u\leq -1. \end{cases} NEWLINE\]NEWLINE An Ambrosetti-Rabinowitz type condition is also required, that is, there exist \(\theta>p\) and \(M\in\mathbb{R}\) such that NEWLINE\[NEWLINE uf(u)\geq \theta F(u)+M,\quad F(u)=\int_0^u f(s)ds. NEWLINE\]NEWLINE The proofs rely on mountain pass arguments, comparison and regularity principles and \textit{a~priori} estimates.