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Infinite semipositone problems with asymptotically linear growth forcing terms. (English) Zbl 1274.35089

Let \(\Omega \) be a bounded domain in \(\mathbb {R}^N\), let \(p\in (1,\infty )\), \(\alpha \in (0,1)\) and \(\lambda >0\), and let \(f:[0,\infty [\rightarrow \mathbb {R}\) be a continuous function asymptotically \(p\)-linear at \(\infty \). Consider the following boundary value problem involving a singular elliptic equation \[ \Delta _p u=\lambda f(u)-u^{-\alpha }\;\text{ in } \Omega ,\; \; u_{\mid \partial \Omega }=0, \] where \(\Delta _p\) is the \(p\)-Laplacian operator.
Using a sub-supersolutions method, the authors prove that there exists a compact interval \(I\subset (0,\infty )\) such that for every \(\lambda \in I\), the above problem admits at least one positive solution. The same result is also established in the case in which \(\Omega \) is the exterior of a ball, and the nonlinearity \(\lambda f(u)-u^{-\alpha }\) is multiplied by a continuous radially symmetric function \(K\: \Omega \rightarrow \mathbb {R}\) which vanishes at \(\infty \). In this case, the authors prove the existence of a positive radially symmetric solution.
Finally, with some additional conditions on the nonlinearities, the above results are extended to systems.

MSC:

35J25 Boundary value problems for second-order elliptic equations
35J75 Singular elliptic equations
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
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