Combined effects in nonlinear singular elliptic problems in a bounded domain. (English) Zbl 1277.31016

Summary: We establish an existence result of positive solutions to the following boundary value problem: \[ \Delta u + a_1(x)u^{\alpha_1} + a_2(x)u^{\alpha_2} = 0 \text{ in } \Omega, \quad u = 0 \text{ on } \partial \Omega \] where \(\Omega\) is a bounded \(C^{1, 1}\)-domain in \(\mathbb R^{n}, \alpha_1, \alpha_2 < 1\) and \(a_1, a_2\) are nonnegative functions in \(C^\gamma_{\text{loc}}(\Omega)\), \(0 < \gamma < 1\), satisfying some appropriate assumptions related to Karamata regular variation theory. We give estimates on such solutions where appear the combined effects of singular and sublinear terms in the nonlinearity.


31C15 Potentials and capacities on other spaces
34B27 Green’s functions for ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
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