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

Chemmam, Rym; Mâagli, Habib; Masmoudi, Syrine; Zribi, Malek

Adv. Nonlinear Anal. 1, No. 4, 301-318 (2012).

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.

MSC:

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

Keywords:

Green function; asymptotic behavior; Dirichlet problem; subsolution; supersolution

References:

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