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Existence and boundary behaviour of solutions for a nonlinear Dirichlet problem in the annulus. (English) Zbl 1413.31004

Summary: In this paper, we mainly study the following semilinear Dirichlet problem \(-\Delta u=q(x)f(u)\), \(u>0\), \(x\in\Omega\), \(u_{|\partial\Omega}=0\), where \(\Omega\) is an annulus in \(\mathbb{R}^n\), \((n\geq 2)\). The function \(f\) is nonnegative in \(\mathcal{C}^1(0,\infty)\) and \(q\in\mathcal{C}_{loc}^{\gamma}(\Omega)\), \((0<\gamma <1)\), is positive and satisfies some required hypotheses related to Karamata regular variation theory. We establish the existence of a positive classical solution to this problem. We also give a global boundary behavior of such solution.

MSC:

31B25 Boundary behavior of harmonic functions in higher dimensions
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B27 Green’s functions for ordinary differential equations
35J61 Semilinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35B09 Positive solutions to PDEs