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Some remarks on a singular elliptic boundary value problem. (English) Zbl 0940.35089
Let \(\Omega\) be a smooth bounded domain in \({\mathbb R}^N\) and let \(\beta\) and \(\gamma\) be positive parameters. The paper deals with the study of positive solutions of the problem \(\Delta u-u^{-\gamma}+\beta f(x)=0\) in \(\Omega\), under the Dirichlet boundary condition \(u=0\) on \(\partial\Omega\), where \(f\geq 0\) is a smooth non-trivial function. A first result of this paper asserts that the above problem has no solution, provided that \(\gamma\geq 1\).
J. I. Diaz, J. M. Morel and L. Oswald [Commun. Partial Differ. Equ. 12, 1333-1344 (1987; Zbl 0634.35031)] have proved that there exists \(\beta^*>0\) such that a solution of this problem exists for any \(\beta >\beta^*\), and the problem has no solution if \(\beta <\beta^*\). The authors also establish several results in the critical case \(\beta =\beta^*\). It is proved that the above problem has a solution if \(N\leq 2\). In the case \(N=1\), \(\gamma =1/2\) and \(f\) is a constant, the authors show that the solution is unique. However, multiple solutions are constructed when \(\gamma <1/3\). The last section of the paper contains several interesting open problems. The proofs are elementary and they are based on monotonicity methods for elliptic boundary problems.

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35B32 Bifurcations in context of PDEs
37K50 Bifurcation problems for infinite-dimensional Hamiltonian and Lagrangian systems
58J32 Boundary value problems on manifolds
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[1] Diaz, J.I.; Morel, J.M.; Oswald, L., An elliptic equation with singular nonlinearity, Comm. P.D.E., 12, 12, 1333-1344, (1987) · Zbl 0634.35031
[2] Amann, H.; Hess, P., A multiplicity result for a class of elliptic boundary value problems, (), 145-151 · Zbl 0416.35029
[3] Zhang, Z., On a Dirichlet problem with a singular nonlinearity, J. of mathematical analysis and applications, 194, 103-113, (1995) · Zbl 0834.35054
[4] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1983), Springer-Verlag · Zbl 0691.35001
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