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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.

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
Full Text: DOI
[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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