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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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##### References:
 [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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