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Radial solutions of an elliptic equation with singular nonlinearity. (English) Zbl 1163.35016
Summary: For the equation
$-\Delta u+u^{-\beta}= u^p, \qquad u>0\quad\text{in }B_R, \qquad u=0\quad\text{on }\partial B_R,$
where $$B_R\subseteq\mathbb R^N$$, $$0<\beta<1$$ and $$1<p< \frac{N+2}{N-2}$$ if $$N\geq 3$$, $$1<p<+\infty$$ if $$N=2$$, we show that there is $$\overline{R}>0$$ such that a radial solution $$u_R$$ exists if and only if $$0<R\leq\overline{R}$$. It is unique in the class of radial solutions and $$u_R'(R)<0$$ if $$R<\overline{R}$$, while $$u_{\overline{R}}'(\overline{R})=0$$. We also give a variational characterization of $$u_{\overline{R}}$$.

##### MSC:
 35J60 Nonlinear elliptic equations 35J20 Variational methods for second-order elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
##### Keywords:
 [1] Chen, H., On a singular nonlinear elliptic equation, Nonlinear anal., 29, 337-345, (1997) · Zbl 0882.35050 [2] Chen, C.C.; Lin, C.S., Uniqueness of the ground state solutions of $$\operatorname{\Delta} u + f(u) = 0$$ in $$\mathbf{R}^n$$, $$n \geqslant 3$$, Comm. partial differential equations, 16, 8-9, 1549-1572, (1991) · Zbl 0753.35034 [3] Coffman, C.V., Uniqueness of the ground state solution for $$\operatorname{\Delta} u - u + u^3 = 0$$ and a variational characterization of other solutions, Arch. ration. mech. anal., 46, 81-95, (1972) · Zbl 0249.35029 [4] Cortázar, C.; Elgueta, M.; Felmer, P., On a semilinear elliptic problem in $$\mathbf{R}^N$$ with a non-Lipschitzian nonlinearity, Adv. differential equations, 1, 2, 199-218, (1996) · Zbl 0845.35031 [5] García-Huidobro, M.; Manásevich, R.; Serrin, J.; Tang, M.; Yarur, C.S., Ground states and free boundary value problems for the n-Laplacian in n dimensional space, J. funct. anal., 172, 1, 177-201, (2000) · Zbl 0996.35019 [6] Gidas, B.; Ni, W.M.; Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. math. phys., 68, 3, 209-243, (1979) · Zbl 0425.35020 [7] Hirano, N.; Shioji, N., Existence of positive solutions for singular Dirichlet problems, Differential integral equations, 14, 1531-1540, (2001) · Zbl 1021.35031 [8] Kawohl, B., Rearrangements and convexity of level sets in PDE, Lecture notes in math., vol. 1150, (1985), Springer-Verlag Berlin · Zbl 0593.35002 [9] Kolodner, I.I., Heavy rotating string—A nonlinear eigenvalue problem, Comm. pure appl. math., 8, 395-408, (1955) · Zbl 0065.17202 [10] Kwong, M.K., Uniqueness of positive solutions of $$\operatorname{\Delta} u - u + u^p = 0$$ in $$\mathbf{R}^n$$, Arch. ration. mech. anal., 105, 3, 243-266, (1989) · Zbl 0676.35032 [11] Kwong, M.K.; Zhang, L.Q., Uniqueness of the positive solution of $$\operatorname{\Delta} u + f(u) = 0$$ in an annulus, Differential integral equations, 4, 3, 583-599, (1991) · Zbl 0724.34023 [12] McLeod, K.; Serrin, J., Uniqueness of positive radial solutions of $$\operatorname{\Delta} u + f(u) = 0$$ in $$\mathbf{R}^n$$, Arch. ration. mech. anal., 99, 2, 115-145, (1987) · Zbl 0667.35023 [13] Ni, W.; Nussbaum, R.D., Uniqueness and nonuniqueness for positive radial solutions of $$\operatorname{\Delta} u + f(u, r) = 0$$, Comm. pure appl. math., 38, 1, 67-108, (1985) · Zbl 0581.35021 [14] T. Ouyang, J. Shi, M. Yao, Exact multiplicity and bifurcation of solutions of a singular equation, preprint [15] Pucci, P.; García-Huidobro, M.; Manásevich, R.; Serrin, J., Qualitative properties of ground states for singular elliptic equations with weights, Ann. mat. pura appl. (4), 185, Suppl., S205-S243, (2006) · Zbl 1115.35050 [16] Serrin, J.; Tang, M., Uniqueness of ground states for quasilinear elliptic equations, Indiana univ. math. J., 49, 3, 897-923, (2000) · Zbl 0979.35049 [17] Yanagida, E., Uniqueness of positive radial solutions of $$\operatorname{\Delta} u + g(r) u + h(r) u^p = 0$$ in $$\mathbf{R}^n$$, Arch. ration. mech. anal., 115, 3, 257-274, (1991) · Zbl 0737.35026