# zbMATH — the first resource for mathematics

Combined effects of singular and superlinear nonlinearities in some singular boundary value problems. (English) Zbl 1109.35344
Summary: This paper deals with a class of singular semilinear elliptic Dirichlet boundary value problems $\begin{cases} \Delta u+\lambda u^{\beta}+p(x)u^{-\gamma}=0 \text{ in } \Omega,\\ u >0\text{ in }\Omega,\\ u =0 \text{ on } \partial\Omega, \end{cases}$ where $$\Omega$$ is a bounded domain in $$\mathbb R^{N}$$ with smooth boundary $$\partial\Omega$$, $$N\geq 3$$, $$p(x)$$ is a nontrival nonnegative $$L^2(\Omega)$$ function, and $$\beta,\gamma$$ are constants satisfying $$0<\gamma<1<\beta<2^*-1$$ with $$2^*:=2N/(N-2)$$ where the combined effects of a superlinear and a singular term allow us to establish some existence and multiplicity results.

##### MSC:
 35J60 Nonlinear elliptic equations 35J20 Variational methods for second-order elliptic equations 35J65 Nonlinear boundary value problems for linear elliptic equations
Full Text:
##### References:
 [1] Aubin, J.P.; Ekeland, I., Applied nonlinear anal., pure and applied mathematics, (1984), Wiley Interscience New York [2] Crandall, M.G.; Rabinowitz, P.H.; Tartar, L., On a Dirichlet problem with a singular nonlinearity, Comm. partial differential equations, 2, 193-222, (1977) · Zbl 0362.35031 [3] Struwe, M., Variational methods, (1990), Springer-Verlag New York/Berlin [4] Lair, A.V.; Shaker, A.W., Classical and weak solutions of a singular semilinear elliptic problem, J. math. anal. appl., 211, 193-222, (1997) [5] Lazer, A.C.; Mckenna, P.J., On a singular nonlinear elliptic boundary value problem, Proc. amer. math. soc., 111, 721-730, (1991) · Zbl 0727.35057 [6] del Pino, M.A., A global estimate for the gradient in a singular elliptic boundary value problem, Proc. roy. soc. Edinburgh sect. A, 122, 341-352, (1992) · Zbl 0791.35046 [7] Agarwal, R.P.; O’Regan, D., Singular boundary value problems for superlinear second order ordinary and differential equations, J. differential equations, 130, 333-355, (1996) · Zbl 0863.34022 [8] Tarantello, G., On nonhomogeneous elliptic equations involving critical soblev exponent, Ann. inst. H. Poincaré anal. non lineaire, 9, 281-304, (1992) · Zbl 0785.35046 [9] Shi, J.; Yao, M., On a singular nonlinear semilinear elliptic problem, Proc. roy. soc. Edinburgh sect. A, 128, 1389-1401, (1998) · Zbl 0919.35044 [10] Coclite, M.M.; Palmieri, G., On a singular nonlinear Dirichlet problem, Comm. partial differential equation, 14, 1315-1327, (1989) · Zbl 0692.35047 [11] Wiegner, M., A degenerate diffusion equation with a nonlinear source term, Nonlinear anal., 28, 1977-1995, (1997) · Zbl 0874.35061 [12] Kusano, T.; Swanson, C.A., Entire position solutions of singular semilinear elliptic equations, Japan J. math., 11, 145-155, (1985) · Zbl 0585.35034 [13] Edelson, A., Entire solutions of singular elliptic equations, J. math. anal. appl., 139, 523-532, (1989) · Zbl 0679.35003 [14] Shaker, A.W., On singular semilinear elliptic equations, J. math. anal. appl., 173, 222-228, (1993) · Zbl 0785.35032 [15] Diaz, J.I.; Morel, J.M.; Oswald, L., An elliptic equation with singular nonlinearity, Comm. partial differential equations, 12, 1333-1344, (1987) · Zbl 0634.35031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.