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The existence and non-existence of positive solutions to a singular quasilinear elliptic problem in \(\mathbb R^N\). (English) Zbl 1168.35354

Summary: This paper deals with the existence and nonexistence of entirely positive solutions of the quasilinear elliptic equation \(-\varDelta _pu=\rho a(x)g(u)+\lambda b(x)f(u)\) in \(\mathbb R^N,1<p<N,N\geq 3\), with \(u(x)\rightarrow 0\) as \(|x|\rightarrow \infty \). Here, either \(g\) or \(f\) (or both of them) are singular at 0 in the sense that \(g(t), f(t)\rightarrow \infty \) as \(t\rightarrow 0\). By using a perturbation method which eliminates the singularity on a ball with radius \(R\) and then letting \(R\) tend to infinity, with help of the bounded super-solution of the original problem, we obtain the existence of a weak solution of the problem. The main results of this paper improve the corresponding results of D.-P. Covei [Nonlinear Anal., Theory Methods Appl. 69, No. 8 (A), 2615–2622 (2008; Zbl 1157.35366)] and J. V. Goncalves and C. A. Santos [Nonlinear Anal., Theory Methods Appl. 66, No. 9 (A), 2078–2090 (2007; Zbl 1162.35030)].

MSC:

35J60 Nonlinear elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35J20 Variational methods for second-order elliptic equations
35B20 Perturbations in context of PDEs
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