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Non-existence and existence of entire solutions for a quasi-linear problem with singular and super-linear terms. (English) Zbl 1189.35104
Summary: We establish results concerning non-existence and existence of entire positive solutions for the nonlinear elliptic problem $$\cases -\Delta_pu= a(x)u^m+\lambda b(x)u^n \quad\text{in }\Bbb R^N,\\ u>0\quad\text{in }\Bbb R^N, \qquad u(x)@>|x|\to\infty>>0, \endcases$$ where $-\infty<m<p-1<n$ with $1<p<N$; $a,b\ge 0$, $a,b\ne 0$ are locally Hölder continuous functions and $\lambda\ge 0$ is a real parameter. The main purpose of this paper, in short, is to complement the principal theorem of {\it B. Xu} and {\it Z. Yang} [Bound. Value Probl. 2007, Article ID 16407, 8 p. (2007; Zbl 1139.35348)] showing existence and non-existence of solutions for the above problem for $\lambda>0$ appropriately.

##### MSC:
 35J61 Semilinear elliptic equations 35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian 35J25 Second order elliptic equations, boundary value problems 35J20 Second order elliptic equations, variational methods 35J67 Boundary values of solutions of elliptic equations 35B09 Positive solutions of PDE
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