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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
\[ \begin{cases} -\Delta_pu= a(x)u^m+\lambda b(x)u^n \quad\text{in }\mathbb R^N,\\ u>0\quad\text{in }\mathbb R^N, \qquad u(x)@>|x|\to\infty>>0, \end{cases} \]
where \(-\infty<m<p-1<n\) with \(1<p<N\); \(a,b\geq 0\), \(a,b\neq 0\) are locally Hölder continuous functions and \(\lambda\geq 0\) is a real parameter. The main purpose of this paper, in short, is to complement the principal theorem of B. Xu and 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 Boundary value problems for second-order elliptic equations
35J20 Variational methods for second-order elliptic equations
35J67 Boundary values of solutions to elliptic equations and elliptic systems
35B09 Positive solutions to PDEs

Citations:

Zbl 1139.35348
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References:

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