Positive solution for the elliptic problems with sublinear and superlinear nonlinearities.

*(English)*Zbl 1216.35056Summary: This paper deals with the existence of positive solutions for the elliptic problems with sublinear and superlinear nonlinearities \(-\Delta u=\lambda a(x)u^p+ b(x)u^q\) in \(\Omega\), \(u>0\) in \(\Omega\), \(u=0\) on \(\partial\Omega\), where \(\lambda>0\) is a real parameter, \(0<p<1<q\). \(\Omega\) is a bounded domain in \(\mathbb R^N\) \((N\geq 3)\), and \(a(x)\) and \(b(x)\) are some given functions. By means of variational method and super-subsolution method, we obtain some results about existence of positive solutions.

##### MSC:

35J61 | Semilinear elliptic equations |

35J91 | Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian |

35B09 | Positive solutions to PDEs |

35A01 | Existence problems for PDEs: global existence, local existence, non-existence |

35D30 | Weak solutions to PDEs |

35J20 | Variational methods for second-order elliptic equations |

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\textit{C. Yuan} et al., Math. Probl. Eng. 2010, Article ID 640841, 10 p. (2010; Zbl 1216.35056)

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