Title: | Classification of positive solutions of $p$-Laplace equation with a growth term (English) |

Author: | Franca, Matteo |

Language: | English |

Journal: | Archivum Mathematicum |

ISSN: | 0044-8753 (print) |

ISSN: | 1212-5059 (online) |

Volume: | 40 |

Issue: | 4 |

Year: | 2004 |

Pages: | 415-434 |

Summary lang: | English |

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Category: | math |

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Summary: | We give a structure result for the positive radial solutions of the following equation: \[ \Delta _{p}u+K(r) u|u|^{q-1}=0 \] with some monotonicity assumptions on the positive function $K(r)$. Here $r=|x|$, $x \in {\mathbb R}^n$; we consider the case when $n>p>1$, and $q >p_* =\frac{n(p-1)}{n-p}$. We continue the discussion started by Kawano et al. in [KYY], refining the estimates on the asymptotic behavior of Ground States with slow decay and we state the existence of S.G.S., giving also for them estimates on the asymptotic behavior, both as $r \rightarrow 0$ and as $r \rightarrow \infty $. We make use of a Emden-Fowler transform which allow us to give a geometrical interpretation to the functions used in [KYY] and related to the Pohozaev identity. Moreover we manage to use techniques taken from dynamical systems theory, in particular the ones developed in [JPY2] for the problems obtained by substituting the ordinary Laplacian $\Delta $ for the $p$-Laplacian $\Delta _{p}$ in the preceding equations. (English) |

Keyword: | $p$-Laplace equations |

Keyword: | radial solution |

Keyword: | regular/singular ground state |

Keyword: | Fowler inversion |

Keyword: | invariant manifold |

MSC: | 35B05 |

MSC: | 35H30 |

MSC: | 35J60 |

MSC: | 37D10 |

idZBL: | Zbl 1110.37018 |

idMR: | MR2129963 |

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Date available: | 2008-06-06T22:44:45Z |

Last updated: | 2012-05-10 |

Stable URL: | http://hdl.handle.net/10338.dmlcz/107925 |

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Reference: | [1] Battelli F., Johnson R.: Singular ground states of the scalar curvature equation in ${\mathbb{R}}^n$.Differential Integral Equations 14 (2000), 123–139. MR 1797383 |

Reference: | [2] Battelli F., Johnson R.: On positive solutions of the scalar curvature equation when the curvature has variable sign.Nonlinear Anal. 47 (2001), 1029–1037. Zbl 1042.34550, MR 1970715 |

Reference: | [3] Battelli F., Johnson R.: On transversal smoothness of invariant manifold.Comm. Appl. Nonlinear Anal. 5 (2001), 383–401. MR 1847301 |

Reference: | [4] Damascelli L., Pacella F., Ramaswamy M.: Symmetry of ground states of $p$-Laplace equations via the Moving Plane Method.Arch. Rational Mech. Anal. 148 (1999), 291–308. Zbl 0937.35050, MR 1716666 |

Reference: | [5] Franca M.: Some results on the $m$-Laplace equations with two growth terms.preprint. Zbl 1097.34020, MR 2157785 |

Reference: | [6] Gazzola F.: Critical exponents which relate embedding inequalities with quasilinear elliptic operator.preprint. |

Reference: | [7] Gazzola F., Serrin J., Tang M.: Existence of ground states and free boundary problem for quasilinear elliptic operators.Adv. Differential Equations 5 (2000), 1–30. MR 1734535 |

Reference: | [8] Johnson R.: Concerning a theorem of Sell.J. Differential Equations 30 (1978), 324–339. MR 0521857 |

Reference: | [9] Johnson R., Pan X. B., Yi Y. F.: The Melnikov method and elliptic equation with critical exponent.Indiana J. Math. 43 (1994), 1045–1077. MR 1305959 |

Reference: | [10] Johnson R., Pan X. B., Yi Y. F.: Singular ground states of semilinear elliptic equations via invariant manifold theory.Nonlinear Anal. 20 (1993), 1279–1302. MR 1220836 |

Reference: | [11] Kawano N., Yanagida N., Yotsutani S.: Structure theorems for positive radial solutions to $\operatorname{div}(|Du|^{m-2}Du)+K(|x|)u^q=0$ in $\mathbb{R}^n$.J. Math. Soc. Japan 45 (1993), 719–742. MR 1239344 |

Reference: | [12] Ni W. M., Serrin J.: Nonexistence theorems for quasilinear partial differential equations.Rend. Circ. Mat. Palermo (2) Suppl. 8 (1985), 171–185. MR 0881397 |

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