# Article

 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 . Category: math . 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 . Date available: 2008-06-06T22:44:45Z Last updated: 2012-05-10 Stable URL: http://hdl.handle.net/10338.dmlcz/107925 . 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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