Classification of positive solutions of p -Laplace equation with a growth term

Matteo Franca

Archivum Mathematicum (2004)

  • Volume: 040, Issue: 4, page 415-434
  • ISSN: 0044-8753

Abstract

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We give a structure result for the positive radial solutions of the following equation: Δ p u + K ( r ) u | u | q - 1 = 0 with some monotonicity assumptions on the positive function K ( r ) . Here r = | x | , x ∈ ℝ n ; we consider the case when n > p > 1 , and q > p * = 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 → 0 and as r → ∞ . 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 Δ for the p -Laplacian Δ p in the preceding equations.

How to cite

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Franca, Matteo. "Classification of positive solutions of $p$-Laplace equation with a growth term." Archivum Mathematicum 040.4 (2004): 415-434. <http://eudml.org/doc/249305>.

@article{Franca2004,
abstract = {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.},
author = {Franca, Matteo},
journal = {Archivum Mathematicum},
keywords = {$p$-Laplace equations; radial solution; regular/singular ground state; Fowler inversion; invariant manifold; radial solution; Fowler inversion; invariant manifold},
language = {eng},
number = {4},
pages = {415-434},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Classification of positive solutions of $p$-Laplace equation with a growth term},
url = {http://eudml.org/doc/249305},
volume = {040},
year = {2004},
}

TY - JOUR
AU - Franca, Matteo
TI - Classification of positive solutions of $p$-Laplace equation with a growth term
JO - Archivum Mathematicum
PY - 2004
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 040
IS - 4
SP - 415
EP - 434
AB - 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.
LA - eng
KW - $p$-Laplace equations; radial solution; regular/singular ground state; Fowler inversion; invariant manifold; radial solution; Fowler inversion; invariant manifold
UR - http://eudml.org/doc/249305
ER -

References

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  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. (1993) MR1220836
  11. Kawano N., Yanagida N., Yotsutani S., Structure theorems for positive radial solutions to div ( | D u | m - 2 D u ) + K ( | x | ) u q = 0 in ℝ n , J. Math. Soc. Japan 45 (1993), 719–742. (1993) MR1239344
  12. Ni W. M., Serrin J., Nonexistence theorems for quasilinear partial differential equations, Rend. Circ. Mat. Palermo (2) Suppl. 8 (1985), 171–185. (1985) MR0881397

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