WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 19, 2020
Existence and Global Behavior of Decaying Solutions of a Nonlinear Elliptic Equation
Authors: ,
Abstract: This paper is devoted to study the following radial equation $$(|u ′ |^ {p−2}u′)′ + \\frac{N − 1}{r} |u′|^{p−2}u′+α|u|^{q−1}u+βr(|u|^{q−1}u)′= 0, r > 0.$$ where $$p > 2, q > 1, N ≥ 1, α > 0 and β > 0$$. Our purpose is to give existence results of decaying solutions of the above equation and their asymptotic behavior near infinity. The study depends strongly of the sign of $$Nβ − α $$ and the comparison between $$\frac{α}{qβ}, \frac{p}{q + 1 − p} $$ and $$\frac{N − p}{p − 1}$$ More precisely, we prove that if $$Nβ −α > 0$$, there is a positive solution $$u$$ which has one of the following behaviors near infinity:
$$(i) u(r) _{+∞}^{∼}Lr^{− \frac{α}{qβ}} where L > 0 $$
$$(ii) u(r) _{+∞}^{∼} ( ( \frac{p − 1} {qβ})(q + 1 − p)(\frac{N − p} {p − 1} − \frac{α} {qβ}) (\frac {α} {qβ})^{p−1} )^ {\frac{1} {q+1−p}} r ^{− \frac{α} {qβ}} (ln r) ^ {\frac{1} {q+1−p}} .$$
$$(iii) u(r) _{+∞}^{∼} (\frac{(p-1)(\frac{p}{q + 1 − p})^{p−1}(\frac{N − p}{p − 1}-\frac{p}{q + 1 − p})}{α − qβ\frac{p}{q + 1 − p}})^{\frac{1}{q+1-p}}r^{\frac{−p}{q+1−p}}$$.
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Keywords: Porous medium equation, fast diffusion equation, radial self-similar solutions, shooting method, decaying solutions, energy function.
Pages: 662-675
DOI: 10.37394/23206.2020.19.73