Blow-up rates of radially symmetric large solutions. (English) Zbl 1163.35015
Summary: This paper adapts a technical device going back to [J. López-Gómez,
J. Differ. Equations 224, No. 2, 385–439 (2006; Zbl 1208.35036)] to ascertain the blow-up rate of the (unique) radially symmetric large solution given through the main theorem of [J. López-Gómez, Discrete Contin. Dyn. Syst., Suppl. 2007, 677–686 (2007; Zbl 1163.35352)]. The requested underlying estimates are based upon the main theorem of [S. Cano-Casanova and J. López-Gómez, J. Differ. Equations 244, No. 12, 3180–3203 (2008; Zbl 1149.34020)]. Precisely, we show that if \(\Omega\) is a ball, or an annulus, \(f\in{\mathcal C}[0,\infty)\) is positive and non-decreasing, \(V\in {\mathcal C}[0,\infty)\cap{\mathcal C}^2(0,\infty)\) satisfies \(V(0)=0\), \(V'(u)>0\), \(V''(u)\geq 0\), for every \(u>0\), and \(V(u)\sim Hu^{p-1}\) as \(u\uparrow\infty\), for some \(H>0\) and \(p>1\), then, for each \(\lambda\geq 0\),
\[ -\Delta u=\lambda u- f\big(\text{dist}(x,\partial\Omega)\big)V(u)u \]
possesses a unique positive large solution in \(\Omega\), \(L\), which must be radially symmetric, by uniqueness, and we can estimate the exact blow-up rate of \(L(x)\) at \(\partial\Omega\) in terms of \(p\), \(H\) and \(f\).
\[ -\Delta u=\lambda u- f\big(\text{dist}(x,\partial\Omega)\big)V(u)u \]
possesses a unique positive large solution in \(\Omega\), \(L\), which must be radially symmetric, by uniqueness, and we can estimate the exact blow-up rate of \(L(x)\) at \(\partial\Omega\) in terms of \(p\), \(H\) and \(f\).
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