IUMJ

Title: Critical exponents for uniformly elliptic extremal operators

Authors: Patricio L. Felmer and Alexander Quaas

Issue: Volume 55 (2006), Issue 2, 593-630

Abstract:

In this article we present the analysis of critical exponents for a large class of extremal operators, in the case of radially symmetric solutions. More precisely, for such an operator $\mathcal{M}$, we consider the nonlinear equation $$\mathcal{M}(D^2 u) + u^p = 0,\quad u > 0 \ \mbox{in }\mathbb{R}^N\tag(*)$$ and we prove the existence of a critical exponent $p^*$ that determines the range of $p > 1$ for which we have existence or non-existence of a positive radial solution to (*). In the case of maximal operators, we define two dimension-like numbers $N_{\infty}$ and $N_0$, depending on $\mathcal{M}$ and $N$, that satisfy $0 < N_{\infty} \le N_{0}$. We prove that our critical exponent satisfies $$\max\left\{{{N_{\infty}}\over{N_{\infty}-2}},p_0\right\} \le p^* \le p_{\infty},$$ where $p_0 = (N_0 + 2)/(N_0 - 2)$ and $p_{\infty} = (N_{\infty} + 2)/(N_{\infty} - 2)$. In the non-trivial case, $N_{\infty} < N_0$ and both inequalities above are strict.