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On critical exponents for the Pucci’s extremal operators. (English) Zbl 1274.35115
Summary: In this article we study some results on the existence of radially symmetric non-negative solutions for the nonlinear elliptic equation
\[ \mathcal{M}_{\lambda,\Lambda}^+(D^2u)+u^p=0\quad\text{ in }\;\mathbb{R}^N.\qquad\qquad\qquad\qquad(*) \]
Here \(N\geq 3\), \(p>1\) and \(\mathcal{M}_{\lambda,\Lambda}^+\) denotes the Pucci’s extremal operators with parameters \(0<\lambda\leq\Lambda\). The goal is to describe the solution set in function of the parameter \(p\). We find critical exponents \(1<p_*^+<p_+^p\) that satisfy: (i) If \(1<p<p_+^*\) then there is no non-trivial radial solution of \((*)\). (ii) If \(p=p_+^*\) then there is a unique fast decaying radial solution of \((*)\). (iii) If \(p_+^*<p\leq p_+^p\) then there is a unique pseudo-slow decaying radial solution to \((*)\). (iv) If \(p_+^p<p\) then there is a unique slow decaying radial solution to \((*)\). Similar results are obtained for the operator \(\mathcal{M}_{\lambda,\Lambda}^-\).

MSC:
35J60 Nonlinear elliptic equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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