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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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