Estimates for eigenvalues of nonlinear elliptic systems in certain domains of $$\mathbb {R}^N$$

Abstract This paper establishes lower and upper bounds for the radial eigenvalues of nonlinear elliptic systems defined in appropriate annular domains of $$\mathbb {R}^N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:math> . For the problem, we will prove a result in the sense of a conjecture proposed by Nápoli and Pinasco in the paper Estimates for eigenvalues of quasilinear elliptic systems , J. Diff. Eq. 227 (2006), 102–115, when we are in certain domains of $$\mathbb {R}^N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:math> . Moreover, we obtain a hyperbolic type function defining a region that contains all the generalized eigenvalues (whether variational or not), and with general hypotheses, the proof is carried out via ABP estimate for the p-Laplacian operator.