# Existence of solutions of semilinear systems with gradient dependence via eigenvalue criteria

## TL;DR

This paper develops new eigenvalue-based criteria for the existence of positive radial solutions to semilinear elliptic systems with gradient dependence, expanding the theoretical understanding of such systems.

## Contribution

It introduces novel eigenvalue criteria linked to nonlinearities and linear operators, providing a new approach to establish solution existence for gradient-dependent elliptic systems.

## Key findings

- Established new eigenvalue criteria for solution existence.
- Linked nonlinear bounds to principal eigenvalues of integral operators.
- Provided conditions ensuring solutions in specific function cones.

## Abstract

In this paper new criteria are established for the existence of positive radial solutions of a semilinear elliptic system depending on the gradient. These criteria are determined by some relationships between the upper and lower bounds on suitable stripes of $\mathbb R^n$ of the nonlinearities of the system and the principal characteristic values of some associated linear Hammerstein integral operators. Moreover, using smoothing tools, the totality of the involved cone is established.