On the Lane–Emden conjecture with convolution part

Abstract We study the Hartree type Lane–Emden conjecture, which states the nonexistence of the positive classical solutions for the following Hartree type system where , and satisfy We use elliptic methods to provide partially positive answers as an extension of the classical Lane–Emden conjecture. We also identify the regions where nonexistence occurs in relation to this nonlocal problem. The theorems in the present paper extend the results for the famous Lane–Emden conjecture studied in the references (Mitidieri, Comm. Partial Differential Equations , 18 (1993), 125–151; Serrin and Zou, Differential Integral Equations , 9(1996), 635–653).