Necessary Conditions for Existence Results of Some Integral System
Abstract
In this paper, we give some necessary conditions for the existence of positive solutions for integral systems.
1. Introduction
Theorem 1. Suppose that λ > 0 and problem (1) possesses a C1 positive solution; then
As for λ < 0, we have the following nonexistence result.
Theorem 2. If −N < λ < 0, then problem (1) possesses no C1 positive solution provided that p < 0 or q < 0.
This paper is organized as follows. We prove Theorem 1 in Section 2. The proof of Theorem 2 is completed in Section 3.
2. Proof of Theorem 1
We first claim that u(x) ∈ Lq+1(ℝN) and v(x) ∈ Lp+1(ℝN).
3. Proof of Theorem 2
Acknowledgments
The first author is supported by SZPP (no. KQCX20120802140634893), Guangdong S&T Major Project (no. 2012A080104014), and SRF for ROCS, SEM. Xiaohui Yu is supported by NSFC no. 11101291. The authors would like to thank the anonymous referees for their valuable suggestions.