New result on Lane-Emden conjecture with exponents in a disk

Abstract

We obtain new results on Lane-Emden conjecture to prove a Liouville theorem for the Lane-Emden system $$\{\begin{array}{ll}-\Delta u=v^p,& x\in {\mathbb{R}}^N,\\ -\Delta v=u^q,& x\in {\mathbb{R}}^N,\end{array}$$ with $N\ge 3$, $p,q>0$, $pq>1$, and \[ \left(p-\frac2{N-2}\right)^2+ \left(q-\frac2{N-2}\right)^2\le\frac{2N^2}{(N-2)^2}, \] the system has no positive classical solution as $(p,q)$ in the disk unless $p=q=(N+2)/(N-2)$, no growth or decay assumptions are imposed, and the disk is tangent to the critical hyperbola. Especially, for $N\ge 5$, the disk contains a nonempty open set not covered by the criteria of Busca-Manásevich~\cite{BM} and Souplet~\cite{Souplet}, or by the explicit conditions of Li-Li-Wei~\cite{LLW}. After reducing the exponent range by known Liouville criteria, we first construct vector fields to derive a divergence inequality, then apply Sobolev embedding theorem, Gagliardo-Nirenberg interpolation inequality, and Young's inequality to give a uniform local integral bound ${\int }_{B_1}(u^r{v}^{2p}+u^{2q}{v}^s)dx\le C(N,p,q)$ for suitable weights for any positive solution $(u,v)$, finally the nonexistence of positive classical solutions follows by scaling.