Track new results across pure and applied mathematics.
Search papers, labs, and topics across Lattice.
This paper proves a Liouville theorem for the Lane-Emden system $-\Delta u=v^p, -\Delta v=u^q$ in $\mathbb{R}^N$ ($N\ge3$) for exponents $(p,q)$ within a specific disk tangent to the critical hyperbola, showing no positive classical solutions exist unless $p=q=(N+2)/(N-2)$. The proof involves constructing vector fields to derive a divergence inequality, followed by applications of Sobolev embedding, Gagliardo-Nirenberg interpolation, and Young's inequalities to establish a uniform local integral bound for any positive solution, ultimately leading to nonexistence via scaling. This result extends existing criteria for $N\ge5$ by covering a novel open set of exponents.
Positive solutions to the Lane-Emden system vanish for a broad class of exponents previously unaddressed, even without growth or decay assumptions.
We obtain new results on Lane-Emden conjecture to prove a Liouville theorem for the Lane-Emden system $$\begin{cases} -\Delta u=v^p,&x\in \mathbb{R}^N,\\ -\Delta v=u^q,&x\in \mathbb{R}^N,\end{cases}$$ with $N\ge3$, $p,q>0$, $pq>1$, 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 $N\ge5$, the disk contains a nonempty open set not covered by the criteria of Busca-Man\'asevich~\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 $\int_{B_1}(u^rv^{2p}+u^{2q}v^s)dx\le C(N,p,q)$ for suitable weights for any positive solution $(u,v)$, finally the nonexistence of positive classical solutions follows by scaling.