6 problems

Lane–Emden concentration conjecture. As $\beta\to\infty$,

Nonexistence conjecture. For subcritical exponents, positive classical solutions do not exist.

Let $\Omega\subset\mathbb{R}^2$ be a smooth bounded domain, and let $(u,v)$ be a classical solution of the Lane–Emden system with exponents $p,q\geq 1$ satisfying … for some…

Exterior-domain critical-curve conjecture. The curve produced by $\partial\mathcal{C}$ is a critical curve separating the exponent pairs for which fast-decaying exterior-domain sol…

Critical-curve conjecture. The boundary $\partial\mathcal{C}$ is a critical curve in the $(p,q)$-plane separating the exponents for which regular solutions exist in $\mathbb{R}^N$…

Consider the Lane–Emden system and its least energy solutions $(u_{\epsilon},v_{\epsilon})$ in a smooth bounded domain $\Omega\subset\mathbb{R}^n$, with exponent $p$ and the associ…