On a critical curl-curl system of Lane-Emden type

Abstract

We study the system $\nabla\times\nabla\times u = |v|^{q-2}v$, $\nabla\times\nabla\times v = |u|^{p-2}u$ of Lane-Emden type in $\mathbb R^3$, with exponents $p,q>3$ on the critical hyperbola $1/p+1/q=1/3$. We prove the existence of a ground state solution by reducing the system to a quasilinear fourth-order curl-curl equation and minimizing the associated Sobolev-curl quotient. A concentration-compactness argument shows there are finitely many concentration points for a Palais-Smale sequence $(u_n)$. This yields a.e. convergence of $\nabla\times\nabla\times u_n$ to $\nabla\times\nabla\times u$ where $u$ is the weak limit of $(u_n)$. We also prove that the optimal Sobolev-curl constant is attained and is strictly larger than the usual second-order Sobolev constant. If $p=q$, we show $u$ and $v$ are necessarily equal for every solution. In an appendix we show that our concentration argument also applies to a scalar quasilinear fourth order equation.