Asymptotics of nonlocal nonlinear Robin energies

Abstract

We study the free minimization problem for a nonlinear, nonlocal functional associated with Robin-type nonlocal exterior conditions depending on a positive parameter $\alpha$. We prove that, for every $\alpha >0$, the minimizer exists, is unique, and satisfies suitable decay properties at infinity. We also investigate the regularity of the maps $\alpha \mapsto u_{\alpha }$ and $\alpha \mapsto E_{\alpha }$, where~$u_{\alpha }$ denotes the minimizer and $E_{\alpha }$ the corresponding energy. Finally, we derive asymptotic expansions of the energy as $\alpha \to +\infty$ and as $\alpha \to 0^{+}$. The latter regime requires distinguishing between two cases, depending on whether the forcing term has zero mass. In one case, the limiting energy possesses a minimizer and $E_{\alpha }$ converges to its energy, but in the other case $E_{\alpha }$ diverges to $-\infty$.

22 pages