Asymptotics of nonlocal nonlinear Robin energies

We study the free minimization problem for a nonlinear, nonlocal functional associated with Robin-type nonlocal exterior conditions depending on a positive parameter $α$. We prove that, for every $α>0$, the minimizer exists, is unique, and satisfies suitable decay properties at infinity. We also investigate the regularity of the maps $α\mapsto u_α$ and $α\mapsto E_α$, where~$u_α$ denotes the minimizer and $E_α$ the corresponding energy. Finally, we derive asymptotic expansions of the energy as $α\to+\infty$ and as $α\to0^+$. 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 $E_α$ converges to its energy, but in the other case $E_α$ diverges to $-\infty$.