Asymptotic behavior of the solutions of linear and quasilinear elliptic equations on ℝ^{ℕ}
Patrick J. Rabier · Transactions of the American Mathematical Society · 2003
We investigate the relationship between the decay at infinity of the right-hand side f f and solutions u u of an equation L u = f Lu=f when L L is a second order elliptic operator on R N . \mathbb {R}^{N}. It is shown that when L L is Fredholm, u u inherits the type of decay of f f (for instance, exponential, or power-like). In particular, the generalized eigenfunctions associated with all the Fredholm eigenvalues of L , L, isolated or not, decay exponentially. No use is made of spectral theory. The result is next extended when L L is replaced by a Fredholm quasilinear operator. Various generalizations to other unbounded domains, higher order operators or elliptic systems are possible and briefly alluded to, but not discussed in detail.