Asymptotic behavior of the solutions of linear and quasilinear elliptic equations on ℝ^{ℕ}

Rabier, Patrick

doi:10.1090/S0002-9947-03-03234-3

Asymptotic behavior of the solutions of linear and quasilinear elliptic equations on $\mathbb {R}^{N}$
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Trans. Amer. Math. Soc. 356 (2004), 1889-1907 Request permission

Abstract:

We investigate the relationship between the decay at infinity of the right-hand side $f$ and solutions $u$ of an equation $Lu=f$ when $L$ is a second order elliptic operator on $\mathbb {R}^{N}.$ It is shown that when $L$ is Fredholm, $u$ inherits the type of decay of $f$ (for instance, exponential, or power-like). In particular, the generalized eigenfunctions associated with all the Fredholm eigenvalues of $L,$ isolated or not, decay exponentially. No use is made of spectral theory. The result is next extended when $L$ 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.

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