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Asymptotic behaviour via the Harnack inequality. (English) Zbl 0840.35011
Ambrosetti, A. (ed.) et al., Nonlinear analysis. A tribute in honour of Giovanni Prodi. Pisa: Scuola Normale Superiore, Quaderni. Universitá di Pisa. 135-144 (1991).
Let $$Lu= a_{ij} u_{ij}+ b_i u_i+ cu$$ be uniformly elliptic with $$L^\infty$$ coefficients. The authors investigate solutions of $$Lu= 0$$ on the semi-infinite cylinder $$[0, \infty)\times \omega$$, $$\omega\subset \mathbb{R}^{n- 1}$$, with $$\partial u/\partial\nu= 0$$ on $$[0, \infty)\times \partial\omega$$. They show that if $$u$$, $$v$$ are positive solutions with $$u, v\to 0$$ as $$x_1\to \infty$$, and if $$c(x)\leq 0$$ then, for some constant $$A> 0$$, $$v(x_1, y)/u(x_1, y)\to A$$ as $$x_1\to \infty$$, uniformly in $$\omega$$. The same estimate is proved when $$v$$ is as before and $$u$$ satisfies the semilinear equation $$Lu= f(x, u)$$, provided $$|f(x, u)|\leq Cu^{1+ \delta}$$ for some $$\delta> 0$$, $$0< u$$ small, and $$c(x)\leq - m< 0$$. As a corollary, a similar asymptotic estimate is proved for solutions in $$\mathbb{R}^n$$ when $$|x|b_i(x)$$ and $$|x|^2 c(x)$$ are bounded for $$|x|\geq 1$$.
For the entire collection see [Zbl 0830.00011].
Reviewer: G.Porru (Cagliari)

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35J25 Boundary value problems for second-order elliptic equations 35J65 Nonlinear boundary value problems for linear elliptic equations