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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. Universit√° 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)

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