Asymptotic behaviour of large solutions of quasilinear elliptic problems

Abstract

Abstract. The paper deals with the large solutions of the problems u = u p and u = e u . They blow up at the boundary. It is well-known that the first term in their asymptotic behaviour near the boundary is independent of the geometry of the boundary. We determine the second term which depends on the mean curvature of the nearest point on the boundary. The computation is based on suitable upper and lower solutions and on estimates given in [4]. In the last section these estimates are used together with the P -function to establish the asymptotic behaviour of the gradients. Mathematics Subject Classification (2000). 35B45, 35J25, 35J60.Keywords. Nonlinear boundary value problems, upper and lower solutions, boundary blowup, maximal solutions.

Second-order estimates for boundary blowup solutions of special elliptic equations

“…Moreover, if δ = δ(x) denotes the distance from x to ∂Ω, we have [10] u(x) − log(2/δ 2 (x)) → 0 as x → ∂Ω. Recently, Bandle [4] has improved the previous estimate finding the expansion…”

Uniqueness results and asymptotic behavior of solutions with boundary blow-up for logistic-type porous media equations

“…For studies of other boundary blow-up problems, we also refer the reader to [1,2,5,7,17,18,[20][21][22]25,29,33] and the references therein. [13,15].…”

Exact multiplicity for boundary blow-up solutions

“…The theorem is proved in a similar way to [3] (see, e.g., [15]). Fist we use the following lemma which can be proved by arguments similar to those in [8, Theorem 3].…”