Second order estimates for boundary blow-up solutions of elliptic equations

We investigate blow-up solutions of the equation $\Deltau$ = $f(u)$ in a bounded smooth domain $\Omega \subset R^N$. Under appropriate growth conditions on $f(t)$ as $t$ goes to infinity we show how the mean curvature of the boundary $\partial\Omega$ appears in the second order term of the asymptotic expansion of the solution $u(x)$ as $x$ goes to $\partial\Omega$.


Introduction.
Let Ω ⊂ R N be a bounded smooth domain, and let f (t) be a smooth function, increasing for t ≥ 0, which satisfies f (0) = 0 and the Keller-Osserman condition It is well known [14], [17] that under these conditions the Dirichlet problem Under some additional condition on f , it is possible to show the estimate [7] lim x→∂Ω u(x) φ(δ(x)) = 1, where δ(x) denotes the distance of x from ∂Ω. This means that the main part of the asymptotic behaviour of the solution u(x) near ∂Ω is independent of the geometry of the domain. The behaviour of boundary blow-up solutions near the boundary has been investigated by many researches, see [1], [2], [3], [4], [6], [7], [8], [9], [13], [15]. Let us recall a result of C. Bandle [8]. Let f (t) > 0 and F (t) as in (2); moreover, if G(t) = t 0 F (τ )dτ, suppose there exist a, b, with 1 < a < b such that for large t. Note that (3) implies the Keller-Osserman condition and that F (t)/t 2 is increasing for t large. Under condition (3), in [8] it is proved that where φ is defined as in (2) and C is a suitable constant. A function which satisfies (3) is f (t) = t p , p > 1. In this case we have For this special case C. Bandle [4] has improved the estimates (4) proving the expansion where K(x) denotes the mean curvature of ∂Ω at the point x nearest to x, and o(δ) has the usual meaning. The object of the present paper is to find a similar expansion for a suitable class of functions f . We suppose that where p > 1, β > 0, and O(1) denotes a bounded quantity. In case of 1 < p ≤ 3 we also use the following additional condition: Furthermore, suppose there is a constant M such that for all θ ∈ (1/2, 2) and for t large we have Then we find the estimate where φ is defined as in (2), K(x) is the mean curvature of the surface {x ∈ Ω : δ(x) = constant} and σ > 1 depends on β and p. A typical example which satisfies all the conditions in above is f (t) = t p + P (t), where P (t) has a polynomial growth q with q < p. Note that this special case has been discussed in [3] in a different context. Indeed, in [3] the basic function φ used to give the expansion of the solution u(x) was related to the principal part t p only. Now we take φ defined as in (2), which makes a strong difference when q is close to p.
Results of existence for singular equations in presence of a gradient term are also been discussed, see for example [5], [10]. Also the cases of weighted quasilinear equations as well as p-Laplace equations have been investigated, see [16], [12] and references therein.
Integration by parts on (1, t) yields where Here and in what follows, C is a suitable constant. By (5) and (8) we find where g(t) is not necessarily the same as in (8), but it again satisfies estimate (9). By (10) we find, for some constant C > 1 and t large If φ is defined as in (2), by using the last estimates, for s small we get Proof. Let us write Recall that (5) implies (8). Putting φ(s) = t, and using de l'Hôpital rule, (5) and This estimate and (14) yield (13). The lemma is proved.
where O(1) denotes a bounded quantity.
We look for a sub-solution of the kind where A is the same as before and α is a positive constant to be determined. Instead of (20) now we have This means that we can find suitable constants C i (not necessarily the same as before) such that After α is fixed, we consider only points x ∈ Ω such that Using Taylor's expansion, (9) and (7) we find by (30) and (32) we have ∆v > f (v) (33) Rearranging and using (9) we find which looks like (27) possibly with different values of the constants C i . Therefore we can take δ 0 small and α 0 large in order to satisfy this inequality as well as (31) for δ < δ 0 and α > α 0 with αδ σ ≤ α 0 δ σ 0 . Take α and δ as in above and put ρ = αδ σ . By (29), we can decrease δ (increasing α according to αδ σ = ρ) until Decrease δ 1 again (and increase α) in order to have Aδ 1 < ρ/2 and αδ σ 1 = ρ. Then v(x) < u(x) for δ(x) = δ 1 .
3. Concluding remarks. If the function f (t) has an exponential growth instead of a polynomial growth then the behaviour of the second order term of the solution to problem (1) is quite different. For example, in case of f (t) = e t , C. Bandle [4] has found the expansion u(x) = log 2 δ 2 + (N − 1)K(x)δ + o(δ), where x is the point of ∂Ω nearest to x, and o(δ) has the usual meaning.
More generally, assume f (t) : R → R smooth and such that Suppose there is β > 0 such that, for t > 0, where O(1) is a bounded quantity. Furthermore suppose that, again for t > 0, Finally, we suppose that for some m > 2 there are > 0 and M , t 0 large such that for all t > t 0 . Under these conditions one proves that where Φ(s) is defined as follows: Note that all these conditions hold for f (t) = e t|t| β−1 P (t), where P (t) > 0 has a polynomial growth. The proof of this result will appear in a forthcoming paper.