In this paper, we consider the exact asymptotic behavior of solutions to the following boundary blow-up elliptic problem:
$$ \triangle_{\infty} u =b(x)f(u), \quad u>0, x\in \Omega , u|_{\partial \Omega }=\infty, $$
(1.1)
where the operator \(\triangle_{\infty}\) is the ∞-Laplacian, a highly degenerate elliptic operator given by
$$\triangle_{\infty}u := \bigl\langle D^{2}uDu, Du \bigr\rangle = \sum_{i,j=1} ^{N}D_{i}u D_{ij}uD_{j}u, $$
Ω is a bounded domain with smooth boundary in \(\mathbb{R}^{N}\) (\(N\geq2\)), b satisfies
- (b1):
-
\(b \in C({\bar{\Omega }})\) is positive in Ω,
and f satisfies
- (f1):
-
\(f\in C^{1}([0, \infty))\), \(f(0)=0\), f is increasing on \((0, \infty)\);
- (f2):
-
\(\int_{1}^{\infty}\frac{d\nu}{(f(\nu))^{\frac {1}{3}}}<\infty\);
- (f3):
-
there exists \(C_{f}>0\) such that \(\lim _{s\rightarrow+\infty}f'(s)\int_{s}^{\infty}\frac {d\nu}{f(\nu)}=C_{f}\).
The ∞-Laplacian has been the subject of extensive investigation since the fundamental work of Aronsson [1] in which he established that the equation \(\triangle_{\infty}u= 0\) is the Euler-Lagrange equation for smooth absolute minimizers. As a result of the high degeneracy of the ∞-Laplacian, the associated Dirichlet problems may not have classical solutions. Therefore solutions are understood in the viscosity sense, a concept introduced by Crandall and Lions [2], Crandall et al. [3], and Crandall et al. [4], and to be defined in Section 2. Later, Jensen [5] proved the existence and uniqueness of the viscosity solutions to the Dirichlet problem to the infinity harmonic equation. Since then, the infinity Laplace equation has been attracting considerable attention and we direct the reader to see [6–8] and the references therein.
By a solution to the problem (1.1), we mean a nonnegative function \(u\in C(\Omega)\) that satisfies the equation in the viscosity sense (see Section 2 for definition) and the boundary condition with \(u(x)\rightarrow\infty\) as the distance function \(d(x): =\operatorname{dist}(x,\Omega)\rightarrow0\). Such a solution is called a boundary blow-up solution. Recently, A Mohammed and S Mohammed [9, 10] first supplied a necessary and sufficient condition
$$ \int_{a}^{\infty}\frac{ds}{\sqrt[4]{F(s)}}< \infty,\quad \forall a>0, F(s)=\int_{0}^{s} f(\nu)\,d\nu, $$
(1.2)
for the existence of solutions to problem (1.1).
The investigation of boundary blow-up problems for elliptic equations has a long history. Early studies mainly focused on problems involving the classical Laplace operator Δ, i.e.
$$ \triangle u =b(x)f(u),\quad u>0, x\in \Omega , u|_{\partial \Omega }=\infty, $$
(1.3)
The problem (1.3) arises in Riemannian geometry, mathematical physics or population dynamics, and has been discussed and extended by many authors in many contexts; see, for instance, [11–28] and the references therein.
For \(b \equiv1\) on Ω and f satisfying (f1), Keller and Osserman [21, 26] first supplied the necessary and sufficient condition
$$ \int_{a}^{\infty}\frac{ds}{\sqrt{2F(s)}}< \infty,\quad \forall a>0, F(s)= \int_{0}^{s} f(\nu)\,d\nu, $$
(1.4)
for the existence of solutions to problem (1.3).
Loewner and Nirenberg [24] showed that if \(f(u)=u^{p_{0}}\) with \(p_{0}=(N+2)/(N-2)\), \(N>2\), then problem (1.3) has a unique positive solution u which satisfies
$$\lim_{d(x)\rightarrow0}u(x) \bigl(d(x)\bigr)^{(N-2)/2}=\bigl(N(N-2)/4 \bigr)^{(N-2)/4}. $$
When f satisfies (f1), (f3), and the condition that
- (f4):
-
there exist \(p>1\), \(S_{0}>0\) such that \(f(s)/{s^{p}}\) is increasing on \([S_{0}, \infty)\)
and \(b\in C^{\alpha}(\Omega)\) which is positive in Ω and satisfies
- (b01):
-
there exist \(b_{0}>0\) and \(\sigma\in(0, 2)\) such that
$$\lim_{d(x)\rightarrow0}b(x) \bigl(d(x) \bigr)^{\sigma}=b_{0}, $$
García-Melián [18] showed (by using nonlinear transformations, a perturbation method, and a comparison principle) that:
(i) if \(C_{f}>1\), then for any solution u of problem (1.1)
$$ \lim_{d(x)\rightarrow0}\frac{u(x)}{ \psi (A(d(x))^{2-\sigma} )}=1, $$
(1.5)
where
$$A=\frac{b_{0}}{(2-\sigma) ((2-\sigma)(C_{f}-1)+1 )} $$
and ψ satisfies
$$ \int_{\psi(t)}^{\infty}\frac{ds}{f(s)}=t,\quad \forall t>0; $$
(1.6)
(ii) if \(C_{f} =1\) and \(h(t): =t f'(\psi(t))\geq 1\) for sufficiently small \(t>0\), then (i) still holds.
Now we introduce a class of functions.
Let Λ denote the set of all positive non-decreasing functions \(k\in C^{1}(0, \nu)\) which satisfy
$$ \lim_{t \rightarrow0^{+}} \frac {d}{dt} \biggl( \frac{K(t)}{k(t)} \biggr)=C_{k},\quad \mbox{where }K(t)= \int_{0}^{t} k(s)\,ds. $$
(1.7)
We note that for each \(k\in\Lambda\),
$$\lim_{t \rightarrow0^{+}} \frac{K(t)}{k(t)}=0\quad \mbox{and}\quad C_{k} \in[0,1]. $$
The set Λ was first introduced by Cîrstea and Rǎdulescu. Meanwhile, Cîrstea and Rǎdulescu [12–15] introduced the Karamata regular variation theory to study the boundary behavior and uniqueness of solutions for problem (1.3) and obtained a series of rich and significant information about the boundary behavior of the blow-up solutions.
Inspired by the above works, in this paper, by constructing new comparison functions, we consider the exact asymptotic behavior of the solution u of problem (1.1) near ∂Ω under appropriate conditions on \(b(x)\).
Suppose b also satisfies
- (b2):
-
there exist some \(k\in\Lambda\) and a positive constant \(b_{0} \in\mathbb{R}\) such that
$$\lim_{d(x) \rightarrow0 } \frac{b(x)}{k^{4}(d(x))} =b_{0}. $$
The key to our estimates in this paper is the solution to the problem
$$ \int_{\phi(t)}^{\infty} \frac{ds}{ (f(s) )^{\frac{1}{3}}}=t,\quad t>0. $$
(1.8)
Our main results are summarized as follows.
Theorem 1.1
Let
f
satisfy (f1)-(f3), b
satisfy (b1)-(b2) and
\(1\leq C_{f}<\frac{3}{2}\). Then, for any solution
u
of problem (1.1),
$$ \lim _{d(x)\rightarrow 0}\frac{u(x)}{\phi (K^{\frac{4}{3}}(d(x)) )}=\xi_{0}, $$
(1.9)
where
ϕ
is uniquely determined by (1.8) and
$$ \xi_{0}= \biggl( \biggl(\frac{4}{3} \biggr)^{3}\frac{C_{k}}{b_{0}}+ \biggl(\frac{4}{3} \biggr)^{4}\frac{5C_{f}-3}{b_{0}(3-2C_{f})} \biggr)^{\frac{C_{f}-1}{3-2C_{f}}} . $$
(1.10)
In particular, when
\(C_{f}=1\), u
verifies
$$ \lim_{d(x)\rightarrow0} \frac{u(x)}{\phi (K^{\frac{4}{3}}(d(x)) )}=1. $$
Remark 1.1
For the existence of solutions for problem (1.1), see A. Mohammed and S. Mohammed [9, 10].
The outline of this paper is as follows. In Sections 2 and 3, we give some preparation that will be used in the next section. The proof of Theorem 1.1 will be given in Section 4.