Open Access

Existence and boundary behavior of solutions to p-Laplacian elliptic equations

Boundary Value Problems20162016:119

https://doi.org/10.1186/s13661-016-0627-2

Received: 13 May 2016

Accepted: 14 June 2016

Published: 24 June 2016

Abstract

Under appropriate conditions on \(b(x)\) and \(g(u)\), we consider the singular Dirichlet problems \(-\Delta_{p} u=b(x)g(u)\), \(u>0\), \(x \in \Omega \), \(u\vert_{\partial \Omega }=0\). These problems are shown to admit weak solutions, and we analyze their exact asymptotic behavior near the boundary. As the main tools, we use Karamata regular variation theory and the method of upper and lower solutions.

Keywords

singular Dirichlet problemexistence of solutionsexact asymptotic behaviorupper and lower solutions

1 Introduction and the main results

The purpose of this paper is to investigate the existence and exact asymptotic behavior of the solution near the boundary to the following problems:
$$ -\Delta_{p} u =b(x)g(u), \quad u>0, x\in \Omega , u\vert _{\partial \Omega }=0, $$
(1.1)
where \(\Delta_{p}u:= \operatorname {div}(\vert \nabla u\vert ^{p-2}\nabla u)\) stands for the p-Laplacian operator with \(p>1\), Ω is a bounded domain with smooth boundary in \(\mathbb {R}^{N}\) (\(N\geq2\)), b satisfies the condition
\((\mathrm{b}_{1})\)

\(b \in C^{\alpha }(\bar{\Omega })\) for some \(\alpha \in(0,1)\) and is positive in Ω,

and g satisfies the following conditions:
\((\mathrm{g}_{1})\)

\(g\in C^{1} ((0,\infty),(0,\infty) )\), \(\lim_{s\rightarrow0^{+}}g(s)=\infty\), and \(g'(s)\leq0\) for all \(s>0\);

\((\mathrm{g}_{2})\)

\(\int_{0}^{1} \frac{d\nu}{(g(\nu ))^{q/p}}<\infty\);

\((\mathrm{g}_{3})\)

there exists \(C_{g}>0\) such that \(\lim_{s\rightarrow0}\frac{q}{p g^{1-\frac{q}{p}}(s)}g'(s)\int_{0}^{s} g^{-q/p}(\nu)\,d\nu=-C_{g}\),

where q stands for the Hölder conjugate of p.
A solution of (1.1) is meant as a positive function \(u\in C^{1}(\Omega)\) with \(u(x)\rightarrow0\) as \(d(x):=\operatorname {dist}(x, \partial\Omega)\rightarrow0\) and
$$\int_{\Omega} \vert \nabla u\vert ^{p-2}\nabla u \nabla \phi \,dx= \int_{\Omega}b(x)g(u)\phi \,dx, \quad \forall\phi\in C_{0}^{\infty}(\Omega). $$

The class of problems (1.1) appears in many nonlinear phenomena, for instance, in the theory of quasi-regular and quasi-conformal mappings [13], in the generalized reaction-diffusion theory [4], in the turbulent flow of a gas in a porous medium, and in the non-Newtonian fluid theory [5].

The investigation of problem (1.1) has a long history. Early studies mainly focused on problems involving the classical Laplace operator Δ, that is,
$$ -\Delta u =b(x)g(u),\quad u>0, x\in \Omega , u\vert_{\partial \Omega }=0. $$
(1.2)
For \(b\equiv1\) and \(g(u)=u^{-\gamma}\) with \(\gamma>1\), in 1977, Crandall, Rabinowitz, and Tartar [6] have derived that problem (1.2) has a unique solution \(u\in C^{2+\alpha}(\Omega) \cap C(\bar{\Omega})\). This paper is the starting point on semilinear problem with singular nonlinearity. Moreover, the following result was established: there exist positive constants \(c_{1}\) and \(c_{2}\) such that
$$ c_{1} \bigl(d(x) \bigr)^{2/(1+\gamma)}\leq u(x)\leq c_{2} \bigl(d(x) \bigr)^{2/(1+\gamma)} \quad \text{near } \partial \Omega. $$
(1.3)

Lazer and McKenna [7] showed that (1.3) continues to hold on Ω̄, and instead of \(b \equiv1\) on Ω, they assumed that \(0< b_{1}\leq b(x) (d(x) )^{\lambda}\leq b_{2}\) for all \(x\in\bar{\Omega}\), where \(b_{1}\), \(b_{2}\) are positive constants, and \(\lambda\in(0,2)\). Later, a lot of work has been done related to the existence and asymptotic behavior of the solutions to problem (1.2); we refer to [816] and the references therein.

It is worth pointing out that Cîrstea and Rǎdulescu [1719], Cîrstea and Du [20], and Repovs̆ [21] introduced the Karamata regular variation theory to study the boundary behavior and uniqueness of solutions for boundary blow-up elliptic problems and obtained a series of significant information about the qualitative behavior of large solutions in a general framework.

Recently, by using the Karamata regular variation theory Zhang and Li [22], Zhang [23], Zhang and Cheng [24], and Mi and Liu [25] further studied the boundary behavior of the solutions to problem (1.2).

Now, let us return to problem (1.1). When \(b(x)\equiv1\) and \(g(u)=u^{m}\), the first results concerning (1.1) (\(0 < p - 1 < m\)) have been obtained by Ni and Serrin [26, 27], who gave a priori estimates near a singularity. In particular, they show that \(m = N(p - 1)/(N - p) \) is a critical value. They also obtained nonexistence results for positive solutions in an exterior domain for \(p - 1 < m < N(p - 1 )/(N - p)\). Guedda and Veron [28] give Ni and Serrin’s estimates under a slightly weaker hypothesis. Later, Bognara and Drabekb [29] deals with the existence and multiplicity results for radial symmetric solutions of problem (1.1) for a more general nonlinearity \(g(u)\). In recent years, the existence and uniqueness of positive solutions for the general quasilinear elliptic problem \(-\Delta_{p} u = \lambda h(x,u, \nabla u)\), \(u>0\), \(x\in \Omega \), \(u\vert_{\partial \Omega }=0\), has been studied by many authors. Some sufficient conditions on h and Ω have been proposed to ensure the existence or nonexistence of solutions; see [3038] and the reference therein. However, to the best of our knowledge, up to now, few papers have addressed the boundary behavior of solutions to problem (1.1) for more general nonlinear terms g.

Inspired by the works mentioned, in this paper, by using Karamata regular variation theory and the method of upper and lower solutions, we show the existence of a solution to problem (1.1) and provide some asymptotic boundary estimates under appropriate conditions on \(b(x)\) and \(g(u)\).

In order to present our main results, we introduce the following two kinds of functions.

Let Λ denote the set of positive nondecreasing functions \(k\in C^{1}(0, \nu)\) that satisfy
$$ \lim_{t \rightarrow0^{+}} \frac {d}{dt} \biggl( \frac{K(t)}{k(t)} \biggr)=C_{k}, \quad \mbox{where } K(t)= \int_{0}^{t} k(s)\,ds. $$
(1.4)
We note that, for each \(k\in\Lambda\),
$$\lim_{t \rightarrow0^{+}} \frac{K(t)}{k(t)}=0 \quad \text{and} \quad C_{k}\in[0,1]. $$

The set Λ was first introduced by Cîrstea and Rǎdulescu.

Next, we denote by Θ the set of all Karamata functions that are normalized slowly varying at zero (see the definition in Section 2) defined on \((0, \eta]\) for some \(\eta>0\) by
$$ \hat{L}(s)=c_{0}\exp \biggl( \int_{s}^{\eta}\frac{y(\tau)}{\tau}\,d\tau \biggr), \quad s\in(0, \eta], $$
(1.5)
where \(c_{0}>0\), and the function \(y\in C([0, \eta])\) with \(y(0)= 0\).
The key to our estimates in this paper is the solution to the problem
$$ \int_{\phi(t)}^{\infty} \frac {ds}{ (f(s) )^{\frac{1}{p-1}}}=t,\quad t>0. $$
(1.6)

Our main results are summarized as follows.

Theorem 1.1

Let g satisfy \((\mathrm{g}_{1})\)-\((\mathrm{g}_{2})\), and b satisfy \((\mathrm{b}_{1})\)-\((\mathrm{b}_{2})\). Suppose that b also satisfies the following condition:
\((\mathrm{b}_{3})\)
the linear problem
$$ -\Delta_{p} u=b(x),\quad u>0, x\in \Omega , u\vert _{\partial \Omega }=0, $$
(1.7)
has a unique solution \(v_{0}\in C^{1,\alpha}(\Omega)\cap C(\bar{\Omega})\) for some \(\alpha\in(0,1)\).
Then, problem (1.1) has at least one solution \(u\in C^{1,\alpha}(\Omega)\cap C(\bar{\Omega})\).

Theorem 1.2

Let g satisfy \((\mathrm{g}_{1})\)-\((\mathrm{g}_{2})\), and b satisfy \((\mathrm{b}_{1})\)-\((\mathrm{b}_{3})\). Suppose that b also satisfies the following condition:
\((\mathrm{b}_{4})\)
there exist \(k\in\Lambda\) and a positive constant \(b_{0} \in\mathbb{R}\) such that
$$\lim_{d(x) \rightarrow0 } \frac{b(x)}{k^{p}(d(x))} =b_{0}. $$
If
$$C_{k}+q C_{g}>q, $$
then any solution u to problem (1.1) satisfies
$$ \lim_{d(x) \rightarrow 0}\frac{u(x)}{\phi(K^{q}(d(x)))}=A_{1}^{1-C_{g}}, $$
(1.8)
where ϕ is uniquely determined by (1.6), q stands for the Hölder conjugate of p, and
$$ A_{1}=\frac{1}{q} \biggl(\frac{b_{0}}{(p-1)(q C_{g}+C_{k}-q)} \biggr)^{\frac{1}{p-1}}. $$

Theorem 1.3

Let g satisfy \((\mathrm{g}_{1})\)-\((\mathrm{g}_{2})\), and b satisfy \((\mathrm{b}_{1})\)-\((\mathrm{b}_{3})\). Suppose that b also satisfies the following condition:
\((\mathrm{b}_{5})\)
there exist \(L\in\Theta\) and a positive constant \(b_{1} \in\mathbb{R}\) such that
$$\lim_{d(x) \rightarrow0 } \frac{b(x)}{(d(x))^{-p}L(d(x))} =b_{1}. $$
Then any solution u to problem (1.1) satisfies
$$ \lim_{d(x) \rightarrow0}\frac{u(x)}{\phi (h(d(x)))}=A_{2}^{1-C_{g}}, $$
(1.9)
where ϕ is uniquely determined by (1.6),
$$ h(t)= \int_{0}^{t}s^{-1} \bigl(L(s) \bigr)^{\frac{1}{p-1}}\,ds, $$
(1.10)
and
$$ A_{2}= \biggl(\frac{b_{1}}{p-1} \biggr)^{\frac{1}{p-1}}. $$

The outline of this paper is as follows. In Sections 2-3, we give some preparation that will be used in the next section. The proofs of Theorems 1.1-1.3 will be given in Sections 4-5.

2 Preliminaries

Our approach relies on Karamata regular variation theory established by Karamata in 1930, which is a basic tool in the theory of stochastic processes (see [3943] and the references therein). In this section, we first give a brief account of the definition and properties of regularly varying functions.

Definition 2.1

A positive measurable function f defined on \([a,\infty)\) for some \(a>0\) is called regularly varying at infinity with index ρ, written as \(f \in\mathit{ RV}_{\rho }\), if for each \(\xi>0\) and some \(\rho \in\mathbb {R}\),
$$ \lim_{s \rightarrow\infty} \frac{f(\xi s)}{f(s)}= \xi^{\rho }. $$
(2.1)
In particular, when \(\rho=0\), f is called slowly varying at infinity.

Clearly, if \(f\in\mathit{ RV}_{\rho}\), then \(L(s):=f(s)/{s^{\rho}}\) is slowly varying at infinity.

Definition 2.2

A positive measurable function f defined on \([a,\infty)\) for some \(a>0\) is called rapidly varying at infinity if for each \(\rho>1\),
$$ \lim_{s \rightarrow\infty} \frac{f(s)}{s^{\rho}}=\infty. $$
(2.2)

We also see that a positive measurable function g defined on \((0,a)\) for some \(a>0\) is regularly varying at zero with index σ (written as \(g \in\mathit{ RVZ}_{\sigma}\)) if \(t\rightarrow g(1/t)\) belongs to \(\mathit{RV}_{-\sigma}\). Similarly, g is called rapidly varying at zero if \(t\rightarrow g(1/t)\) is rapidly varying at infinity.

Proposition 2.1

(Uniform convergence theorem)

If \(f\in\mathit{ RV}_{\rho }\), then (2.1) holds uniformly for \(\xi\in[c_{1}, c_{2}]\) with \(0< c_{1}< c_{2}\). Moreover, if \(\rho<0\), then the uniform convergence holds on intervals of the form \((a_{1}, \infty)\) with \(a_{1}>0\); if \(\rho>0\), then the uniform convergence holds on intervals \((0, a_{1}]\), provided that f is bounded on \((0, a_{1}]\) for all \(a_{1}>0\).

Proposition 2.2

(Representation theorem)

A function L is slowly varying at infinity if and only if it may be written in the form
$$ L(s)=\varphi(s) \exp \biggl( \int_{a_{1}}^{s} \frac{y(\tau)}{\tau} \,d\tau \biggr),\quad s \geq a_{1}, $$
(2.3)
for some \(a_{1}\geq a\), where the functions φ and y are measurable and \(y(s)\rightarrow0\) and \(\varphi(s)\rightarrow c_{0}>0\) as \(s \rightarrow\infty\).
We say that
$$ \hat{L}(s)=c_{0} \exp \biggl( \int_{a_{1}}^{s} \frac{y(\tau)}{\tau} \,d\tau \biggr),\quad s \geq a_{1}, $$
(2.4)
is normalized slowly varying at infinity and
$$ f(s)=c_{0}s^{\rho}\hat{L}(s),\quad s \geq a_{1}, $$
(2.5)
is normalized regularly varying at infinity with index ρ (and written as \(f\in\mathit{ NRV}_{\rho}\)).

Similarly, g is called normalized regularly varying at zero with index σ, written as \(g \in\mathit{ NRVZ}_{\sigma}\) if \(t\rightarrow g(1/t)\) belongs to \(\mathit{NRV}_{-\sigma}\).

A function \(f\in\mathit{ RV}_{\rho}\) belongs to \(\mathit{NRV}_{\rho}\) if and only if
$$ f\in C^{1}[a_{1}, \infty)\quad \text{for some } a_{1}>0 \quad \text{and}\quad \lim_{s \rightarrow\infty} \frac{sf'(s)}{f(s)}=\rho. $$
(2.6)

Proposition 2.3

If functions L, \(L_{1}\) are slowly varying at infinity, then
  1. (i)

    \(L^{\sigma}\) for every \(\sigma\in\mathbb {R}\), \(c_{1} L+c_{2} L_{1}\) (\(c_{1}\geq0\), \(c_{2}\geq0\) with \(c_{1}+c_{2}>0\)), \(L\circ L_{1}\) (if \(L_{1}(t)\rightarrow+\infty\) as \(t\rightarrow +\infty\)) are also slowly varying at infinity.

     
  2. (ii)

    For every \(\theta>0\), \(t^{\theta} L(t)\rightarrow+\infty\) and \(t^{-\theta}L(t)\rightarrow0\) as \(t\rightarrow+\infty\),

     
  3. (iii)

    For \(\rho\in\mathbb {R}\), \(\frac{\ln(L(t))}{\ln t}\rightarrow0\) and \(\frac{\ln(t^{\rho}L(t))}{\ln t}\rightarrow\rho\) as \(t\rightarrow+\infty\).

     

Proposition 2.4

  1. (i)

    If \(f_{1}\in\mathit{{R}V}_{\rho_{1}}\) and \(f_{2}\in\mathit{ {R}V}_{\rho_{2}} \) with \(\lim_{t\rightarrow\infty} f_{2} (t)=\infty\), then \(f_{1}\circ f_{2}\in\mathit{{R}V}_{\rho_{1} \rho_{2}}\).

     
  2. (ii)

    If \(f\in\mathit{ RV}_{\rho}\), then \(f^{\alpha}\in\mathit{ RV}_{\rho\alpha}\) for every \(\alpha\in\mathbb{R}\).

     

Proposition 2.5

If a function L defined on \((0, \eta]\) is slowly varying at zero, then
$$ \lim_{t \rightarrow0^{+}} \frac{L(t)}{\int_{t}^{\eta }\frac{L(s)}{s}\,ds}=0. $$
(2.7)
If, moreover, \(\int_{0}^{\eta}\frac{L(s)}{s}\,ds\) converges, then
$$ \lim_{t \rightarrow0^{+}} \frac{L(t)}{\int_{0}^{t}\frac {L(s)}{s}\,ds}=0. $$
(2.8)

Proposition 2.6

(Asymptotic behavior)

If a function L is slowly varying at zero, then, for \(a>0\) and \(t\rightarrow0^{+}\),
  1. (i)

    \(\int_{0}^{t} s^{\rho}L(s)\,ds\cong (\rho+1)^{-1} t^{1+\rho} L(t)\) for \(\rho>-1\);

     
  2. (ii)

    \(\int_{t}^{a} s^{\rho}L(s)\,ds\cong (-\rho-1)^{-1} t^{1+\rho} L(t)\) for \(\rho<-1\).

     

Proposition 2.7

(Proposition 2.6 in [44])

Let \(Z \in C^{1}(0, \eta]\) be positive and \(\lim_{t \rightarrow0^{+}}\frac{sZ'(s)}{Z(s)} = +\infty\). Then Z is rapidly varying to zero at zero.

Proposition 2.8

(Proposition 2.7 in [44])

Let \(Z \in C^{1}(0, \eta)\) be positive and \(\lim_{t \rightarrow0^{+}}\frac{sZ'(s)}{Z(s)} = -\infty\). Then Z is rapidly varying to infinity at zero.

3 Some auxiliary results

In this section, we collect some useful results.

Lemma 3.1

Let \(k\in\Lambda\). Then
  1. (i)

    \(\lim_{t\rightarrow0^{+}}\frac{K(t)}{k(t)}=0\), \(\lim_{t \rightarrow0^{+}} \frac{tk(t)}{K(t)}=C_{k}^{-1}\), i.e., \(K\in\mathit{ NRVZ}_{C_{k}^{-1}}\);

     
  2. (ii)

    \(\lim_{t \rightarrow0^{+}} \frac{tk'(t)}{k(t)}=\frac{1-C_{k}}{C_{k}}\), i.e., \(k\in\mathit{ NRVZ}_{(1-C_{k})/{C_{k}}}\); \(\lim_{t \rightarrow0^{+}} \frac{K(t)k'(t)}{k^{2}(t)}=1-C_{k}\).

     

Proof

The proof is similar to that of Lemma 2.1 in [23]; so we omit it. □

Lemma 3.2

Let
$$ a(t)=t^{-p}L(t) $$
and
$$ h(t)= \int_{0}^{t}s^{-1} \bigl(L(s) \bigr)^{\frac{1}{p-1}}\,ds, $$
where \(t\in(0, \delta_{0})\), \(\int_{0}^{\eta}s^{-1}(L(s))^{\frac{1}{p-1}}\,ds<\infty\) for some \(\eta >0\), and \(L(s)\in\Theta\). Then
  1. (i)

    \(\lim_{t\rightarrow0^{+}}\frac{(h'(t))^{p}}{h(t)a(t)}=0 \) and \(\lim_{t\rightarrow0^{+}}\frac{t h'(t)}{h(t)}=0 \);

     
  2. (ii)

    \(\lim_{t\rightarrow0^{+}}\frac{t h''(t)}{h'(t)}=-1\);

     
  3. (iii)

    \(\lim_{t\rightarrow0^{+}}\frac{(h'(t))^{p-2}h''(t)}{a(t)}=-1\).

     

Proof

(i) Since \(h'(t)=t^{-1}(L(t))^{\frac{1}{p-1}}\), we have
$$ \frac{(h'(t))^{p}}{h(t)a(t)}= \frac{t^{-p}L^{\frac {p}{p-1}}(t)}{t^{-p}L(t)\int_{0}^{t} s^{-1}L^{\frac{1}{p-1}}(s)\,ds}= \frac{L^{\frac{1}{p-1}}(t)}{\int_{0}^{t} s^{-1}L^{\frac{1}{p-1}}(s)\,ds} $$
and
$$ \frac{t h'(t)}{h(t)}= \frac{L^{\frac{1}{p-1}}(t)}{\int_{0}^{t} s^{-1}L^{\frac{1}{p-1}}(s)\,ds} $$
Hence, by Proposition 2.5 we get \(\lim_{t\rightarrow 0^{+}}\frac{(h'(t))^{p}}{h(t)a(t)}=\lim_{t\rightarrow 0^{+}}\frac{t h'(t)}{h(t)}=0\).
(ii) By a direct computation we get
$$h''(t)=-t^{-2}\bigl(L(t) \bigr)^{\frac{1}{p-1}}+\frac{1}{p-1}t^{-1} \bigl(L(t) \bigr)^{\frac{1}{p-1}-1}L'(t) $$
and
$$\frac{t h''(t)}{h'(t)}=\frac{1}{p-1}\frac{t L'(t)}{L(t)}-1. $$
Since \(L\in\Theta\), it follows that \(\lim_{t\rightarrow 0^{+}}\frac{t L'(t)}{L(t)}=0\). Hence,
$$\lim_{t\rightarrow0^{+}}\frac{t h''(t)}{h'(t)}=-1. $$
(iii) Since
$$\frac{(h'(t))^{p-2}h''(t)}{a(t)}= \frac{t h''(t)}{h'(t)}\frac{(h'(t))^{p-1}}{t a(t)}=\frac{t h''(t)}{h'(t)}, $$
by (ii) we get
$$\lim_{t\rightarrow 0^{+}}\frac{(h'(t))^{p-2}h''(t)}{a(t)}=-1. $$
 □

Lemma 3.3

Let g satisfy \((\mathrm{g}_{1})\)-\((\mathrm{g}_{2})\).
  1. (i)

    If g satisfies \((\mathrm{g}_{3})\), then \(C_{g}\leq1\);

     
  2. (ii)

    \((\mathrm{g}_{3})\) holds for \(C_{g}\in(0,1)\) if and only if \(g\in\mathit{ NRV}_{-p C_{g}/(q(1-C_{g}))}\);

     
  3. (iii)

    \((\mathrm{g}_{3})\) holds for \(C_{g}=0\) if and only if g is normalized slowly varying at zero;

     
  4. (iv)

    if \((\mathrm{g}_{3})\) holds with \(C_{g}=1\), then g is rapidly varying to infinity at zero.

     

Proof

Since g satisfies \((\mathrm{g}_{1})\) and is strictly decreasing on \((0, S_{0})\), we see that
$$0< \int_{0}^{s} \frac{1}{g^{q/p}(\nu)}\,d\nu< \frac{s}{g^{q/p}(s)},\quad \forall s\in(0, S_{0}), $$
that is,
$$ 0< g^{q/p}(s) \int_{0}^{s} \frac{1}{g^{q/p}(\nu)}\,d\nu< s, \quad \forall s\in (0, S_{0}), $$
(3.1)
and
$$ \lim_{s\rightarrow0} g^{q/p}(s) \int_{0}^{s} \frac{1}{g^{q/p}(\nu)}\,d\nu=0. $$
(3.2)
(i) Let
$$I(s) =-\frac{q}{p g^{1-\frac{q}{p}}(s)}g'(s) \int_{0}^{s} g^{-q/p}(\nu)\,d\nu, \quad \forall s\in(0, s_{0}). $$
Integrating \(I(t)\) from 0 to s and integrating by parts, we obtain by (3.2) that
$$\int_{0}^{s}I(t)\,dt=-g^{q/p}(s) \int_{0}^{s} \frac{1}{g^{q/p}(\nu)}\,d\nu+s, \quad \forall s\in(0, s_{0}), $$
that is,
$$0< \frac{g^{q/p}(s)}{s} \int_{0}^{s} \frac{1}{g^{q/p}(\nu)}\,d\nu=1- \frac{\int_{0}^{s}I(t)\,dt}{s},\quad \forall s\in(0, s_{0}). $$
It follows by l’Hospital’s rule that
$$ 0\leq\lim_{s\rightarrow0^{+}} \frac{g^{q/p}(s)}{s} \int_{0}^{s} \frac{1}{g^{q/p}(\nu)}\,d\nu=1-\lim _{s\rightarrow 0^{+}}I(s)=1-C_{g}. $$
(3.3)
So (i) holds.
(ii) When \((\mathrm{g}_{3})\) holds with \(C_{g} \in(0, 1)\), it follows by (3.3) that
$$ \lim_{s\rightarrow0^{+}}\frac{g(s)}{s g'(s)}= \lim _{s\rightarrow0^{+}}\frac{\frac{q}{p}g^{q/p}(s)\int_{0}^{s} \frac{1}{g^{q/p}(\nu)}\,d\nu}{\frac{q}{p}sg'(s)\int_{0}^{s} \frac{1}{g^{q/p}(\nu)}\,d\nu g^{\frac{q}{p}-1}(s)}=-\frac{q(1-C_{g})}{p C_{g}}, $$
(3.4)
that is, \(g\in\mathit{ NRV}_{-p C_{g}/(q(1-C_{g}))}\).
Conversely, when \(g\in\mathit{ NRV}_{-\gamma}\) with \(\gamma> 0\), that is, \(\lim_{s\rightarrow0^{+}}\frac{s g'(s)}{ g(s)}=-\gamma\) and there exist a positive constant η and \(\hat{L}\in\Theta\) such that \(g(s)=c_{0}s^{-\gamma} \hat{L}(s)\), \(s\in(0, \eta]\), it follows by (2.6) and Proposition 2.6(i) that
$$\begin{aligned} -\lim_{s\rightarrow0^{+}}\frac{q}{p g^{1-\frac{q}{p}}(s)}g'(s) \int_{0}^{s} g^{-q/p}(\nu)\,d\nu =&- \frac{q}{p}\lim_{s\rightarrow0^{+}}\frac{s g'(s)}{g(s)}\lim _{s\rightarrow0^{+}}\frac{g^{q/p}(s)}{s} \int_{0}^{s} g^{-q/p}(\nu)\,d\nu \\ =& \frac{q\gamma}{p}\lim_{s\rightarrow 0^{+}}s^{-\frac{q\gamma}{p}-1} \bigl(\hat{L}(s) \bigr)^{\frac{q}{p}} \int_{0}^{s} \nu^{\frac{q\gamma}{p}} \bigl(\hat{L}(\nu) \bigr)^{-\frac{q}{p}} \,d\nu \\ =&\frac{q\gamma}{p+q\gamma}. \end{aligned}$$
(iii) By \(C_{g} = 0\) and the proof of (ii) we can see that
$$\begin{aligned} \lim_{s\rightarrow0^{+}}\frac{s g'(s)}{g(s)} =& \lim_{s\rightarrow0^{+}} \frac{\frac{q}{p}sg'(s)\int_{0}^{s} \frac{1}{g^{q/p}(\nu)}\,d\nu g^{\frac{q}{p}-1}(s)}{\frac{q}{p}g^{q/p}(s)\int_{0}^{s} \frac{1}{g^{q/p}(\nu)}\,d\nu} \\ =&\frac{p}{q} \biggl(\lim_{s\rightarrow 0^{+}}\frac{g^{q/p}(s)}{s} \int_{0}^{s} \frac{1}{g^{q/p}(\nu)}\,d\nu \biggr)^{-1}\lim_{s\rightarrow0^{+}} \frac{q}{p g^{1-\frac{q}{p}}(s)}g'(s) \int_{0}^{s} g^{-q/p}(\nu)\,d\nu \\ =&0, \end{aligned}$$
that is, g is normalized slowly varying at zero.
Conversely, when g is normalized slowly varying at zero, that is, \(\lim_{s\rightarrow0^{+}}\frac{s g'(s)}{g(s)}=0\), it follows by (3.3) that
$$\lim_{s\rightarrow0^{+}}\frac{q}{p g^{1-\frac{q}{p}}(s)}g'(s) \int_{0}^{s} g^{-q/p}(\nu)\,d\nu=\lim _{s\rightarrow0^{+}}\frac{q}{p}\frac{s g'(s)}{g(s)} \frac{g^{q/p}(s)}{s} \int_{0}^{s} \frac{1}{g^{q/p}(\nu)}\,d\nu=0. $$

(iv) By \(C_{g} = 1\) and the proof of (ii) we see that \(\lim_{s\rightarrow0^{+}}\frac{g(s)}{s g'(s)}=0\), that is, \(\lim_{s\rightarrow0^{+}}\frac{s g'(s)}{g(s)}=-\infty\), and by Proposition 2.8 we get that g is rapidly varying to infinity at zero. □

Lemma 3.4

Let g satisfy \((\mathrm{g}_{1})\)-\((\mathrm{g}_{3})\), and ϕ be the solution to the problem
$$\int_{0}^{\phi(t)}\frac{ds}{(g(s))^{\frac{1}{p-1}}}=t, \quad \forall t>0. $$
Then
  1. (i)

    \(\phi'(t)= (g(\phi(t)) )^{\frac{1}{p-1}}\), \(\phi(t)>0\), \(t>0\), \(\phi(0)=0\), and \(\phi''(t)=\frac{q}{p} (g(\phi(t)) )^{\frac{2q-p}{p}}g'(\phi(t))\), \(t>0\);

     
  2. (ii)

    \(\phi\in\mathit{ NRVZ}_{1-C_{g}}\) and \(\phi' \in\mathit{ NRVZ}_{-C_{g}}\);

     
  3. (iii)

    when \(C_{k} + qC_{g} > q\) and \(k \in \Lambda\), \(\lim_{t\rightarrow0^{+}} \frac{t}{\phi(\xi K^{q}(t))}= 0\) uniformly for \(\xi\in[c_{1}, c_{2}]\) with \(0 < c_{1} < c_{2}\), where q stands for the Hölder conjugate of p;

     
  4. (iv)

    \(\lim_{t\rightarrow0^{+}} \frac{t}{\phi(\xi h(t))}= 0\) uniformly for \(\xi\in[c_{1}, c_{2}]\) with \(0 < c_{1} < c_{2}\), where h is given as in (1.10).

     

Proof

By the definition of ϕ and a direct calculation we show that (i) holds.

(ii) It follows from (i), (3.4), and \((\mathrm{g}_{3})\) that
$$\begin{aligned} \lim_{t\rightarrow0^{+}}\frac{t\phi'(t)}{\phi(t)} =& \lim_{t\rightarrow0^{+}} \frac {t (g(\phi(t)))^{\frac{1}{p-1}}}{\phi(t)} \\ =&\lim_{s\rightarrow0}\frac{(g(s))^{\frac{1}{p-1}}\int_{0}^{s} \frac {d\nu}{(g(\nu))^{\frac{1}{p-1}}}}{s}=1-C_{g}, \end{aligned}$$
that is, \(\phi\in\mathit{ NRVZ}_{1-C_{g}}\), and
$$\begin{aligned} \lim_{t\rightarrow0^{+}}\frac{t\phi''(t)}{\phi'(t)} =& \frac {q}{p}\lim _{t\rightarrow0^{+}}\frac{g'(\phi(t)) (g(\phi(t)))^{\frac{q}{p}}\int_{0}^{\phi(t)} (g(\nu))^{-\frac{1}{p-1}}\,d\nu}{g(\phi(t))} \\ =&\frac{q}{p}\lim_{s\rightarrow0^{+}}\frac{g'(s) (g(s))^{\frac{q}{p}}\int_{0}^{s} (g(\nu))^{-\frac{1}{p-1}}\,d\nu}{g(s)} \\ =&-C_{g}. \end{aligned}$$

(iii) By Lemma 3.1(i) we see that \(K\in\mathit{ NRVZ}_{C_{k}^{-1}}\). It follows by Proposition 2.4 that \(\phi\circ K^{q} \in\mathit{ NRVZ}_{\frac{q(1-C_{g})}{C_{k}}}\). Since \(C_{k} + qC_{g} > q\), the result follows by Proposition 2.3(ii).

(iv) As in the proof of (iii), by Lemma 3.2(i) we see \(h\in\mathit{ NRVZ}_{0}\). It follows by Proposition 2.4 that \(\phi\circ h \in\mathit{ NRVZ}_{0}\). Then the result follows by Proposition 2.3(ii). □

4 Existence of solutions to problem (1.1)

In this section, we prove Theorem 1.1.

Proof of Theorem 1.1

Let
$$H(u) = \int_{0}^{u} \frac{1 }{(g(s))^{\frac{1}{p-1}}}\,ds\quad \mbox{for } u>0. $$
It follows that \(H : [0,\infty)\rightarrow[0,\infty)\) is strictly increasing and
$$H'(u) = \frac{1 }{(g(u))^{\frac{1}{p-1}}} \quad \mbox{for } u>0. $$
Let \(\bar{u}(x):= H^{-1}(v_{0}(x))\), \(x \in\Omega\), where \(H^{-1}\) denotes the inverse function of H, and \(v_{0}\) is the unique classical solution of problem (1.7). We see that \(u\vert_{\partial\Omega} = 0\) and
$$-\Delta_{p}\bar{u}+\frac{g'(\bar{u})\vert \nabla\bar {u}\vert ^{p}}{g(\bar{u})}= b(x)g(\bar{u}), \quad x\in \Omega. $$
It follows by \((\mathrm{g}_{1})\) that
$$-\Delta_{p}\bar{u}\geq b(x)g(\bar{u}),\quad x\in\Omega, $$
that is, \(\bar{u} = H^{-1}(v_{0})\) is a supersolution of problem (1.1).
On the other hand, hypothesis \((\mathrm{g}_{1})\) implies that \(\lim_{s\rightarrow0^{+}} g(s)\in(0, \infty]\), so that
$$\lim_{s\rightarrow0^{+}}\frac{g(s)}{s}=+\infty \quad \text{and}\quad \lim_{s\rightarrow 0^{+}}\frac{(g(s))^{\frac{1}{p-1}}}{s}=+\infty. $$
There then exists \(c_{0} \in(0, 1)\) such that
$$\frac{g(c_{0}\vert v_{0}\vert _{\infty})}{c_{0}}\geq1 \quad \text{and}\quad \frac{ (g(c_{0}\vert v_{0}\vert _{\infty}) )^{\frac {1}{p-1}}}{c_{0}}\geq1 . $$
Let \(\underline{u} =c_{0}v_{0}\). It follows that
$$-\Delta_{p}\underline{u}= c_{0}b(x)\leq b(x) g \bigl(c_{0}\vert v_{0}\vert _{\infty}\bigr) \leq b(x) g(\underline{u}), \quad x\in\Omega, $$
that is, \(\underline{u} =c_{0}v_{0}\) is a subsolution of problem (1.1). Moreover, we see that
$$H\bigl(c_{0}v_{0}(x)\bigr) = \int_{0}^{c_{0}v_{0}(x)} \frac{1 }{(g(s))^{\frac{1}{p-1}}}\,ds\leq \frac{c_{0}v_{0}(x)}{ (g(c_{0}\vert v_{0}\vert _{\infty}))^{\frac{1}{p-1}}}\leq v_{0}(x), \quad x\in\Omega, $$
that is, \(\underline{u}\leq\bar{u}\) on Ω. Therefore, by the lower and upper theorem the claim follows. □

5 Boundary behaviors of solutions to problem (1.1)

In this section, we prove Theorems 1.2-1.3.

First, we need the following comparison principle for weak solutions to quasilinear equations (see [45] for a proof).

Lemma 5.1

(Weak comparison principle)

Let \(D \subset\mathbb {R}^{N}\) be a bounded domain, \(G: D \times\mathbb {R} \rightarrow\mathbb {R}\) be nonincreasing in the second variable and continuous. Let \(u, w\in W^{1,p}(D)\) satisfy the respective inequalities
$$\begin{aligned} \int_{D}\vert \nabla u\vert ^{p-2}\nabla u \cdot \nabla\phi \leq& \int_{D}G(x,u) \phi \quad \textit{and} \\ \int_{D}\vert \nabla w\vert ^{p-2}\nabla w \cdot \nabla\phi \geq& \int_{D}G(x,w) \phi \end{aligned}$$
for all nonnegative \(\phi\in W_{0}^{1,p}(D)\). Then the inequality \(u \leq w\) on ∂D implies \(u \leq w\) in D.
Fix \(\varepsilon>0\). For any \(\delta>0\), we define \(\Omega_{\delta}=\{x\in\Omega: 0< d(x)<\delta\}\). Since Ω is \(C^{2}\)-smooth, choose \(\delta_{1}\in(0, \delta_{0})\) such that \(d\in C^{2}(\Omega_{\delta_{1}}) \) and
$$ \bigl\vert \nabla d(x) \bigr\vert = 1,\quad\quad \Delta d(x) =-(N-1)H( \bar{x})+o(1),\quad \forall x\in\Omega_{\delta_{1}}, $$
(5.1)
where, for \(x\in\Omega_{\delta_{1}}\), denotes the unique point of the boundary such that \(d(x) = \vert x - \bar{x} \vert \), and \(H(\bar{x}) \) denotes the mean curvature of the boundary at that point.

5.1 Proof of Theorem 1.2

Define \(r=d(x)\) and
$$\begin{aligned}& I_{1\pm}(r)=(A_{1}\pm\varepsilon)^{p-1}(p-1)q^{p-1} \biggl(p \frac {(A_{1}\pm\varepsilon)K^{q}(r)\phi''((A_{1}\pm \varepsilon)K^{q}(r))}{\phi'((A_{1}\pm \varepsilon)K^{q}(r))}+1+\frac{p}{q}\frac {K(r)k'(r)}{k^{2}(r)} \biggr) ; \\& I_{2}(x)=(A_{1}\pm\varepsilon)^{p-1}q^{p-1} \frac {K(r)}{k(r)}\Delta d(x)+\frac{b(x)}{k^{p}(r)}\frac {g (\phi((A_{1}\pm\varepsilon)K^{q}(r)) )}{ (\phi'((A_{1}\pm\varepsilon)K^{q}(r)) )^{p-1}}. \end{aligned}$$
By Lemmas 3.1 and 3.4, combined with the choices of \(A_{1}\) in Theorem 1.2, we get the following lemma.

Lemma 5.2

Suppose that g satisfies \((\mathrm{g}_{1})\)-\((\mathrm{g}_{3})\) and b satisfies \((\mathrm{b}_{1})\)-\((\mathrm{b}_{4})\). Then
  1. (i)

    \(\lim_{r\rightarrow0} I_{1\pm}(r)=(A_{1}\pm \varepsilon)^{p-1}(p-1)q^{p-1}(q-q C_{g}-C_{k})\);

     
  2. (ii)

    \(\lim_{d(x) \rightarrow0} I_{2 }(x)=b_{0}=-A_{1}^{p-1}(p-1)q^{p-1}(q-q C_{g}-C_{k})\);

     
  3. (iii)

    \(\lim_{d(x)\rightarrow0} ( I_{1\pm }(r)+I_{2}(x) )=(p-1)q^{p-1}(q-q C_{g}-C_{k}) ((A_{1}\pm \varepsilon)^{p-1}-A_{1}^{p-1} )\).

     

Proof of Theorem 1.2

Let \(v \in C^{1+\alpha} (\Omega) \cap C^{1}(\bar{\Omega})\) be the unique solution of the problem
$$ -\Delta_{p} v=1,\quad v>0, x \in\Omega, v\vert _{\partial \Omega}=0. $$
(5.2)
Then, we see that
$$ \nabla v (x) \neq0, \quad \forall x \in\partial\Omega\quad \text{and}\quad c_{3} d(x)\leq v(x) \leq c_{4} d(x), \quad \forall x \in\Omega, $$
(5.3)
where \(c_{3}\), \(c_{4}\) are positive constants.
By Lemma 5.2, since \(K\in C[0, \delta_{0})\) with \(K(0)=0\), we see that there exist \(\delta_{1\varepsilon}, \delta_{2\varepsilon}\in (0, \min\{1, \delta_{0}\} )\) (which corresponds to ε) sufficiently small such that
  1. (i)

    \(0\leq K^{q}(r)\leq\delta_{1\varepsilon}\), \(r\in(0, \delta_{2\varepsilon})\);

     
  2. (ii)

    \(I_{1+}(r)+I_{2}(x)\leq0\), \(\forall (x,r)\in \Omega_{\delta_{1\varepsilon}}\times(0, \delta_{2\varepsilon}) \);

     
  3. (iii)

    \(I_{1-}(r)+I_{2}(x)\geq0\), \(\forall (x,r)\in \Omega_{\delta_{1\varepsilon}}\times(0, \delta_{2\varepsilon})\).

     
Now we define
$$ \bar{u}_{\varepsilon}=\phi \bigl((A_{1}+\varepsilon )K^{q} \bigl(d(x) \bigr) \bigr), \quad x\in\Omega_{\delta_{1\varepsilon}}. $$
Before we prove the theorem, let us note the following. Suppose that z is a \(C^{2}\) function on a domain Ω in \(\mathbb {R}^{N}\) and \(v = \phi(z)\), where ϕ is uniquely determined by (1.6). A direct computation shows that
$$\begin{aligned} \Delta_{p}v = &(p-1) \bigl\vert \phi'(z) \bigr\vert ^{p-2}\phi''(z)\vert \nabla z \vert ^{p}+ \bigl\vert \phi'(z) \bigr\vert ^{p-2}\phi'(z)\Delta_{p}z. \end{aligned}$$
(5.4)
Hence, by (5.4), Lemma 5.2, and a direct calculation we see that, for \(x\in\Omega_{\delta_{1\varepsilon}}\)
$$\begin{aligned}& \Delta_{p}\bar{u}_{\varepsilon}(x)+b(x) g \bigl( \bar{u}_{\varepsilon}(x) \bigr) \\& \quad = \bigl( \phi' \bigl(K^{q} \bigl(d(x) \bigr) \bigr) \bigr)^{p-1}k^{p} \bigl(d(x) \bigr) \bigl( I_{1+}(r)+I_{2}(x) \bigr) \leq0, \end{aligned}$$
where \(r=d(x)\), that is, \(\bar{u}_{\varepsilon}\) is a supersolution of problem (1.1) in \(\Omega_{\delta_{1\varepsilon}}\).
In a similar way, we show that
$$ \underline{u}_{\varepsilon}=\phi \bigl((A_{1}-\varepsilon )K^{q} \bigl(d(x) \bigr) \bigr),\quad x\in\Omega_{\delta_{1\varepsilon}}, $$
is a subsolution of problem (1.1) in \(\Omega_{\delta_{1\varepsilon}}\).
Let \(u\in C(\bar{\Omega})\cap C^{1,\alpha}(\Omega)\) be the unique solution to problem (1.1). We assert that there exists M large enough such that
$$ u(x)\leq M v (x)+\bar{u}_{\varepsilon}(x),\quad\quad \underline{u}_{\varepsilon}(x)\leq u(x)+M v(x), \quad x\in\Omega_{\delta _{1\varepsilon}}, $$
(5.5)
where v is the solution of problem (5.2).
In fact, we can choose M large enough such that
$$u(x)\leq \bar{u}_{\varepsilon}(x)+M v(x) \quad \text{and}\quad \underline{u}_{\varepsilon}(x)\leq u(x)+M v(x) \quad \text{on } \bigl\{ x\in \Omega: d(x)=\delta_{1\varepsilon}\bigr\} . $$
We see by \((\mathrm{g}_{1})\) that \(\bar{u}_{\varepsilon}(x)+M v(x)\) and \(u(x)+M v(x)\) are also supersolutions of problem (1.1) in \(\Omega_{\delta_{1\varepsilon}}\). Since \(u= \bar{u}_{\varepsilon}+M v=u+Mv=\underline{u}_{\varepsilon}=0 \) on Ω, (5.5) follows by \((\mathrm{g}_{1})\) and the weak comparison principle (Lemma 5.1). Hence, for \(x\in \Omega_{\delta_{1\varepsilon}}\)
$$\frac{u(x)}{\phi((A_{1}+\varepsilon) K^{q}(d(x)))}\leq\frac{M v(x)}{\phi((A_{1}+\varepsilon) K^{q}(d(x)))}+1 $$
and
$$1-\frac{M v(x)}{\phi((A_{1}-\varepsilon) K^{q}(d(x)))}\leq\frac{u(x)}{\phi((A_{1}-\varepsilon) K^{q}(d(x)))}. $$
Consequently, by (5.3) and Lemma 3.4(iii),
$$1\leq\liminf_{d(x) \rightarrow0 } \frac{u(x)}{\phi((A_{1}-\varepsilon) K^{q}(d(x)))} \leq \limsup _{d(x) \rightarrow0 } \frac{u(x)}{\phi((A_{1}+\varepsilon) K^{q}(d(x)))} \leq1 $$
and
$$\lim_{d(x) \rightarrow0 } \frac{\phi((A_{1}-\varepsilon) K^{q}(d(x)))}{\phi( K^{q}(d(x)))}=(A_{1}- \varepsilon)^{1-C_{g}}. $$
Thus, letting \(\varepsilon\rightarrow0\), we obtain (1.8). □

5.2 Proof of Theorem 1.3

As before, fix \(\varepsilon>0\). For any \(\delta>0\), we define \(\Omega_{\delta}=\{x\in\Omega: 0< d(x)<\delta\}\). Since Ω is \(C^{2}\)-smooth, choose \(\delta_{1}\in(0, \delta_{0})\) such that \(d\in C^{2}(\Omega_{\delta_{1}})\) and (5.1) holds.

Define \(r=d(x)\) and
$$\begin{aligned} &I_{1\pm}(r)=(A_{2}\pm\varepsilon)^{p-1}(p-1) \biggl( \frac{(A_{2}\pm \varepsilon)h(r)\phi''((A_{2}\pm\varepsilon)h(r))}{\phi'((A_{1}\pm \varepsilon)h(r))}\frac{(h'(r))^{p}}{h(r)r^{-p}L(r)} \\ &\hphantom{I_{1\pm}(r)=}{}+ \frac {(h'(r))^{p-2}h''(r)}{r^{-p}L(r)} \biggr) , \\ &I_{2}(x)=(A_{2}\pm\varepsilon)^{p-1} \frac{(h'(r))^{p-1}}{r^{-p}L(r)}\Delta d(x)+\frac {b(x)}{k^{p}(r)}\frac{g (\phi((A_{2}\pm \varepsilon)h(r)) )}{ (\phi'((A_{2}\pm \varepsilon)h(r)) )^{p-1}}. \end{aligned}$$
By Lemmas 3.2 and 3.4, combined with the choices of \(A_{2}\) in Theorem 1.3, we get the following lemma.

Lemma 5.3

Suppose that g satisfies \((\mathrm{g}_{1})\)-\((\mathrm{g}_{3})\), b satisfies \((\mathrm{b}_{1})\)-\((\mathrm{b}_{3})\), and \((\mathrm{b}_{5})\) holds. Then
  1. (i)

    \(\lim_{r\rightarrow0} I_{1\pm }(r)=-(p-1)(A_{2}\pm \varepsilon)^{p-1}\);

     
  2. (ii)

    \(\lim_{d(x) \rightarrow0} I_{2 }(x)=b_{1}=(p-1)A_{2}^{p-1}\);

     
  3. (iii)

    \(\lim_{d(x)\rightarrow0} ( I_{1\pm }(r)+I_{2}(x) )=-(p-1) ((A_{2}\pm \varepsilon)^{p-1}-A_{2}^{p-1} )\).

     

Proof of Theorem 1.3

By Lemma 5.3, since \(h\in C[0, \delta_{0})\) with \(h(0)=0\), we see that there exist \(\delta_{1\varepsilon}, \delta_{2\varepsilon}\in (0, \min\{1, \delta_{0}\} )\) (which corresponds to ε) sufficiently small such that
  1. (i)

    \(0\leq h(r)\leq\delta_{1\varepsilon}\), \(r\in(0, \delta_{2\varepsilon})\);

     
  2. (ii)

    \(I_{1+}(r)+I_{2}(x)\leq0\), \(\forall (x,r)\in \Omega_{\delta_{1\varepsilon}}\times(0, \delta_{2\varepsilon}) \);

     
  3. (iii)

    \(I_{1-}(r)+I_{2}(x)\geq0\), \(\forall (x,r)\in \Omega_{\delta_{1\varepsilon}}\times(0, \delta_{2\varepsilon})\).

     
As in the proof of Theorem 1.2, we define
$$ \bar{u}_{\varepsilon}=\phi \bigl((A_{1}+\varepsilon)h \bigl(d(x) \bigr) \bigr), \quad x\in\Omega_{\delta_{1\varepsilon}}, $$
where
$$ h(t)= \int_{0}^{t}s^{-1} \bigl(L(s) \bigr)^{\frac{1}{p-1}}\,ds. $$
By (5.4), Lemma 5.3, and a direct calculation we see that, for \(x\in\Omega_{\delta_{1\varepsilon}}\)
$$\begin{aligned}& \Delta_{p}\bar{u}_{\varepsilon}(x)+b(x) g \bigl( \bar{u}_{\varepsilon}(x) \bigr) \\& \quad = \bigl( \phi' \bigl(h(r) \bigr) \bigr)^{p-1}r^{-p}L(r) \bigl( I_{1+}(r)+I_{2}(x) \bigr) \leq0, \end{aligned}$$
where \(r=d(x)\), that is, \(\bar{u}_{\varepsilon}\) is a supersolution of problem (1.1) in \(\Omega_{\delta_{1\varepsilon}}\).
In a similar way, we show that
$$ \underline{u}_{\varepsilon}=\phi \bigl((A_{2}-\varepsilon)h \bigl(d(x) \bigr) \bigr), \quad x\in\Omega_{\delta_{1\varepsilon}}, $$
is a subsolution of problem (1.1) in \(\Omega_{\delta_{1\varepsilon}}\).
As in the proof of Theorem 1.2, we obtain, for \(x\in \Omega_{\delta_{1\varepsilon}}\)
$$\frac{u(x)}{\phi((A_{2}+\varepsilon) h(d(x)))}\leq\frac{M v(x)}{\phi((A_{2}+\varepsilon) h(d(x)))}+1 $$
and
$$1-\frac{M v(x)}{\phi((A_{2}-\varepsilon) h(d(x)))}\leq\frac{u(x)}{\phi((A_{2}-\varepsilon)h(d(x)))}. $$
Consequently, by (5.3) and Lemma 3.4(iv),
$$1\leq\liminf_{d(x) \rightarrow0 } \frac{u(x)}{\phi((A_{2}-\varepsilon) h(d(x)))} \leq\limsup _{d(x) \rightarrow0 } \frac{u(x)}{\phi((A_{2}+\varepsilon) h(d(x)))} \leq 1 $$
and
$$\lim_{d(x) \rightarrow0 } \frac{\phi((A_{2}-\varepsilon) h(d(x)))}{\phi( h(d(x)))}=(A_{2}- \varepsilon)^{1-C_{g}}. $$
Thus, letting \(\varepsilon\rightarrow0\), we obtain (1.9). □

Declarations

Acknowledgements

The author is thankful to the honorable reviewers for their valuable suggestions and comments, which improved the paper. This work was partially supported by NSF of China (Grant no. 11301250), NSF of Shandong Province (Grant no. ZR2013AQ004), the Applied Mathematics Enhancement Program (AMEP) of Linyi University, and PhD research startup foundation of Linyi University (Grant no. LYDX2013BS049 ).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Science, Linyi University

References

  1. Mikljukov, V: On the asymptotic properties of subsolutions of quasilinear equations of elliptic type and mappings with bounded distortion. Sb. Math. 111, 42-66 (1980) (in Russian) MathSciNetGoogle Scholar
  2. Reshetnyak, Y: Index boundedness condition for mappings with bounded distortion. Sib. Math. J. 9, 281-285 (1968) View ArticleMATHGoogle Scholar
  3. Uhlenbeck, K: Regularity for a class of non-linear elliptic systems. Acta Math. 138, 219-240 (1977) MathSciNetView ArticleMATHGoogle Scholar
  4. Herrero, M, Vásquez, J: On the propagation properties of a nonlinear degenerate parabolic equation. Commun. Partial Differ. Equ. 7(12), 1381-1402 (1982) MathSciNetView ArticleMATHGoogle Scholar
  5. Esteban, R, Vásquez, J: On the equation of turbulent filtration in one-dimensional porous media. Nonlinear Anal. 10(11), 1303-1325 (1986) MathSciNetView ArticleMATHGoogle Scholar
  6. Crandall, M, Rabinowitz, P, Tartar, L: On a Dirichlet problem with a singular nonlinearity. Commun. Partial Differ. Equ. 2, 193-222 (1977) MathSciNetView ArticleMATHGoogle Scholar
  7. Lazer, AC, McKenna, PJ: On a singular elliptic boundary value problem. Proc. Am. Math. Soc. 111, 721-730 (1991) MathSciNetView ArticleMATHGoogle Scholar
  8. Fulks, W, Maybee, J: A singular nonlinear elliptic equation. Osaka J. Math. 12, 1-19 (1960) MathSciNetMATHGoogle Scholar
  9. Stuart, C: Existence and approximation of solutions of nonlinear elliptic equations. Math. Z. 147, 53-63 (1976) MathSciNetView ArticleMATHGoogle Scholar
  10. Lazer, A, McKenna, P: On a singular elliptic boundary value problem. Proc. Am. Math. Soc. 111, 721-730 (1991) MathSciNetView ArticleMATHGoogle Scholar
  11. Anedda, C: Second-order boundary estimates for solutions to singular elliptic equations. Electron. J. Differ. Equ. 2009, 90 (2009) MathSciNetMATHGoogle Scholar
  12. Anedda, C, Porru, G: Second-order boundary estimates for solutions to singular elliptic equations in borderline cases. Electron. J. Differ. Equ. 2011, 51 (2011) MathSciNetView ArticleMATHGoogle Scholar
  13. Berhanu, S, Gladiali, F, Porru, G: Qualitative properties of solutions to elliptic singular problems. J. Inequal. Appl. 3, 313-330 (1999) MathSciNetMATHGoogle Scholar
  14. Berhanu, S, Cuccu, F, Porru, G: On the boundary behaviour, including second order effects, of solutions to elliptic singular problems. Acta Math. Sin. Engl. Ser. 23, 479-486 (2007) MathSciNetView ArticleMATHGoogle Scholar
  15. Giarrusso, E, Porru, G: Boundary behaviour of solutions to nonlinear elliptic singular problems. In: Misra, JC (ed.) Appl. Math. in the Golden Age, pp. 163-178. Narosa Publishing House, New Dalhi (2003) Google Scholar
  16. Ghergu, M, Rǎdulescu, VD: Bifurcation and asymptotics for the Lane-Emden-Fowler equation. C. R. Math. Acad. Sci. Paris, Sér. I 337, 259-264 (2003) MathSciNetView ArticleMATHGoogle Scholar
  17. Cîrstea, F, Rǎdulescu, V: Uniqueness of the blow-up boundary solution of logistic equations with absorbtion. C. R. Math. Acad. Sci. Paris, Sér. I 335, 447-452 (2002) MathSciNetView ArticleMATHGoogle Scholar
  18. Cîrstea, F, Rǎdulescu, V: Asymptotics for the blow-up boundary solution of the logistic equation with absorption. C. R. Math. Acad. Sci. Paris, Sér. I 336, 231-236 (2003) MathSciNetView ArticleMATHGoogle Scholar
  19. Cîrstea, F, Rǎdulescu, V: Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach. Asymptot. Anal. 46, 275-298 (2006) MathSciNetMATHGoogle Scholar
  20. Cîrstea, F, Du, Y: General uniqueness results and variation speed for blow-up solutions of elliptic equations. Proc. Lond. Math. Soc. 91, 459-482 (2005) MathSciNetView ArticleMATHGoogle Scholar
  21. Repovs̆, D: Asymptotics for singular solutions of quasilinear elliptic equations with an absorption term. J. Math. Anal. Appl. 395, 78-85 (2012) MathSciNetView ArticleGoogle Scholar
  22. Zhang, Z, Li, B: The boundary behavior of the unique solution to a singular Dirichlet problem. J. Math. Anal. Appl. 391, 278-290 (2012) MathSciNetView ArticleMATHGoogle Scholar
  23. Zhang, Z: The second expansion of the solution for a singular elliptic boundary value problems. J. Math. Anal. Appl. 381, 922-934 (2011) MathSciNetView ArticleMATHGoogle Scholar
  24. Zhang, Z, Cheng, J: Existence and optimal estimates of solutions for singular nonlinear Dirichlet problems. Nonlinear Anal. 57, 473-484 (2004) MathSciNetView ArticleMATHGoogle Scholar
  25. Mi, L, Liu, B: The second order estimate for the solution to a singular elliptic boundary value problem. Appl. Anal. Discrete Math. 6, 194-213 (2012) MathSciNetView ArticleMATHGoogle Scholar
  26. Ni, W, Serrin, J: Existence and nonexistence theorems for ground states of quasilinear partial differential equations: the anomalous case. Acad. Naz. Lincei 77, 231-257 (1986) Google Scholar
  27. Ni, W, Serrin, J: Nonexistence theorems for singular solutions of quasilinear partial differential equations. Commun. Pure Appl. Math. 38, 379-399 (1986) MathSciNetView ArticleMATHGoogle Scholar
  28. Guedda, M, Veron, L: Local and global properties of solutions of quasilinear elliptic equations. J. Differ. Equ. 76, 159-189 (1988) MathSciNetView ArticleMATHGoogle Scholar
  29. Bognara, G, Drabekb, P: The p-Laplacian equation with superlinear and supercritical growth, multiplicity of radial solutions. Nonlinear Anal. 60, 719-728 (2005) MathSciNetView ArticleGoogle Scholar
  30. Prashanth, S, Sreenadh, K: Multiplicity of positive solutions for p-Laplace equation with superlinear-type nonlinearity. Nonlinear Anal. 56, 867-878 (2004) MathSciNetView ArticleMATHGoogle Scholar
  31. Reichel, W, Walter, W: Radial solutions of equations and inequalities involving the p-Laplacian. J. Inequal. Appl. 1, 47-71 (1997) MathSciNetMATHGoogle Scholar
  32. Santos, C: Non-existence and existence of entire solutions for a quasi-linear problem with singular and super-linear terms. Nonlinear Anal. 72, 3813-3819 (2010) MathSciNetView ArticleMATHGoogle Scholar
  33. Serrin, J, Zou, H: Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta Math. 189, 79-142 (2002) MathSciNetView ArticleMATHGoogle Scholar
  34. Mâagli, H, Giacomoni, J, Sauvy, P: Existence of compact support solutions for a quasilinear and singular problem. Differ. Integral Equ. 25(7-8), 629-656 (2012) MathSciNetMATHGoogle Scholar
  35. Masmoudi, S, Zermani, S: Existence and asymptotic behavior of solutions to nonlinear radial p-Laplacian equations. Electron. J. Differ. Equ. 2015, 171 (2015) MathSciNetView ArticleMATHGoogle Scholar
  36. Xu, B, Yang, Z: Entire bounded solutions for a class of quasilinear elliptic equations. Bound. Value Probl. 2007 Article ID 16407 (2007) MathSciNetView ArticleMATHGoogle Scholar
  37. Cencelj, M, Repovš, D, Virk, Z̆: Multiple perturbations of a singular eigenvalue problem. Nonlinear Anal. 119, 37-45 (2015) MathSciNetView ArticleMATHGoogle Scholar
  38. Chetri, M, Drabek, P, Shivaji, R: Analysis of positive solutions for classes of quasilinear singular problems on exterior domains. Adv. Nonlinear Anal. (2016). doi:10.1515/anona-2015-0143 Google Scholar
  39. Karamata, J: Sur un mode de croissance régulière de fonctions. Théorèmes fondamentaux. Bull. Soc. Math. Fr. 61, 55-62 (1933) MathSciNetMATHGoogle Scholar
  40. Bingham, N, Goldie, C, Teugels, J: Regular Variation. Encyclopedia of Mathematics and Its Applications, vol. 27. Cambridge University Press, Cambridge (1987) View ArticleMATHGoogle Scholar
  41. Maric, V: Regular Variation and Differential Equations. Lecture Notes in Math., vol. 1726. Springer, Berlin (2000) MATHGoogle Scholar
  42. Resnick, SI: Extreme Values, Regular Variation, and Point Processes. Springer, New York (1987) View ArticleMATHGoogle Scholar
  43. Seneta, R: Regular Varying Functions. Lecture Notes in Math., vol. 508. Springer, Berlin (1976) View ArticleMATHGoogle Scholar
  44. Zhang, Z, Li, B, Li, X: The exact boundary behavior of solutions to singular nonlinear Lane-Emden-Fowler type boundary value problems. Nonlinear Anal., Real World Appl. 21, 34-52 (2015) MathSciNetView ArticleMATHGoogle Scholar
  45. Tolksdorf, P: On the Dirichlet problem for quasilinear equations in domains with conical boundary points. Commun. Partial Differ. Equ. 8, 773-817 (1983) MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Mi 2016