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Abstract and Applied Analysis
VolumeΒ 2010, Article IDΒ 971268, 20 pages
http://dx.doi.org/10.1155/2010/971268
Research Article

Existence and Asymptotic Behavior of Boundary Blow-Up Solutions for Weighted 𝑝(π‘₯)-Laplacian Equations with Exponential Nonlinearities

Li Yin,1Β Yunrui Guo,2Β Jing Yang,1Β Bibo Lu,3Β and Qihu Zhang1,4

1Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China
2Department of Mathematics, Henan Institute of Science and Technology, Xinxiang, Henan 453003, China
3School of Computer Science and Technology, Henan Polytechnic University, Jiaozuo, Henan 454000, China
4School of Mathematics and Statistics, Huazhong Normal University, Wuhan, Hubei 430079, China

Received 21 March 2010; Accepted 3 October 2010

Academic Editor: PavelΒ DrΓ‘bek

Copyright Β© 2010 Li Yin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper investigates the following 𝑝(π‘₯)-Laplacian equations with exponential nonlinearities: βˆ’Ξ”π‘(π‘₯)𝑒+𝜌(π‘₯)𝑒𝑓(π‘₯,𝑒)=0 in Ξ©, 𝑒(π‘₯)β†’+∞ as 𝑑(π‘₯,πœ•Ξ©)β†’0, where βˆ’Ξ”π‘(π‘₯)𝑒=βˆ’div(|βˆ‡π‘’|𝑝(π‘₯)βˆ’2βˆ‡π‘’) is called 𝑝(π‘₯)-Laplacian, 𝜌(π‘₯)∈𝐢(Ξ©). The asymptotic behavior of boundary blow-up solutions is discussed, and the existence of boundary blow-up solutions is given.

1. Introduction

The study of differential equations and variational problems with nonstandard 𝑝(π‘₯)-growth conditions is a new and interesting topic. On the background of this class of problems, we refer to [1–3]. Many results have been obtained on this kind of problems, for example, [4–18]. On the regularity of weak solutions for differential equations with nonstandard 𝑝(π‘₯)-growth conditions, we refer to [4, 5, 8]. On the existence of solutions for 𝑝(π‘₯)-Laplacian equation Dirichlet problems in bounded domain, we refer to [7, 9, 15, 18]. In this paper, we consider the following 𝑝(π‘₯)-Laplacian equations with exponential nonlinearities βˆ’Ξ”π‘(π‘₯)𝑒+𝜌(π‘₯)𝑒𝑓(π‘₯,𝑒)=0,inΞ©,𝑒(π‘₯)⟢+∞,as𝑑(π‘₯,πœ•Ξ©)⟢0,(P)

where βˆ’Ξ”π‘(π‘₯)𝑒=βˆ’div(|βˆ‡π‘’|𝑝(π‘₯)βˆ’2βˆ‡π‘’) and Ξ©=𝐡(0,𝑅)βŠ‚β„π‘ is a bounded radial domain (𝐡(0,𝑅)={π‘₯βˆˆβ„π‘βˆ£|π‘₯|<𝑅}). Our aim is to give the asymptotic behavior and the existence of boundary blow-up solutions for problem (P).

Throughout the paper, we assume that 𝑝(π‘₯), 𝜌(π‘₯), and 𝑓(π‘₯,𝑒) satisfy the following.

(H1) 𝑝(π‘₯)∈𝐢1(Ξ©) is radial and satisfies 1<π‘βˆ’β‰€π‘+<+∞,whereπ‘βˆ’=infΩ𝑝(π‘₯),𝑝+=supΩ𝑝(π‘₯).(1.1)

(H2) 𝑓(π‘₯,𝑒) is radial with respect to π‘₯, 𝑓(π‘₯,β‹…) is increasing, and 𝑓(π‘₯,0)=0 for any π‘₯∈Ω.

(H3)β€‰π‘“βˆΆΞ©Γ—β„β†’β„ is continuous and satisfies||||𝑓(π‘₯,𝑑)≀𝐢1+𝐢2|𝑑|𝛾(π‘₯),,βˆ€(π‘₯,𝑑)βˆˆΞ©Γ—β„(1.2)

where 𝐢1, 𝐢2 are positive constants and 0β‰€π›ΎβˆˆπΆ(Ξ©).

(H4) 𝜌(π‘₯)∈𝐢(Ξ©) is a radial nonnegative function, and there exists a constant 𝜎∈[𝑅/2,𝑅) such that 𝜌0(π‘…βˆ’π‘Ÿ)βˆ’π›½(π‘Ÿ)β‰€πœŒ(π‘Ÿ)β‰€πœŒ1(π‘…βˆ’π‘Ÿ)βˆ’π›½1(π‘Ÿ)for[π‘ŸβˆˆπœŽ,𝑅)uniformly,(1.3)

where 𝜌0 and 𝜌1 are positive constants and 𝛽(π‘Ÿ) and 𝛽1(π‘Ÿ) are Lipschitz continuous on [𝜎,𝑅], which satisfy 𝛽(π‘Ÿ)≀𝛽1(π‘Ÿ)<𝑝(π‘Ÿ) for any π‘Ÿβˆˆ[𝜎,𝑅].

The operator βˆ’Ξ”π‘(π‘₯)𝑒=βˆ’div(|βˆ‡π‘’|𝑝(π‘₯)βˆ’2βˆ‡π‘’) is called 𝑝(π‘₯)-Laplacian. Specifically, if 𝑝(π‘₯)≑𝑝 (a constant), (P) is the well-known 𝑝-Laplacian problem. If 𝑓(π‘₯,𝑒) can be represented as β„Ž(π‘₯)𝑓(𝑒), on the boundary blow-up solutions for the following 𝑝-Laplacian equations (𝑝 is a constant):βˆ’Ξ”π‘π‘’+β„Ž(π‘₯)𝑓(𝑒)=0,inΞ©,(1.4)

we refer to [19–26], and the following generalized Keller-Osserman condition is crucialξ€œβˆž11(𝐹(𝑑))1/𝑝𝑑𝑑<+∞,whereξ€œπΉ(𝑑)=𝑑0𝑓(𝑠)𝑑𝑠,(1.5)

but the typical form of 𝑝(π‘₯)-Laplacian equation isβˆ’Ξ”π‘(π‘₯)𝑒+|𝑒|π‘ž(π‘₯)βˆ’2𝑒=0,inΞ©,(1.6)

and there are some differences between the results of (1.4) and (1.6) (see [16]).

On the boundary blow-up solutions for the following 𝑝-Laplacian equations with exponential nonlinearities (𝑝 is a constant):βˆ’Ξ”π‘π‘’+π‘’β„Ž(π‘₯)𝑓(𝑒)=0,inΞ©,(1.7)

we refer to [20–22], but the results on the boundary blow-up solutions for 𝑝(π‘₯)-Laplacian equations are rare (see [16]).

In [16], the present author discussed the existence and asymptotic behavior of boundary blow-up solutions for the following 𝑝(π‘₯)-Laplacian equations: βˆ’Ξ”π‘(π‘₯)𝑒+𝑓(π‘₯,𝑒)=0,inΞ©,𝑒(π‘₯)⟢+∞,as𝑑(π‘₯,πœ•Ξ©)⟢0,(1.8)

on the condition that 𝑓(π‘₯,β‹…) satisfies polynomial growth condition.

If 𝑝(π‘₯) is a function, the typical form of (P) is the following:βˆ’Ξ”π‘(π‘₯)𝑒+𝜌(π‘₯)𝑒|𝑒|π‘ž(π‘₯)βˆ’2𝑒=0,(1.9)

and the method to construct subsolution and supersolution in [16] cannot give the exact asymptotic behavior of solutions for (P). Our results partially generalized the results of [20–22].

Because of the nonhomogeneity of 𝑝(π‘₯)-Laplacian, 𝑝(π‘₯)-Laplacian problems are more complicated than those of 𝑝-Laplacian ones (see [10]); another difficulty of this paper is that 𝑓(π‘₯,𝑒) cannot be represented as β„Ž(π‘₯)𝑓(𝑒).

2. Preliminary

In order to deal with 𝑝(π‘₯)-Laplacian problems, we need some theories on the spaces 𝐿𝑝(π‘₯)(Ξ©), π‘Š1,𝑝(π‘₯)(Ξ©) and properties of 𝑝(π‘₯)-Laplacian, which we will use later (see [6, 11]). Let𝐿𝑝(π‘₯)(Ξ©)=π‘’βˆ£π‘’isameasurablereal-valuedfunction,∫Ω||||𝑒(π‘₯)𝑝(π‘₯).𝑑π‘₯<∞(2.1)

We can introduce the norm on 𝐿𝑝(π‘₯)(Ξ©) by|𝑒|𝑝(π‘₯)ξ‚»ξ€œ=infπœ†>0∣Ω|||𝑒(π‘₯)πœ†|||𝑝(π‘₯)ξ‚Ό.𝑑π‘₯≀1(2.2)

The space (𝐿𝑝(π‘₯)(Ξ©), |β‹…|𝑝(π‘₯)) becomes a Banach space. We call it generalized Lebesgue space. The space (𝐿𝑝(π‘₯)(Ξ©), |β‹…|𝑝(π‘₯)) is a separable, reflexive, and uniform convex Banach space (see [6, Theorems 1.10, 1.14] ).

The space π‘Š1,𝑝(π‘₯)(Ξ©) is defined byπ‘Š1,𝑝(π‘₯)ξ€½(Ξ©)=π‘’βˆˆπΏπ‘(π‘₯)||||(Ξ©)βˆ£βˆ‡π‘’βˆˆπΏπ‘(π‘₯)ξ€Ύ,(Ξ©)(2.3)

and it can be equipped with the norm ‖𝑒‖=|𝑒|𝑝(π‘₯)+||||βˆ‡π‘’π‘(π‘₯),βˆ€π‘’βˆˆπ‘Š1,𝑝(π‘₯)(Ξ©).(2.4)

π‘Š01,𝑝(π‘₯)(Ξ©) is the closure of 𝐢∞0(Ξ©) in π‘Š1,𝑝(π‘₯)(Ξ©). π‘Š1,𝑝(π‘₯)(Ξ©) and π‘Š01,𝑝(π‘₯)(Ξ©) are separable, reflexive, and uniform convex Banach spaces (see [6, Theorem 2.1]).

If π‘’βˆˆπ‘Š1,𝑝(π‘₯)loc(Ξ©)∩𝐢(Ξ©), 𝑒 is called a blow-up solution of (P) when it satisfiesξ€œπ‘„||||βˆ‡π‘’π‘(π‘₯)βˆ’2ξ€œβˆ‡π‘’βˆ‡π‘žπ‘‘π‘₯+π‘„πœŒ(π‘₯)𝑓(π‘₯,𝑒)π‘žπ‘‘π‘₯=0,βˆ€π‘žβˆˆπ‘Š01,𝑝(π‘₯)(𝑄),(2.5)

for any domain 𝑄⋐Ω, and max(π‘˜βˆ’π‘’,0)βˆˆπ‘Š01,𝑝(π‘₯)(Ξ©) for every positive integer π‘˜.

Let π‘Š1,𝑝(π‘₯)0,loc(Ξ©)={π‘’βˆ£ there is an open domain 𝑄⋐Ω such that π‘’βˆˆπ‘Š01,𝑝(π‘₯)(𝑄)}, and define π΄βˆΆπ‘Š1,𝑝(π‘₯)loc(Ξ©)∩𝐢(Ξ©)β†’(π‘Š1,𝑝(π‘₯)0,loc(Ξ©))βˆ— asξ€œβŸ¨π΄π‘’,πœ™βŸ©=Ξ©ξ‚€||||βˆ‡π‘’π‘(π‘₯)βˆ’2βˆ‡π‘’βˆ‡πœ™+𝜌(π‘₯)𝑒𝑓(π‘₯,𝑒)πœ™ξ‚π‘‘π‘₯,βˆ€π‘’βˆˆπ‘Š1,𝑝(π‘₯)loc(Ξ©)∩𝐢(Ξ©),βˆ€πœ™βˆˆπ‘Š1,𝑝(π‘₯)0,loc(Ξ©).(2.6)

Lemma 2.1 (see [9, Theorem 3.1]). Let β„Žβˆˆπ‘Š1,𝑝(π‘₯)(Ξ©)∩𝐢(Ξ©), and 𝑋=β„Ž+π‘Š01,𝑝(π‘₯)(Ξ©)∩𝐢(Ξ©). Then, π΄βˆΆπ‘‹β†’(π‘Š1,𝑝(π‘₯)0,loc(Ξ©))βˆ— is strictly monotone.

Letting π‘”βˆˆ(π‘Š1,𝑝(π‘₯)0,loc(Ξ©))βˆ—, if βŸ¨π‘”,πœ™βŸ©β‰₯0,forallπœ™βˆˆπ‘Š1,𝑝(π‘₯)0,loc(Ξ©) with πœ™β‰₯0 a.e. in Ξ©, then denote 𝑔β‰₯0 in (π‘Š1,𝑝(π‘₯)0,loc(Ξ©))βˆ—; correspondingly, if βˆ’π‘”β‰₯0 in (π‘Š1,𝑝(π‘₯)0,loc(Ξ©))βˆ—, then denote 𝑔≀0 in (π‘Š1,𝑝(π‘₯)0,loc(Ξ©))βˆ—.

Definition 2.2. Let π‘’βˆˆπ‘Š1,𝑝(π‘₯)loc(Ξ©)∩𝐢(Ξ©). If 𝐴𝑒β‰₯0(𝐴𝑒≀0) in (π‘Š1,𝑝(π‘₯)0,loc(Ξ©))βˆ—, then 𝑒 is called a weak supersolution (weak subsolution) of (P).

Copying the proof of [14], we have the following.

Lemma 2.3 (comparison principle). Let 𝑒,π‘£βˆˆπ‘Š1,𝑝(π‘₯)loc(Ξ©)∩𝐢(Ξ©) satisfy π΄π‘’βˆ’π΄π‘£β‰₯0,inξ‚€π‘Š01,𝑝(π‘₯)(Ξ©)βˆ—.(2.7) Let πœ™(π‘₯)=min{𝑒(π‘₯)βˆ’π‘£(π‘₯),0}. If πœ™(π‘₯)βˆˆπ‘Š01,𝑝(π‘₯)(Ξ©) (i.e., 𝑒β‰₯𝑣 on πœ•Ξ©), then 𝑒β‰₯𝑣 a.e. in Ξ©.

Lemma 2.4 (see [8, Theorem 1.1]). Under the conditions (H1) and (H3), if π‘’βˆˆπ‘Š1,𝑝(π‘₯)(Ξ©) is a bounded weak solution of βˆ’Ξ”π‘(π‘₯)𝑒+𝜌(π‘₯)𝑒𝑓(π‘₯,𝑒)=0 in Ξ©, then π‘’βˆˆπΆ1,πœ—loc(Ξ©), where πœ—βˆˆ(0,1) is a constant.

3. Asymptotic Behavior of Boundary Blow-Up Solutions

If 𝑒 is a radial solution for (P), then (P) can be transformed intoξ‚€π‘Ÿπ‘βˆ’1||π‘’ξ…ž||𝑝(π‘Ÿ)βˆ’2π‘’ξ…žξ‚ξ…ž=π‘Ÿπ‘βˆ’1𝜌(π‘Ÿ)𝑒𝑓(π‘Ÿ,𝑒),π‘Ÿβˆˆ(0,𝑅),𝑒(0)=𝑒0,π‘’ξ…ž(0)=0,π‘’ξ…ž(π‘Ÿ)β‰₯0,for0<π‘Ÿ<𝑅.(3.1)

It means that 𝑒(π‘Ÿ) is increasing.

Theorem 3.1. If 𝑓(π‘Ÿ,𝑒) satisfies 𝑓(π‘Ÿ,𝑒)β‰₯𝛼𝑒𝑠(asπ‘’βŸΆ+∞)for[π‘ŸβˆˆπœŽ,𝑅)uniformly,(3.2) where 𝜎 is defined in (H4) and Ξ± and 𝑠 are positive constants, then there exists a supersolution Ξ¦1(π‘₯) which satisfies Ξ¦1(π‘₯)β†’+∞ (as 𝑑(π‘₯,πœ•Ξ©)β†’0), such that for every solution 𝑒 of problem (P), one has 𝑒(π‘₯)≀Φ1(π‘₯).

Proof. Define the function 𝑔(π‘Ÿ,π‘Ž,πœ†) on [0,π‘…πœ†) as ⎧βŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽ©ξ‚΅1𝑔(π‘Ÿ,π‘Ž,πœ†)=π‘Žln(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ‚Άβˆ’πœ†1/𝑠+π‘˜,𝑅0β‰€π‘Ÿ<π‘…πœ†,ξ€œπ‘˜βˆ’π‘…0π‘ŸβŽ‘βŽ’βŽ’βŽ£π‘Ž1/𝑠(1βˆ’πœƒ)π‘…βˆ’π‘…0ξ€Έβˆ’πœƒπ‘ ξ‚€ξ€·π‘…βˆ’π‘…0ξ€Έ1βˆ’πœƒξ‚ξƒ©1βˆ’πœ†lnξ€·π‘…βˆ’π‘…0ξ€Έ1βˆ’πœƒξƒͺβˆ’πœ†(1/𝑠)βˆ’1⎀βŽ₯βŽ₯⎦(𝑝(𝑅0)βˆ’1)/(𝑝(𝑑)βˆ’1)×𝑅0ξ€Έπ‘βˆ’1π‘‘π‘βˆ’1ξƒ­sinπœ–(π‘‘βˆ’πœŽ)1/(𝑝(𝑑)βˆ’1)+1π‘‘π‘‘π‘Žlnξ€·π‘…βˆ’π‘…0ξ€Έ1βˆ’πœƒξƒͺβˆ’πœ†1/𝑠,𝜎<π‘Ÿ<𝑅0,ξ€œπ‘˜βˆ’π‘…0πœŽβŽ‘βŽ’βŽ’βŽ£π‘Ž1/𝑠(1βˆ’πœƒ)π‘…βˆ’π‘…0ξ€Έβˆ’πœƒπ‘ ξ‚€ξ€·π‘…βˆ’π‘…0ξ€Έ1βˆ’πœƒξ‚ξƒ©1βˆ’πœ†lnξ€·π‘…βˆ’π‘…0ξ€Έ1βˆ’πœƒξƒͺβˆ’πœ†(1/𝑠)βˆ’1⎀βŽ₯βŽ₯⎦(𝑝(𝑅0)βˆ’1)/(𝑝(𝑑)βˆ’1)×𝑅0ξ€Έπ‘βˆ’1π‘‘π‘βˆ’1ξƒ­sinπœ–(π‘‘βˆ’πœŽ)1/(𝑝(𝑑)βˆ’1)+1π‘‘π‘‘π‘Žlnξ€·π‘…βˆ’π‘…0ξ€Έ1βˆ’πœƒξƒͺβˆ’πœ†1/𝑠,π‘Ÿβ‰€πœŽ,(3.3) where πœƒ<𝛽(𝑅)/𝑝(𝑅), π‘Ž>(1/𝛼)sup|π‘₯|β‰₯𝑅0𝑝(π‘₯) are constants, 𝑅0∈(𝜎,𝑅), π‘…βˆ’π‘…0 is small enough, parameter πœ†βˆˆ[0,(π‘…βˆ’π‘…0)1βˆ’πœƒ/2], π‘…πœ† satisfies (π‘…βˆ’π‘…πœ†)1βˆ’πœƒβˆ’πœ†=0, πœ–=πœ‹/2(𝑅0βˆ’πœŽ)ξƒ¬π‘˜=2𝑝+||𝛽||((1+𝑠)/𝑠+1/(1βˆ’πœƒ))++/(1βˆ’πœƒ)𝛼2lnξ€·π‘…βˆ’π‘…0ξ€Έ(1βˆ’πœƒ)ξƒ­1/𝑠+ξ€œπ‘…0𝜎⎑⎒⎒⎣2π‘Ž1/𝑠(1βˆ’πœƒ)π‘ ξ€·π‘…βˆ’π‘…02lnξ€·π‘…βˆ’π‘…0ξ€Έ1βˆ’πœƒξƒͺ(1/𝑠)βˆ’1⎀βŽ₯βŽ₯⎦(𝑝(𝑅0)βˆ’1)/(𝑝(𝑑)βˆ’1)×𝑅0ξ€Έπ‘βˆ’1π‘‘π‘βˆ’1ξƒ­sinπœ–(π‘‘βˆ’πœŽ)1/(𝑝(𝑑)βˆ’1)𝑑𝑑.(3.4)
Obviously, for any positive constant π‘Ž, we have 𝑔(π‘Ÿ,π‘Ž,πœ†)∈𝐢1[0,π‘…πœ†).
When 𝑅0<π‘Ÿ<π‘…πœ†<𝑅, we have π‘”ξ…ž=π‘”ξ…žπ‘Ž(π‘Ÿ,π‘Ž,πœ†)=1/𝑠𝑠1ln(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ‚Άβˆ’πœ†(1/𝑠)βˆ’1(1βˆ’πœƒ)(π‘…βˆ’π‘Ÿ)βˆ’πœƒ(π‘…βˆ’π‘Ÿ)1βˆ’πœƒ,||π‘”βˆ’πœ†ξ…ž||𝑝(π‘Ÿ)βˆ’2π‘”ξ…ž=ξ‚Έ(1βˆ’πœƒ)π‘Ž1/𝑠𝑠𝑝(π‘Ÿ)βˆ’1ξ‚΅1ln(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ‚Άβˆ’πœ†((1/𝑠)βˆ’1)(𝑝(π‘Ÿ)βˆ’1)(π‘…βˆ’π‘Ÿ)βˆ’πœƒ(𝑝(π‘Ÿ)βˆ’1)ξ€Ί(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ€»βˆ’πœ†π‘(π‘Ÿ)βˆ’1,ξ‚€π‘Ÿπ‘βˆ’1||π‘”ξ…ž||𝑝(π‘Ÿ)βˆ’2π‘”ξ…žξ‚ξ…ž=π‘Ÿπ‘βˆ’1ξ‚Έ(1βˆ’πœƒ)π‘Ž1/𝑠𝑠𝑝(π‘Ÿ)βˆ’1ξ‚΅1ln(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ‚Άβˆ’πœ†((1/𝑠)βˆ’1)(𝑝(π‘Ÿ)βˆ’1)Γ—(𝑝(π‘Ÿ)βˆ’1)(π‘…βˆ’π‘Ÿ)βˆ’πœƒπ‘(π‘Ÿ)ξ€Ί(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ€»βˆ’πœ†π‘(π‘Ÿ)[(],1βˆ’πœƒ)+Ξ (π‘Ÿ)(3.5)
where ξ‚†π‘ŸΞ (π‘Ÿ)=π‘βˆ’1ξ€Ί(1βˆ’πœƒ)π‘Ž1/𝑠/𝑠𝑝(π‘Ÿ)βˆ’1ξ‚‡ξ…ž(𝑝(π‘Ÿ)βˆ’1)π‘Ÿπ‘βˆ’1ξ€Ί(1βˆ’πœƒ)π‘Ž1/𝑠/𝑠𝑝(π‘Ÿ)βˆ’1(π‘…βˆ’π‘Ÿ)1βˆ’πœƒβˆ’πœ†(π‘…βˆ’π‘Ÿ)1βˆ’πœƒ(π‘…βˆ’π‘Ÿ)+((1/𝑠)βˆ’1)(1βˆ’πœƒ)ξ€·ξ€·ξ€·ln1/(π‘…βˆ’π‘Ÿ)1βˆ’πœƒ+βˆ’πœ†ξ€Έξ€Έξ€Έ(π‘…βˆ’π‘Ÿ)1βˆ’πœƒβˆ’πœ†(π‘…βˆ’π‘Ÿ)1βˆ’πœƒ(π‘…βˆ’π‘Ÿ)((1/𝑠)βˆ’1)π‘ξ…ž(π‘Ÿ)ξ‚Έ1(𝑝(π‘Ÿ)βˆ’1)lnln(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ‚Ή+βˆ’πœ†πœƒπ‘ξ…ž(π‘Ÿ)(𝑝(π‘Ÿ)βˆ’1)(π‘…βˆ’π‘Ÿ)1βˆ’πœƒβˆ’πœ†(π‘…βˆ’π‘Ÿ)1βˆ’πœƒ1(π‘…βˆ’π‘Ÿ)ln(π‘…βˆ’π‘Ÿ)+πœƒ(π‘…βˆ’π‘Ÿ)1βˆ’πœƒβˆ’πœ†(π‘…βˆ’π‘Ÿ)1βˆ’πœƒ+βˆ’π‘ξ…ž(π‘Ÿ)(𝑝(π‘Ÿ)βˆ’1π‘…βˆ’π‘Ÿ)(π‘…βˆ’π‘Ÿ)1βˆ’πœƒβˆ’πœ†(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ€Ί(lnπ‘…βˆ’π‘Ÿ)1βˆ’πœƒξ€».βˆ’πœ†(3.6)
If (π‘…βˆ’π‘…0) is small enough, it is easy to see that ||||1Ξ (π‘Ÿ)≀ln(π‘…βˆ’π‘Ÿ)1βˆ’πœƒ,βˆ’πœ†forξƒ¬ξ€·πœ†βˆˆ0,π‘…βˆ’π‘…0ξ€Έ1βˆ’πœƒ2ξƒ­uniformly,(3.7)
and then ξ‚€π‘Ÿπ‘βˆ’1||π‘”ξ…ž||𝑝(π‘Ÿ)βˆ’2π‘”ξ…žξ‚ξ…žβ‰€π‘Ÿπ‘βˆ’1ξ‚Έ(1βˆ’πœƒ)π‘Ž1/𝑠𝑠𝑝(π‘Ÿ)βˆ’1ξ‚΅1ln(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ‚Άβˆ’πœ†((1/𝑠)βˆ’1)(𝑝(π‘Ÿ)βˆ’1)+1Γ—(𝑝(π‘Ÿ)βˆ’1)(π‘…βˆ’π‘Ÿ)βˆ’πœƒπ‘(π‘Ÿ)ξ€Ί(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ€»βˆ’πœ†π‘(π‘Ÿ)𝑅,βˆ€π‘Ÿβˆˆ0,π‘…πœ†ξ€Έ.(3.8)
Thus, when 0<π‘…βˆ’π‘…0 is small enough, from (3.5) and (3.8), for πœ†βˆˆ[0,(π‘…βˆ’π‘…0)1βˆ’πœƒ/2] uniformly, we have ξ‚€π‘Ÿπ‘βˆ’1||π‘”ξ…ž||𝑝(π‘Ÿ)βˆ’2π‘”ξ…žξ‚ξ…žβ‰€2π‘Ÿπ‘βˆ’1ξ‚Έ(1βˆ’πœƒ)π‘Ž1/𝑠𝑠𝑝(π‘Ÿ)βˆ’1ξ‚΅1ln(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ‚Άβˆ’πœ†((1/𝑠)βˆ’1)(𝑝(π‘Ÿ)βˆ’1)+1(𝑝(π‘Ÿ)βˆ’1)(π‘…βˆ’π‘Ÿ)βˆ’πœƒπ‘(π‘Ÿ)ξ€Ί(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ€»βˆ’πœ†π‘(π‘Ÿ)β‰€π‘Ÿπ‘βˆ’1ξ‚΅1𝜌(π‘Ÿ)(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ‚Άβˆ’πœ†π›Όπ‘Ž=π‘Ÿπ‘βˆ’1𝜌(π‘Ÿ)π‘’π›Όπ‘”π‘ β‰€π‘Ÿπ‘βˆ’1𝜌(π‘Ÿ)𝑒𝑓(π‘Ÿ,𝑔)𝑅,βˆ€π‘Ÿβˆˆ0,π‘…πœ†ξ€Έ.(3.9)
Thus, when 0<π‘…βˆ’π‘…0 is small enough, the following inequality is valid for πœ†βˆˆ[0,(π‘…βˆ’π‘…0)1βˆ’πœƒ/2] uniformly: ξ‚€π‘Ÿπ‘βˆ’1||π‘”ξ…ž||𝑝(π‘Ÿ)βˆ’2π‘”ξ…žξ‚ξ…žβ‰€π‘Ÿπ‘βˆ’1ξ€·π‘…πœŒ(π‘Ÿ)𝑓(π‘Ÿ,𝑔),βˆ€π‘Ÿβˆˆ0,π‘…πœ†ξ€Έ.(3.10)
Obviously, if π‘…βˆ’π‘…0 is small enough, then 𝑔β‰₯[((2𝑝+((𝑠+1)/𝑠+1/(1βˆ’πœƒ))+|𝛽|+/(1βˆ’πœƒ))/𝛼)ln(2/(π‘…βˆ’π‘…0)1βˆ’πœƒ)]1/𝑠 is large enough. Since πœ†βˆˆ[0,(π‘…βˆ’π‘…0)1βˆ’πœƒ/2], ξ‚€π‘Ÿπ‘βˆ’1||π‘”ξ…ž||𝑝(π‘Ÿ)βˆ’2π‘”ξ…žξ‚ξ…žξ€·π‘…=πœ–0ξ€Έπ‘βˆ’1βŽ‘βŽ’βŽ’βŽ£π‘Ž1/𝑠(ξ€·1βˆ’πœƒ)π‘…βˆ’π‘…0ξ€Έβˆ’πœƒπ‘ ξ‚€ξ€·π‘…βˆ’π‘…0ξ€Έ1βˆ’πœƒξ‚ξƒ©1βˆ’πœ†lnξ€·π‘…βˆ’π‘…0ξ€Έ1βˆ’πœƒξƒͺβˆ’πœ†(1/𝑠)βˆ’1⎀βŽ₯βŽ₯⎦(𝑝(𝑅0)βˆ’1)𝑅cos(πœ–(π‘Ÿβˆ’πœŽ))β‰€πœ–0ξ€Έπ‘βˆ’1βŽ‘βŽ’βŽ’βŽ£π‘Ž1/𝑠(1βˆ’πœƒ)π‘…βˆ’π‘…0ξ€Έβˆ’πœƒξ€·π‘ (1/2)π‘…βˆ’π‘…0ξ€Έ1βˆ’πœƒξƒ©2lnξ€·Rβˆ’π‘…0ξ€Έ1βˆ’πœƒξƒͺ(1/𝑠)+1⎀βŽ₯βŽ₯⎦(𝑝(𝑅0)βˆ’1)ξ€·π‘…β‰€πœ–0ξ€Έπ‘βˆ’1⎑⎒⎒⎣2π‘Ž1/𝑠(1βˆ’πœƒ)π‘ ξ€·π‘…βˆ’π‘…02ξ€·π‘…βˆ’π‘…0ξ€Έ1βˆ’πœƒξƒͺ(1/𝑠)+1⎀βŽ₯βŽ₯⎦(𝑝(𝑅0)βˆ’1)ξ€·π‘…β‰€πœ–0ξ€Έπ‘βˆ’12π‘Ž1/𝑠(1βˆ’πœƒ)𝑠2π‘…βˆ’π‘…0ξ‚Ά((𝑠+1)/𝑠)(1βˆ’πœƒ)+1𝑝+β‰€π‘Ÿπ‘βˆ’1𝜌(π‘Ÿ)π‘’π›Όπ‘”π‘ β‰€π‘Ÿπ‘βˆ’1𝜌(π‘Ÿ)𝑒𝑓(π‘Ÿ,𝑔),𝜎<π‘Ÿ<𝑅0.(3.11)
Thus, ξ‚€π‘Ÿπ‘βˆ’1||π‘”ξ…ž||𝑝(π‘Ÿ)βˆ’2π‘”ξ…žξ‚ξ…žβ‰€π‘Ÿπ‘βˆ’1𝜌(π‘Ÿ)𝑒𝑓(π‘Ÿ,𝑔),𝜎<π‘Ÿ<𝑅0.(3.12)
Obviously, ξ‚€π‘Ÿπ‘βˆ’1||π‘”ξ…ž||𝑝(π‘Ÿ)βˆ’2π‘”ξ…žξ‚ξ…ž=0β‰€π‘Ÿπ‘βˆ’1𝜌(π‘Ÿ)𝑒𝑓(π‘Ÿ,𝑔),0β‰€π‘Ÿ<𝜎.(3.13)
Since 𝑔(π‘₯,π‘Ž,πœ†)=𝑔(|π‘₯|,π‘Ž,πœ†) is a 𝐢1 function on 𝐡(0,π‘…πœ†), if 0<π‘…βˆ’π‘…0 is small enough (𝑅0 depends on 𝑅, 𝑝, 𝑠, Ξ±), from (3.10), (3.12), and (3.13), for any πœ†βˆˆ[0,(π‘…βˆ’π‘…0)1βˆ’πœƒ/2], we can see that 𝑔(|π‘₯|,π‘Ž,πœ†) is a supersolution for (P) on 𝐡(0,π‘…πœ†), and then 𝑔(|π‘₯|,π‘Ž,0) is a supersolution for (P).
Defining the function π‘”π‘š(|π‘₯|,π‘Žβˆ’πœ€)=𝑔(π‘Ÿ,π‘Žβˆ’πœ€,1/π‘š) on [0,𝑅1/π‘š), where π‘Žβˆ’πœ€>(1/𝛼)sup|π‘₯|β‰₯𝑅0𝑝(π‘₯), then π‘”π‘š(|π‘₯|,π‘Žβˆ’πœ€) is a supersolution for (P) on 𝐡(0,π‘…βˆ’(1/π‘š)). If 𝑒 is a solution for (P), according to the comparison principle, we get that π‘”π‘š(|π‘₯|,π‘Žβˆ’πœ€)β‰₯𝑒(π‘₯) for any π‘₯∈𝐡(0,𝑅1/π‘š). For any π‘₯∈𝐡(0,𝑅)⧡𝐡(0,𝑅0), we have π‘”π‘š(|π‘₯|,π‘Žβˆ’πœ€)β‰₯π‘”π‘š+1(|π‘₯|,π‘Žβˆ’πœ€), when π‘š is large enough. Thus 𝑒(π‘₯)≀limπ‘šβŸΆ+βˆžπ‘”π‘šξ€·(|π‘₯|,π‘Žβˆ’πœ€),βˆ€π‘₯∈𝐡(0,𝑅)⧡𝐡0,𝑅0ξ€Έ.(3.14)
When 𝑑(π‘₯,πœ•Ξ©)>0 is small enough, we have limπ‘šβŸΆ+βˆžπ‘”π‘šξ‚΅1(|π‘₯|,π‘Žβˆ’πœ€)<π‘Žln(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ‚Ά1/𝑠+π‘˜β‰€π‘”(|π‘₯|,π‘Ž,0).(3.15)
According to the comparison principle, we get that 𝑔(|π‘₯|,π‘Ž,0)β‰₯𝑒(π‘₯),forallπ‘₯∈𝐡(0,𝑅); then Ξ¦1(π‘₯)=Ξ¦1(|π‘₯|)=𝑔(|π‘₯|,π‘Ž,0) is a radial upper control function of all of the solutions for (P), and Ξ¦1(π‘₯)=Ξ¦1(|π‘₯|) is a radial supersolution for (P). The proof is completed.

Theorem 3.2. If 𝑓(π‘Ÿ,𝑒) satisfies 𝑓(π‘Ÿ,𝑒)βŸΆβˆ’βˆž(as)π‘’βŸΆβˆ’βˆžfor[)π‘ŸβˆˆπœŽ,𝑅uniformly,𝑓(π‘Ÿ,𝑒)≀𝛿𝑒𝑠(asπ‘’βŸΆ+∞)for[π‘ŸβˆˆπœŽ,𝑅)uniformly,(3.16) where 𝜎 is defined in (H4) and 𝛿 and 𝑠 are positive constants, then there exists a subsolution Ξ¦2(π‘₯) which satisfies Ξ¦2(π‘₯)β†’+∞ (as 𝑑(π‘₯,πœ•Ξ©)β†’0), such that for every solution 𝑒(π‘₯) for problem (P), one has 𝑒(π‘₯)β‰₯Ξ¦2(π‘₯).

Proof. We will prove this theorem in the following two cases.(i)𝛽1(𝑅)>0. (ii)𝛽1(𝑅)≀0. Case 1 (𝛽1(𝑅)>0). Let 𝑧1 be a radial solution of βˆ’Ξ”π‘(π‘₯)𝑧1(π‘₯)=βˆ’πœ‡,inΞ©1=𝐡(0,𝜎),𝑧1=0,onπœ•Ξ©1,(3.17)
where πœ‡>2(maxπ‘Ÿβˆˆ[0,𝑅0]𝜌(π‘Ÿ)+1)2(𝑝+βˆ’1)/(π‘βˆ’βˆ’1) is a positive constant. We denote 𝑧1=𝑧1(π‘Ÿ)=𝑧1(|π‘₯|). Then, 𝑧1 satisfies βˆ’ξ‚€π‘Ÿπ‘βˆ’1||π‘§ξ…ž1||𝑝(π‘Ÿ)βˆ’2π‘§ξ…ž1ξ‚ξ…ž=βˆ’π‘Ÿπ‘βˆ’1πœ‡,𝑧1(𝜎)=0,π‘§ξ…ž1𝑧(0)=0,ξ…ž1=|||π‘Ÿπœ‡π‘|||1/(𝑝(π‘Ÿ)βˆ’1),𝑧1ξ€œ=βˆ’πœŽπ‘Ÿ|||π‘Ÿπœ‡π‘|||1/(𝑝(π‘Ÿ)βˆ’1)π‘‘π‘Ÿ.(3.18)
Denote β„Žπ‘(π‘Ÿ,πœ†) on [𝜎,𝑅0] as β„Žπ‘ξ€œ(π‘Ÿ,πœ†)=𝑅0π‘Ÿξƒ―ξ€·π‘…0ξ€Έπ‘βˆ’1π‘‘π‘βˆ’1π‘‘βˆ’πœŽπ‘…0βŽ‘βŽ’βŽ’βŽ£ξ€·βˆ’πœŽπ‘(1βˆ’πœƒ)π‘…βˆ’π‘…0ξ€Έβˆ’πœƒπ‘ ξ‚€ξ€·π‘…βˆ’π‘…0ξ€Έ1βˆ’πœƒξ‚ξƒ©1+πœ†π‘lnξ€·π‘…βˆ’π‘…0ξ€Έ1βˆ’πœƒξƒͺ+πœ†(1/𝑠)βˆ’1⎀βŽ₯βŽ₯βŽ¦π‘(𝑅0)βˆ’1+(𝜎)π‘βˆ’1π‘‘π‘βˆ’1𝑅0βˆ’π‘‘π‘…0|||βˆ’πœŽπœŽπœ‡π‘|||ξ‚Ό1/(𝑝(𝑑)βˆ’1)𝑑𝑑.(3.19)
It is easy to see that βˆ’β„Žξ…žπ‘(𝜎,πœ†)=π‘§ξ…ž1|||(𝜎)=πœŽπœ‡π‘|||1/(𝑝(𝜎)βˆ’1),βˆ’β„Žξ…žπ‘ξ€·π‘…0ξ€Έ=ξ€·,πœ†π‘(1βˆ’πœƒ)π‘…βˆ’π‘…0ξ€Έβˆ’πœƒπ‘ ξ‚€ξ€·π‘…βˆ’π‘…0ξ€Έ1βˆ’πœƒξ‚ξƒ©1+πœ†π‘lnξ€·π‘…βˆ’π‘…0ξ€Έ1βˆ’πœƒξƒͺ+πœ†(1/𝑠)βˆ’1.(3.20) Define the function 𝑣(π‘Ÿ,𝑏,πœ†) on [0,𝑅) as ⎧βŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺβŽ©ξ‚΅1𝑣(π‘Ÿ,𝑏,πœ†)=𝑏ln(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ‚Ά+πœ†1/π‘ βˆ’π‘˜βˆ—,𝑅01β‰€π‘Ÿ<𝑅,𝑏lnξ€·π‘…βˆ’π‘…0ξ€Έ1βˆ’πœƒξƒͺ+πœ†1/π‘ βˆ’π‘˜βˆ—βˆ’β„Žπ‘(π‘Ÿ,πœ†),𝜎<π‘Ÿ<𝑅0,βˆ’ξ€œπœŽπ‘Ÿ|||π‘Ÿπœ‡π‘|||1/(𝑝(π‘Ÿ)βˆ’1)1π‘‘π‘Ÿ+𝑏lnξ€·π‘…βˆ’π‘…0ξ€Έ1βˆ’πœƒξƒͺ+πœ†1/π‘ βˆ’π‘˜βˆ—βˆ’β„Žπ‘(𝜎,πœ†),π‘Ÿβ‰€πœŽ,(3.21)where πœƒβˆˆ(𝛽1(𝑅)/𝑝(𝑅),1), π‘βˆˆ(0,(1/𝛿)inf|π‘₯|β‰₯𝑅0𝑝(π‘₯)) are constants, 𝑅0∈(𝜎,𝑅),π‘…βˆ’π‘…0 is small enough, parameter πœ†βˆˆ[0,(π‘…βˆ’π‘…0)1βˆ’πœƒ/2], and π‘˜βˆ—ξƒ©1=𝑀+𝑏lnξ€·π‘…βˆ’π‘…0ξ€Έ1βˆ’πœƒξƒͺ1/𝑠,(3.22)where 𝑀 satisfies (𝜎)π‘βˆ’11𝑅0βˆ’πœŽβ‰₯π‘Ÿπ‘βˆ’1𝜌(π‘Ÿ)𝑒𝑓(π‘Ÿ,𝑦)ξ€Ί,βˆ€π‘¦β‰€βˆ’π‘€,βˆ€π‘Ÿβˆˆ0,𝑅0ξ€».(3.23)
Obviously, for any positive constant 𝑏, 𝑣(π‘Ÿ,𝑏,πœ†)∈𝐢1[0,𝑅).
By computation, when π‘Ÿβˆˆ(𝑅0,𝑅), we have π‘£ξ…ž=π‘£ξ…žπ‘(π‘Ÿ,𝑏,πœ†)=1/𝑠𝑠1ln(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ‚Ά+πœ†1/π‘ βˆ’1(1βˆ’πœƒ)(π‘…βˆ’π‘Ÿ)βˆ’πœƒ(π‘…βˆ’π‘Ÿ)1βˆ’πœƒ,||𝑣+πœ†ξ…ž||𝑝(π‘Ÿ)βˆ’2π‘£ξ…ž=ξ‚Έ(1βˆ’πœƒ)𝑏1/𝑠𝑠𝑝(π‘Ÿ)βˆ’1ξ‚΅1ln(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ‚Ά+πœ†(1/π‘ βˆ’1)(𝑝(π‘Ÿ)βˆ’1)(π‘…βˆ’π‘Ÿ)βˆ’πœƒ(𝑝(π‘Ÿ)βˆ’1)ξ€Ί(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ€»+πœ†π‘(π‘Ÿ)βˆ’1,ξ‚€π‘Ÿπ‘βˆ’1||π‘£ξ…ž||𝑝(π‘Ÿ)βˆ’2π‘£ξ…žξ‚ξ…ž=π‘Ÿπ‘βˆ’1ξ‚Έ(1βˆ’πœƒ)𝑏1/𝑠𝑠𝑝(π‘Ÿ)βˆ’1ξ‚΅1ln(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ‚Ά+πœ†(1/π‘ βˆ’1)(𝑝(π‘Ÿ)βˆ’1)Γ—(𝑝(π‘Ÿ)βˆ’1)(π‘…βˆ’π‘Ÿ)βˆ’πœƒ(𝑝(π‘Ÿ)βˆ’1)βˆ’1ξ€Ί(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ€»+πœ†π‘(π‘Ÿ)βˆ’1(πœƒ+Ξ›(π‘Ÿ)),(3.24)
where ξ‚†π‘ŸΞ›(π‘Ÿ)=π‘βˆ’1ξ€Ί(1βˆ’πœƒ)𝑏1/𝑠/𝑠𝑝(π‘Ÿ)βˆ’1ξ‚‡ξ…ž(𝑝(π‘Ÿ)βˆ’1)π‘Ÿπ‘βˆ’1ξ€Ί(1βˆ’πœƒ)𝑏1/𝑠/𝑠𝑝(π‘Ÿ)βˆ’1(π‘…βˆ’π‘Ÿ)+(1/π‘ βˆ’1)(1βˆ’πœƒ)ξ€·ξ€·ξ€·ln1/(π‘…βˆ’π‘Ÿ)1βˆ’πœƒ+πœ†ξ€Έξ€Έξ€Έξ€Ί(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ€»+πœ†Γ—(π‘…βˆ’π‘Ÿ)1βˆ’πœƒ+(1/π‘ βˆ’1)π‘ξ…ž(π‘Ÿ)ξ‚Έ1(𝑝(π‘Ÿ)βˆ’1)(π‘…βˆ’π‘Ÿ)lnln(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ‚Ή++πœ†πœƒπ‘ξ…ž(π‘Ÿ)1(𝑝(π‘Ÿ)βˆ’1)(π‘…βˆ’π‘Ÿ)ln+(π‘…βˆ’π‘Ÿ)(1βˆ’πœƒ)ξ€Ί(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ€»+πœ†(π‘…βˆ’π‘Ÿ)1βˆ’πœƒ+βˆ’π‘ξ…ž(π‘Ÿ)𝑝(π‘Ÿ)βˆ’1(π‘…βˆ’π‘Ÿ)ln(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ€».+πœ†(3.25)
By computation, when π‘…βˆ’π‘…0 is small enough, for πœ†βˆˆ[0,(π‘…βˆ’π‘…0)1βˆ’πœƒ/2] uniformly, we have ξ‚€π‘Ÿπ‘βˆ’1||π‘£ξ…ž||𝑝(π‘Ÿ)βˆ’2π‘£ξ…žξ‚ξ…žβ‰₯π‘Ÿπ‘βˆ’1ξ‚Έ(1βˆ’πœƒ)𝑏1/𝑠𝑠𝑝(π‘Ÿ)βˆ’1ξ‚΅1ln(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ‚Ά+πœ†(1/π‘ βˆ’1)(𝑝(π‘Ÿ)βˆ’1)Γ—(𝑝(π‘Ÿ)βˆ’1)(π‘…βˆ’π‘Ÿ)βˆ’πœƒ(𝑝(π‘Ÿ)βˆ’1)βˆ’1ξ€Ί(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ€»+πœ†π‘(π‘Ÿ)βˆ’1πœƒξ‚€11βˆ’2β‰₯πœƒ2π‘Ÿπ‘βˆ’1ξ‚Έ(1βˆ’πœƒ)𝑏1/𝑠𝑠𝑝(π‘Ÿ)βˆ’1ξ‚΅1ln(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ‚Ά+πœ†(1/π‘ βˆ’1)(𝑝(π‘Ÿ)βˆ’1)Γ—(𝑝(π‘Ÿ)βˆ’1)(π‘…βˆ’π‘Ÿ)βˆ’πœƒ(𝑝(π‘Ÿ)βˆ’1)βˆ’1ξ€Ί(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ€»+πœ†π‘(π‘Ÿ)(π‘…βˆ’π‘Ÿ)1βˆ’πœƒβ‰₯πœƒ2π‘Ÿπ‘βˆ’1ξ‚Έ(1βˆ’πœƒ)𝑏1/𝑠𝑠𝑝(π‘Ÿ)βˆ’1ξ‚΅1ln(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ‚Ά+πœ†(1/π‘ βˆ’1)(𝑝(π‘Ÿ)βˆ’1)(𝑝(π‘Ÿ)βˆ’1)(π‘…βˆ’π‘Ÿ)βˆ’πœƒπ‘(π‘Ÿ)ξ€Ί(π‘…βˆ’π‘Ÿ)1βˆ’πœƒξ€»+πœ†π‘(π‘Ÿ)β‰₯π‘Ÿπ‘βˆ’1𝜌1(π‘…βˆ’π‘Ÿ)βˆ’π›½1(π‘Ÿ)𝑒𝛿𝑣𝑠β‰₯π‘Ÿπ‘βˆ’1𝜌(π‘Ÿ)𝑒𝑓(π‘Ÿ,𝑣)𝑅,βˆ€π‘Ÿβˆˆ0ξ€Έ.,𝑅(3.26)
Then, for πœ†βˆˆ[0,(π‘…βˆ’π‘…0)1βˆ’πœƒ/2] uniformly, we have ξ‚€π‘Ÿπ‘βˆ’1||π‘£ξ…ž||𝑝(π‘Ÿ)βˆ’2π‘£ξ…žξ‚ξ…žβ‰₯π‘Ÿπ‘βˆ’1𝜌(π‘Ÿ)𝑒𝑓(π‘Ÿ,𝑣)𝑅,βˆ€π‘Ÿβˆˆ0ξ€Έ.,𝑅(3.27)
When π‘…βˆ’π‘…0 is small enough, forallπ‘Ÿβˆˆ(𝜎,𝑅0), since π‘£β‰€βˆ’π‘€, it is easy to see that ξ‚€π‘Ÿπ‘βˆ’1||π‘£ξ…ž||𝑝(π‘Ÿ)βˆ’2π‘£ξ…žξ‚ξ…žβ‰₯ξ‚€π‘Ÿπ‘βˆ’1||β„Žξ…ž||𝑝(π‘Ÿ)βˆ’2β„Žξ…žξ‚ξ…ž=𝑅0ξ€Έπ‘βˆ’11𝑅0βŽ‘βŽ’βŽ’βŽ£ξ€·βˆ’πœŽπ‘(1βˆ’πœƒ)π‘…βˆ’π‘…0ξ€Έβˆ’πœƒπ‘ ξ‚€ξ€·π‘…βˆ’π‘…0ξ€Έ1βˆ’πœƒξ‚ξƒ©1+πœ†π‘lnξ€·π‘…βˆ’π‘…0ξ€Έ1βˆ’πœƒξƒͺ+πœ†1/π‘ βˆ’1⎀βŽ₯βŽ₯βŽ¦π‘(𝑅0)βˆ’1βˆ’(𝜎)π‘βˆ’11𝑅0|||βˆ’πœŽπœŽπœ‡π‘|||β‰₯(𝜎)π‘βˆ’11𝑅0βˆ’πœŽβ‰₯π‘Ÿπ‘βˆ’1𝜌(π‘Ÿ)𝑒𝑓(π‘Ÿ,𝑣),(3.28)
Then, ξ‚€π‘Ÿπ‘βˆ’1||π‘£ξ…ž||𝑝(π‘Ÿ)βˆ’2π‘£ξ…žξ‚ξ…žβ‰₯π‘Ÿπ‘βˆ’1𝜌(π‘Ÿ)𝑒𝑓(π‘Ÿ,𝑣)ξ€·,βˆ€π‘ŸβˆˆπœŽ,𝑅0ξ€Έ.(3.29)
Obviously, ξ‚€π‘Ÿπ‘βˆ’1||π‘£ξ…ž||𝑝(π‘Ÿ)βˆ’2π‘£ξ…žξ‚ξ…ž=π‘Ÿπ‘βˆ’1πœ‡β‰₯π‘Ÿπ‘βˆ’1𝜌(π‘Ÿ)𝑒𝑓(π‘Ÿ,𝑣),βˆ€π‘Ÿβˆˆ(0,𝜎).(3.30)
Combining (3.27), (3.29), and (3.30), when π‘…βˆ’π‘…0 is large enough, for any πœ†βˆˆ[0,(π‘…βˆ’π‘…0)1βˆ’πœƒ/2], one can see that 𝑣(π‘Ÿ,π‘Ž,πœ†) is a subsolution for (P).
Define the function π‘£π‘š(π‘Ÿ,𝑏+πœ€) on 𝐡(0,𝑅) as π‘£π‘š(π‘Ÿ,𝑏+πœ€)=π‘£π‘šξ‚€1π‘Ÿ,𝑏+πœ€,π‘šξ‚,(3.31)
where πœ€ is a small enough positive constant such that (𝑏+πœ€)<(1/𝛿)inf|π‘₯|β‰₯𝑅0𝑝(π‘₯).
For any π‘š=1,2,…, we can see that π‘£π‘š(π‘Ÿ,𝑏+πœ€)∈𝐢1([0,𝑅)) is a subsolution for (P) on 𝐡(𝑅0,𝑅). According to the comparison principle, we get that π‘£π‘š(π‘Ÿ,𝑏+πœ€)≀𝑒(π‘₯) for any π‘₯∈𝐡(0,𝑅). For any π‘₯∈𝐡(0,𝑅)⧡𝐡(0,𝑅0), we have π‘£π‘š(|π‘₯|,𝑏+πœ€)β‰€π‘£π‘š+1(|π‘₯|,𝑏+πœ€). Thus 𝑒(π‘₯)β‰₯limπ‘šβŸΆ+βˆžπ‘£π‘šξ€·(|π‘₯|,𝑏+πœ€),βˆ€π‘₯∈𝐡(0,𝑅)⧡𝐡0,𝑅0ξ€Έ.(3.32)
When 𝑑(π‘₯,πœ•Ξ©) is small enough, we havelimπ‘šβ†’+βˆžπ‘£π‘š(|π‘₯|,𝑏+πœ€)>𝑣(|π‘₯|,𝑏,0).
According to the comparison principle, we get that 𝑣(|π‘₯|,𝑏,0)≀𝑒(π‘₯),βˆ€π‘₯∈𝐡(0,𝑅); then Ξ¦2(π‘₯)=Ξ¦2(|π‘₯|)=𝑣(|π‘₯|,𝑏,0) is a radial lower control function of all of the solutions for (P), and Ξ¦2(π‘₯) is a radial subsolution for (P).
Case 2 (𝛽1(𝑅)≀0). Let πœ‡>2(maxπ‘Ÿβˆˆ[0,𝑅0]𝜌(π‘Ÿ)+1)2(𝑝+βˆ’1)/(π‘βˆ’βˆ’1) be a positive constant. Denote πœ›π‘(π‘Ÿ,πœ†) on [𝜎,𝑅0] as πœ›π‘ξ€œ(π‘Ÿ,πœ†)=𝑅0π‘Ÿξƒ―ξ€·π‘…0ξ€Έπ‘βˆ’1π‘‘π‘βˆ’1π‘‘βˆ’πœŽπ‘…0ξƒ¬π‘βˆ’πœŽπ‘ ξ€·π‘…+πœ†βˆ’π‘…0𝑏ln𝑅+πœ†βˆ’π‘…0ξ€Έβˆ’11/π‘ βˆ’1𝑝(𝑅0)βˆ’1+(𝜎)π‘βˆ’1π‘‘π‘βˆ’1𝑅0βˆ’π‘‘π‘…0|||βˆ’πœŽπœŽπœ‡π‘|||ξ‚Ό1/(𝑝(𝑑)βˆ’1)𝑑𝑑.(3.33)
It is easy to see that βˆ’πœ›ξ…žπ‘(𝜎,πœ†)=π‘§ξ…ž1|||(𝜎)=πœŽπœ‡π‘|||1/(𝑝(𝜎)βˆ’1),βˆ’πœ›ξ…žπ‘ξ€·π‘…0ξ€Έ=𝑏,πœ†π‘ ξ€·π‘…+πœ†βˆ’π‘…0𝑏ln𝑅+πœ†βˆ’π‘…0ξ€Έβˆ’11/π‘ βˆ’1.(3.34)
Define the function πœ‚(π‘Ÿ,𝑏,πœ†) on 𝐡(0,𝑅) as ⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξ€·πœ‚(π‘Ÿ,𝑏,πœ†)=𝑏ln(𝑅+πœ†βˆ’π‘Ÿ)βˆ’1ξ€Έ1/π‘ βˆ’π‘˜βˆ—,𝑅0ξ‚€ξ€·β‰€π‘Ÿ<𝑅,𝑏ln𝑅+πœ†βˆ’π‘…0ξ€Έβˆ’11/π‘ βˆ’π‘˜βˆ—βˆ’πœ›π‘(π‘Ÿ,πœ†),𝜎<π‘Ÿ<𝑅0,βˆ’ξ€œπœŽπ‘Ÿ|||π‘Ÿπœ‡π‘|||1/(𝑝(π‘Ÿ)βˆ’1)ξ‚€ξ€·π‘‘π‘Ÿ+𝑏ln𝑅+πœ†βˆ’π‘…0ξ€Έβˆ’11/π‘ βˆ’π‘˜βˆ—βˆ’πœ›π‘(𝜎,πœ†),π‘Ÿβ‰€πœŽ,(3.35)where π‘βˆˆ(0,(1/𝛿)inf|π‘₯|β‰₯𝑅0[𝑝(π‘₯)βˆ’π›½1(π‘₯)]) is a constant, 𝑅0∈(𝜎,𝑅),π‘…βˆ’π‘…0 is small enough, parameter πœ†βˆˆ[0,(π‘…βˆ’π‘…0)/2], and π‘˜βˆ—ξ‚΅1=𝑀+𝑏lnπ‘…βˆ’π‘…0ξ‚Ά1/𝑠,(3.36)where 𝑀 is defined in (3.23).
Obviously, for any positive constant 𝑏, πœ‚(π‘Ÿ,𝑏,πœ†)∈𝐢1[0,𝑅).
Similar to the proof of Case 1, when π‘…βˆ’π‘…0 is small enough, we have ξ‚€π‘Ÿπ‘βˆ’1||πœ‚ξ…ž||𝑝(π‘Ÿ)βˆ’2πœ‚ξ…žξ‚ξ…žβ‰₯π‘Ÿπ‘βˆ’1𝑏1/𝑠𝑠𝑝(π‘Ÿ)βˆ’1(𝑝(π‘Ÿ)βˆ’1)(𝑅+πœ†βˆ’π‘Ÿ)βˆ’π‘(π‘Ÿ)ξ€·ln(𝑅+πœ†βˆ’π‘Ÿ)βˆ’1ξ€Έ(1/π‘ βˆ’1)(𝑝(π‘Ÿ)βˆ’1)ξ‚€11βˆ’2β‰₯π‘Ÿπ‘βˆ’1𝜌(π‘Ÿ)𝑒𝑓(π‘Ÿ,πœ‚)𝑅,βˆ€π‘Ÿβˆˆ0ξ€Έ.,𝑅(3.37)
When π‘…βˆ’π‘…0 is small enough, forallπ‘Ÿβˆˆ(𝜎,𝑅0), from the definition of π‘˜βˆ—, it is easy to see that ξ‚€π‘Ÿπ‘βˆ’1||πœ‚ξ…ž||𝑝(π‘Ÿ)βˆ’2πœ‚ξ…žξ‚ξ…žβ‰₯(𝜎)π‘βˆ’11𝑅0βˆ’πœŽβ‰₯π‘Ÿπ‘βˆ’1𝜌(π‘Ÿ)𝑒𝑓(π‘Ÿ,πœ‚).(3.38)
Obviously ξ‚€π‘Ÿπ‘βˆ’1||πœ‚ξ…ž||𝑝(π‘Ÿ)βˆ’2πœ‚ξ…žξ‚ξ…ž=π‘Ÿπ‘βˆ’1πœ‡β‰₯π‘Ÿπ‘βˆ’1𝜌(π‘Ÿ)𝑒𝑓(π‘Ÿ,πœ‚),βˆ€π‘Ÿβˆˆ(0,𝜎).(3.39)
Combining (3.37), (3.38), and (3.39), when π‘…βˆ’π‘…0 is large enough, for any πœ†βˆˆ[0,(π‘…βˆ’π‘…0)/2], one can see that πœ‚(π‘Ÿ,π‘Ž,πœ†) is a subsolution for (P).
Define the function πœ‚π‘š(π‘Ÿ,𝑏+πœ–) on 𝐡(0,𝑅) as πœ‚π‘šξ‚€1(π‘Ÿ,𝑏+πœ–)=πœ‚π‘Ÿ,𝑏+πœ–,π‘šξ‚,(3.40)
where πœ– is a small enough positive constant such that (𝑏+πœ–)<(1/𝛿)inf|π‘₯|β‰₯𝑅0𝑝(π‘₯).
We can see that πœ‚π‘š(π‘Ÿ,𝑏+πœ–)∈𝐢1[0,𝑅) is a subsolution for (P) for any π‘š=1,2…. According to the comparison principle, we get that πœ‚π‘š(π‘Ÿ,𝑏+πœ–)≀𝑒(π‘₯) for any π‘₯∈𝐡(0,𝑅). For any π‘₯∈𝐡(0,𝑅)⧡𝐡(0,𝑅0), we have πœ‚π‘š(|π‘₯|,𝑏+πœ–)β‰€πœ‚π‘š+1(|π‘₯|,𝑏+πœ–). Then, 𝑒(π‘₯)β‰₯limπ‘šβ†’+βˆžπœ‚π‘šξ€·(|π‘₯|,𝑏+πœ–),βˆ€π‘₯∈𝐡(0,𝑅)⧡𝐡0,𝑅0ξ€Έ.(3.41)
When 𝑑(π‘₯,πœ•Ξ©) is small enough, we have limπ‘šβŸΆ+βˆžπœ‚π‘š(|π‘₯|,𝑏+πœ–)>πœ‚(|π‘₯|,𝑏,0).(3.42)
According to the comparison principle, we get that πœ‚(|π‘₯|,𝑏,0)≀𝑒(π‘₯),βˆ€π‘₯∈𝐡(0,𝑅); then Ξ¦2(π‘₯)=Ξ¦2(|π‘₯|)=πœ‚(|π‘₯|,𝑏,0) is a radial lower control function of all of the solutions for (P), and Ξ¦2(π‘₯)=Ξ¦2(|π‘₯|) is a radial subsolution for (P).

Theorem 3.3. If 𝑓(π‘Ÿ,𝑒) satisfies limπ‘’βŸΆ+βˆžπ‘“(π‘Ÿ,𝑒)𝑒𝑠=𝛿(asπ‘’βŸΆ+∞)for[π‘ŸβˆˆπœŽ,𝑅)uniformly,(3.43) where 𝜎 is defined in (H4), 𝛿 and 𝑠 are positive constants, 𝜌(π‘Ÿ)=𝜌0(π‘…βˆ’π‘Ÿ)βˆ’π›½(π‘Ÿ), where 𝛽(𝑅)<𝑝(𝑅), then each solution 𝑒(π‘₯) for (P) satisfies lim|π‘₯|βŸΆπ‘…π‘’(π‘₯)ξ€·ξ€·(𝑝(𝑅)/𝛿)ln1/(π‘…βˆ’|π‘₯|)1βˆ’πœƒξ€Έξ€Έ1/𝑠=1,whereπœƒ=𝛽(𝑅).𝑝(𝑅)(3.44)

Proof. It is easy to be seen from Theorems 3.1 and 3.2

4. The Existence of Boundary Blow-Up Solutions

Theorem 4.1. If infπ‘₯βˆˆΞ©π‘(π‘₯)>𝑁 and 𝑓(π‘Ÿ,𝑒) satisfies 𝑓(π‘Ÿ,𝑒)β‰₯π‘Žπ‘’π‘ (as𝑒→+∞)for[π‘ŸβˆˆπœŽ,𝑅)uniformly,(4.1) where 𝜎 is defined in (H4), π‘Ž and 𝑠 are positive constants, then (P) possesses a boundary blow-up solution.

Proof. In order to deal with the existence of boundary blow-up solutions, let us consider the problem βˆ’Ξ”π‘(π‘₯)𝑒+𝜌(π‘Ÿ)𝑒𝑓(π‘₯,𝑒)=0,inΞ©0,𝑒(π‘₯)=𝑐,forπ‘₯βˆˆπœ•Ξ©0,(4.2) where 𝑐 is a positive constant and Ξ©0⋐Ω is a radial subdomain of Ξ©. Since infπ‘₯βˆˆΞ©π‘(π‘₯)>𝑁, then W1,𝑝(π‘₯)(Ξ©0)β†ͺ𝐢𝛼(Ξ©0), where π›Όβˆˆ(0,1). The relative functional of (4.2) is ξ€œπœ™=Ξ©01||||𝑝(π‘₯)βˆ‡π‘’(π‘₯)𝑝(π‘₯)ξ€œπ‘‘π‘₯+Ξ©0𝐹(π‘₯,𝑒)𝑑π‘₯,(4.3) where ∫𝐹(π‘₯,𝑒)=𝑒0𝑒𝑓(π‘₯,𝑑)𝑑𝑑. Since πœ™ is coercive in π‘‹βˆΆ=𝑐+π‘Š01,𝑝(π‘₯)(Ξ©0), then πœ™ possesses a nontrivial minimum point 𝑒. So, problem (4.2) possesses a weak solution 𝑒.
Since π‘Žπ‘’π‘ β‰€π‘“(π‘Ÿ,𝑒)≀𝐢1+𝐢2|𝑒|𝛾(π‘₯), from Theorems 3.1 and 3.2, we get that (P) possesses a supersolution π‘”βˆ—(π‘₯) and a subsolution π‘”βˆ—(π‘₯), which satisfy π‘”βˆ—(π‘₯)β‰₯π‘”βˆ—(π‘₯), when 𝑑(π‘₯,πœ•Ξ©) (the distance from π‘₯ to πœ•Ξ©) is small enough. According to the comparison principle, we get that π‘”βˆ—(π‘₯)β‰₯π‘”βˆ—(π‘₯) for any π‘₯∈Ω.
Denote 𝐷𝑗={π‘₯∣|π‘₯|<1βˆ’1/(𝑗+1)𝑅} (𝑗=1,2,…). Let us consider the problem βˆ’Ξ”π‘(π‘₯)𝑒𝑗+𝜌(π‘₯)𝑒𝑓(π‘₯,𝑒𝑗)=0,in𝐷𝑗,𝑒𝑗(π‘₯)=π‘”βˆ—(π‘₯),forπ‘₯βˆˆπœ•π·π‘—,(4.4)
and the relative functional is ξ€œπœ™=𝐷𝑗1||𝑝(π‘₯)βˆ‡π‘’π‘—||(π‘₯)𝑝(π‘₯)ξ€œπ‘‘π‘₯+π·π‘—ξ€·πœŒ(π‘₯)𝐹π‘₯,𝑒𝑗𝑑π‘₯.(4.5)
Let π‘”βˆ—π‘—(π‘₯)=π‘”βˆ—(π‘₯)βˆ£π·π‘—. Since the functional πœ™ is coercive in 𝑋𝑗=π‘”βˆ—π‘—(π‘₯)+π‘Š01,𝑝(π‘₯)(𝐷𝑗), then πœ™ has a nontrivial minimum point 𝑒𝑗. Therefore, problem (4.4) has a weak solution 𝑒𝑗.
According to the comparison principle, we get that π‘”βˆ—(π‘₯)≀𝑒𝑗(π‘₯) for any π‘₯βˆˆπ·π‘— (𝑗=1,2,… ). Since 𝑒𝑗(π‘₯)=π‘”βˆ—(π‘₯) for any π‘₯βˆˆπœ•π·π‘—, then 𝑒𝑗(π‘₯)≀𝑒𝑗+1(π‘₯) for any π‘₯βˆˆπœ•π·π‘— (𝑗=1,2,…). According to the comparison principle, we get that 𝑒𝑗(π‘₯)≀𝑒𝑗+1(π‘₯) for any π‘₯βˆˆπ·π‘— (𝑗=1,2,…).
Since π‘”βˆ—(π‘₯) is a supersolution and π‘”βˆ—(π‘₯)β‰₯π‘”βˆ—(π‘₯) for any π‘₯∈Ω, so we have 𝑒𝑗(π‘₯)=π‘”βˆ—(π‘₯)β‰€π‘”βˆ—(π‘₯) for any π‘₯βˆˆπœ•π·π‘— (𝑗=1,2,…). According to the comparison principle, we get that 𝑒𝑗(π‘₯)β‰€π‘”βˆ—(π‘₯) for any π‘₯βˆˆπ·π‘— (𝑗=1,2,…).
Since π‘”βˆ—(π‘₯) and π‘”βˆ—(π‘₯) are locally bounded, from Lemma 2.4, each weak solution of (4.4) is a 𝐢1,𝛼loc function. The 𝐢1,𝛼 interior regularity result implies that the sequences {𝑒𝑗} and {βˆ‡π‘’π‘—} are equicontinuous in 𝐷2, and hence we can choose a subsequence, which we denoted by {𝑒1𝑗}, such that 𝑒1𝑗→𝑀1 and βˆ‡π‘’1π‘—β†’πœ›1 uniformly on 𝐷1 for some 𝑀1∈𝐢(𝐷1) and πœ›1∈(𝐢(𝐷1))𝑁. In fact, πœ›1=βˆ‡π‘€1 on 𝐷1, and from the interior 𝐢1,𝛼 estimate, we conclude that βˆ‡π‘€1∈(𝐢𝛼(𝐷1))𝑁 for some 0<𝛼<1. Thus, 𝑀1βˆˆπ‘Š1,𝑝(π‘₯)(𝐷1)∩𝐢1,𝛼(𝐷1). From the 𝐢1,𝛼 interior regularity result, we see that |βˆ‡π‘’π‘—|π‘βˆ’1|βˆ‡πœ™|≀𝐢|βˆ‡πœ™| on 𝐷1, and since the function πœ‰β†’|πœ‰|π‘βˆ’2πœ‰ is continuous on ℝ𝑁, it follows that |βˆ‡π‘’1𝑗(π‘₯)|π‘βˆ’2βˆ‡π‘’1𝑗(π‘₯)β‹…βˆ‡πœ™(π‘₯)β†’|βˆ‡π‘€1(π‘₯)|π‘βˆ’2βˆ‡π‘€1(π‘₯)β‹…βˆ‡πœ™(π‘₯) for π‘₯∈𝐷1. Thus, by the dominated convergence theorem, we have ξ€œπ·1||βˆ‡π‘’1𝑗||(π‘₯)π‘βˆ’2βˆ‡π‘’1π‘—ξ€œ(π‘₯)β‹…βˆ‡πœ™(π‘₯)𝑑π‘₯⟢𝐷1||βˆ‡π‘€1||(π‘₯)π‘βˆ’2βˆ‡π‘€1(π‘₯)β‹…βˆ‡πœ™(π‘₯)𝑑π‘₯,βˆ€πœ™βˆˆπ‘Š01,𝑝(π‘₯)𝐷1ξ€Έ.(4.6)
Furthermore, since 0≀𝑓(𝑒1𝑗)≀𝑓(𝑒1𝑗+1) and 𝑓(𝑒1𝑗(π‘₯))→𝑓(𝑀1(π‘₯)) for each π‘₯∈𝐷1, by the monotone convergence theorem, we obtain ξ€œπ·1πœŒπ‘’π‘“(𝑒1𝑗)ξ€œπ‘žπ‘‘π‘₯⟢𝐷1πœŒπ‘’π‘“(𝑀1)π‘žπ‘‘π‘₯,βˆ€π‘žβˆˆπ‘Š01,𝑝(π‘₯)𝐷1ξ€Έ.(4.7)
Therefore, it follows that ξ€œπ·1||βˆ‡π‘€1||(π‘₯)π‘βˆ’2βˆ‡π‘€1ξ€œ(π‘₯)β‹…βˆ‡π‘ž(π‘₯)𝑑π‘₯+𝐷1πœŒπ‘’π‘“(𝑀1)π‘žπ‘‘π‘₯=0,βˆ€π‘žβˆˆπ‘Š01,𝑝(π‘₯)𝐷1ξ€Έ,(4.8)
and hence 𝑀1 is a weak solution for βˆ’Ξ”π‘(π‘₯)𝑀1+πœŒπ‘’π‘“(𝑀1)=0 on 𝐷1.
Thus, there exists a subsequence of {𝑒𝑗} which we denote it by{𝑒1𝑗}, such that 𝑒1𝑗→𝑀1 in 𝐷1 (as π‘—β†’βˆž), where 𝑀1βˆˆπ‘Š1,𝑝(π‘₯)(𝐷1)∩𝐢1,𝛼1(𝐷1) and satisfies ξ€œπ·1||βˆ‡π‘€1||𝑝(π‘₯)βˆ’2βˆ‡π‘€1ξ€œβˆ‡π‘žπ‘‘π‘₯+𝐷1𝜌(π‘₯)𝑒𝑓(π‘₯,𝑀1)π‘žπ‘‘π‘₯=0,βˆ€π‘žβˆˆπ‘Š01,𝑝(π‘₯)𝐷1ξ€Έ.(4.9)
Similarly, we can prove that there exists a subsequence of {𝑒1𝑗} which we denote by{𝑒2𝑗}, such that 𝑒2𝑗→𝑀2 in 𝐷2 (as π‘—β†’βˆž), where 𝑀2βˆˆπ‘Š1,𝑝(π‘₯)(𝐷2)∩𝐢1,𝛼2(𝐷2) satisfies 𝑀1=𝑀2∣𝐷1 and ξ€œπ·2||βˆ‡π‘€2||𝑝(π‘₯)βˆ’2βˆ‡π‘€2ξ€œβˆ‡π‘žπ‘‘π‘₯+𝐷2𝜌(π‘₯)𝑒𝑓(π‘₯,𝑀2)π‘žπ‘‘π‘₯=0,βˆ€π‘žβˆˆπ‘Š01,𝑝(π‘₯)𝐷2ξ€Έ.(4.10)
Repeating the above steps, we can get a subsequence of {π‘’π‘–π‘—βˆ£π‘—=1,2,…} which we denote by {𝑒𝑗𝑖+1βˆ£π‘—=1,2,…} (𝑖=1,2,…) and satisfies the following.(10) For any fixed 𝑖, {𝑒𝑗𝑖+1} is a subsequence of{𝑒𝑖𝑗}.(20) For any fixed 𝑖, 𝑒𝑗𝑖+1→𝑀𝑖+1 in 𝐷𝑖+1 (as π‘—β†’βˆž), where 𝑀𝑖+1βˆˆπ‘Š1,𝑝(π‘₯)(𝐷𝑖+1)∩𝐢1,𝛼𝑖+1(𝐷𝑖+1) satisfies 𝑀𝑖=𝑀𝑖+1βˆ£π·π‘–.(30)For any fixed 𝑖, 𝑀𝑖 satisfies ξ€œπ·π‘–||βˆ‡π‘€π‘–||𝑝(π‘₯)βˆ’2βˆ‡π‘€π‘–ξ€œβˆ‡π‘žπ‘‘π‘₯+π·π‘–πœŒ(π‘₯)𝑒𝑓(π‘₯,𝑀𝑖)π‘žπ‘‘π‘₯=0,βˆ€π‘žβˆˆπ‘Š01,𝑝(π‘₯)𝐷𝑖.(4.11)
Thus, we can conclude that
(i){𝑒𝑗𝑗}is a subsequence of{𝑒𝑗},(ii) there exists a function π‘€βˆˆπ‘Š1,𝑝(π‘₯)loc(Ξ©)∩𝐢1,𝛼loc(Ξ©) such that 𝑀𝑖=π‘€βˆ£π·π‘–, and for any π‘₯∈Ω, there exists a constant 𝑗π‘₯ such that when 𝑗β‰₯𝑗π‘₯, 𝑒𝑗𝑗(π‘₯) is defined at π‘₯, and limπ‘—β†’βˆžπ‘’π‘—π‘—(π‘₯)=𝑀(π‘₯),(iii)ξ€œΞ©||||βˆ‡π‘€π‘(π‘₯)βˆ’2ξ€œβˆ‡π‘€βˆ‡π‘žπ‘‘π‘₯+Ω𝜌(π‘₯)𝑒𝑓(π‘₯,𝑀)π‘žπ‘‘π‘₯=0,βˆ€π‘žβˆˆπ‘Š1,𝑝(π‘₯)0,loc(Ξ©).(4.12)
Obviously, 𝑀 is a boundary blow-up solution for (P).
This completes the proof.

In Theorem 4.1, when infπ‘₯βˆˆΞ©π‘(π‘₯)>𝑁, the existence of solutions for (P) is given. In the following, we will consider the existence of solutions for (P) in the general case 1<infπ‘₯βˆˆΞ©π‘(π‘₯)≀supπ‘₯βˆˆΞ©π‘(π‘₯)<∞. We need to do some preparation. Let us consider ξ‚€π‘Ÿπ‘βˆ’1||π‘’ξ…ž||𝑝(π‘Ÿ)βˆ’2π‘’ξ…žξ‚ξ…ž=π‘Ÿπ‘βˆ’1𝜌(π‘Ÿ)𝑒𝑓(π‘Ÿ,𝑒)ξ€·,π‘Ÿβˆˆ0,π‘…πœ†ξ€Έ,π‘’ξ…ž(𝑅0)=0,π‘’πœ†ξ€Έ=𝑑,(I)

where π‘…πœ†βˆˆ(0,𝑅) and 𝑑 is a constant.

Lemma 4.2. If Ξ¦2(π‘…πœ†)≀𝑑≀Φ1(π‘…πœ†), where Ξ¦1 and Ξ¦2 are defined in Theorems 3.13.2, respectively, then (4.13) has a solution 𝑒 satisfying Ξ¦2(π‘Ÿ)≀𝑒(π‘Ÿ)≀Φ1ξ€Ί(π‘Ÿ),βˆ€π‘Ÿβˆˆ0,π‘…πœ†ξ€».(4.13)

Proof. Denote ⎧βŽͺ⎨βŽͺβŽ©π‘’β„Ž(π‘Ÿ,𝑒)=𝑓(π‘Ÿ,Ξ¦1(π‘Ÿ))ξ€·+arctan𝑒(π‘Ÿ)βˆ’Ξ¦1ξ€Έ(π‘Ÿ),𝑒(π‘Ÿ)>Ξ¦1𝑒(π‘Ÿ),𝑓(π‘Ÿ,𝑒),Ξ¦2(π‘Ÿ)≀𝑒(π‘Ÿ)≀Φ1𝑒(π‘Ÿ),𝑓(π‘Ÿ,Ξ¦2(π‘Ÿ))ξ€·+arctan𝑒(π‘Ÿ)βˆ’Ξ¦2ξ€Έ(π‘Ÿ),𝑒(π‘Ÿ)<Ξ¦2(π‘Ÿ).(4.14)
Let 𝜌𝐸(𝑑)=𝜌(|𝑑|),andβ„ŽπΈ(𝑑,𝑒)=β„Ž(|𝑑|,𝑒),forallπ‘‘βˆˆ[βˆ’π‘…πœ†,π‘…πœ†]. Let us consider the even solutions of the following ξ‚€|𝑑|π‘βˆ’1||π‘’ξ…ž||𝑝(|𝑑|)βˆ’2π‘’ξ…žξ‚ξ…ž=|𝑑|π‘βˆ’1𝜌𝐸(𝑑)β„ŽπΈξ€·(𝑑,𝑒),π‘‘βˆˆβˆ’π‘…πœ†,π‘…πœ†ξ€Έ,π‘’ξ€·βˆ’π‘…πœ†ξ€Έξ€·π‘…=𝑑,π‘’πœ†ξ€Έ=𝑑.(II)
It is easy to see that 𝑒 is an even solution for (4.15) if and only if 𝑒 is even and ξ€œπ‘’=π‘‘βˆ’π‘…πœ†π‘Ÿξ‚Έ|𝑑|1βˆ’π‘ξ€œπ‘‘0|𝑠|π‘βˆ’1πœŒξ‚Ή(𝑠)β„Ž(𝑠,𝑒(𝑠))𝑑𝑠1/(𝑝(𝑑)βˆ’1)𝑑𝑑,βˆ€π‘Ÿβˆˆ0,π‘…πœ†ξ€».(4.15)
Denote Ξ¨(𝑒,πœ‡)=πœ‡βˆ«π‘‘βˆ’πœ‡π‘…πœ†π‘Ÿ[|𝑑|1βˆ’π‘βˆ«π‘‘0|𝑠|π‘βˆ’1𝜌(𝑠)β„Ž(𝑠,𝑒(𝑠))𝑑𝑠]1/(𝑝(𝑑)βˆ’1)𝑑𝑑. Similar to the proof of Lemma 2.3 of [18], for any πœ‡βˆˆ[0,1], it is easy to see that Ξ¨(𝑒,πœ‡) is compact continuous and bounded from 𝐢1𝐸[0,