Peng and Chen Boundary Value Problems (2016) 2016:125 DOI 10.1186/s13661-016-0632-5

RESEARCH

Open Access

Existence and multiplicity of positive solutions for p-Laplacian elliptic equations Zhen Peng1 and Guanwei Chen2* *

Correspondence: [email protected] 2 School of Mathematical Sciences, University of Jinan, Jinan, Shandong Province 250022, P.R. China Full list of author information is available at the end of the article

Abstract We study a p-Laplacian elliptic equation with Hardy term and Hardy-Sobolev critical exponent, where the nonlinearity is (p – 1)-sublinear near zero and (p∗ (s) – 1)-sublinear is the Hardy-Sobolev critical exponent). By using variational near infinity (p∗ (s) = p(N–s) N–p methods and some analysis techniques, we obtain the existence and multiplicity of positive solutions for the p-Laplacian elliptic equation. To the best of our knowledge, no result has been published concerning the existence and multiplicity of positive solutions for the p-Laplacian elliptic equation. MSC: 35A15; 35J91 Keywords: p-Laplacian elliptic equations; Hardy term; Hardy-Sobolev critical exponent; variational methods

1 Introduction and main results In this paper, we will study the existence and multiplicity of positive solutions for the following p-Laplacian elliptic equation: ⎧ ⎨– u – μ |u|p– u = p |x|p ⎩u = ,

∗

|u|p (s)– u + λf (x, u), |x|s

x ∈  \ {},

(.)

x ∈ ∂.

Here,  ⊂ RN (N ≥ ) is an open bounded domain with smooth boundary ∂ and  ∈ , p ∈ (, N), s ∈ [, p), λ, μ ∈ R+ , p u := div(|∇u|p– ∇u) is the p-Laplacian differential operator, p∗ (s) =

p(N–s) N–p

is the Hardy-Sobolev critical exponent, p∗ = p∗ () =

Np N–p

is the Sobolev

critical exponent, and we have the function f :  × R → R. Let

u :=

    p |u|p |∇u|p – μ p dx , |x| 

,p

u ∈ W (), ,p

which is well defined on the Sobolev space W () by the Hardy inequality []. From [], ,p

we know u is comparable with the standard Sobolev norm of W (), but it is not a norm since the triangle inequality or subadditivity may fail. The following best Hardy© 2016 Peng and Chen. 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.

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Sobolev constant will be useful in this paper: Aμ,s () :=

inf

,p

u∈W ()\{}

(



u p

∗ |u|p (s)  |x|s

p

dx) p∗ (s)

.

(.)

In recent decades, there were many authors [, –] who have studied the existence or ) ). multiplicity of solutions for elliptic equations with the operator – – |x|μ ( ≤ μ < ( N–  But most of the authors only considered the case s = . Next we only state some most related results of (.). Han [] obtained the existence of multiplicity of positive solutions for the following equation: ⎧ ⎨– u – μ |u|p– u = Q(x)|u|p∗ – u + λ|u|p– u, p |x|p ⎩u = ,

x ∈ ,

(.)

x ∈ ∂,

where Q(x) ≥  is a bounded function on . The authors [] only studied (.) in the special cases where Q(x) ≡  and μ = . The authors [] studied the following equation: ⎧ ⎨– u – μ up– = p |x|p ⎩u ∈ Dp (RN ),

∗

|u|p (s)– |x|s

∗ –

+ |u|p

, x ∈ RN ,

(.)



where D (RN ) is defined as the completion of Cc∞ (RN ), and they obtained a positive sop lution u ∈ D (RN ) ∩ C  (RN \ {}) for any  < s < p and μ ∈ (–∞, μ ), where μ := ( N–p )p . p Later, the authors [] obtained a nontrivial solution of a more general case than (.) by the ideas in []. Kang [] obtained one positive solution for the following equation: p

⎧ ⎨– u – μ |u|p– u = p |x|p ⎩u = ,

∗

q– |u|p (s)– u + λ |u||x|t u , |x|s

x ∈  \ {},

(.)

x ∈ ∂,

where  ≤ t < p, p ≤ q < p∗ (t). Inspired by the above results, we shall study the existence and multiplicity of positive solutions for (.) with the nonlinearity f being (p – )-sublinear at zero and (p∗ (s) – )sublinear at infinity (see the following (A )), which is different from the above results. ∗ ,p ,p Due to the lack of compactness of the embeddings in W () → Lp (), W () → ∗ ,p Lp (, |x|–p dx), and W () → Lp (s) (, |x|–s dx), we cannot use the standard variational argument directly. The corresponding energy functional fails to satisfy the classical Palais,p Smale ((PS)) condition in W (). But we can establish a local (PS) condition in a suitable range, so the existence result can be obtained by constructing a minimax level within this range and the mountain pass lemma in [, ]. t Let · p be the norm in Lp () and F(x, t) :=  f (x, s) ds, x ∈ , t ∈ R. Let a(μ) and b(μ) be zeros of the function  f (t) = (p – )t p – (N – p)t p– + μ, satisfying  ≤ a(μ) < assumptions:

N–p p

t ≥ ,  ≤ μ < μ :=

N –p p

p

< b(μ); see []. To state our results, we make the following

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(A ) f ∈ C( × R+ , R), f (x, ) ≡ , and f (x, t) = +∞, t p–

lim

t→+

lim

f (x, t)

∗ t→∞ t p (s)–

=  uniformly for x ∈ .

(A ) f :  × R+ → R is nondecreasing with respect to the second variable. (A )  ≤ p < N , N < min{pb(μ), p( + p)} and  ≤ s ≤ N – (N–p)(+p) . p (A )  ≤ p < N , pb(μ) ≤ N < p +

p b(μ) +p

and N – pb(μ) < s ≤ N –

(N–p)(+p) . p

Remark . In (A ) and (A ), we can easily check that N < p( + p) implies N – (N–p)(+p) > p , N < p +

p b(μ) +p

implies N – pb(μ) < N –

(N–p)(+p) . p

Besides, N –

(N–p)(+p) p

< p holds.

Now our results read as follows. Theorem . If N ≥ ,  ≤ s < p,  ≤ μ < μ ,  < p < N and (A ) hold, then there exists λ∗ >  such that (.) has at least one nontrivial positive solution uλ for any λ ∈ (, λ∗ ). Theorem . If N ≥ ,  ≤ s < p,  ≤ μ < μ , (A ), (A ) and ((A ) or (A )) hold, then there exists λ∗ >  such that (.) has at least two nontrivial positive solutions for every λ ∈ (, λ∗ ). Remark . We should mention that the above p-Laplacian problems studied in [, – ] are all not (p – )-sublinear at zero. Besides, our nonlinearity f is more general. To the best of our knowledge, our Theorems . and . are new. ∗

Let D,p (RN ) := {u ∈ Lp (RN ); |∇u| ∈ Lp (RN )}. A typical model of (.) is the following equation: ⎧ up– p∗ – ⎪ , in RN \ {}, ⎪ ⎨ –p u – μ |x|p = u ⎪ ⎪ ⎩

in RN \ {},

u > , u ∈ D,p (RN ),

μ ∈ [, μ ).

From [], we see that this problem has radially symmetric ground states, Vε (x) = ε–

N–p p

Up,μ

    N–p x |x| = ε– p Up,μ , ε ε

∀ε > ,

and they satisfy   p N



∗

Vε (x) p dx = A p ,

∇Vε (x) p – μ |Vε (x)| dx = μ, |x|p RN RN



where Up,μ (x) = Up,μ (|x|) is the unique radial solution of this problem, satisfying  Up,μ () =

N(μ – μ) N –p



 p∗ –p

.

Moreover,Up,μ has the following properties: lim ra(μ) Up,μ (r) = c > ,

r→

lim rb(μ) Up,μ (r) = c > ,

r→+∞

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 lim ra(μ)+ Up,μ (r) = c a(μ) ≥ ,

 lim rb(μ)+ Up,μ (r) = c b(μ) > ,

r→+∞

r→

where c and c are positive constants depending on p and N ; a(μ) and b(μ) are zeros of the function f (t) = (p – )t p – (N – p)t p– + μ, satisfying  ≤ a(μ) < tion (.).

N–p p

t ≥ ,  ≤ μ < μ ,

< b(μ); see []. The above results are useful in studying equa-

Remark . As μ =  and s = , then b(μ) = b() = N–p . When p =  and  ≤ μ < μ := p– √ √ √ √ N–  (  ) , it is well known that a(μ) = μ – μ – μ and b(μ) = μ + μ – μ. In Section , we will give the proof of Theorem .. In Section , we first of all give some preliminary lemmas, and then we will complete the proof of Theorem ..

2 Proof of Theorem 1.1 ,p Let X := W () and u± := max{±u, }. Note that the values of f (x, t) for t <  are irrelevant in Theorems .-., so we define f (x, t) ≡ ,

x ∈ , t ≤ .

The functional corresponding of (.) is

I(u) =

    ∗  (u+ )p (s) |u|p |∇u|p – μ p dx – ∗ dx |x| p (s)  |x|s    ,p – λ F x, u+ dx, u ∈ W ().

 p



,p

By (A ) and the Hardy inequalities (see []), we have I ∈ C  (W (), R). Now it is well known that there is a one-to-one correspondence between the weak solutions of (.) and ,p ,p the critical points of I on W (). More precisely, we say u ∈ W () is a weak solution of (.) if 

 I (u), v =

     ∗  |u|p– uv (u+ )p (s)– v dx – λ f x, u+ v dx |∇u|p– ∇u∇v – μ dx – p s |x| |x|   

= ,p

for any v ∈ W (). Proof of Theorem . By the Sobolev and Hardy-Sobolev inequalities, we get  u pp ≤ C u p , p∗ u p∗

p∗

≤ C u ,



∗

|u|p (s) ∗ dx ≤ C u p (s) s |x|

∀u ∈ X,

and (.)

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and it follows from (A ) that



∃δ >  such that F(x, t) <

∗

t p (s) p∗ (s)|xs |



∃M >  such that F(x, t) ≤ M ,

for t > δ, ∀t ∈ (, δ],

uniformly for all x ∈  \ {}. Thus, we get



F(x, t) ≤ M +

∗

t p (s) , ∗ p (s)|x|s

∀t ∈ R, x ∈  \ {}.

(.)

By (.) and (.), we have   I(u) = u p – ∗ p p (s)

 

∗

(u+ )p (s) dx – λ |x|s

for all λ ∈ (, ] and some C = I(u) >  if u = ρ

Cμ , p∗ (s)



  ∗ F x, u+ dx ≥ u p – C u p (s) – λM || p 

so there are ρ >  and λ∗ ∈ (, ] such that

and I(u) ≥ –C

if u ≤ ρ

∗

for any  < λ < λ∗ , where C = C ρ p (s) + λ∗ M ||. We choose u ∈ W () ∩ L∞ () such p that u+ = . Let M := u p /(λ u+ p ). By (A ), there is δ such that



F(x, t) ≥ M |t|p , p

,p

 < t < δ .

Hence, we get 

 ∗  (u+ )p (s) dx – λ F x, ru+ dx s |x|   p p p   r r r p λM u+ p = – u p <  ≤ u p – p p p ∗

I(ru ) =

rp rp (s) u p – ∗ p p (s)

for any  < λ < λ∗ and  < r < min{ρ, δ / u+ ∞ }. So there is u small enough such that I(u) < . We deduce that inf I(u) <  < u∈Bρ ()

I(u).

inf u∈∂Bρ ()

By Ekeland’s variational principle in [], there is a minimizing sequence {un } ⊂ Bρ () such that  I(un ) ≤ inf I(u) + , n u∈Bρ ()

 I(ω) ≥ I(un ) – ω – un , n

So, we have   I (un ) →  and

I(un ) → cλ

as n → ∞,

ω ∈ Bρ ().

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where cλ stands for the infimum of I(u) on Bρ (). Note that {un } is bounded and Bρ () is ,p a closed convex set, so there is uλ ∈ Bρ () ⊂ W (). By [], we have ,p

un uλ

weakly in W (),

un → uλ

strongly in Lγ (),  < γ < p∗ ,

un → uλ

a.e. in ,

∇un → ∇uλ

a.e. in ,

uλ un

weakly in Lp (), x x   ∗ ∗ |un |p (s)– un |uλ |p (s)– uλ v dx → v dx, |x|s |x|s  

,p

∀v ∈ W ().

Thus, passing to the limit in I  (un ), v, as n → ∞, we have   |∇uλ |p– ∇uλ ∇v – μ 

   ∗  |uλ |p– uλ v (u+λ )p (s)– v dx – dx – λ f x, u+λ v dx =  p s |x| |x|  

for all v ∈ W (). That is, I  (uλ ), v = . Therefore, uλ is a critical point of I. Since u–λ p = –I  (uλ ), u–λ  = , uλ = u+λ ≥ . Moreover, by (A ) and the boundedness of , we have ,p



 t p∗ (s)– for t > M , ∃M >  such that f (x, t) < λ |x|s



∃δ ∈ (, M ) such that f (x, t) >  for  < t < δ ,



∃M >  such that f (x, t) ≤ M for all t ∈ [δ , M ] for all x ∈  \ {}. Therefore, we deduce that ∗

f (x, t) ≥ –

 t p (s)– – M tδ– , λ |x|s

∀t ∈ R+ , x ∈  \ {}.

(.)

From (.) and (.), we have –p uλ + λM δ– uλ ≥ . By the strong maximum principle, we have uλ > . So the proof of Theorem . is finished. 

3 Proof of Theorem 1.2 In this section, we will look for the second positive solution by a translated functional as in []. For fixed λ ∈ (, λ∗ ), we will look for the second solution of (.) of the form u = uλ + v, where uλ is the first positive solution obtained in the previous section. The corresponding equation for v is ⎧ p– ⎪ – p v – μ |v||x|p v ⎪ ⎪ ⎨ p∗ (s)– = (uλ +v) – |x|s ⎪ ⎪ ⎪ ⎩v = ,

p∗ (s)–

uλ |x|s

+ λf (x, uλ + v) – λf (x, uλ ), x ∈  \ {}, x ∈ ∂.

(.)

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Let us define ⎧ ⎨ (uλ +t)p∗ (s)– g(x, t) =

|x|s

⎩, 

p∗ (s)–

–

uλ |x|s

+ λf (x, uλ + t) – λf (x, uλ ), t ≥ ,

(.)

t < ,

t

G(x, t) =

g(x, s) ds, 

and      |v|p p |∇v| – μ p dx – G x, v+ dx |x|     p∗ (s) p∗ (s)– +  + p∗ (s)  uλ uλ (uλ + v )  v p ∗ – – p (s) dx = v – ∗ s s s p p (s)  |x| |x| |x|    F x, uλ + v+ – F(x, uλ ) – f (x, uλ )v+ dx. –λ

 J(v) = p

 ,p

Now, we have one-to-one correspondence between critical points of J in W () and ,p solutions of (.). That is, if v ∈ W (), v ≡  is a critical point of J, then v is a solution of (.). Since v– p = –J  (v), v–  = , v = v+ ≥ . Besides, by the maximum principle, v >  in . Here, u = uλ + v is a positive solution of (.) and u = uλ . If v =  is the only critical ,p point of J in W (), we will get a contradiction. Then the existence of the second positive solution of (.) can be proved. ,p

Lemma . v =  is a local minimum of J in W (). ,p

Proof For any v ∈ W (), we write v = v+ – v– . By J and direct computation, we have   p  J(v) = v–  + I uλ + v+ – I(uλ ). p

(.) ,p

Since uλ is a local minimizer of I in W (), we have J(v) ≥ p v– p for v ≤ ε with ε being small enough.  Lemma . Suppose that  < p < N , (A ) and (A ) hold, moreover, v =  is the only critical point of J. Let {vn } be a (PS)c sequence with  < c <

N–s

p–s p–s A p(N–s) μ,s

, then we have

,p

vn →  in W () as n → ∞. ,p

Proof Let {vn } be a sequence in W () such that J(vn ) → c <

N–s p–s p–s Aμ,s p(N – s)

 ,p ∗ and J  (vn ) →  in W () .

(.)

By (.) and (.), we have   p  J(vn ) = v–n  + I uλ + v+n – I(uλ ) = c + o(), p

(.)

Peng and Chen Boundary Value Problems (2016) 2016:125

  J (vn ), uλ + v+n =

 

Page 8 of 15



– p– –  



∇v ∇v ∇uλ dx + I  uλ + v+ , uλ + v+ = o()uλ + v+ , n n n n n

which yields    J (vn ), uλ + v+n p    

– p– –  

   – p + + 



vn – ∇vn ∇uλ dx – I uλ + vn , uλ + vn + I uλ + v+n – I(uλ ) ∇vn = p     ≤ c +  + o() uλ + v+n .

J(vn ) –

Therefore, we have      ∗ 



  (uλ + v+n )p (s) v– p – ∇v– p– ∇v– ∇uλ dx +  –  dx n n n p p p∗ (s)  |x|s         + + + f x, uλ + vn uλ + vn – F x, uλ + vn dx +λ  p   ≤ I(uλ ) + c +  + o()uλ + v+n .

(.)

By (A ) and the boundedness of , for any ε > , there is M = M (ε) >  such that p∗ (s)



f (x, t)t ≤ ε |t| for x ∈  \ {} and |t| > M , |x|s



f (x, t)t ≤ C (ε) for x ∈  and |t| ∈ [, M ]; p∗ (s)



F(x, t) ≤ ε |t| for x ∈  \ {} and |t| > M , p |x|s



F(x, t) ≤ C (ε) for x ∈  and |t| ∈ [, M ],

where C (ε), C (ε) > . Thus, we have p∗ (s)



f (x, t)t ≤ C (ε) + ε |t| , |x|s

 (x, t) ∈  \ {} × R,

(.)

p∗ (s)



F(x, t) ≤ C (ε) + ε |t| , p |x|s

 (x, t) ∈  \ {} × R.

(.)

Let C(ε) = p C (ε) + C (ε), by (.) and (.), we have ∗

ε |t|p (s)  , F(x, t) – f (x, t)t ≤ C(ε) + p p |x|s

 (x, t) ∈  \ {} × R.

By (.) and (.), we have 



∗

(uλ + v+n )p (s) dx |x|s   p–     p ≤ λC(ε)|| – v–n  + C v–n  + C + o()uλ + v+n , p

p–s λε – p(N – s) p

(.)

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where C = p uλ and C = I(uλ ) + c + . Let ε =  

∗ (s)

(uλ + v+n )p |x|s

where C = we have

p(N–s) C p–s

p–s , (N–s)λ

then we have

 p–   dx ≤ C v–n  + C + o()uλ + v+n , and C =

p(N–s) (λC(ε)|| + C ). p–s

Together with (.), (.), and (.),

       –ε v– p +  ( – ε)v+ p – Cε uλ p – ( – ε)v+ p– n n n p p  p    p   ≤ v–n  + ( – ε)v+n  – Cε uλ p p p p   p    ≤ v–n  + v+n  – uλ

p p p   p   ≤ v–n  + uλ + v+n  p p   ∗   (uλ + v+n )p (s) = ∗ dx + λ F x, uλ + v+n dx + J(vn ) + I(uλ ) + o() s p (s)  |x|   – p–   + ≤ C vn  + C + o()uλ + vn , where the second inequality is due to the elementary inequality |a – b|t ≥ ( – ε)at – Cε bt ,

t ≥ , a, b > .

)C and C = λC (ε)|| + ( p∗(s) + λε )C + I(uλ ) + c + o(). Since v–n p + Here, C = ( p∗(s) + λε p p v+n p = vn p , we get  p–     – p– vn ≤ C + o() uλ , vn p – C v+n  – C  = where C =  + o(), C

C p , –ε

C =

Cε uλ p +pC . –ε

So we get

vn p – C vn p– ≤ C + o() uλ ,  . It shows that {vn } is bounded in W (), going if necessary to a where C = C + C subsequence, we have ,p

,p

vn v

weakly in W (),

vn → v

strongly in Lγ (),  < γ < p∗ ,

vn → v

a.e. in ,

(.)

as n → ∞. ,p Since vn is bounded in W (), it follows from the Sobolev embedding theorem that p∗ (s) there is M >  such that uλ + v+n p∗ (s) ≤ M . Let meas E denote the measure of E. By (A ), for any ε > , there is C (ε) >  such that



f (x, t)t ≤ C (ε) + ε |t|p∗ (s) , M

(x, t) ∈  × R.

Peng and Chen Boundary Value Problems (2016) 2016:125

Let δ =

ε C (ε)

Page 10 of 15

> , if E ⊂ , meas E < δ, we have













f x, uλ + v+ uλ + v+ dx ≤ f x, uλ + v+ uλ + v+ dx n n n n



E E  

∗ ε

uλ + v+ p (s) dx ≤ C (ε) dx + n M E E ε ≤ C (ε) meas E + < ε.  By the Vitali theorem, we have  



  f x, uλ + v+n uλ + v+n dx →



  f x, uλ + v+ uλ + v+ dx

as n → ∞.

Hence,  

 f x, uλ + v+n (uλ + vn ) dx =

 

  f x, uλ + v+n uλ + v+n dx –





→ 

+

f x, uλ + v (uλ + v ) dx

 

 f (x, uλ ) v–n dx

as n → ∞.

(.)

By the same method, we get  f









x, uλ + v+n





 ω dx → 

+

F x, uλ + vn dx →





 f x, uλ + v+ ω dx, (.)

 F x, uλ + v+ dx

,p

as n → ∞ for ω ∈ W (). Similar to the proof of Theorem ., we have

   = lim J  (vn ), ω = J  (v ), ω n→∞

for ω ∈ W (), which implies that J  (v ) = . Therefore, v is a critical point of J in ,p W (). By the assumption that v =  is the only critical point of J, we have v =. Now, we ,p want to prove v →  strongly in W (). By (.), (.), and the Brezis-Leib Lemma (see []), we have ,p

  p    J(vn ) = v–n  + I uλ + v+n – I(uλ ) = vn p – ∗ p p p (s)



∗



(v+n )p (s) dx + o(). |x|s

Therefore,

  J (vn ), vn = vn p –



∗



(v+n )p (s) dx + o() → . |x|s

In fact, vn p →  as n → ∞. If not, then there is a subsequence (still denoted by vn ) such that  lim vn p = k,

n→∞

lim

n→∞ 

∗

(v+n )p (s) dx = k, |x|s

k > .

Peng and Chen Boundary Value Problems (2016) 2016:125

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By (.), we get 

∗

vn ≥ Aμ,s p



(v+n )p (s) dx |x|s



p p∗ (s)

for all n ∈ N.

,

N–s p–s

p

Then k ≥ Aμ,s k p∗ (s) , i.e., k ≥ Aμ,s . Thus, we have   c = o() + J(vn ) = vn p – ∗ p p (s) p–s = k + o() p(N – s) ≥



∗



(v+n )p (s) dx + o() |x|s

N–s p–s p–s Aμ,s . p(N – s)

,p

It is a contradiction. So vn →  strongly in W () as n → ∞.



Lemma . [] If  < p < N ,  ≤ s < p and  ≤ μ < μ , then the limiting problem ⎧ up– ⎪ ⎪–p u – μ |x|p = ⎨ ⎪ ⎪ ⎩

∗

up (s)– , |x|s

in RN \ {}, in RN \ {},

u > ,

(P)

u ∈ D,p (RN ),

has radially symmetric ground states, ε (x) := ε– V

N–p p

p,μ U

    N–p x |x| – p  Up,μ =ε , ε ε

∀ε > ,

and it satisfies    N–s ε (x)|p∗ (s)  (x)|p



|V |V p–s

∇ V ε (x) p – μ ε dx = dx = A , μ,s p s N N |x| |x| R R



p,μ (x) = U p,μ (|x|) is the unique radial solution of (P), satisfying where U p,μ () = U



(N – s)(μ – μ) N –p



 p∗ (s)–p

.

p,μ has the following properties: Moreover, U p,μ (r) = c > , lim ra(μ) U

r→

p,μ (r) = c > , lim rb(μ) U

r→+∞

 p,μ (r) = c a(μ) ≥ , lim ra(μ)+ U

r→

 p,μ lim rb(μ)+ U (r) = c b(μ) > ,

r→+∞

where c and c are positive constants depending on p and N ; a(μ) and b(μ) are zeros of the function f (t) = (p – )t p – (N – p)t p– + μ, satisfying  ≤ a(μ) <

N–p p

< b(μ) <

N–p . p–

t ≥ ,  ≤ μ < μ ,

Peng and Chen Boundary Value Problems (2016) 2016:125

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Since uλ >  is a solution of (.), similar to the proof of Theorem . in [], there are constants R >  and r >  such that BR () ⊂  and  < r ≤ uλ (x),

∀x ∈ BR () \ {}.

(.)

Let ϕ ∈ C∞ () such that  ≤ ϕ(x) ≤  and

ϕ(x) :=

⎧ ⎨,

|x| ≤ R, ⎩, |x| ≥ R,

ε (x), ε > , where V ε (x) is defined in Lemma .. Then where BR () ⊂ . Set vε (x) = ϕ(x)V we can get the following results by the method used in []: N–s  p–s vε p = Aμ,s + O εb(μ)p+p–N ,  ∗ N–s  |vε |p (s) ∗ p–s dx = Aμ,s + O εb(μ)p (s)+s–N , s |x|   N –s  |vε |r dx = O εp–s , < r < p∗ (s). s |x| b(μ) 

(.) (.) (.)

Lemma . For γ ≥ ,  ≤ t ≤ γ – , ∀a, b > , there exists a positive constant C such that (a + b)γ ≥ aγ + bγ + Caγ –t bt . Proof To prove this lemma, we only need to prove ( + x)γ ≥  + xγ + Cxt ,

 < x < ∞.

Let γ = k + θ , t = m + η, where k ≥ ,  ≤ m ≤ k –  are integral numbers and  ≤ η ≤ θ <  are real numbers. Clearly,  ( + x)γ = ( + x)k+θ = ( + x)k ( + x)θ ≥  + xk + Cxm ( + x)θ ≥  + xk+θ + Cxm ( + x)θ ≥  + xk+θ + Cxm xη =  + xγ + Cxt .



Lemma . If N ≥ ,  ≤ s < p,  ≤ μ < μ ,  < p < N , (A ), (A ), (A ) (or (A )), and ,p f (x, ) ≡  hold, then there is v∗ ∈ W (), v∗ ≡ , such that sup J(tv∗ ) < t≥

N–s p–s p–s Aμ,s . p(N – s)

Proof By (.), (A ), and Lemma ., we have ∗

g(x, l) ≥

p∗ (s)–p

lp (s)– lp– uλ +C s |x| |x|s

.

Peng and Chen Boundary Value Problems (2016) 2016:125

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By (A ) or (A ), we have p ≥  and s ≤ N – p –  ≤ (p∗ (s) – ) – . Therefore, p∗ (s)

∗

(N–p)(+p) , p

which imply p∗ (s) –  ≥  and  ≤

p p∗ (s)–p

t p (s) vε Ct p vε uλ G(x, tvε ) ≥ ∗ + p (s) |x|s p |x|s

.

From (A ) (or (A )), we have s > N – Pb(μ), which implies p > (.)-(.), we have tp J(tvε ) = vε p – p

so (.) holds. So by

 G(x, tvε ) dx  ∗

t p (s) tp ≤ vε p – ∗ p p (s) =

N–s , b(μ)

 

∗

|vε |p (s) dx – C t p |x|s p∗ (s)

 

p

vε dx |x|s

t     t ∗ Aμ,s + O εb(μ)p+p–N – ∗ Aμ,s + O εb(μ)p (s)+s–N – C t p O εp–s , p p (s) p

N–s p–s

N–s p–s

p∗ (s)–p

where C =

Cr

. Let

p

∗

t p (s)  N–s    t p  N–s ∗ p–s p–s Aμ,s + O εb(μ)p+p–N – ∗ Aμ,s + O εb(μ)p (s)+s–N – C t p O εp–s . Q(t) := p p (s) Clearly, the following equation:  N–s    N–s    ∗ ∗ p–s p–s  = Q (t) = t p– Aμ,s + O εb(μ)p+p–N – O εp–s – t p (s)– Aμ,s + O εb(μ)p (s)+s–N has only a positive root  tε =

N–s p–s

Aμ,s + O(εb(μ)p+p–N ) – O(εp–s )



 p∗ (s)–p

N–s p–s

.

Aμ,s + O(εb(μ)p∗ (s)+s–N )

We have p∗ (s)

 tε  N–s   tε  N–s ∗ p–s p–s Aμ,s + O εb(μ)p+p–N – O εp–s – ∗ Aμ,s + O εb(μ)p (s)+s–N Q(tε ) = p p (s) p

 =

 N–s  N–s  p∗ (s) p–s  p–s  b(μ)p∗ (s)+s–N  Aμ,s  + O(εb(μ)p+p–N ) – O(εp–s ) p∗ (s)–p  – Aμ,s + O ε N–s p p∗ (s) p–s Aμ,s + O(εb(μ)p∗ (s)+s–N )

=

– N–p  N–s   N–s   p – s  N–s ∗ p–s p–s p–s Aμ,s + O εb(μ)p+p–N – O εp–s p–s Aμ,s + O εb(μ)p (s)+s–N p(N – s)

=

N–s    p–s ∗ p–s Aμ,s + O εb(μ)p (s)+s–N + O εb(μ)p+p–N – O εp–s . p(N – s)

By s > N – pb(μ) (see (A ) or (A )), we have b(μ)p + p – N > p – s.

Peng and Chen Boundary Value Problems (2016) 2016:125

Since b(μ) >

N–p p

Page 14 of 15

implies b(μ)p∗ (s) + s – N > b(μ)p + p – N , we have

b(μ)p∗ (s) + s – N > p – s. Since Q() =  and limt→+∞ Q(t) = –∞, we have sup Q(t) = Q(tε ) < t≥

N–s p–s p–s Aμ,s p(N – s)

for ε >  sufficiently small. So we get sup J(tvε ) ≤ sup Q(t) < t≥

t≥

N–s p–s p–s Aμ,s , p(N – s)

for ε >  sufficiently small. It completes the proof if we let v∗ = vε with ε >  being sufficiently small.  ,p

Proof of Theorem . If v =  is the only critical point of J in W (). By Lemma ., we ,p know there is α >  such that J(v) > α, ∀v ∈ ∂Bρ = {v ∈ W (), v = ρ}, where ρ >  is ,p small enough. Lemma . implies that there is v∗ ∈ W () and v∗ ≡  such that N–s p–s p–s Aμ,s . p(N – s)

sup J(tv∗ ) < t≥

By (.), we get limt→∞ J(tv∗ ) → –∞. Hence, we can choose t >  such that t v∗ > ρ ,p and J(t v∗ ) < . By the mountain pass lemma in [], there is a sequence {vn } ⊂ W () satisfying J(vn ) → c ≥ α

and J  (vn ) → ,

where  c = inf max J h(t) , h∈ t∈[,]

    = h ∈ C [, ], X | h() = , h() = t v∗ . We have   < α ≤ c = inf max J h(t) ≤ max J(tt v∗ ) ≤ sup J(tv∗ ) < h∈ t∈[,]

t∈[,]

t≥

N–s p–s p–s Aμ,s , p(N – s)

,p

and this together with Lemma . implies that vn →  strongly in W () as n → ∞. Hence, we have  = J() = limn→∞ J(vn ) = c ≥ α > , a contradiction. So, Theorem . holds.  Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Peng and Chen Boundary Value Problems (2016) 2016:125

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Author details 1 School of Mathematics and Statistics, Anyang Normal University, Anyang, Henan Province 455000, P.R. China. 2 School of Mathematical Sciences, University of Jinan, Jinan, Shandong Province 250022, P.R. China. Acknowledgements We would like to thank the referee for his/her valuable comments, which have led to an improvement of the presentation of this paper. Research supported by National Natural Science Foundation of China (No. 11401011). Received: 27 April 2016 Accepted: 27 June 2016 References 1. Ghoussoub, N, Yuan, C: Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents. Trans. Am. Math. Soc. 352, 5703-5743 (2000) 2. Filippucci, R, Pucci, P, Robert, F: On a p-Laplace equation with multiple critical nonlinearities. J. Math. Pures Appl. 91, 156-177 (2009) 3. Ambrosetti, A, Brezis, H, Cerami, G: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122, 519-543 (1994) 4. Abdellaoui, B, Peral, I: Some results for semilinear elliptic equations with critical potential. Proc. R. Soc. Edinb. A 132(1), 1-24 (2002) 5. Cao, DM, Han, PG: Solutions for semilinear elliptic equations with critical exponents and Hardy potential. J. Differ. Equ. 205(2), 521-537 (2004) 6. Ding, L, Tang, C-L: Existence and multiplicity of solutions for semilinear elliptic equations with Hardy terms and Hardy-Sobolev critical exponents. Appl. Math. Lett. 20, 1175-1183 (2007) 7. Ding, L, Tang, C-L: Existence and multiplicity of positive solutions for a class of semilinear elliptic equations involving Hardy term and Hardy-Sobolev critical exponents. J. Math. Anal. Appl. 339, 1073-1083 (2008) 8. Ferrero, A, Gazzola, F: Existence of solutions for singular critical growth semi-linear elliptic equations. J. Differ. Equ. 177(2), 494-522 (2001) 9. Garcia Azorero, JP, Peral Alonso, I: Hardy inequalities and some critical elliptic and parabolic problems. J. Differ. Equ. 144(2), 441-476 (1998) 10. Ghoussoub, N, Kang, XS: Hardy-Sobolev critical elliptic equations with boundary singularities. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 21(6), 767-793 (2004) 11. Han, P: Multiple positive solutions for a critical growth problem with Hardy potential. Proc. Edinb. Math. Soc. 49, 53-69 (2006) 12. Han, P: Multiple solutions to singular critical elliptic equations. Isr. J. Math. 156, 359-380 (2006) 13. Han, P, Liu, Z: Solutions for a singular critical growth problem with a weight. J. Math. Anal. Appl. 327, 1075-1085 (2007) 14. Han, P: Many solutions for elliptic equations with critical exponents. Isr. J. Math. 164, 125-152 (2008) 15. Kang, DS, Peng, SJ: Positive solutions for singular critical elliptic problems. Appl. Math. Lett. 17(4), 411-416 (2004) 16. Kang, DS, Peng, SJ: Solutions for semi-linear elliptic problems with critical Sobolev-Hardy exponents and Hardy potential. Appl. Math. Lett. 18(10), 1094-1100 (2005) 17. Terracini, S: On positive entire solutions to a class of equations with a singular coefficient and critical exponent. Adv. Differ. Equ. 1(2), 241-264 (1996) 18. Han, P: Quasilinear elliptic problems with critical exponents and Hardy terms. Nonlinear Anal. 61, 735-758 (2005) 19. Degiovanni, M, Lancelotti, S: Linking solutions for p-Laplace equations with nonlinearity at critical growth. J. Funct. Anal. 256, 3643-3659 (2009) 20. Xuan, B, Wang, J: Existence of a nontrivial weak solution to quasilinear elliptic equations with singular weights and multiple critical exponents. Nonlinear Anal. 72, 3649-3658 (2010) 21. Kang, DS: On the quasilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy terms. Nonlinear Anal. 68, 1973-1985 (2008) 22. Rabinowitz, PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg. Conf. Series. Math., vol. 65. Am. Math. Soc., Providence (1986) 23. Abdellaoui, B, Felli, V, Peral, I: Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian. Boll. Unione Mat. Ital., B 9(2), 445-484 (2006) 24. Mawhin, J, Willem, M: Critical Point Theory and Hamiltonian Systems. Appl. Math. Sci., vol. 74. Springer, New York (1989) 25. Brezis, H, Lieb, E: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88, 486-490 (1983) 26. Chen, JQ: Multiple positive solutions for a class of nonlinear elliptic equations. J. Math. Anal. Appl. 295(2), 341-354 (2004) 27. Kang, DS, Huang, Y, Liu, S: Asymptotic estimates on the extremal functions of a quasilinear elliptic problem. J. South Cent. Univ. Natl. 27(3), 91-95 (2008)