Monatsh Math DOI 10.1007/s00605-013-0564-4

Multiple positive solutions to a class of quasi-linear elliptic equations involving critical Sobolev exponent Haining Fan · Xiaochun Liu

Received: 27 March 2013 / Accepted: 13 September 2013 © Springer-Verlag Wien 2013

Abstract In this paper, we study the multiplicity results of positive solutions for a quasi-linear problem involving concave-convex nonlinearities and critical Sobolev exponent. With the help of Nehari manifold and a mini-max principle, we prove that problem has at least two or three positive solutions under different conditions. Keywords Nehari manifold · Critical Sobolev exponent · Quasi-linear problem · Mini-max principle · Multiple positive solutions Mathematics Subject Classification (2000)

35J20 · 35J50 · 35J60

1 Introduction and the main result In this paper, we consider the multiplicity result for positive solutions of the following quasi-linear elliptic problem:  (E f,g )

− p u = f (x)|u|q−2 u + g(x)|u| p u = 0,

∗ −2

u, x ∈ , x ∈ ∂,

Communicated by A. Jüngel. Supported by NSFC (Grant No. 11171261, 11371282). H. Fan (B) · X. Liu School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China e-mail: [email protected] X. Liu e-mail: [email protected]

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where  p u = div(|∇u| p−2 ∇u), 1 < q < p < N , p ∗ = NN−pp ,  ⊂ R N is an open bounded domain with smooth boundary and f, g ∈ C() satisfy g + = max{g, 0} ≡ 0 and f + = max{ f, 0} ≡ 0. Moreover, we assume that the domain  satisfies  (H ) Bρ (0)  = ∅ and B 1 (0)\Bρ (0) ⊂  for ρ > 0 sufficiently small, where ρ

Br (0) = {x ∈ R N ; |x| < r }. Under the assumption f ≡ 0 and g ≡ 1, (E f,g ) can be regarded as a perturbation problem of the following equation:  (E)

− p u = |u| p

∗ −2

u, x ∈ ,

u = 0,

x ∈ ∂.

For the case p = 2, it is well known the existence of positive solutions of (E) is affected by the shape of the domain . This has been the focus of a great deal of research by several authors. In particular, the first striking result is due to Pohozaev [19] who proved that if  is star-shaped with respect to some point, (E) has no solution. However, if  is an annulus, Kazdan and Warner [16] proved that (E) admits a positive solution. For a non-contractible domain , Coron [12] showed that (E) has a positive solution. More existence result for “rich topology” domain, we quote [1,6–8,15,17,23] for p = 2 and [2–5,10,18,20,21] for p = 2, and the references therein. When f ≡ 0, the existence and multiplicity of positive solutions for (E f,g ) may be influenced by the concave and convex nonlinearities. Recently, Hsu [13,14] proved that (E f,g ) admits at least two positive solutions if the negative and the positive parts of f are both sufficiently small in  and f, g satisfy some integrability conditions. Similar results can be found in [4,11,25,26]. The idea of them is the Nehari manifold and fibering maps method. In this work we aim to get a better result about the multiplicity of positive solutions of (E f,g ) by using the Nehari manifold and a mini-max principle. We will prove that if  is a non-contractible domain and g ≡ 1 in , (E f,g ) admits at least three positive solutions when f is sufficiently small in . Moreover, for a general domain, we show that (E f,g ) has at least two positive solutions if only the positive part of f is small enough in . The following assumptions are used in this paper: ( f ) There exist β, ρ > 0 and x0 ∈  such that B2ρ (x0 ) ⊂  and f (x) ≥ β for all x ∈ B2ρ (x0 ). −p (g1 ) There exist β0 ≥ Np−1 such that |g|∞ = g(x0 ) = max g(x), g(x) > 0, ∀x ∈ B2ρ (x0 ) and x∈

g(x) = g(x0 ) + O(|x − x0 |β0 ) where x0 , ρ is defined in ( f ). (g2 ) g ≡ 1.

123

as x → x0 ,

Multiple positive solutions to a class of quasi-linear elliptic equations

Throughout the paper by | · |r we denote the L r-norm. Set q ∗ =

1 = S

N ( p−q) q +p p2



p−q ∗ ( p − q)|g + |∞

 (N − p)( p−q)  p2

p∗ p ∗ −q

and

 p∗ − p . p∗ − q

The main results of this paper are concluded in the following theorems and conclusions are new to the best of our knowledge. Theorem 1.1 For | f + |q ∗ < 1 , (E f,g ) has at least one positive solution. Theorem 1.2 Suppose ( f ) and (g1 ) hold. Then if | f + |q ∗ < two positive solutions.

q p 1 ,

(E f,g ) has at least

Theorem 1.3 Suppose (g2 ) and (H ) hold. Then there exists 2 > 0 such that if | f |q ∗ < 2 , (E f,g ) has at least three positive solutions. Associated with (E f,g ), we consider the energy functional J f,g for each u ∈ X := 1, p W0 (),

J f,g (u) =

1 p

 |∇u| p d x − 

1 q

 f |u|q d x − 

1 p∗



∗

g|u| p d x. 

From the assumption, it is easy to prove that J f,g is well defined in X and J f,g ∈ C 1 (X, R). Furthermore, the critical points of J f,g are weak solutions of (E f,g ). This paper is organized as follows. Some preliminaries and properties of the Nehari manifold are established in Sect. 2. Theorems 1.1–1.2 are proved in Sect. 3 and Theorem 1.3 is proved in Sect. 4. For convenience, we will denote positive constant (possibly different) as c from then on.

2 Notations and preliminaries On the space X we consider the norm ⎛

u = ⎝



⎞1

p

|∇u| d x ⎠ . p



Set also  ∂u ∗ ∈ L p (R N ) for i = 1, . . . , N D1, p (R N ) := u ∈ L p (R N ); ∂ xi

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equipped with the norm ⎛ ⎜

u D1, p (R N ) = ⎝

⎞1



p

⎟ |∇u| p d x ⎠ .

RN

We then define the Palais–Smale (simply by (P S)) sequences, (P S)-values, and (P S)-condition in X for J f,g as follows. Definition 2.1 (i) For β ∈ R, a sequence {u k } is a (P S)β -sequence in X for J f,g if J f,g (u k ) = β + o(1) and J f,g (u k ) = o(1) strongly in X ∗ as k → ∞; (ii) β ∈ R is a (P S)-value in X for J f,g if there exists a (P S)β -sequence in X for J f,g ; (iii) J f,g satisfies the (P S)β -condition in X if every (P S)β -sequence in X for J f,g contains a convergent subsequence. By the Lemma 3.1 in Struwe ([22], Chapter III), we have the following Lemma. Lemma 2.1 Let {u k } be a (P S)-sequence of J f,1 (J f,g for g ≡ 1) in X . Then there j j j exists a number n ∈ N, sequences {εk }, {xk }, 1 ≤ j ≤ n of radii εk → 0 (as k → ∞), j and points xk ∈ , a solution u 0 ∈ X ⊂ D1, p (R N ) to (E f,1 )((E f,g ) for g ≡ 1), and nontrivial solutions u j ∈ D1, p (R N ), 1 ≤ j ≤ n to the “limiting problem” of (E f,1 ), such that a subsequence of {u k }, still denoted by {u k }, satisfies     n   j u k − u 0 − uk      j=1

→ 0,

as k → ∞,

D1, p (R N )

j

here u k denotes the scaled function j u k (x)

=

j − N−p (εk ) p u j



j

x − xk j

εk

 , 1 ≤ j ≤ n.

Moreover, J f,1 (u k ) → J f,1 (u 0 ) +

n 

J ∞ (u j ),

as k → ∞,

j=1

where J ∞ (u) =

1 p

 RN

|∇u| p d x −

1 p∗

 RN

∗

|u| p d x.

Remark 2.1 Similarly as the argument of Remark 3.2 in Struwe ([22], Chapter III), we know that (P S)β -condition holds for E f,1 (E f,g for g ≡ 1) for all levels β which cannot be decomposed

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Multiple positive solutions to a class of quasi-linear elliptic equations

β = J f,1 (u 0 ) + ι

1 Np S , N

where ι ∈ N. As J f,g is not bounded below on X , we consider the behaviors of J f,g on the Nehari manifold N f,g = {u ∈ X \{0}; J f,g (u), u = 0}, where ,  denotes the usual duality between X and X ∗ . Clearly, for each u ∈ X \{0}, u ∈ N f,g if and only if 

 |∇u| d x −





p∗

g|u| d x −

p



f |u|q d x = 0. 

Thus, on the Nehari manifold N f,g , we have    1 1 1 ∗ p q J f,g (u) = |∇u| d x − f |u| d x − ∗ g|u| p d x p q p        1 1 1 1 − ∗ − ∗ |∇u| p d x − f |u|q d x = p p q p       1 1 1 1 −q − ∗ u p − − ∗ | f + |q ∗ S p u q , ≥ p p q p

(2.1)

∗

where S is the best constant of the embedding of X → L p (). Hence J f,g is coercive and bounded below on N f,g . We now define ψ f,g (u) := J f,g (u), u =



 |∇u| p d x −



 f |u|q d x −



∗

g|u| p d x. 

Then for u ∈ N f,g , ψ f,g (u), u = ( p − q)



|∇u| p d x − ( p ∗ − q)







= (p − p )

|∇u| d x + ( p − q)

∗

 p



∗

∗

g|u| p d x

(2.2)

 f |u|q d x.

(2.3)



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We split N f,g into three parts:  N+ f,g = {u ∈ N f,g ; ψ f,g (u), u > 0};

N 0f,g = {u ∈ N f,g ; ψ f,g (u), u = 0};  N− f,g = {u ∈ N f,g ; ψ f,g (u), u < 0}.

Then we have the following results. Lemma 2.2 Suppose that u is a local minimizer for J f,g on N f,g and that u ∈ N 0f,g . Then J f,g (u) = 0 in X ∗ .  

Proof See Wu ([27], Lemma 4). Lemma 2.3 We have N 0f,g = ∅ with f + ≡ 0 and | f + |q ∗ < 1 .

Proof Suppose otherwise. Then there exists f ∈ C() with f + ≡ 0 and | f + |q ∗ < 1 such that N 0f,g = ∅. Then for u ∈ N 0f,g , we from (2.2) and (2.3) obtain

u p =

p∗ − q p−q



∗

g|u| p d x

u p =

and



p∗ − q p∗ − p

 f |u|q d x. 

By (g1 ), the Hölder inequality and Sobolev imbedding theorem, we have that 

p−q

u ≥ ∗ ( p − q)|g + |∞



p

p p∗ − p

S

N p

u ≤ S p

and

q q− p

  p ∗ p−q + ∗ p −q | f |q ∗ . p −p

Hence we must have +

| f |q ∗ ≥ S

N ( p−q) q +p p2



p−q ( p ∗ − q)|g + |∞

 (N − p)( p−q)  p2

p∗ − p p∗ − q

which is a contradiction.  ∗ For each u ∈ X with  g|u| p d x > 0, we set  tmax =

 = 1 ,  

  1 ( p − q)  |∇u| p d x p∗ − p  > 0. ∗ ( p ∗ − q)  g|u| p d x

Lemma 2.4 Suppose that | f + |q ∗ ∈ (0, 1 ) and u ∈ X is a function satisfying with  p∗  g|u| d x > 0.  − − (i) If  f |u|q d x ≤ 0, then there exists a unique t − f,g > tmax such that t f,g u ∈ N f,g and J f,g (t − f,g u) = sup J f,g (tu). t≥0

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Multiple positive solutions to a class of quasi-linear elliptic equations

 + (ii) If  f |u|q d x > 0, then there exists a unique t ± f,g such that 0 < t f,g < tmax < + + − − t− f,g , t f,g u ∈ N f,g and t f,g u ∈ N f,g . Moreover, J f,g (t + f,g u) =

inf

0≤t≤tmax

J f,g (tu), J f,g (t − f,g u) = sup J f,g (tu). t≥t + f,g

 

Proof Similarly as Brown–Wu ([9], Lemma 2.6).

Weremark that it follows Lemma 2.3, for 0 < | f + |q ∗ < 1 , we write N f,g = + − N− f,g . Furthermore, by Lemma 2.4, it follows that N f,g and N f,g are nonempty, and by (2.1), we may define N+ f,g

α +f,g = inf J f,g (u); α −f,g = inf J f,g (u). u∈N + f,g

u∈N − f,g

Then we have the following result. Lemma 2.5 (i) α +f,g < 0 for all f ∈ C() with f + ≡ 0 and | f + |q ∗ < 1 . (ii) If 0 < | f + |q ∗ < qp 1 , then α −f,g ≥ d0 for some d0 > 0. + Proof (i) From Lemma 2.4, we know that N + f,g is nonempty. Given u 0 ∈ N f,g , from (2.3) we obtain     1 1 1 1 − ∗ − ∗ J f,g (u 0 ) = |∇u 0 | p d x − f |u 0 |q d x p p q p        1 1 1 p∗ − p 1 − ∗ − − ∗ |∇u 0 | p d x ≤ p p q p p∗ − q    1 p∗ − p 1 − |∇u 0 | p d x < 0. = p∗ p q 

This yields α +f,g = inf J f,g (u) ≤ J f,g (u 0 ) < 0. u∈N + f,g

(ii) For u ∈ N − f,g , by (2.2) and the Sobolev embedding theorem, we get

u p <

∗ p∗ − q + ∗ −p |g |∞ S p u p , p−q

which implies p∗

u p ≥ S p( p∗ − p)



p−q ( p ∗ − q)|g + |∞



1 p∗ − p

.

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H. Fan, X. Liu

Moreover,  1 1 J f,g (u) = |∇u| d x − f |u|q d x − ∗ q p       1 1 1 1 −q − ∗ u p − − ∗ | f + |q ∗ S p u q ≥ p p q p      q 1 1 1 1 q p−q + ∗ −p . − ∗ u − ∗ | f |q S = u − p p q p 

1 1 − ∗ p p

Thus, if 0 < | f + |q ∗ <

q p 1 ,





p

then

α −f,g ≥ d0 for some d0 > 0.   3 Proofs of Theorems 1.1 and 1.2 Lemma 3.1 (i) If 0 < | f + |q ∗ < 1 , then J f,g has a (P S)α + -sequence {u k } ⊂ N + f,g . (ii) If 0 < | f + |q ∗ <

q p 1 ,

f,g

then J f,g has a (P S)α − -sequence {u k } ⊂ N − f,g . f,g

Proof The proof is similar to Hsu [13] and the details are omitted.

 

Now, we establish the existence of a local minimum for J f,g on N f,g . Lemma 3.2 If 0 < | f + |q ∗ < 1 , the functional J f,g has a minimizer u +f,g ∈ N + f,g and it satisfies (i) (ii) (iii) (iv)

J f,g (u +f,g ) = α +f,g ; u +f,g is a positive solution of (E f,g );

u +f,g → 0 as | f + |q ∗ → 0; J f,g (u +f,g ) → 0 as | f + |q ∗ → 0.

Proof By (2.1) and Lemma 3.1 (i), there exists a minimizing sequence {u k } ⊂ N + f,g such that J f,g (u k ) = α +f,g + o(1) and

J f,g (u k ) = o(1) in X ∗ .

(3.1)

Since J f,g is coercive on N f,g [see (2.1)], we get that {u k } is bounded in X . Passing to a subsequence, there exists u +f,g ∈ X such that as k → ∞, u k u +f,g weakly in X and u k → u +f,g strongly in L q ().

123

(3.2)

Multiple positive solutions to a class of quasi-linear elliptic equations

Thus J f,g (u +f,g ) = 0 in X ∗ and the fact {u k } ⊂ N + f,g implies that  f |u|q d x ≥ 

q( p ∗ − p) p∗ q

u k p − ∗ J f,g (u k ). ∗ p( p − q) p −q

(3.3)

Taking k → ∞ in (3.3), by (3.1), (3.2) and the fact α +f,g < 0, we get 

f |u +f,g |q d x ≥ −



p∗ q + α > 0. p ∗ − q f,g

(3.4)

Thus u +f,g is a nontrivial solution of (E f,g ). Next, we prove that u k → u +f,g strongly in X . From Lemma 2.5, (2.1), and the fact u k , u +f,g ∈ N f,g , it follows that α +f,g ≤ J f,g (u +f,g ) =

1 + p p∗ − q

u f,g − N p∗ q

⎛

1 p∗ − q ≤ lim inf ⎝ u k p − k→∞ N p∗ q = lim inf J f,g (u k ) = k→∞

α +f,g ,





f |u +f,g |q d x



⎞

f |u k |q d x ⎠



which implies that J f,g (u +f,g ) = α +f,g and limk→∞ u k p = u +f,g p . Thus u k → u +f,g + strongly in X . Moreover, since N 0f,g = ∅, we obtain u +f,g ∈ N + f,g , and by J f,g (u f,g ) = J f,g (|u +f,g |) = α +f,g , we get |u +f,g | ∈ N + f,g is a local minimizer of J f,g on N f,g . Then, + by Lemma 2.2, we may assume that u f,g is a nontrivial nonnegative solution of (E f,g ). By Harnack inequality due to Trudinger [24], we obtain that u +f,g > 0 in . Finally, by (2.3) and u +f,g ∈ N + f,g , we obtain

u +f,g p


q q− p



p∗ − q + | f |q ∗ p∗ − p



p p−q

,

which implies that u +f,g → 0 as | f + |q ∗ → 0, and so J f,g (u +f,g ) → 0 as | f + |q ∗ → 0.   Note the results in Sect. 2. It can be easily to give a proof of the following Lemma along the same argument of Hsu ([14], Theorem 1.2) and the details are omitted. Lemma 3.3 Assume ( f ) and (g1 ) hold. If | f + |q ∗ < N− f,g and itsatisfies

q p 1 ,

then there exists u −f,g ∈

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H. Fan, X. Liu

(i) J f,g (u −f,g ) = α −f,g ; (ii) u −f,g is a solution of (E f,g ).  

Proof of Theorem 1.1 By Lemma 3.2, we complete the proof.

Proof of Theorem 1.2 Suppose ( f ) and (g1 ) hold. By Lemma 3.3, we complete the proof.   4 Proof of Theorem 1.3 In this Section, we suppose (g2 ) and (H ) hold throughout. Denote V := {u ∈ X ; |u| p∗ = 1}, Jf,1 (u) = max J f,1 (tu) : V → R t≥0

and J0,1 (u) = max J0,1 (tu) : V → R. t≥0

Lemma 4.1 For each u ∈ N − f,1 and b > 0, we have − − (i) There is a unique t0,b such that t0,b u ∈ N0,b and

1 p−N − max J0,b (tu) = J0,b (t0,b u) = b p t≥0 N



∗

u p  p∗  |u| d x

 N−p p

.

− − (ii) For μ ∈ (0, 1), there is a unique t0,1 such that t0,1 u ∈ N0,1 . Moreover, N

− u) − J f,1 (u) ≥ (1 − μ) p J0,1 (t0,1

p p − q q−q p −q μ (| f |q ∗ S p ) p−q pq

and N

− u) + J f,1 (u) ≤ (1 + μ) p J0,1 (t0,1

(iii) α −f,1 →

1 N

p p − q q−q p −q μ (| f |q ∗ S p ) p−q . pq

N

S p as | f |q ∗ → 0.

Proof (i) For each u ∈ N − f,1 , let 1 1 ∗ h(t) = J0,b (tu) = t p u p − ∗ t p p p

123



∗

b|u| p d x. 

Multiple positive solutions to a class of quasi-linear elliptic equations

Then h(t) → −∞ as t → ∞, 

h (t) = t

p−1

u − t p

p ∗ −1



∗

b|u| p d x 

and h  (t) = ( p − 1)t p−2 u p − ( p ∗ − 1)t p

∗ −2



∗

b|u| p d x. 

Let − t0,b

 =

u p p∗  b|u| d x





1 p∗ − p

> 0.

− − − 2− p  − ) = 0, t0,b u ∈ N0,b and (t0,b ) h (t0,b ) = ( p − p ∗ ) u p < 0. Then h  (t0,b − − such that t0,b u ∈ N0,b and Hence there is a unique t0,b

max J0,b (tu) = t≥0

− J0,b (t0,b u)

1 p−N = b p N



∗

u p  p∗  |u| d x

 N−p p

.

(ii) For μ ∈ (0, 1), we have     q    f |u|q d x  ≤ | f |q ∗ S − p u q     

p p p − q q−q p q q −q μ (| f |q ∗ S p ) p−q + (μ p u q ) q p p q p p − q q−q p qμ − μ (| f |q ∗ S p ) p−q +

u p . = p p

≤

Then by part (i),   − J f,1 (u) = max J f,1 (tu) ≥ J f,1 t 1 u 0, 1−μ t≥0  p 1 p − q q−q p 1−μ − ∗ −q p

t 1 u − ∗ |t − 1 u| p d x − μ (| f |q ∗ S p ) p−q ≥ 0, 1−μ 0, 1−μ p p pq    p p − q q−q p −q − μ (| f |q ∗ S p ) p−q = (1 − μ)J0, 1 t 1 u − 0, 1−μ 1−μ pq

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H. Fan, X. Liu

1 = (1 − μ) N



N p

∗

u p  p∗  |u| d x

N

− = (1 − μ) p J0,1 (t0,1 u) −

 N−p p

−

p p − q q−q p −q μ (| f |q ∗ S p ) p−q pq

p p − q q−q p −q μ (| f |q ∗ S p ) p−q . pq

Similarly, we obtain ⎧ ⎨1 + μ

⎫ ⎬



p 1 p − q q−q p ∗ −q μ (| f |q ∗ S p ) p−q |tu| p d x + ∗ ⎭ t≥0 ⎩ p p pq    p p − q q−q p −q μ (| f |q ∗ S p ) p−q = (1 + μ)J0, 1 t − 1 u + 0, 1+μ 1+μ pq q p N p − q −q − μ q− p (| f |q ∗ S p ) p−q . = (1 + μ) p J0,1 (t0,1 u) + pq

J f,1 (u) ≤ max

tu p −

N

(iii) It follows from part (ii) and the fact that α0,1 = N1 S p , where α0,1 =  (u), u = 0}. inf u∈N0,1 J0,1 (u) and N0,1 = {u ∈ X \{0}; J0,1   Lemma 4.2 Suppose {u k } is a non-negative (P S)β -sequence of J f,1 with 1 Np 2 N S < β < S p − ε0 N N for some ε0 > 0. Then there exists 0 > 0 such that for 0 < | f |q ∗ < 0 , {u k } has a convergent subsequence. Proof Since {u k } is bounded in X , we may assume that u k u 0 weakly in X as k → ∞ and u 0 ∈ X . Suppose otherwise, that (P S)β -condition dose not hold. Then by Lemma 2.1 and Remark 2.1, β = J f,1 (u 0 ) + ι

1 Np S , N

where ι ∈ N. Thus 1 N 2 N 1 Np S < J f,1 (u 0 ) + ι S p < S p − ε0 . N N N We assume 0 < | f |q ∗ < have

q p 1 .

Then if u 0 ∈ N + f,1 , by the proof of Lemma 2.4, we

α +f,1 ≤ J f,1 (u 0 ) < 0.

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Multiple positive solutions to a class of quasi-linear elliptic equations

Thus ι ≥ 2. On the other hand, by Lemma 3.2 (iv) and ε0 > 0, we obtain that there exists 0 < λ0 < qp 1 such that for 0 < | f |q ∗ < λ0 , J f,1 (u 0 ) + ι

1 Np 2 N 2 N S ≥ α +f,1 + S p > S p − ε0 , N N N

when ι ≥ 2. This contradicts to our assumption. Hence we obtain that u 0 ∈ N − f,1 and J f,1 (u 0 ) ≥ α −f,1 > 0. However, by Lemma 4.1 (iii), we know that there exists 0 ∈ (0, λ0 ) such that for 0 < | f |q ∗ < 0 , J f,1 (u 0 ) + ι

1 Np 1 N 2 N S ≥ α −f,1 + S p > S p − ε0 , N N N

when ι ≥ 1. This still contradicts to our assumption. Therefore ι = 0 and by Lemma 2.1, we complete the proof.   It is well known that the best Sobolev constant ⎫ ⎧ ⎪ ⎪ ⎬ ⎨ p 1, p N |∇u| d x; u ∈ D (R ), |u| p∗ = 1 S = inf ⎪ ⎪ ⎭ ⎩N R

is attained by the functions  yδ (x) = δ

p p−1

 N

N−p p−1

 p−1  Np−2 p

p

p

δ p−1 + |x| p−1

! p−N p

for any δ > 0. Moreover, the functions yδ (x) are the only positive radial solutions of − p u = |u| p

∗ −2

u

in R N . Hence, ⎛ ⎜ S⎝

 RN

⎞ ∗ ⎟ |yδ | p d x ⎠

p p∗

 =

 |∇ yδ | p d x =

RN

∗

N

|yδ | p d x = S p . RN

Let ϕρ ∈ C0∞ (R N ) be a radially symmetric function such that ⎧ 3ρ ⎪ ⎨ 0, 0 ≤ |x| ≤ 2 , 1 , ϕρ (x) = 1, 2ρ ≤ |x| ≤ 2ρ ⎪ 3 ⎩ 0, |x| ≥ 4ρ

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H. Fan, X. Liu

and  u eδ (x)

= δ

p p−1

 N

N−p p−1

 p−1  Np−2 p

p

p

δ p−1 + |x − (1 − δ)e| p−1

! p−N p

,

where e ∈ S = {x ∈ R N ; |x| = 1} and 0 < δ < 1. Set ωρδ,e (x)

ωρδ,e (x) = ϕρ (x)u eδ (x) and ϑρδ,e (x) =

|ωρδ,e (x)| p∗

∈ X.

Lemma 4.3 ϑρδ,e (x) p → S as δ → 0 uniformly in e ∈ S. Proof It is sufficient to show p∗

N

N

|ωρδ,e | p∗ → S p , ωρδ,e p → S p

(4.1)

uniformly in e ∈ S as δ → 0. In order to show (4.1), we estimate p∗



p∗

|u eδ (x)| p∗ − |ωρδ,e (x)| p∗ =

∗

∗

(1 − ϕρ (x) p )|u eδ (x)| p d x RN



≤c

N

δ p−1 p

B2ρ

p

δ p−1 + |x − (1 − δ)e| p−1 

N

δ p−1

+c R N \B

!N d x

p

p

δ p−1 + |x − (1 − δ)e| p−1

1 2ρ

= o(1) as δ → 0 for all e ∈ S. Similarly, 

|∇ωρδ,e (x)| p d x −



 |∇u eδ (x)| p d x

RN



=



|∇ϕρ (x)u eδ (x) + ϕρ (x)∇u eδ (x)| p d x 





≤

|ϕρ (x) p − 1||∇u eδ (x)| p d x + c 



+c RN

123

−

|∇u eδ (x)| p d x

RN

|∇ϕρ (x)u eδ (x)| p d x

RN

|ϕρ (x)∇u eδ (x)| p−1 |∇ϕρ (x)u eδ (x)|d x

!N d x

Multiple positive solutions to a class of quasi-linear elliptic equations





≤ 

|∇u eδ (x)| p d x + cρ p



R N \B

+ cρ



−p

B

B2ρ

1 2ρ

|u eδ (x)| p d x 3 4ρ

\B

1 2ρ

⎛  |u eδ (x)| p d x

B2ρ

⎜ ⎜ ⎜ +c⎜ ⎜ ⎝

⎞ ⎟ ⎟

 

R N \B



1 2ρ

⎟ |∇u eδ (x)| p d x ⎟ ⎟ ⎠

B2ρ

⎛ ⎜ ⎜ + c ⎜ρ p ⎝

p p−1

⎞1  B

3 4ρ

\B

p



|u eδ (x)| p d x + ρ − p

B2ρ

1 2ρ

⎟ ⎟ |u eδ (x)| p d x ⎟ → 0 ⎠

  ∗ as δ → 0 uniformly in e ∈ S. Note the fact R N |∇u eδ (x)| p d x = R N |u eδ (x)| p d x = N   S p , we complete the proof. Lemma 4.4 There is a ρ0 > 0 such that for 0 < ρ < ρ0 , sup

0<δ<1,e∈S

p

ϑρδ,e (x) p < 2 N S.

Proof The assertion can be verified similarly as in Lemma 4.3.

 

Let us define   : V → R N , (u) =

∗

x|u| p d x, RN

where the function u is extended to R N by setting u = 0 outside . Set A0 = {u ∈ V ; (u) = 0}. Lemma 4.5 Let c0 = inf u∈A0 u p , then S < c0 . Proof Apparently, c0 ≥ S. To show S < c0 , we argue by contradiction. Suppose that c0 = S, then there is a sequence {vk } ⊂ X such that |vk | p∗ = 1, (vk ) = 0 and ∇vk p → S as k → ∞. So the sequence u k = S

N−p p2

J0,1 (u k ) →

vk satisfies 1 Np S N

and

 J0,1 (u k ) → 0 as k → ∞.

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H. Fan, X. Liu

By a result in Struwe [22], there exist {εk }, {xk } ∈ , εk → 0 as k → ∞, for the functions vk = S

− N −2 p p

u εk ,xk (x), where

 u εk ,xk (x) = εk

p p−1

 N

N−p p−1

 p−1  Np−2 p

p

p

εk p−1 + |x − xk | p−1

! p−N p

,

we have (vk ) = xk . From (H ), we know xk = 0 and this contradicts to our assumption that (vk ) = 0.   Hence S < c0 . Lemma 4.6 There holds limδ→0 (ϑρδ,e (x)) = e. Proof Since 

∗

(x − e)|ωρδ,e (x)| p d x

RN



∗

=

(x − e)|u eδ (x)| p d x RN



+

∗

∗

(x − e)(ϕρ (x) p − 1)|u eδ (x)| p d x

RN



= ((1 − δ)e − e) +

∗

∗

(x − e)(ϕρ (x) p − 1)|u eδ (x)| p d x → 0

RN

as δ → 0. Thus 

(ϑρδ,e (x)) − e

∗

δ,e p dx N (x − e)|ωρ (x)| = R →0 δ,e p∗ R N |ωρ (x)| d x

as δ → 0.

 

To pursue further, we need the following result in Willem [28]. Lemma 4.7 Let K be a compact metric space, K 0 ⊂ K be a closed set, X be a Banach space and χ ∈ C(K 0 , X ). Let us define the complete metric space M by M = {k ∈ C(K , X ); k(s) = χ (s) if s ∈ K 0 } with the usual distance. Let ϕ ∈ C 1 (X, R) and let us define c = inf sup ϕ(k(s)),  c = sup ϕ. k∈M s∈K

123

χ (K 0 )

Multiple positive solutions to a class of quasi-linear elliptic equations

If c >  c, then for each ε > 0 and each k ∈ M satisfying sup ϕ(k(s)) ≤ c + ε, s∈K

there exists v ∈ X such that 1

1

c − ε ≤ ϕ(v) ≤ sup ϕ(k(s)), dist (v, k(K )) ≤ ε 2 , ϕ  (v) ≤ ε 2 . s∈K

Let r0 =

1 2

− δ0 and 

B r0 =

      1 1 1 − δ e ∈ R N ;  − δ e ≤ r0 , 0 < δ ≤ , 2 2 2

where δ0 > 0 is small enough. Then we set F = {h ∈ C(B r0 , V ); h|∂ B r = ϑρδ,e (x)} 0

and c1 = inf

h∈F

sup !

1 2 −δ

e∈B r0

Lemma 4.8 For h ∈ F, we have h(B r0 )

    p   1 h −δ e   .  2 

A0 = ∅.

Proof It is equivalent to show that for any h ∈ F, there exist such that     1 # −δ # e = 0.  h 2 Set θ

"" 1 2

(4.2)

"1 2

$ −# δ # e ∈ B r0 , # e∈S

$ $ " "" $ $$ − δ e ≡  h 21 − δ e , we claim that d(θ, Br0 , 0) = d(I, Br0 , 0) = 0.

In fact, if  h

"1 2

$ − δ e ∈ ∂ Br0 , we have

       1 1 − δ e (x) = ϑρδ,e (x),  h − δ e (x) = (ϑρδ,e (x)) = e + o(1) 2 2

as δ → 0. Then we consider the homotopy        1 1 G t, − δ e = (1 − t)θ − δ e + t I, 0 ≤ t ≤ 1. 2 2

123

H. Fan, X. Liu

If

"1 2

$ − δ e ∈ ∂ Br0 ,       1 1 G t, − δ e = (1 − t)(e + o(1)) + t − δ0 e = 0 2 2

as δ → 0. So the claim is proved and there exist

"1 2

$ −# δ # e ∈ B r0 , # e ∈ S such that

    1 # −δ # e = 0.  h 2  

We complete the proof. By Lemmas 4.5 and 4.8, we obtain S < c0 ≤ c1 . On the other hand, it follows from Lemma 4.4 and (4.2) that for 0 < ρ < ρ0 , c1 ≤

ϑρδ,e (x) p ≤

sup !

1 2 −δ

e∈B r0

p

sup

0<δ<1,e∈S

ϑρδ,e (x) p < 2 N S.

Thus p

S < c0 ≤ c1 < 2 N S. Then we define γ f,1 = inf

h∈F

γ0,1 = inf

h∈F

sup !

1 2 −δ

e∈B r0

sup !

1 2 −δ

e∈B r0

    1 −δ e , Jf,1 h 2     1  −δ e . J0,1 h 2

Lemma 4.9 We have 1 Np 1 N 2 N S < c0p ≤ γ0,1 < S p . N N N Proof Since ⎞ ∗p∗ ⎛ p −p  1 ⎝ p  ⎠ J0,1 (u) = max J0,1 (tu) = |∇u| d x t≥0,u∈V N 

123

Multiple positive solutions to a class of quasi-linear elliptic equations

and inf

h∈F

sup !

1 2 −δ

e∈B r0

    1 1  J0,1 h −δ e = inf 2 N h∈F

sup !

1 2 −δ

e∈B r0

    N   1 h −δ e    , 2 p

we obtain the assertion from (4.2) and the fact S < c0 ≤ c1 < 2 N S.

 

Lemma 4.10 1 N Jf,1 (ϑρδ,e (x)) = S p + o(1) N as δ → 0. Proof It is easy to obtain ϑρδ,e (x) 0 weakly in X as δ → 0. Solving d J f,1 (tϑρδ,e (x)) dt 

we see that





=

|∇ωρδ,e (x)| p d x t p−1  δ,e p |ωρ (x)| p∗

 −t

p ∗ −1

−t

q−1 

f |ωρδ,e (x)|q d x

|ωρδ,e (x)| p∗ q

= 0,

f |ωρδ,e (x)|q d x → 0 and t − (ϑρδ,e ) = |ωρδ,e | p∗ + o(1) as δ → 0.

As a result, 1 N Jf,1 (ϑρδ,e (x)) = J f,1 (t − (ϑρδ,e )ϑρδ,e ) = S p + o(1) N

as δ → 0.  

Lemma 4.11 There exists 2 > 0 such that for 0 < | f |q ∗ < 2 , we have 1 1 Np S < N 2N

  N N 2 N c0p + S p < γ f,1 < S p − ε0 N

for some ε0 > 0. Consequently, there exists a positive solution u of (E f,1 ) such that J f,1 (u) = γ f,1 . Proof For u ∈ V , by Lemma 4.1 we obtain p N p − q q−q p −q (1 − μ) p J0,1 (u) − μ (| f |q ∗ S p ) p−q pq p N p − q q−q p −q μ (| f |q ∗ S p ) p−q . ≤ Jf,1 (u) ≤ (1 + μ) p J0,1 (u) + pq

123

H. Fan, X. Liu

Thus p N p − q q−q p −q μ (| f |q ∗ S p ) p−q J0,1 (u) + ((1 − μ) p − 1) J0,1 (u) − pq p N p − q q−q p −q μ (| f |q ∗ S p ) p−q . ≤ Jf,1 (u) ≤ J0,1 (u) + ((1 + μ) p − 1) J0,1 (u) + pq

Hence, for any ε > 0, there exist μ(ε), (ε) > 0 such that if 0 < | f |q ∗ < (ε), we have γ0,1 − ε < γ f,1 < γ0,1 + ε. Fix a small 0 < ε0 <

2 N

S

N p

−γ0,1 . 2

Since

1 Np 1 N 2 N S < c0p ≤ γ0,1 < S p , N N N there exists 0 < 2 < 0 such that if 0 < | f |q ∗ < 2 , we get 1 Np 1 S < N 2N

  N N 2 N c0p + S p < γ f,1 < S p − ε0 . N

By Lemma 4.10, Jf,1 (ϑρδ,e (x)) =

1 N

N

S p + o(1) as δ → 0. Therefore

1 N γ f,1 > Jf,1 (ϑρδ,e (x)) = S p + o(1) N for δ small enough. Applying Lemmas 4.2 and 4.7, we see that γ f,1 is a critical value of Jf,1 . Standard argument show that there is a positive solution u of (E f,1 ) such that   J f,1 (u) = γ f,1 . Proof of Theorem 1.3 We complete the proof by Lemmas 3.2, 3.3, 4.11 and the fact −∞ < α +f,1 < 0 < α −f,1 < if 0 < | f |q ∗ < 2 .

1 2N

  N N 2 N c0p + S p < γ f,1 < S p , N  

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