Existence and concentration of ground states to a quasilinear problem with competing potentials

Nonlinear Analysis 102 (2014) 120–132

Contents lists available at ScienceDirect

Nonlinear Analysis journal homepage: www.elsevier.com/locate/na

Existence and concentration of ground states to a quasilinear problem with competing potentials✩ Wenbo Wang, Xianyong Yang, Fukun Zhao ∗ Department of Mathematics, Yunnan Normal University, Kunming 650092, Yunnan, PR China

article

info

abstract

Article history: Received 6 October 2013 Accepted 28 January 2014 Communicated by S. Carl

In this paper, we are concerned with the existence and concentration behavior of ground states for the following quasilinear problem with competing potentials

1

−ε 2 ∆u + V (x)u − ε2 ∆(u2 )u = P (x)|u|p−1 u + Q (x)|u|q−1 u, 2

Keywords: Quasilinear equation Concentration Competing potential Nehari manifold

where 3 < q < p < 22∗ − 1, 2∗ is the Sobolev critical exponent, V (x) and P (x) are positive and Q (x) may be sign-changing. We show the existence of the ground states via the Nehari manifold method for ε > 0, and these ground states ‘‘concentrate’’ at a global minimum point of the least energy function C (s) as ε → 0+ . © 2014 Elsevier Ltd. All rights reserved.

1. Introduction In this paper, we are concerned with the following quasilinear Schrödinger equation

∂ψ = − h¯ 2 ∆ψ + W (x)ψ − h˜ (|ψ|2 )ψ − h¯ 2 κ ∆ρ(|ψ|2 )ρ ′ (|ψ|2 )ψ, (1.1) ∂t N N + where ψ : R × R → C, h¯ is the Planck constant, W : R → R is a given potential, κ is a real constant, h and ρ : R → R ih¯

are real functions of essentially pure power forms. Such equations arise in various branches of mathematical physics and they have been the object of extensive study in recent years. When one consider the case where ρ(s) = s, κ = 21 and look for the standing wave solutions of type ψ = exp(−iEt /h¯ )u(x), then (1.1) leads to the following equation of elliptic type 1

− ε2 ∆u + V (x)u − ε 2 ∆(u2 )u = h(u), 2

(1.2)

where ε = h¯ , V (x) = W (x) − E and h(u) = h˜ (|u|2 )u. One of the main difficulties of (1.2) is that there is no suitable space on which the energy functional is well defined and belongs to C 1 -class except for N = 1 (see [1]), since the quasilinear and non-convex term − 21 ∆(u2 )u appears. The first existence result involving variational methods due to [1] for the case where ε = 1, N = 1 or V (x) is radially symmetrical for high dimensions by using a constrained minimization argument (see also [2] for the more general case). After then, there are some ideas and approaches were developed to overcome the difficulty. See [3] for a Nehari manifold argument. Transforming the quasilinear problems into semilinear problems by a change of variables is an effective way to deal with (1.2), see [4] for an Orlicz space framework and [5] for a Sobolev space frame. But this method does not work for the general ✩ Supported by NSFC (11101355 and 11361078), Key Project of Chinese Ministry of Education (212162) and NSFY of Yunnan Province (2011CI020) and China Scholarship Council. ∗ Corresponding author. Tel.: +86 15368160621. E-mail address: [email protected] (F. Zhao).

http://dx.doi.org/10.1016/j.na.2014.01.025 0362-546X/© 2014 Elsevier Ltd. All rights reserved.

W. Wang et al. / Nonlinear Analysis 102 (2014) 120–132

121

quasilinear problems (see e.g. [6]). Recently, a perturbation method was developed in [7] (see also [6] for general quasilinear problems). The main idea is adding a regularizing term to recover the smoothness of the energy functional, so the standard minimax theory can be applied. Differ from the semilinear problems, another feather of the quasilinear problem (1.2) is that the critical exponent is not 2∗ but 22∗ (see [3]), where 2∗ is the usual Sobolev exponent. The critical problems similar to (1.2) was considered in [8,9], see [6,10] for more general quasilinear critical problems. The existence of infinitely many solutions was obtained in [11,12] for the periodic quasilinear problems (in [12,13] the authors considered the more general case). There are some recent works which considered the concentration property of solutions as ε → 0. In [14], assuming the potential V is locally Hölder continuous, bounded from below by a positive constant and admitting a bounded domain Ω ⊂ R2 with infΩ V < min∂ Ω V , the authors showed that there are positive ground states for a 2-dimensional and critical problem concentrating around a local minimum point of V as ε → 0 and with exponential decay. Similar results were proved in [15,16] for the case where dimension N ≥ 3, and see [17] for the 1-dimensional case. The multiplicity of semiclassical solutions were established in [18] for the subcritical case and in [19] for the critical case. In this paper, motivated by Wang and Zeng [20], we consider a quasilinear problem with competing potentials. More precisely, we are devoted to study the existence and concentration of positive ground states to the following quasilinear equation 1

− ε 2 ∆u + V (x)u − ε 2 ∆(u2 )u = P (x)|u|p−1 u + Q (x)|u|q−1 u,

( Pε ) 2 where 3 < q < p < 22∗ − 1, V and P are continuous and positive functions, Q is a bounded and continuous function. A simple model of (Pε ) is the case of Q = 0, V has a global minimum and P has a global maximum, there is possibly a competition between V and P in the following sense: V (x) would attract ground states to its minimum point but P (x) would attract ground states to its maximum point. The competition will become more complex provided Q ̸= 0, and this makes finding the concentration points become more delicate. To the best of our knowledge, there is no work concerning this case. When P (x) ≡ λ > 0 and Q ≡ 0, a result in [4, Theorem 1.1] implies the existence of positive ground states of (Pε ) for any ε > 0. But this result cannot be applied directly to (Pε ) when Q ̸= 0. However, the Nehari manifold is still well defined, even Q is sign-changing (see Lemma 2.1). So we follow [4] with some modifications to obtain a positive ground state for (Pε ) via the Nehari manifold method for ε > 0, and then we find that these ground states concentrate at a global minimum point of the least energy function C (s) as ε → 0+ via a concentration-compactness argument similar to [20]. Throughout this paper, we always make the following assumptions: (V ) V (x) ∈ C 1 (RN , R) and V0 := infx∈RN V (x) > 0. (P ) P (x) ∈ C 1 (RN , R) ∩ L∞ (RN , R) is a positive function. (Q ) Q (x) ∈ C 1 (RN , R) ∩ L∞ (RN , R). To state our main result, we need two auxiliary problems. For each s ∈ RN , consider the following problems with parameter s ∈ RN 1

− ∆u + V (s)u − ∆(u2 )u = P (s)|u|p−1 u + Q (s)|u|q−1 u.

2 Denote the corresponding energy functional by I s and the corresponding least energy by

(Ps )

C (s) := c (V (s), P (s), Q (s)) = inf{I s (u)|u is a nontrivial solution of (Ps )}. Also, we consider a ‘‘limit’’ problem 1

− ∆u + V∞ u − ∆(u2 )u = P∞ |u|p−1 u + Q∞ |u|q−1 u,

2 and denote the corresponding least energy by c∞ = c (V∞ , P∞ , Q∞ ), where V∞ := lim inf V (x),

P∞ := lim sup P (x)

|x|→∞

|x|→∞

(P∞ )

and Q∞ := lim sup Q (x). |x|→∞

Our main result is the following. Theorem 1.1. Suppose that (V ), (P ) and (Q ) are satisfied, 3 < q < p < 22∗ − 1. If inf C (s) < c∞ ,

(1.6)

s∈RN

then there is an ε0 > 0 such that (1) (Pε ) possesses a positive ground state solution uε for all ε ∈ (0, ε0 ). (2) uε possesses at most one local (hence global) maximum point xε in RN such that lim C (xε ) = inf C (s).

ε→0+

s∈RN

(3) there exist C1 , C2 > 0 such that uε (x) ≤ C1 e−C2 |

x−xε

ε

|

.

122

W. Wang et al. / Nonlinear Analysis 102 (2014) 120–132

Remark 1.2. In fact, (1.6) will hold in some cases. For example, suppose that there exists a point s0 ∈ RN such that V∞ ≥ V (s0 ),

K∞ ≤ K (s0 ) and Q∞ ≤ Q (s0 ),

with one of the above inequalities being strict. Then it is easy to check that (1.6) hold. See [20, Corollary 2.8] for more cases. 2. Existence of ground states 2.1. Preliminaries Set

 X :=

u ∈ H 1 (RN ) :

 RN

V (x)u2 dx < ∞,

In fact, X = {u ∈ H (R ) : u ∈ H (R )} and that ε = 1, then Pε becomes 1

N

2

1

N





u2 |∇ u|2 dx < ∞ .

RN



RN

V (x)u2 dx < ∞ (see [21]). Without loss of generality, we may assume

1

− ∆u + V (x)u − ∆(u2 )u = P (x)|u|p−1 u + Q (x)|u|q−1 u.

2 A function u ∈ X is called a weak of (2.1), if for all φ ∈ C0∞ (RN ), it holds



(2.1)

  (1 + u2 )∇ u∇φ dx + |∇ u|2 uφ dx + V (x)uφ dx RN RN RN   = P (x)|u|p−1 uφ dx + Q (x)|u|q−1 uφ dx. RN

RN

We formally formulated problem (2.1) in a variational way as     1 1 1 1 (1 + u2 )|∇ u|2 dx + V (x)u2 dx − P (x)|u|p+1 dx − Q (x)|u|q+1 dx. I ( u) = 2 RN 2 RN p + 1 RN q + 1 RN Given u ∈ X and φ ∈ C0∞ (RN ), recall the derivative of I in the direction φ at u is defined as I (u + t φ) − I (u) ⟨I ′ (u), φ⟩ = lim t →0 t    2 = (1 + u )∇ u∇φ dx + |∇ u|2 uφ dx + V (x)uφ dx N RN RN  R − P (x)|u|p−1 uφ dx − Q (x)|u|q−1 uφ dx. RN

RN

Let

N := {u ∈ X \ {0}|γ (u) = 0}

(2.2)

be the Nehari manifold, where

γ (u) = ⟨I (u), u⟩ = ′



(1 + 2u )|∇ u| dx + 2

RN



2

P (x)|u|

p+1

− RN

 RN

 dx − RN

V (x)u2 dx

Q (x)|u|q+1 dx,

u ∈ X.

(2.3)

Set c = inf I (u). u∈N

Lemma 2.1. Suppose that u ̸= 0 and 3 < q < p < 22∗ − 1. Then there is a unique t = t (u) > 0 such that tu ∈ N and I (ru) < I (tu) if r ̸= t. Proof. Set f (t ) := I (tu) =

At 2 2

+

Bt 4 2

−

Ct p+1

−

p+1

Dt q+1 q+1

,

where

 A= RN

 C = RN

(|∇ u|2 + V (x)u2 )dx, P (x)|u|p+1 dx and

 B= RN

 D= RN

u2 |∇ u|2 dx,

Q (x)|u|q+1 dx.

W. Wang et al. / Nonlinear Analysis 102 (2014) 120–132

123

Then f ′ (t ) = At + 2Bt 3 − Ct p − Dt q , and hence f ′ (t ) > 0 for t small and f ′ (t ) < 0 for t large. Hence there is t = t (u) > 0 such that f ′ (t ) = 0. Thus

γ (tu) = tf ′ (t ) = 0, which implies tu ∈ N . The uniqueness follows from the fact that the equation A + 2Bt 2 − Ct p−1 − Dt q−1 = 0 has a unique positive solution.



Lemma 2.2. Suppose {un } ⊂ X satisfy limn→∞ RN P (x)|un |p+1 dx = l > 0, limn→∞ RN Q (x)|un |q+1 dx = d, l + d > 0 and limn→∞ γ (un ) = 0. Then there is a sequence {tn } such that tn un ∈ N and limn→∞ tn = 1.





Proof. The proof is similar to [4]. Here we only check that limn→∞ tn = 1. Set

 an = RN

 ln = RN



(|∇ un |2 + V (x)u2n )dx, P (x)|un |p+1 dx,

bn =

 dn = RN

RN

u2n |∇ un |2 dx

Q (x)|un |q+1 dx.

We have ln → l > 0. Passing to a subsequence, an → a, bn → b, dn → d. Since γ (un ) = an + 2bn − ln − dn → 0, we have a + 2b − l − d = 0. We claim that a > 0. Suppose to the contrary that an → 0. By the interpolation inequality and the (p−1)(N −2) Sobolev inequality, we have with θ = 2(N +2)

 | un |

p+1

1−θ 

 dx ≤ C1

RN

RN

u2n dx

| un |

≤ C2 RN

u2n dx

θ dx

RN

1−θ 



4N N −2

RN

u2n

2

 Nθ−N2

|∇ un | dx

,

(2.4)

which implies that ln → 0, a contradiction. By Lemma 2.1, for n sufficiently large, we can find tn > 0 such that

γ (tn un ) = an tn2 + 2bn tn4 − ln tnp+1 − dn tnq+1 = 0. A simple estimate shows that 0 < T1 ≤ tn ≤ T2 < ∞. So let tn → t∗ . Since 1 is the unique positive solution of the equation at 2 + 2t 4 − lt p+1 − dt q+1 = 0, it is easy to obtain t∗ = 1. If it has another positive solution t0 ̸= 1, we have 2b ≥ l + d. Since a + 2b − l − d = 0 and a > 0, we have 2b < l + d. It is a contradiction.



Similar to [4, Lemma 2.5], there holds Lemma 2.3. Suppose that u ∈ N and I (u) = c, then u is a weak solution of (2.1). Set c ∗ = inf{I (u)|u is a nontrivial solution of (2.1)} and c ∗∗ =

inf max I (t v).

v∈X \{0} t ≥0

The following results are essentially known (see e.g. [22,20]). Lemma 2.4. c = c ∗ = c ∗∗ > 0. Proof. For any u ∈ X \ {0}, by Lemma 2.1, the ray Rt = {tu : t ≥ 0} intersects the Nehari manifold N once and only once at t (u)u, where t (u) is given in Lemma 2.1. This implies that c = c ∗∗ . Next, we show that c = c ∗ . Obviously, c ≤ c ∗ . On the other hand, when u is a nontrivial solution of (2.1), it holds that I (u) ≤ maxt ≥0 I (tu), so c ∗ ≤ c ∗∗ . It remains to show that c > 0. Denote

ρ 2 = ρ 2 (u) =

 RN

(1 + u2 )|∇ u|2 dx +

 RN

V (x)u2 dx.

124

W. Wang et al. / Nonlinear Analysis 102 (2014) 120–132

By the Hölder inequality and the Sobolev inequality we have with θ1 =



P (x)|u|

p+1

RN



1−θ1 

2

dx ≤ C1

|un |

u dx RN



θ1 dx

RN

1−θ1 

2

≤ C2

4N N −2

(p−1)(N −2) , 2(N +2)

2

 Nθ1−N2

2

u |∇ u| dx

u dx RN

RN N

≤ C2 ρ 2(1−θ1 ) ρ 2θ1 N −2 = C2 ρ 2+ similarly, with θ2 =

 RN

2(p−1) N +2

,

(q−1)(N −2) , 2(N +2)

Q (x)|u|q+1 dx ≤ C3 ρ

2(q−1) 2+ N +2

.

Then I ( u) =

≥

1 2 1 2

 RN

(1 + u2 )|∇ u|2 dx +

ρ 2 − C2 ρ 2+

2(p−1) N +2

1



2

− C3 ρ 2+

RN

V (x)u2 dx −

2(q−1) N +2

1 p+1

 RN

P (x)|u|p+1 dx −

1 q+1

 RN

Q (x)|u|q+1 dx

.

Choose ρ > 0 so small that 1 2

ρ 2 − C2 ρ 2+

2(p−1) N +2

− C3 ρ 2+

2(q−1) N +2

≥

1 4

ρ2.

Take λ > 0 small such that ρ(λu) satisfies the above estimate. By Lemma 2.1, for u ∈ N , I (u) ≥ I (λu) ≥ that c > 0. 

1 4

ρ 2 , which yields

2.2. Existence results in bounded domains In order to obtain existence results, firstly, we consider the corresponding problem in bonded domains since the Sobolev embedding in not compact in the whole space. Define cR = inf I (u), u∈NR

where NR := N ∩ H01 (BR ) and BR is the ball in RN centered in 0 with radius R. Lemma 2.5. cR is decreasing in R and limR→∞ cR = c. Proof. Obviously, cR is decreasing in R, and cR ≥ c. By the definition of c, for ϵ > 0, there exists u ∈ N such that I (u) < c +ϵ . Let uR = ηR u, where ηR is a cut-off function satisfies

ηR =

  =1,

|x| ≤

 =0,

|x| ≥

1 4 3 4

R, R.

We have



P (x)|uR |p+1 dx = BR

 RN

P (x)|uR |p+1 dx →

 RN

P (x)|u|p+1 dx > 0

and



Q (x)|uR |q+1 dx =

 RN

BR

Q (x)|uR |q+1 dx →

 RN

Q (x)|u|q+1 dx

as R → ∞. By Lemmas 2.1 and 2.2, there exists tR > 0 such that tR uR ∈ N and tR → 1 as R → ∞, and c ≤ cR ≤ I (tR uR )

≤ I (tR u) + C

 RN \B R 4

[(1 + u2 )|∇ u|2 + V (x)u2 + P (x)|u|p+1 + Q (x)|u|q+1 ]dx

W. Wang et al. / Nonlinear Analysis 102 (2014) 120–132

125

≤ I (u) + o(1) < c + ϵ + o(1), where o(1) → 0 as R → ∞. Thus, limR→∞ cR = c.



Lemma 2.6. Suppose {un } ⊂ NR satisfied cR = limn→∞ I (un ). Then

 BR

|un |p+1 dx has a positive lower bound with respect to n.

Proof. Suppose the conclusion is false. Passing to a subsequence if necessary, we have that Lp+1 (BR ) and Lq+1 (BR ) norms of un converge to 0 as n → ∞. Now we are led to





1 cR = lim I (un ) − ⟨IR′ (un ), un ⟩ n→∞ 2

= lim

n→∞

   1 1 − u2n |∇ un |2 dx + − 2

2

BR



1 p+1

P (x)|un |p+1 dx +



BR

which implies cR ≤ 0. This is impossible because cR ≥ c > 0.

1 2

−

1





q+1

Q (x)|un |q+1 dx , BR



Lemma 2.7. cR is achieved. Proof. Let {un } ⊂ NR be a minimizing sequence, i.e. I (un ) → cR . We claim that





V (x)|un | dx, 2

BR

u2n

2



|∇ un |2 dx are bounded.

|∇ un | dx and

BR

BR

In fact, it follows from γ (un ) = 0 that I (un ) = I (un ) −

1

⟨I ′ (un ), un ⟩     1 1 1 2 2 2 = (|∇ un | + V (x)un )dx + u2n |∇ un |2 dx − − 2 q+1 2 q+1 BR BR   1 1 P (x)|un |p+1 dx, + − q+1 p+1 BR q+1

(2.5)

which implies the claim since limn→∞ I (un ) → cR . Thus {un } is bounded in H01 (BR ). Without loss of generality, we assume un ⇀ u in H01 (BR ). By the Sobolev embedding theorem (see e.g. [23]), we may assume un → u in Lp+1 (BR ) and Lq+1 (BR ). It follows from Lemma 2.6 that u ̸≡ 0. Combining with Lemma 2.1, there is t > 0 such that γ (tu) = 0 and hence cR ≤ I (tu) ≤ lim inf I (tun ) ≤ lim inf I (un ) = lim I (un ) = cR . n→∞

This completes the proof.

n→∞

n→∞



Similar to Lemma 2.3, we have Lemma 2.8. Suppose that uR ∈ NR and I (uR ) = cR . Then uR is a weak solution of (2.1) in BR . Remark 2.9. Since v ∈ NR implies |v| ∈ NR , without loss of generality, we may assume that the solution uR obtained in Lemma 2.8 is nonnegative. Furthermore, uR > 0. If fact, a Moser iteration argument implies that uR ∈ Ls (RN ) for 2 ≤ s ≤ ∞ (see [6, Proposition 2.2]). It follows from [24, 1,β Corollary 4.23 and Theorem 4.24] that uR ∈ Cloc (RN ) for some β > 0. Consequently, by Schauder estimate (see [25]), 2,β uR ∈ Cloc (RN ). Following [3,5], set vR = f −1 (uR ), where f is defined by

1 f ′ (t ) =  1 + f 2 (t )

on [0, ∞) and f (t ) = −f (−t ) on (−∞, 0].

(2.6)

See [3,5] for more properties of f . Then vR satisfies the equation

−∆vR + c (x)vR = (V (x)f ′ (vR ) + Q − (x)|f (vR )|q )(vR − f (vR )) + (P (x)|f (vR )|p + Q + (x)|f (vR )|q )f ′ (vR ) ≥ 0, where c (x) := V (x)f ′ (vR ) + Q − (x)|f (vR (x))|q > 0 and Q ± := max{±Q , 0}. Applying the strong maximum principle (see [25]), we have vR > 0. Thus uR > 0.

126

W. Wang et al. / Nonlinear Analysis 102 (2014) 120–132

2.3. Existence result Theorem 2.10. Under the assumptions of Theorem 1.1, (2.1) has a positive ground state. Let {Rn } ⊂ R satisfy Rn → ∞ as n → ∞ and un := uRn be the solution obtained in Lemma 2.8. Thus un ∈ NRn and I (un ) = cRn . In view of Lemma 2.5, cRn → c as n → ∞. It follows from (2.5) that

 RN



(1 + u2n )|∇ un |2 dx,

RN

V (x)u2n dx and

 RN

P (x)|un |p+1 dx are all bounded.

Therefore, going to a subsequence if necessary, we assume un ⇀ u in H 1 (RN ), ∇(u2n ) ⇀ ∇(u2 ) in L2 (RN ) and un → u in Ltloc (RN ) for t ∈ [2, 22∗ ). We need to apply the concentration-compactness principle (see [26] or [27]) to ρn (x) := |un (x)|p+1 . In view of Lemma 2.6, we can assume that

 RN

ρn (x)dx → α > 0.

(2.7)

Lemma 2.11. {ρn (x)} is not vanishing. Proof. By Lion’s vanishing lemma (see [26] and [23, Lemma 1.21]), we see that un → 0 in Ls (RN ) for s ∈ (2, 22∗ ) (in our case by interpolation inequality the range of s can be extended), since {u2n } is bounded in H 1 (RN ). Note that 1 ′ ⟨I (un ), un ⟩ 2     1 1 1 1 u2n |∇ un |2 dx + − P (x)|un |p+1 dx + − Q (x)|un |q+1 dx. 2 p+1 2 q+1 RN RN

cn = I (un ) = In (un ) − 1

=−



2

RN

Letting n → ∞, by Lemma 2.5 we have cn → c ≤ 0, which contradicts Lemma 2.4.



Next, we show that dichotomy does not occur. Suppose to the contrary that there exist a subsequence of {ρn }, still denote

{ρn }, β ∈ (0, 1] and {xn } ⊂ RN such that for each ϵ > 0, there exists rϵ > 0 satisfies r ′ ≥ r ≥ rϵ ,  lim inf ρn (x)dx ≥ αβ − ϵ n→∞

(2.8)

Br (xn )

and

 lim inf n→∞

RN \Br ′ (xn )

ρn (x)dx ≥ (1 − β)α − ϵ,

(2.9)

where α is the same as in (2.7). Lemma 2.12. β = 1, i.e., the dichotomy does not occur. Proof. If β < 1, we claim that lim inf I (un ) ≥ 2c . n→∞

In fact, in (2.8) and (2.9), we choose two sequences ϵn → 0 and rn → ∞ as n → ∞. Let φn be a cut-off function such that φn ≡ 1 in Brn (xn ) and φn ≡ 0 in RN \ B2rn (xn ). Write un = φn un + (1 − φn )un =: vn + wn . By (2.8) and (2.9), we have



p+1

|vn | RN

dx ≥ βα − ϵn ,

I (un ) = I (vn ) + I (wn ) + o(1),

 RN

|wn |p+1 dx ≥ (1 − β)α − ϵn ,

γ (vn ) = o(1) and γ (wn ) = o(1). It follows from Lemma 2.2 that there are two sequences {sn } and {tn } with tn → 1 and sn → 1 such that tn vn and sn wn ∈ N . Thus we obtain c = lim inf I (un ) ≥ lim inf I (tn vn ) + lim inf I (sn wn ) ≥ c + c = 2c . n→∞

n→∞

It is impossible since c > 0.

n→∞



Let I∞ be the energy functional corresponding to (P∞ ). Observe that the problem (P∞ ) is invariant under translation and a result similar to Lemma 2.4 holds for c∞ . Thus one can check that, in a standard way, there is a bounded and nonvanishing minimizing sequence for c∞ . So, c∞ is achieved by some u∞ ∈ N∞ up to a translation (see e.g. [2]). Thus I∞ (u∞ ) = c∞ and ′ γ∞ (u∞ ) = ⟨I∞ (u∞ ), u∞ ⟩ = 0. Moreover, similar to Remark 2.9, u∞ > 0.

W. Wang et al. / Nonlinear Analysis 102 (2014) 120–132

127

Proof of Theorem 2.10. By Lemmas 2.11–2.12, the compactness occurs, i.e., there exists a sequence {xn } ⊂ RN such that for any ϵ > 0, there is a r > 0 such that

 B(xn ,r )

|un |p+1 dx ≥ α − ϵ.

(2.10)

Following the same spirits as in the proof of Lemmas 3.1 and 3.4, {xn } is bounded in RN . Therefore, un → u ≥ 0 in Lp+1 (RN ). Similarly, un → u in Lq+1 (RN ). Moreover, (2.10) implies u ̸≡ 0, and hence γ (u) = 0. Therefore, c ≤ I (u) ≤ lim inf I (un ) = lim cRn = c . n→∞

n→∞

It means that c is achieved by u. Similar to Remark 2.9, we have u(x) > 0. Thus we complete the proof.



3. Concentration of ground states In order to study the concentration behavior of the ground state solution uε of (1.2), we shall consider vε (x) = uε (ε x) which is a ground state solution of the following problem 1

− ∆u + V (εx)u − ∆(u2 )u = h(εx, u),

(3.1)

2

where h(x, u) := P (x)|u|p−1 u + Q (x)|u|q−1 u. Our aim is to study the behavior of the points where holds the maximum of vε . Let Iε be the energy functional associated with (3.1) and Nε be the corresponding Nehari manifold and set cε = infv∈Nε Iε (v). Lemma 3.1. There exists C > 0 independent with ε such that cε ≥ C . On the other hand, lim sup cε ≤ inf C (s).

(3.2)

s∈RN

ε→0+

Proof. Since cε ≥ c (inf V , ∥P ∥L∞ , ∥Q ∥L∞ ) > 0, we only need to prove (3.2). We exploit the proof of Lemma 2.2 in [20] (see also [28]). Take a sequence {yk } such that C (yk ) → L := infs∈RN C (s). Let u0 be a ground state of 1

− ∆u + VL u − ∆(u2 )u = PL |u|p−1 u + QL |u|q−1 u,

(3.3)

2

where VL , PL and QL are constants depending on L, which means V (yk ) → VL ,

P (yk ) → PL

and

Q (yk ) → QL

as k → ∞.

Let I be the energy functional associated with (3.3) and u0 be a ground state of (3.3). For any R > 0, take a cut-off function ϕR with ϕR = 1 on BR (0) and ϕR = 0 in BcR+1 . Set vR = ϕR u0 and w(x) = vR (x − yεk ). Then there exists a unique θ > 0 such that θw ∈ Nε and θ → 1 as R → ∞, k → ∞ and ε → 0+ . Since L

cε = inf Iε (v) ≤ Iε (θ w) v∈Nε

  θ p+1 θ q+1 (((1 + w 2 )|∇w|2 + V (εx)w 2 )dx − P (ε x)|w|p+1 dx − Q (ε x)|w|q+1 dx 2 RN p + 1 RN q + 1 RN    1 1 − θ p−1 = θ 2 I L (w) + (V (εx) − VL w 2 )dx + P (ε x)|w|p+1 dx 2 RN p+1 RN     1 − θ q−1 1 1 q +1 p+1 q +1 + Q (ε x)|w| dx + (PL − P (εx))|w| dx + (QL − Q (εx)|w| )dx q+1 p + 1 RN q + 1 RN RN    1 1 − θ p−1 = θ 2 I L (vR ) + (V (εx + yk ) − VL )vR2 )dx + P (ε x + yk )|vR |p+1 dx 2 RN p+1 RN   1 1 − θ q −1 q +1 Q (ε x + yk )|vR | dx + (PL − P (εx + yk ))|vR |p+1 dx + q+1 p + 1 RN RN   1 + (QL − Q (εx + yk ))|vR |q+1 dx . q + 1 RN

=

θ2



Letting R → ∞ and k → ∞ in the above inequality, we have lim sup cε ≤ inf C (s).  ε→0+

s∈RN

128

W. Wang et al. / Nonlinear Analysis 102 (2014) 120–132

Next we shall prove that there exists a sequence {yεk } in RN such that:

(i) most of the ‘‘mass’’ of vεk is contained in a ball (of fixed size) centered at yεk ; (ii) {εk yεk } is bounded. The basic ideal is derived from [20]. Define a measure µε by



   1 1 2 − [|∇vε |2 + V (εx)vε2 ]dx + − vε2 |∇vε |2 dx 2 q+1 2 q+1 Ω Ω   1 1 + − P (ε x)|vε |p+1 dx. q+1 p+1 Ω By Lemma 3.1, passing to a subsequence if necessary, as ε → 0+ µε (Ω ) =

1

µε (RN ) = cε → c˜ ≤ inf C (s).

(3.4)

s∈RN

Using the concentration-compactness lemma again, one of the following three conclusions holds: (1) (Compactness) There exists a sequence {yεk } satisfies the following: for any ϵ > 0, there is a ρ > 0 such that



dµεk ≥ c˜ − ϵ.

(3.5)

Bρ(yε ) k

(2) (Vanishing) There exists a sequence {εk } such that for all ρ > 0,

 lim sup

εk →0+ y∈RN

Bρ (y)

dµεk = 0.

(3) (Dichotomy) There exist a constant 0 < c˜ ′ < c˜ , Rεk → ∞, {yεk } ⊂ RN and two nonnegative measures µ1εk and µ2εk such that 0 ≤ µ1εk + µ2εk ≤ µεk , supp(µ1εk ) ⊂ Bρεk (yεk ),

µ1εk (RN ) → c˜ ′ ,

supp(µ2εk ) ⊂ Bc2ρε (yεk ), k

µ2εk (RN ) → c˜ − c˜ ′ .

Lemma 3.2. Vanishing does not occur. Proof. Suppose to the contrary that there exists a sequence εk → 0 such that for all ρ > 0,

 lim sup

k→∞

Bρ (y)

y∈RN

dµεk = 0.

This and the Sobolev embedding imply that

 lim sup

k→∞

y∈RN

Bρ (y)

vε2k dx = 0.

By Lion’s vanishing lemma (see [26]), we see that vεk → 0 in Ls (RN ) for s ∈ (2, 22∗ ) (in our case by interpolation inequality the range of s can be extended). Note that cεk = Iεk (vεk ) = Iεk (vεk ) −

=−

1 2

 RN

1 ′ ⟨Iεk (vεk ), vεk ⟩

2

vε2k |∇vεk |2 dx +



1 2

−

1 p+1

 RN

P (εk x)|vεk |p+1 dx +

Letting εk → 0+ , we have 0 < lim inf cεk = − εk →0+

1

 lim inf

2 εk →0+

RN

vε2k |∇vεk |2 dx ≤ 0.

Now the conclusion follows from the above contradiction.



Lemma 3.3. Dichotomy does not occur. Proof. Otherwise, take φε = 1 in Bρε (yε ) and φε = 0 in Bc2ρε (yε ). Write

vε = φε vε + (1 − φε )vε := v1ε + v2ε . By a standard argument as in the proof of Lemma 3.3 in [20], we have Iεk (vεk ) = Iεk (v1εk ) + Iεk (v2εk ) + o(1),

γ (v1εk ) = o(1),

γ (v2εk ) = o(1).



1 2

−

1 q+1

 RN

Q (εk x)|vεk |q+1 dx.

W. Wang et al. / Nonlinear Analysis 102 (2014) 120–132

129

By Lemma 2.3, there are two sequences {t1εk } and {t2εk } such that t1εk v1εk , t2εk v2εk ∈ Nεk and limεk →0+ t1εk = 1, limεk →0+ t2εk = 1. Thus we have c˜ = lim Iεk (vεk ) ≥ lim inf Iεk (t1εk v1εk ) + lim inf Iεk (t2εk v2εk ) ≥ 2c˜ . εk →0+

εk →0+

εk →0+

Now the desired conclusion follows from the above contradiction.



Let

wk (x) := vεk (x + yεk ) = uεk (εk x + εk yεk ), where yεk is obtained in (3.5). So wk (x) is a positive ground state solution of 1

− ∆wk + V (εk x + εk yεk )wk − ∆(wk2 )wk = h(εk x + εk yεk , wk ).

(3.6)

2

Lemma 3.4. Suppose c∞ > infs∈RN C (s). Then {εk yεk } is bounded as εk → 0. Proof. Suppose to the contrary that, if necessary going to a subsequence, εk yεk → ∞. According to Lemma 3.1, {cεk } is bounded, so is {wk } in H 1 (RN ). Therefore, passing to a subsequence, we may assume wk ⇀ w0 ≥ 0 in H 1 (RN ), wk → w0 in Lsloc (RN ) for s ∈ [2, 22∗ ). By (3.5), for any ϵ > 0, there is a ρ > 0 such that



1 2

−



1 q+1

[|∇vε |2 + V (εx)vε2 ]dx ≤ µεk (Bcρ (yεk )) < ϵ.

Bcρ

Jointly with the Sobolev embedding, we have

wk → w0 in Ls (RN ) for s ∈ [2, 22∗ ).

(3.7)

Since V , P and Q are C , in a similar way to Lemma 2.3 in [20], C (s) is continuous. So we can choose ϵ > 0 small such that 1

C ϵ := C (V∞ − ϵ, P∞ + ϵ, Q∞ + ϵ) > inf C (s).

(3.8)

s∈RN

Let N ϵ be the corresponding Nehari manifold for the equation 1

−∆u + (V∞ − ϵ)u − ∆(u2 )u = (P∞ + ϵ)|u|p−1 u + (Q∞ + ϵ)|u|−1 u. 2

By Lemma 3.1, we see that



   1 1 |w0 |p+1 dx + − Q∞ |w0 |p+1 dx N N 2 p+1 2 q + 1 R R      1 1 1 1 p+1 ≥ lim − P (εk x + εk yεk )|wk | dx + − Q (εk x + εk yεk )|wk |p+1 2 p+1 2 q+1 εk →0+ RN RN    1 = lim I (wk ) + u2 |∇ u|2 dx 1



1

−



P∞

2

εk →0+

RN

≥ lim cεk ≥ C > 0, εk →0+

which yields that w0 ̸= 0. Thus there is a θ > 0 such that θ w0 ∈ N ϵ . Therefore, ϵ

θ2



θ4



|∇w0 | + (V∞ − ϵ)w + w02 |∇w0 |2 dx 2 RN   t t q+1 − (P∞ + ϵ)|w0 |p+1 dx − (Q∞ + ϵ)|w0 |q+1 dx p + 1 RN q + 1 RN  2  θ θ4 (|∇wk |2 + V (εk x + εk yεk )wk2 )dx + wk2 |∇wk |2 dx ≤ lim inf

C ≤

2

2

RN p+1

2

εk →0+

−

θ

p+1

RN

P (εk x + εk yεk )|wk |p+1 dx −

  ≤ lim inf εk →0+

1

2

2

RN



p+1

2 0 dx

RN

θ

q +1

q+1

(|∇wk |2 + V (εk x + εk yεk )wk2 )dx +

RN



 RN

1 2

Q (εk x + εk yεk )|wk |q+1 dx

 RN

wk2 |∇wk |2 dx

130

W. Wang et al. / Nonlinear Analysis 102 (2014) 120–132



1

−

P (εk x + εk yεk )|wk |p+1 dx −

p + 1 RN = lim inf cεk ≤ inf C (s). εk →0+

1 q+1

 RN



Q (εk x + εk yεk )|wk |q+1 dx

s∈RN

It contradicts (3.8). The desired conclusion follows from this contradiction.



Without loss of generality, we may assume that εk yεk → x0 as k → ∞. By a Moser iteration argument, we see that wk → w0 in Ls (RN ) for 2 ≤ s ≤ ∞ (see [6, Proposition 2.2] and [29, Theorem 3.1]). It follows from [24, Corollary 4.23 and 1,β Theorem 4.24] that wk → w0 in Cloc (RN ) for some β > 0. Consequently, by Schauder estimate (see [25]), wk → w0 in 2,β Cloc (RN ), and hence 1

−∆w0 + V (x0 )w0 − ∆(w02 )w0 = P (x0 )w0p + Q (x0 )w0q . 2

Lemma 3.5. C (x0 ) = infs∈RN C (s). Furthermore, wk → w0 in H 1 (RN ). Proof. Fatou’s lemma and (3.2) imply that inf C (s) ≤ C (x0 )

s∈RN



   1 2 1 − (|∇w0 |2 + V (x0 )w02 )dx + − w02 |∇w0 |2 dx 2 q+1 2 q+1 RN RN   1 1 p+1 − P (x0 )w0 dx + q+1 p+1 RN     1 1 2 1 2 2 − (|∇wε | + V (ε x + ε yε )wε )dx + − wε2 |∇wε |2 dx ≤ lim inf 2 q+1 2 q+1 ε→0+ RN RN    1 1 − P (ε x + ε yε )wεp+1 dx + q+1 p+1 RN = lim inf cε =

1

ε→0+

≤ inf C (s). s∈RN

It follows from the above inequalities and (3.7) imply that



(|∇wε | + V (εx + ε yε )wε )dx → 2

RN

2

which yields that wk → w0 in H 1 (RN ).

 RN

(|∇w0 |2 + V (x0 )w02 )dx as k → ∞,



Finally, we will discuss the decay properties of wk at infinity by using a change of variables. Following [4,5], set Uk = f −1 (wk ), where f is defined as in (2.6). It is known that (see for example [4,14]) f (t )f ′ (t ) 1. t f (t ) limt →0 t 1. f (t ) limt →∞ √2|t | 1.

=

(i) limt →0 (ii) (iii)

=

=

Then (3.6) can be rewritten as

−∆Uk + V (εk + εk yεk )f (Uk )f ′ (Uk ) = gk (x, Uk ), where gk (x, Uk ) := [P (εk x + εk yεk )f p (Uk ) + Q (εk x + εk yεk )f q (Uk )]f ′ (Uk ). Thus

− ∆Uk ≤ [P (εk x + εk yεk )f p (Uk ) + Q + (εk x + εk yεk )f q (Uk )]f ′ (Uk ).

(3.9)

By [25, Theorem 8.17], for t > N, we have sup Uk (x) ≤ C (∥Uk ∥L2 (B2 (y)) + ∥gk ∥

x∈B1 (y)

t

L 2 (B2 (y))

),

(3.10)

which implies that ∥Uk ∥L∞ (RN ) is uniformly bounded. Since wk → w0 in H 1 (RN ), we have



|f (Uk )|s dx = 0 for s ∈ [2, ∞) uniformly in k.

lim

R→∞

|x|≥R

(3.11)

W. Wang et al. / Nonlinear Analysis 102 (2014) 120–132

131

By the properties of f , we have f (t ) ≥



c | t |,

|t |,

c

|t | ≤ 1 |x| ≥ 1,

and hence t 2 ≤ C (|f (t )|2 + |f (t )|4 ). Combining with (3.10) and (3.11), we have lim Uk (x) = 0

uniformly in k.

|x|→∞

(3.12)

It follows from (3.12) that all the local maximum points of Uk stay in a finite ball in RN . The standard arguments imply that we have {Uk } is bounded in H 1 (RN ) and L∞ (RN ) (see e.g. [8,16]), and hence Uε ⇀ U in H 1 (RN ). It follows from the 2 elliptic regularity that Uε → U in Cloc (RN ). Then by well known result in [30] we have that U is radially symmetric and has a unique local maximum at zero. Now by Lemma 4.2 in [31], for large k, Uk has no critical points other than the origin. In a similar to [8, Lemma 5.8], there holds Lemma 3.6. There exists ε0 > 0 such that lim|x|→∞ Uε (x) = 0 uniformly in ε ∈ (0, ε0 ). The last step is to show the exponential decay properties of {wk }. Lemma 3.7. There exist constants C1 > 0 and C2 > 0 such that Uε (x) ≤ C1 e−C2 |x|

for all |x| ≥ R1 .

Furthermore, uε (x) ≤ C1 e−C2 |

x−xε

ε

|

.

Proof. It is easy to check lim =

gε (x, Uε )

= 0.

Uε

Uε →0

(3.13)

By (3.12), the property (i) of f and (3.13), there is a R1 > 0 such that f ′ (Uk )f (Uε ) ≥ gε (x, Uε ) ≤

1 4

3 4

for |x| ≥ R1 ,

Uε

(3.14)

V0 Uε .

(3.15)

Fix ψ = C1 e−C2 |x| with C22 ≤

∆ψ ≤ ξ 2 ψ ≤

V0 2

V0 2

and C1 e−C2 R1 ≥ Uε . It is obtained that

ψ.

(3.16)

Hence 3

1

4

4

− ∆Uε + V0 Uε ≤ −∆Uε + V (εx + ε yε )f ′ (Uε )f (Uε ) = gε (x, Uε ) ≤

V0 U ε .

(3.17)

Define ψε = ψ − Uε , using (3.16) and (3.17), we get

 V0   −∆ψε + ψε ≥ 0 if |x| ≥ R1 , 2 ψε ≥ 0 if |x| = R1 ,    lim ψε (x) = 0. |x|→∞

The maximum principle implies that ψε ≥ 0 in |x| ≥ R1 and we conclude that Uε (x) ≤ C1 e−C2 |x| ,

for all |x| ≥ R1 and ∀ε ∈ (0, ε0 ).

By the monotonicity of f , we have that wε has unique maximum point pε . Then vε has unique maximum point pε + yε and uε has unique maximum point xε = ε(pε + yε ). It is well known that

wε (x) = f (Uε ) ≤ Uε (x). Thus, we have uε (x) = vε

x ε

= wε (ε−1 x − yε ) = wε (ε−1 x − ε −1 xε + pε ) ≤ C1 e−C2 |

x−xε

ε

|

. 

(3.18)

132

W. Wang et al. / Nonlinear Analysis 102 (2014) 120–132

Acknowledgment F. Zhao thanks the College of William and Mary for the invitation and the hospitality during his visit. References [1] M. Poppenberg, K. Schmitt, Z.-Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations 14 (2002) 329–344. [2] J.-Q Liu, Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Proc. Amer. Math. Soc. 131 (2003) 441–448. [3] J.-Q. Liu, Y. Wang, Z.-Q. Wang, Solutions for quasilinear Schrödinger equations, II, J. Differential Equations 187 (2003) 473–793. [4] J.-Q. Liu, Y. Wang, Z.-Q. Wang, Solutions for quasilinear Schrödinger equations via Nehari method, Comm. Partial Differential Equations 29 (2004) 879–901. [5] M. Colin, L. Jeanjean, Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Anal. 56 (2004) 213–226. [6] X.-Q. Liu, J.-Q. Liu, Z.-Q. Wang, Quasilinear elliptic equations with critical growth via perturbation method, J. Differential Equations 254 (2013) 102–124. [7] X.-Q. Liu, J.-Q. Liu, Z.-Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc. 141 (2013) 253–263. [8] J.M do ó, O.H. Miyagaki, S.H.M. Soares, Soliton soluyions for quasilinear Schrödinger equations with critical growth, J. Differential Equations 248 (2010) 722–744. [9] E.A.B. Silva, G.F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations 39 (2010) 1–33. [10] X.-Q. Liu, J.-Q. Liu, Z.-Q. Wang, Ground states for quasilinear Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations 46 (2013) 641–669. [11] X. Fang, A. Szulkin, Multiple solutions for a quasilinear Schrödinger equation, J. Differential Equations 254 (2013) 2015–2032. [12] J.-Q. Liu, Z.-Q. Wang, Y. Guo, Multibump solutions for quasilinear elliptic equations, J. Funct. Anal. 262 (2012) 4040–4102. [13] J.-Q Liu, Z.-Q. Wang, X. Wu, Multibump solutions for quasilinear elliptic equations with critical growth, J. Math. Phys. 54 (2013) 121–501. [14] J.M do ó, U. Severo, Solitary waves for a class of quasilinear Schrödinger equations in dimension two, Calc. Var. Partial Differential Equations 38 (2010) 275–315. [15] E. Gloss, Existence and concentration of positive solutions for a quasilinear equation in RN , J. Math. Anal. Appl. 371 (2010) 465–484. [16] Y. Wang, W. Zou, Bound states to critical quasilinear schrödinger equations, NoDEA Nonlinear Differential Equations Appl. 19 (2012) 19–47. [17] E. Gloss, Elisandra Existence and concentration of positive solutions for a quasilinear elliptic equation in R, Electron. J. Differential Equations 61 (2010) 1–23. [18] M. Yang, Y. Ding, Existence and multiplicity of semiclassical states for a quasilinear Schrödinger equation in RN , Commun. Pure Appl. Anal. 12 (2013) 429–449. [19] M. Yang, Y. Ding, Existence of semiclassical states for a quasilinear Schrödinger equation with critical exponent in RN , Ann. Mat. 192 (2013) 783–804. [20] X. Wang, B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal. 28 (1997) 633–655. [21] D. Ruiz, G. Siciliano, Existence of ground states for a modified nonlinear Schrödinger equation, Nonlinearity 23 (2010) 1221–1233. [22] P.H. Rabinowitz, On a class of nonlinear Schrdinger equations, Z. Angew. Math. Phys. 43 (1992) 270–291. [23] M. Willem, Minimax theorems, in: Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. [24] Q. Han, F.-H. Lin, Elliptic Partial Differential Equations, in: Courant Institute Lecture. Notes, vol. 1, American Mathematical Society, Providence, RI, 2000. [25] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equtions of Second Order, Springer-verlag, Berlin, New York, 2001. [26] P.L. Lions, The concentration-compactness principle in the calcus of variations. The loacally compact case parts 1 and parts 2, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1894) 109–145. 223–283. [27] M. Struwe, Varitional Methods, Springer-verlag, Berlin, New York, 1990. [28] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Commun, Math. Phy. 153 (1993) 229–244. [29] X. Wu, K. Wu, Existence of positive solutions, negative solutions and high energy solutions for quasi-linear elliptic equations on RN , Nonlinear Anal. RWA 16 (2014) 48–64. [30] B. Gidas, W.M. Ni, L. Nirenberg, Symmetry of positive solutions of nonlinear equations in RN , Adv. Math., Suppl. Studies A 7 (1981) 369–402. [31] W.M. Ni, I. Takagi, On the shape of least-energy solutions to asemilinear Neumann problem, Comm. Pure Appl. Math. XLIV (1991) 819–851.