Existence of solutions for a quasilinear Schrödinger equation with vanishing potentials

J. Math. Anal. Appl. 416 (2014) 924–946

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Existence of solutions for a quasilinear Schrödinger equation with vanishing potentials José F.L. Aires ∗ , Marco A.S. Souto Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, 58429-970, Campina, PB, Brazil

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Article history: Received 31 December 2013 Available online 12 March 2014 Submitted by M. del Pino Keywords: Quasilinear Schrödinger equation Quasicritical Vanishing potentials

In this paper we investigate the existence of positive solutions for the quasilinear Schrödinger equation:   −Δu + V (x)u − Δ u2 u = g(u), in RN , where N  3, g has a quasicritical growth and V is a nonnegative potential, which can vanish at infinity. © 2014 Elsevier Inc. All rights reserved.

1. Introduction In this article, we study the existence of a solution for the quasilinear elliptic problem  −Δu + V (x)u − Δ(u2 )u = g(u), in RN , u ∈ D1,2 (RN ),

u > 0 in RN ,

(P )

where V : RN → R and g : R → R are continuous functions with V being a nonnegative function and g having a quasicritical growth at infinity. The solutions of (P ) are related to the existence of standing wave solutions for quasilinear Schrödinger equations of the form        i∂t z = −Δz + W (x)z − l |z|2 z − κ Δρ |z|2 ρ |z|2 z,

(1.1)

where z : R × RN → C, W : RN → R is a given potential, κ is a real constant and l, ρ are real functions. Here we consider the case ρ(s) = s and κ = 1 and our special interest is the standing wave solutions, i.e., * Corresponding author. E-mail addresses: [email protected] (J.F.L. Aires), [email protected] (M.A.S. Souto). http://dx.doi.org/10.1016/j.jmaa.2014.03.018 0022-247X/© 2014 Elsevier Inc. All rights reserved.

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solutions of the type z(t, x) = e−iξt u(x), where ξ ∈ R and u > 0. Note that z satisfies (1.1) if, and only if, the function u(x) is a solution for (P ) with V (x) = W (x) − ξ. Quasilinear equations of the form (1.1) appear more naturally in mathematical physics and have been derived as models of several physical phenomena corresponding to various types of ρ. We refer the reader to the introduction of [11] and references therein for more information on the subject. Taking into account the behavior of the potential V and the types of nonlinearities, we find in the literature several works in order to obtain a solution to the problem (P ). Regarding subcritical nonlinearities we cite the papers of the Liu, Wang and Wang [11,12], Ruiz and Siciliano [16], Silva and Vieira [18], Yang [21], Colin and Jeanjean [6], Poppenberg, Schmitt and Wang [15] and Fang and Szulkin [10]. For concave and convex nonlinearities we cite the papers of do Ó and Severo [8] and for critical nonlinearities we cite the papers of the Liu, Liu and Wang [13], Wang and Zou [20], Silva and Vieira [19], Yang, Wang and Abdelgadir [23] and do Ó, Miyagaki and Soares [9]. All these works, among other conditions on the potential V (x), assumed that (V1 ) lim inf |x|→+∞ V (x) > 0. Colin and Jeanjean in [6] also studied problem (P ) with the potential V ≡ 0, known in the literature as zero mass potential. Moameni in [14] studied problem (P ) considering the critical exponent case and a radial potential V , that is V (x) = V (|x|) satisfying the following conditions: There exist 0 < R1 < r1 < r2 < R2 and α > 0 such that (V2 ) V (x) = 0, for all x ∈ Ω = {x ∈ RN : r1 < |x| < r2 }, (V3 ) V (x)  α, for all x ∈ Λc , where Λ = {x ∈ RN : R1 < |x| < R2 }. In Yang and Ding [22] assumed the conditions: (V4 ) V ∈ C(RN ) and there is b > 0 such that the set ν b = {x ∈ RN ; V (x)  b} has finite Lebesgue measure; and (V5 ) 0 = V (0) = min V  V (x) < M . ∗

They considered the nonlinearity g of the form g(x, s) = K(x)|s|2(2 )−2 s + (x, s) where N  3, K(x) is a bounded positive function, and (x, s) is superlinear but a subcritical function. Results about p-Laplacian case can be found in the papers of the Alves, Figueiredo and Severo [2,3] and Severo [17], where they have used variational methods. We point out that these results happen under the (V1 ) condition and subcritical growth for the nonlinearity. Note that the conditions (V1 ), (V2 )–(V3 ) and (V4 )–(V5 ) does not imply that lim

|x|→+∞

V (x) = 0.

Motivated by the above articles, and specially by Alves and Souto in [1], in the present article we show the existence of solution for (P ) by considering a new set of hypotheses on potential V , namely: (V0 ) V (x)  0, for all x ∈ RN , (V∞ ) V (x)  V∞ , for all x ∈ RN ,

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(VΛ ) there are Λ > 0 and R > 1 such that 1 inf |x|4 V (x)  Λ. R4 |x|R Remark 1.1. We emphasize that these hypotheses, unlike the articles mentioned above, enables the potential (V ) vanishes at infinity. The following is an example of a potential which verifies the above hypotheses: ⎧ α , ⎪ ⎨ 2 V (x) = (|x| − R + α1 ) + α2 , ⎪ ⎩ (α1 +α2 )Rσ , |x|σ

if |x|  R − α1 , if R − α1  |x|  R, if |x|  R,

with 0 < α1 < R, α2  0 and 0 < σ  4. As for as we know, there is not any reference about quasilinear problems with vanishing potentials. Related to function g, we assume that: sg(s) s2∗ sg(s) lims→+∞ s22∗ =

(g1 ) lim sups→0+

< +∞; where 2∗ =

2N N −2

and N  3.

(g2 ) 0. (g3 ) There exists θ > 2 such that 2θG(s)  sg(s),

∀s > 0.

The statement of our main result is the following. Theorem 1.2. Suppose that V satisfies (V0 ), (V∞ ), (VΛ ), and g satisfies (g1 )–(g3 ). Then, there exists a constant Λ∗ = Λ∗ (θ, V∞ , c0 ) > 0 such that the problem (P ) possesses a positive solution for all Λ  Λ∗ . As an immediate corollary of the proof of Theorem 1.2 we have. Theorem 1.3. Suppose that V satisfies (V0 ), (V∞ ) and (V˜Λ ) there are Λ > 0 and R > 1 such that 1 inf |x|N +2 V (x)  Λ; RN +2 |x|R and g satisfies (g2 ), (g3 ) and (˜ g1 ) lims→+∞

sg(s) s22∗

= 0.

Then, there exists a constant Λ∗ = Λ∗ (θ, V∞ , c0 ) > 0 such that the problem (P ) possesses a positive solution for all Λ  Λ∗ . g1 ), but (V˜Λ ) is weaker than (VΛ ). In fact, it is enough to check Remark 1.4. Condition (g1 ) is weaker than (˜ ˜ that (VΛ ) is equivalent to lim inf |x|4 V (x) > 0. |x|→∞

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The organization of this paper is as follows: In Section 2, with a convenient change of variable, we reformulate the original problem. In Section 3, we adapt a method explored by Del Pino and Felmer in [7] (see also [1]) to modify the reformulated problem. In Section 4, we verify that the functional energy associated to the problem satisfies the geometric conditions of the Mountain Pass Theorem and show the boundedness of the Cerami sequences associated with the minimax level. In Section 5 we provide an estimate involving the L∞ -norm of a solution of the modified problem and we prove Theorems 1.2 and 1.3. Notation. In this paper we make use of the following notation: • • • •

C, C0 , C1 , C2 , . . . denote positive (possibly different) constants. BR denotes the open ball centered at the origin with radius R > 0. C0∞ (RN ) denotes the space of the infinitely differentiable functions with compact support. For 1  s  ∞, the usual Lebesgue space is endowed with the norm  uLs (RN ) :=

1/s |u| dx s

.

RN ∗

• D1,2 (RN ) = {u ∈ L2 (RN ); |∇u| ∈ L2 (RN )} endowed with the norm ∇uL2 (RN ) . • S denotes the best constant that verifies   u2L2∗ (RN )  S |∇u|2 dx, ∀u ∈ D1,2 RN . RN

• • • •

By ·, · we denote the duality pairing between E and its dual E  . We denote the weak convergence in a space E and in E  by and the strong convergence by →. ωN denotes the volume of the unitary ball in RN . [|x|  a] := {x ∈ RN ; |x|  a}, a ∈ R.

2. Reformulation of the problem It should be pointed out that we may not apply directly the variational method to study (P ) since the natural associated functional    1  1 2 2 2 J(u) = 1 + 2u |∇u| dx + V (x)u dx − G u(x) dx (2.2) 2 2 RN

RN

RN

 is not well defined in general, because, RN u2 |∇u|2 dx is not finite, for all u ∈ D1,2 (RN ). In order to overcome this difficulty we use the following change of variables introduced by Colin and Jeanjean in [6] and by Liu, Wang and Wang in [11] (see also [8]): w = f −1 (u), where f is defined by 1 f  (t) =  1 + 2f 2 (t)

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

and verifies the following properties, which statement may be found in [6] and [8]. Lemma 2.1. The function f satisfies the following properties: (1) f is uniquely defined, is C ∞ and invertible;

(2.3)

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(2) |f  (t)|  1, ∀t ∈ R; (3) |f (t)|  |t|, ∀t ∈ R; (4) f (t) t → 1 as t → 0; 1 f (t) (5) √t → 2 4 as t → +∞  (6) f (t) 2  tf (t)  f (t), ∀t  0; 1 1 (7) |f (t)|  2 4 |t| 2 , ∀t ∈ R; 2 (8) f 2(t)  tf (t)f  (t)  f 2 (t), ∀t ∈ R; (9) there exists a positive constant C such that

  f (t)  (10) |f (t)f  (t)| 

√1 , 2



C|t|, 1 2

C|t| ,

if |t|  1 if |t|  1;

∀t ∈ R.

At this moment, we need to detach that the properties (4)–(5) of Lemma 2.1, (g1 ) and (g2 ) imply that there exists c0 > 0 such that    f (s)g f (s)   c0 |s|2∗ ,

∀s ∈ R.

(2.4)

Indeed, from assumption (g1 ) and Lemma 2.1-(4), we obtain f (s)g(f (s)) < +∞ s2∗

lim sup s→0+

and using (g2 ) and Lemma 2.1-(5), it follows that lim

s→+∞

f (s)g(f (s)) = 0. s2∗

These limits imply inequality (2.4). From assumption (V0 ), we can introduce the following subspace  E=

  w ∈ D1,2 RN ;



V (x)w2 dx < +∞ RN

of D1,2 (RN ) endowed with the inner product defined by (u, w) =

  ∇u∇w + V (x)uw dx

RN

and associated norm w = 2



 |∇w|2 + V (x)w2 dx.

RN

Remark 2.2. Note that the functional Ψ : E → R, given by Ψ (w) = RN

  |∇w|2 + V (x)f 2 (w) dx,

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is a C 1 function and Ψ (w)  w2 ,

for all w ∈ E.

Furthermore, there is a constant β > 0 such that  2∗ /2 βw2  Ψ (w) + Ψ (w) ,

for all w ∈ E.

Indeed, from condition (V0 ) and Lemma 2.1-(3), it follows that Ψ (w)  w2 ,

for all w ∈ E.

On the other hand, from condition (V∞ ), Lemma 2.1-(9) and the Sobolev Embedding Theorem, we find a constant C > 0 such that 1 1 2 V (x)w dx  2 V (x)f 2 (w) dx  2 Ψ (w) C C [|w(x)|1]

[|w(x)|1]

and



 2∗ /2 ∗ |w|2 dx  V∞ C1 Ψ (w) .

V (x)w2 dx  V∞ [|w(x)|1]

[|w(x)|1]

Thus, V (x)w2 dx 

 2∗ /2 1 Ψ (w) + V∞ C1 Ψ (w) , 2 C

RN

that is,  2∗ /2 βw2  Ψ (w) + Ψ (w) , where β = 1δ and δ = max{1 + C12 , V∞ C1 }. After the change of variables (u = f (w)), we can write J(u) as 1 2

I(w) =

|∇w|2 dx + RN

1 2





  G f (w) dx,

V (x)f 2 (w) dx − RN

(2.5)

RN

which is well defined in E and is of C 1 class by assumptions (V0 ), (g1 ) and (g2 ), and properties (1), (2) and (3) of Lemma 2.1 assures that 

 I (w), ϕ = 



∇w∇ϕ dx +

RN





V (x)f (w)f (w)ϕ dx −

RN

  g f (w) f  (w)ϕ dx,

(2.6)

RN

for all w, ϕ ∈ E. We note that the critical points of I are weak solutions of the problem 

−Δw + V (x)f (w)f  (w) = g(f (w))f  (w), in RN , w ∈ D1,2 (RN ),

w > 0, in RN

(AP1)

930

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Also note that if w ∈ C 2 (RN ) ∩ D1,2 (RN ) is a critical point of the functional I, then u = f (w) is a classical solution of (P ) (see [6]). Remark 2.3. Since we intend to prove the existence of positive solutions of the problem (P ), we can consider g(t) = 0,

for all t  0.

Hereafter, we denote by B the unitary ball in RN , that is, B = B1 (0) and by I0 : H01 (B) → R the functional   1 1 I0 (w) = |∇w|2 dx + V (x)f 2 (w) dx − G f (w) dx. (2.7) 2 2 B

B

B

Moreover, we denote by d the mountain level associated with I0 , that is,   d = inf max I0 γ(t) ,

(2.8)

    Γ = γ ∈ C [0, 1], H01 (B) ; γ(0) = 0 and γ(1) = e ,

(2.9)

γ∈Γ t∈[0,1]

where

with e ∈ H01 (B)\{0} verifying I0 (e) < 0. 3. The auxiliary problem In order to obtain positive solution we will adapt for our case a method explored by Del Pino and Felmer in [7] (see also [1]), which consists in a modification of problem (AP1). To this end we will establish some considerations. 2θ For k > θ−2 (k > 1) and R > 1, let us consider ⎧ g(t), ⎪ ⎨ h(x, t) = g(t), ⎪ ⎩ V (x) k

if |x|  R if |x| > R

and g(t) 

t, if |x| > R

and g(t) >

V (x) k t V (x) k t.

The auxiliary problem that we will consider is the following 

−Δw + V (x)f (w)f  (w) = h(x, f (w))f  (w), w ∈ D1,2 (RN ),

in RN ,

w > 0, in RN .

(AP2)

A direct computation shows that for all t ∈ R, the inequalities below hold: h(x, t)  g(t),

for all x ∈ RN ,

V (x) t, for all |x|  R, k H(x, t) = G(t), if |x|  R, h(x, t) 

H(x, t)  and

V (x) 2 t , 2k

if |x| > R

(3.10) (3.11) (3.12) (3.13)

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1 th(x, t) − H(x, t)  2θ



 1 1 V (x) 2 − t , 2θ 2 k

for all x ∈ RN .

931

(3.14)

Associated to problem (AP2) we define, on E, the Euler–Lagrange functional Φ(w) =

1 2

|∇w|2 dx + RN

1 2



V (x)f 2 (w) dx −

RN

  H x, f (w) dx.

(3.15)

RN

From the assumptions on g and V , it follows that Φ is C 1 with Gateaux derivative 

 Φ (w), ϕ = 



∇w∇ϕ dx +

RN





V (x)f (w)f (w)ϕ dx −

RN

  h x, f (w) f  (w)ϕ dx,

(3.16)

RN

for all w, ϕ ∈ E, and its critical points correspond to weak solutions of (AP2). 4. The Mountain Pass Geometry and the boundedness of the Cerami sequences In this section, we state a version of the Mountain Pass Theorem due to Ambrosetti and Rabinowitz [4], which is an essential tool in this article, then it is shown that the (modified) functional associated to problem (AP2) satisfies the geometric properties of this theorem. Afterwards it is shown the boundedness of the Cerami sequences. We conclude this section showing that the weak limit of the Cerami sequence is a non-trivial solution to problem (AP2). Let X be a real Banach space and I : X → R a functional of class C 1 . We recall that (wn ) ⊂ X is a Cerami sequence for I at level c (we denote it by (Ce)c ) if (wn ) satisfies:    (i) I(wn ) → c and (ii) 1 + wn  I  (wn )X  → 0 as n → +∞. The functional I satisfies the Cerami condition at c, if any Cerami sequence possesses a convergent subsequence. Theorem 4.1. (See [18,19].) Let X be a real Banach space and let I ∈ C 1 (X, R). Let Σ be a closed subset of X which disconnects (archwise) X into distinct connected components X1 and X2 . Suppose further that I(0) = 0 and (I1 ) 0 ∈ X1 and there is α > 0 such that I|Σ  α > 0, (I2 ) there is e ∈ X2 such that I(e) < 0. Then I possesses a (Ce)c sequence with c  α > 0 given by   c = inf max I γ(t) ,

(4.17)

      Γ = γ ∈ C [0, 1], X ; γ(0) = 0 and I γ(1) < 0 .

(4.18)

γ∈Γ t∈[0,1]

where

The following lemma implies that Φ possesses the Mountain Pass Geometry. Lemma 4.2. Suppose that (V0 ), (g1 ), (g2 ) and (g3 ) are satisfied. Then the functional Φ defined by (3.15) satisfies conditions I1 and I2 of Theorem 4.1.

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Proof. First note that Φ(0) = 0. Now, for every ρ > 0, define   Σρ = w ∈ E; Ψ (w) = ρ2 , where Ψ was defined in Remark 2.2. Since the functional Ψ is continuous, it follows that Σρ is a closed subset which disconnects the space E. Using (3.10), (g3 ), (2.4), the Sobolev–Gagliardo–Nirenberg inequality and (V0 ), we have



  H x, f (w) dx  C3

RN

  |∇w|2 + V (x)f 2 (w) dx

2∗ /2 ,

RN

from which it follows that, Φ(w) 

∗ 1 2 ρ − C3 ρ2 , 2

∀w ∈ Σρ .

Since 2∗ > 2, it implies that for ρ > 0 sufficiently small, there exists α > 0 such that Φ(w)  α > 0,

∀w ∈ Σρ ,

thus condition (I1 ) is satisfied. Now let us show that there is ϕ ∈ E such that Φ(tϕ) → −∞ as t → +∞.

(4.19)

Indeed, consider ϕ ∈ C0∞ (RN ) satisfying supp(ϕ) = B1 and 0  ϕ(x)  1, ∀x ∈ B1 . Note, by property (3) of Lemma 2.1, that we get 1 Φ(tϕ)  t2 2



|∇ϕ| dx + 2

RN

2

V (x)ϕ dx

RN

−

  H x, f (tϕ) dx.

(4.20)

B1

By (3.12), it follows that H(x, t) = G(t) in B1 . By hypothesis (g3 ), there exist positive constants C1 and C2 such that G(t)  C1 |t|2θ − C2 ,

∀t ∈ R.

Hence, it follows that 1 Φ(tϕ)  t2 2

 |∇ϕ| dx + RN

Since the function

f (t) t

2

2

V (x)ϕ dx

RN

− C1

  f (tϕ)2θ dx + C3 .

B1

is decreasing for t > 0 and, 0  tϕ(x)  t,

∀x ∈ B1 and t > 0,

we have   f (t)ϕ(x)  f tϕ(x) ,

∀x ∈ B1 and t > 0,

(4.21)

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which together with (4.21) implies Φ(tϕ)  t2

   1  C3 f 2θ (t) |ϕ|2θ dx + 2 . |∇ϕ|2 + V (x)ϕ2 dx − C1 2 2 t t RN

(4.22)

B1

By properties (9) of Lemma 2.1 and provided that θ > 2, we conclude that f 2θ (t) = +∞, t→+∞ t2 lim

and hence (4.19) is proved, and consequently (I2 ) is satisfied. 2 As a consequence of Theorem 4.1 and Lemma 4.2, we have: Corollary 4.3. Suppose that (V0 ), (g1 ), (g2 ) and (g3 ) are satisfied. Then the functional Φ possesses a (Ce)c sequence, with c given by (4.17). Remark 4.4. Once that Φ(w)  I0 (w) for all w ∈ D1,2 (RN ), from the definition of the numbers c and d, we derive that c  d.

(4.23)

We now show the boundedness of the Cerami sequences. Lemma 4.5. Suppose that (V0 ), (V∞ ), (g1 ), (g2 ) and (g3 ) hold. If (wn ) ⊂ E is a Cerami sequence of Φ, then (wn ) is bounded in E. Proof. Let (wn ) be a (Ce)c sequence. Then,    1 + wn  Φ (wn )E  → 0

Φ(wn ) → c and

as n → +∞,

this is, Φ(wn ) =

1 2





 |∇wn |2 + V (x)f 2 (wn ) dx −

RN



  H x, f (wn ) dx = c + on (1)

(4.24)

RN

and 

  1 + wn  Φ (wn )E  = on (1)

(4.25)

For ϕ ∈ E, we have  Φ (wn ), ϕ =





∇wn ∇ϕ dx +

RN

Taking ϕ = ϕn =

f (wn ) f  (wn ) ,

V (x)f (wn )f  (wn )ϕ dx −

RN

RN

by properties (6) of Lemma 2.1, we have |ϕn |  2|wn |

and thus ϕn ∈ E and



and |∇ϕn |  2|∇wn |,

  h x, f (wn ) f  (wn )ϕ dx.

(4.26)

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934

ϕn   2wn . From (4.25) 

 Φ (wn ), ϕn = on (1),

(4.27)

this is,  Ψ (wn ) + RN

   2f 2 (wn ) 2 |∇w | dx − h x, f (w ) f (wn ) dx = on (1). n n 1 + 2f 2 (wn )

(4.28)

RN

From (4.24) and (4.28) it follows that Φ(wn ) −

 1  Φ (wn ), ϕn = c + on (1), 2θ

and thus 

      1 1 1  − Ψ (wn ) + h x, f (wn ) f (wn ) − H x, f (wn ) dx  c + on (1). 2 θ 2θ

(4.29)

RN

From (3.14) we have 

      1  1 1 1 h x, f (wn ) f (wn ) − H x, f (wn ) dx  − V (x)f 2 (wn ) dx. 2θ 2θ 2 k

RN

Noting that k >

(4.30)

RN 2θ θ−2 ,

using (4.29) and (4.30), it follows that 

  k−1 1  Φ (wn ), ϕn = c + on (1). Ψ (wn )  Φ(wn ) − 2 k 2θ

(4.31)

Therefore, Ψ (wn ) is bounded and consequently from Remark 2.2 the sequence (wn ) is bounded in E.

2

Since the sequence (wn ) given by Corollary 4.3 is a bounded sequence in E, there exists w ∈ E and a subsequence, still denoted by wn , such that wn w

in E,

wn → w

  in Lsloc RN for s ∈ [1, 2∗ )

and wn (x) → w(x) a.e. on RN .

Lemma 4.6. Assuming the hypotheses of Lemma 4.5, then the following statements hold: (i) For each  > 0 there exists r > R such that lim sup n→+∞

  |∇wn |2 + V (x)f 2 (wn ) dx < .

|x|2r

(ii)

lim

n→+∞ RN

2

V (x)f 2 (w) dx.

V (x)f (wn ) dx = RN

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(iii) lim

n→+∞ RN



  h x, f (wn ) f (wn ) dx =

  h x, f (w) f (w) dx.

RN

(iv) The weak limit w is a critical point for functional Φ. Proof. (i) Consider r > R and a function η = ηr ∈ C0∞ (Brc ) such that c η ≡ 1 in B2r ,

η≡0

0η1

in Br ,

and |∇η| 

2 , r

for all x ∈ RN .

n) As (wn ) is bounded in E, the sequence (ηϕn ), where ϕn = ( ff(w (wn ) ), is also bounded. Hence, from (4.27) we have



 Φ (wn ), ηϕn = on (1),

that is 

 2f 2 (wn ) 2 1+ |∇wn | η dx + V (x)f 2 (wn )η dx 1 + 2f 2 (wn )

RN



f (wn ) ∇wn ∇η dx + f  (wn )

=− RN



RN

  h f (wn ) f (wn )η dx + on (1).

RN

Once that η ≡ 0 in Br , the last equality combined with (3.11) yields 

1 1− k





 |∇wn |2 + V (x)f 2 (wn ) η dx  2

[|x|r]

|wn ||∇wn ||∇η| dx + on (1),

[|x|r]

that is,  1−

1 k





 4 |∇wn |2 + V (x)f 2 (wn ) η dx  r

[|x|r]

|wn ||∇wn | dx + on (1).

(4.32)

[r|x|2r]

By Hölder’s inequality, 

|wn ||∇wn | dx 

1/2  |∇wn |2 dx

wn2 dx

RN

[r|x|2r]

1/2



.

[r|x|2r]

Since wn → w in L2 (B2r \Br ) and (wn ) is bounded in E, it follows that 

lim sup n→+∞ [r|x|2r]

1/2



|wn ||∇wn | dx  C

2

w dx [r|x|2r]

,

(4.33)

J.F.L. Aires, M.A.S. Souto / J. Math. Anal. Appl. 416 (2014) 924–946

936

for some constant C > 0. On the other hand, using again Hölder’s inequality 

1/2





1/2∗



∗



w2 dx [r|x|2r]

|w|2 dx

|B2r \Br |1/N .

(4.34)

[r|x|2r]

Noting that |B2r \Br |  |B2r | = ωN (2r)N , from (4.33) and (4.34), we have 

|wn ||∇wn | dx 

lim sup n→+∞

1/2∗



1/N 2rCωN

∗

w2 dx

[r|x|2r]

,

(4.35)

[r|x|2r]

and from (4.32) and (4.35), it follows that lim sup n→+∞

 −1    1 1/N 2 2 1− |∇wn | + V (x)f (wn ) dx  8CωN k

[|x|2r]

1/2∗

w

2∗

dx

.

(4.36)

[r|x|2r]

Thus, for every  > 0, we choose r > R such that  1/N

8CωN

1 1− k

−1 

1/2∗



∗

w2 dx

< ,

[r|x|2r]

and this concludes part (i) of the proof. (ii) Note first that from part (i), for each  > 0, there exists r > R such that V (x)f 2 (wn ) dx <

lim sup n→+∞

 4

[|x|2r]

and consequently V (x)f 2 (w) dx 

 . 4

[|x|2r]

Hence,      2      2  V (x) f (wn ) − f (w) dx  +    2  RN

  2   2 V (x) f (wn ) − f (w) dx. 



(4.37)

[|x|2r]

Since wn → w in L2 (B2r ), using Lebesgue Dominated Convergence Theorem, it follows that

lim

n→+∞ [|x|2r]

V (x)f 2 (wn ) dx =

V (x)f 2 (w) dx. [|x|2r]

From (4.37) and (4.38), we have     2     2 lim sup V (x) f (wn ) − f (w) dx  ,  2 n→+∞  RN

(4.38)

J.F.L. Aires, M.A.S. Souto / J. Math. Anal. Appl. 416 (2014) 924–946

937

for every  > 0. Therefore,

2

lim

V (x)f 2 (w) dx.

V (x)f (wn ) dx =

n→+∞ RN

RN

(iii) It follows by (3.11) and by part (i) that, for each  > 0, there exists r > R such that

   h x, f (wn ) f (wn ) dx < 4

lim sup n→+∞ [|x|2r]

and

   h x, f (w) f (w) dx  . 4

[|x|2r]

Therefore,            h x, f (wn ) f (wn ) − h x, f (w) f (w) dx    RN

    + 2 



        h x, f (wn ) f (wn ) − h x, f (w) f (w) dx. 

(4.39)

[|x|<2r]

Since wn (x) → w(x) a.e. on RN ,

h(·, f (s))f (s) → 0 as s → +∞ |f (s)|22∗

and sup n

  ∗ f (wn )22 < +∞,

RN

it follows from the Compactness Lemma of Strauss [5] that lim

n→+∞ [|x|<2r]

  h x, f (wn ) f (wn ) dx =



  h x, f (w) f (w)] dx.

(4.40)

[|x|<2r]

From (4.39) and (4.40), the result follows. This completes the proof of part (iii). (iv) As in the proof of (iii), it is easy to deduct that

  V (x) f (wn )f  (wn ) − f (w)f  (w) φ dx → 0

as n → +∞,

(4.41)

RN

and RN

for all φ ∈ C0∞ (RN ).

      h x, f (w) f  (w) − h x, f (wn ) f  (wn ) φ dx → 0

as n → +∞,

(4.42)

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J.F.L. Aires, M.A.S. Souto / J. Math. Anal. Appl. 416 (2014) 924–946

Moreover, since wn w we have ∇(wn − w)∇φ dx → 0

as n → +∞.

(4.43)

RN

Combining (4.41), (4.42) and (4.43) its proved that 

lim

n→+∞

  for all φ ∈ C0∞ RN .

   Φ (wn ), φ = Φ (w), φ ,

This completes the proof of part (iv). 2 As a consequence of Lemma 4.6, we conclude that Corollary 4.7. Under the hypotheses of Lemma 4.5 we have that w is non-trivial critical point of Φ and Φ(w) = c. Furthermore, the functional Φ satisfies the (Ce)c condition. Proof. Firstly, we observe that proceeding like in the proof of Lemma 4.6-(ii)–(iii), we deduce the following limits lim V (x)f (wn )f  (wn )wn dx = V (x)f (w)f  (w)w dx. (4.44) n→+∞ RN



lim

n→+∞ RN

RN



  h x, f (wn ) f  (wn )wn dx =

  h x, f (w) f  (w)w dx

(4.45)

RN

and lim

n→+∞ RN

  H x, f (wn ) dx =



  H x, f (w) dx.

(4.46)

RN

We know that Φ (w) = 0. Let us show that w = 0. Suppose that w ≡ 0, by Lemma 4.6-(ii), we have lim

n→+∞ RN

V (x)f 2 (wn ) dx = 0.

Now, by limits (4.44) and (4.45), we deduce that

V (x)f (wn )f  (wn )wn dx = 0

lim

n→+∞ RN

and lim

n→+∞ RN

  h x, f (wn ) f  (wn )wn dx = 0.

In particular, by definition of h we also get lim

n→+∞ RN

  H x, f (wn ) dx = 0

J.F.L. Aires, M.A.S. Souto / J. Math. Anal. Appl. 416 (2014) 924–946

939

and since

|∇wn | dx + RN





  h x, f (wn ) f  (wn )wn dx → 0,

V (x)f (wn )f (wn )wn dx −

2

RN

RN

we conclude that |∇wn |2 dx → 0. RN

Hence, Φ(wn ) =



1 2



 |∇wn |2 + V (x)f 2 (wn ) dx −

RN



  H x, f (wn ) dx → 0,

RN

which is a contradiction, because Φ(wn ) → c > 0. Therefore, w = 0. Now, we will show that Φ(w) = c. Passing to the limit in the following expression

|∇wn | dx = − RN





2

  h x, f (wn ) f  (wn )wn dx + o(1),

V (x)f (wn )f (wn )wn dx +

RN

RN

and using the limits (4.44) and (4.45) together with Φ (w)w = 0, we obtain

|∇wn | dx =

|∇w|2 dx.

2

lim

n→+∞ RN

(4.47)

RN

From Lemma 4.6-(ii), (4.46) and (4.47), we conclude that 1 Φ(wn ) = 2









|∇wn | + V (x)f (wn ) dx − 2

2

RN

  H x, f (wn ) dx → Φ(w)

RN

it results that Φ(w) = c. To show that the functional Φ satisfies the (Ce)c condition, it remains to show that wn − w → 0. From Remark 2.2, we have   2∗ /2  wn − w2  C Ψ (wn − w) + Ψ (wn − w) , where Ψ (wn − w) = remains to show that

 RN

|∇wn − ∇w|2 dx +

 RN

V (x)f 2 (wn − w) dx. To conclude that wn → w in E, it

lim

n→+∞ RN

V (x)f 2 (wn − w) dx = 0.

This last limit follows from Proposition 2.1-(3) in [11] (see the following remark). Hence, Φ satisfies the Cerami condition. 2 Remark 4.8. In the proof of the item (3) of Proposition 2.1 in [11], the authors have proved the limit under ∗ condition (V1 ). However, we observe that this condition can be replaced by L2 (RN )-boundedness of wn . In

940

J.F.L. Aires, M.A.S. Souto / J. Math. Anal. Appl. 416 (2014) 924–946

fact, they have used that condition on V to show that the measure of the set An,2 = {x ∈ A; |wn − w| > a} ∗ goes to zero as a → ∞. Since (wn ) is a bounded sequence in L2 (RN ), we can obtain ∗



a2 |An,2 | 

∗

|wn − w|2 dx  C An,2

and thus |An,2 | → 0 as a → ∞. 5. L∞ estimate of the solution of (AP2) and the existence result for problem (P ) In this section, we will establish an L∞ estimate for solution w obtained in Corollary 4.7 and we prove Theorem 1.2. Lemma 5.1. For R > 1, any positive solution w of the problem (AP2) satisfies the estimate w2L2∗ (RN ) 

dSk 2 . k−1

Proof. Repeating (4.31) to w we have 

  1  k−1 Ψ (w)  Φ(w) − Φ (w), w = c. 2 k 2θ

Thus, the estimate follows directly from (4.23) and from Sobolev embeddings. 2 Remark 5.2. In the previous lemma wL2∗ (RN ) is bounded by a constant that does not depend on R > 1. Now we enounce a result of Brézis Kato type which is crucial to our arguments, because it establishes an important estimate involving L∞ (RN ) norm of a solution w of problem (AP2). The proof of the result can be found in Alves and Souto [1]. Proposition 5.3. Let ∈ Lq (RN ), 2q > N , and w ∈ E ⊂ D1,2 (RN ) be a weak solution of problem −Δw + b(x)w = L(x, w)

in RN ,

(5.48)

where L : RN × R → R is a continuous function verifying   L(x, s)  (x)|s|,

for all s > 0,

and b is a nonnegative function in RN . Then there exists a constant M = M (q,  Lq (RN ) ) > 0 such that wL∞ (RN )  M wL2∗ (RN ) . As a consequence we will show that if w is bounded in the Lr (RN ) norm, for some r > 2∗ , then wL∞ (RN ) is bounded, even in the quasicritical case. Corollary 5.4. Let N > 2, r > 2∗ and   L(x, s)  C0 |s|2∗ −1 ,

for all s ∈ R, x ∈ RN .

Then, there exists a constant M = M (C0 , wLr (RN ) ) > 0 such that

J.F.L. Aires, M.A.S. Souto / J. Math. Anal. Appl. 416 (2014) 924–946

941

wL∞ (RN )  M wL2∗ (RN ) , for any w ∈ E ∩ Lr (RN ) weak solution of the problem (5.48). Proof. This proof is a direct application of Proposition 5.3, considering: L(x, w)  (x)|w|

in RN ,

with    (x) = C0 |w|2∗ −2 ,

   

∈ Lq RN for q = r/ 2∗ − 2

and 2q > N

if, and only if,

r > 2∗ .

2

In order to obtain the boundedness in the L∞ norm, we consider for a solution w of the problem (AP2), the following functions  L(x, t) =

g(f (t))f  (t),

if |x| < R

or g(f (t)) 

0,

if |x|  R

and

V (x) k f (t) g(f (t)) > V k(x) f (t)

(5.49)

and  b(x) =

1 wV

(x)f (w)f  (w),

if |x| < R

(1 − k1 ) w1 V (x)f (w)f  (w), if |x|  R

V (x) k f (w) g(f (w)) > V k(x) f (w).

or g(f (w))  and

Observe that w satisfies an analogous problem to (5.48). From property (6) of Lemma 2.1 and (2.4), we derive that   L(x, t)  C1 |t|2∗ −1 , for some constant C1 > 0. To apply Corollary 5.4 it remains to show the boundedness in the Lr (RN ) norm, for some r > 2∗ . Lemma 5.5. Let N > 2 and β = N/(N − 2). Then there exists a constant C = Cε > 0, such that wL2∗ β (RN )  CwL2∗ (RN ) . Proof. Proceeding as in the proof of Proposition 5.3 (see Alves and Souto [1]) let w be a positive solution of (5.48), and for each m ∈ N consider the sets Am = {x ∈ RN ; |w|β−1  m} and Bm = RN \ Am . Let us define  v|v|2(β−1) , in Am , wm = m2 v, in Bm , and  zm =

v|v|β−1 ,

in Am ,

mv,

in Bm .

J.F.L. Aires, M.A.S. Souto / J. Math. Anal. Appl. 416 (2014) 924–946

942

2 Using wm as a test function and since 0  b(x)zm = b(x)wwm in RN and β > 1, we deduce that



  2 |∇zm |2 + b(x)zm dx  β 2

RN

L(x, w)wm dx.

(5.50)

RN

Note that the function L defined in (5.49) verifies the following conditions: ∗

(L1 ) |L(x, t)|  C0 |t|2 −1 , for t sufficiently small, L(x,t) (L2 ) lims→+∞ |t| 2∗ −1 = 0. Certainly conditions (L1 ) and (L2 ) imply that, for each ε > 0, there is C = Cε > 0 such that   L(x, t)  ε|t|2∗ −1 + Cε |t|,

for all x ∈ RN , t ∈ R.

Using this inequality in (5.50) we have



 2 |∇zm |2 + b(x)zm dx  β 2 ε

RN



∗

|w|2

−1

|wm | dx + Cβ 2

RN

2 zm dx.

(5.51)

RN

Note that

2∗ −1

|w|

|wm | dx 

RN



∗ |w|2 −2 zn2

dx 

2∗ −2 2∗

zm 2L2∗ (RN )

RN

|w|

dx

,

RN

that is,

∗

|w|2

−1

∗

2 −2 |wn | dx  SwL 2∗ (RN )

RN

|∇zm |2 dx,

RN

that substituted in (5.51) results in



 2∗ −2 2 |∇zm |2 + b(x)zm dx  β 2 εSwL 2∗ (RN )

RN



|∇zm |2 dx + Cβ 2

RN

RN

By Lemma 5.1, we can choose ε > 0 such that ∗

2 −2 εβ 2 wL 2∗ (RN ) S <

1 , 2

from which it follows that

  2 |∇zm |2 + b(x)zm dx  2Cβ 2

RN

2 zm dx.

RN

Using the Sobolev embeddings we have 

2/2∗ 2∗

|zm | Am

dx

S

|∇zm |2 dx  2SCβ 2

RN

RN

2 zm dx.

2 zm dx.

(5.52)

J.F.L. Aires, M.A.S. Souto / J. Math. Anal. Appl. 416 (2014) 924–946

943

Since |zm | = |w|β in Am and |zm |  |w|β in RN , it follows that 

1/2∗ β |w|

∗

2 β



 2SCβ

dx

 2 1/2β



1/2β |w|

2β

dx

.

RN

Am

By the Monotone Convergence Theorem, letting m → +∞, we have  1/2β wL2∗ β (RN )  2SCβ 2 wL2∗ (RN ) .

2

It follows from this lemma that   w ∈ Lr RN , with r = 2∗ β > 2∗ . Applying Corollary 5.4 it results that there exists a constant M = M (Cε , wLr (RN ) ) > 0 such that wL∞ (RN )  M wL2∗ (RN ) , for any w ∈ E ∩ Lr (RN ) weak solution of the problem (5.48). Hence, any weak solution w of the problem (AP2) satisfies the estimate 

wL∞ (RN )

dSk 2 M k−1

1/2 .

(5.53)

Lemma 5.6. For R > 1, any positive solution w of the problem (AP2) satisfies w(x) 

RN −2 wL∞ (RN ) RN −2 M [S dk 2 (k − 1)−1 ]1/2  , N −2 |x| |x|N −2

∀|x|  R.

Proof. Consider u the C ∞ (RN \ {0}) harmonic function given by u(x) =

RN −2 M [S dk 2 (k − 1)−1 ]1/2 . |x|N −2

By estimate (5.53), we have w(x)  u(x) for |x| = R. It follows that, (w − u)+ = 0 for |x| = R, and the function given by  φ=

(w − u)+ ,

if |x|  R

0,

if |x| < R

belongs to D1,2 (RN ). Moreover, φ ∈ E. Employing φ as the test function and using the fact that w is positive solution of (AP2), we have

J.F.L. Aires, M.A.S. Souto / J. Math. Anal. Appl. 416 (2014) 924–946

944



∇w∇φ dx + RN





V (x)f (w)f (w)φ dx =

RN

  h x, f (w) f  (w)φ dx.

(5.54)

∇u∇φ dx,

(5.55)

RN

On the other hand, by definition of φ it follows that

|∇φ|2 dx = RN

∇w∇φ dx −

A

A

where A = {x ∈ RN ; |x|  R and w(x) > u(x)}. Since Δu = 0

  in RN \BR (0) ,

φ = 0 for |x| = R and φ  0,

we have ∇u∇φ dx = 0. A

Thus using (5.54) and (5.55) it follows that

|∇φ|2 dx = RN

  h x, f (w) f  (w)φ dx −

A



V (x)f (w)f  (w)φ dx,

A

and from (3.11), we conclude that 

|∇φ| dx  2



1 −1 k

RN

V (x)f (w)f  (w)φ dx  0.

A

Hence, we have φ = 0, in RN , which implies that (w − u)+ = 0, in |x|  R. From this we conclude that w  u in |x|  R and the lemma is proved. 2 Proof of Theorem 1.2. It is enough to show that   V (x) f (w) in |x|  R, g f (w)  k for all solution w of the problem (AP2). By inequality (2.4) we have ∗

g(f (w))f (w) g(f (w)) |w|2 =  c , 0 f (w) f 2 (w) f 2 (w)

for all x ∈ RN .

Once wL∞ (RN ) is uniformly bounded and using Lemma 2.1-(9), we deduce that ∗ g(f (w))  C1 |w|2 −2 , f (w)

for all x ∈ RN .

By Lemma 5.6 it follows that g(f (w)) R4 (M [S dk2 (k − 1)−1 ]1/2 )4/(N −2)  C1 , f (w) |x|4

in |x|  R.

J.F.L. Aires, M.A.S. Souto / J. Math. Anal. Appl. 416 (2014) 924–946

945

Fixing Λ∗ = kM1 , where M1 = C1 (M [S dk2 (k − 1)−1 ]1/2 )4/(N −2) and Λ  Λ∗ , it implies that 1 1 R4 g(f (w)) R4  Λ∗ 4  Λ 4 . f (w) k |x| k |x| It follows from the hypothesis (VΛ ) that V (x) g(f (w))  f (w) k

in |x|  R,

which concludes the proof. 2 Proof of Theorem 1.3. Since the condition (g1 ) is weaker than (˜ g1 ), the proof is the same until Lemma 5.6. To conclude the proof we can use property (7) of Lemma 2.1 and (˜ g1 ) to obtain ∗ g(f (w))  C1 |w|2 −1 , f (w)

for all x ∈ RN .

Proceeding as before and under condition (V˜Λ ) we conclude the proof. 2 Acknowledgments Marco A.S. Souto was partially supported by INCT-MAT, casadinho/PROCAD, CNPq/Brazil 552.464/ 2011-2 and 304.652/2011-3. References [1] C.O. Alves, M.A.S. Souto, Existence of solutions for a class of elliptic equations in RN with vanishing potentials, J. Differ. Equ. 252 (2012) 5555–5568. [2] C.O. Alves, G.M. Figueiredo, U.B. Severo, Multiplicity of positive solutions for a class of quasilinear problems, Adv. Differential Equations 252 (2009) 911–942. [3] C.O. Alves, G.M. Figueiredo, U.B. Severo, A result of multiplicity of solutions for a class of quasilinear equations, Proc. Edinb. Math. Soc. 55 (2012) 291–309. [4] A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381. [5] H. Berestycki, P.L. Lions, Nonlinear scalar field equations, I – existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983) 313–346. [6] M. Colin, L. Jeanjean, Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Anal. 56 (2004) 213–226. [7] M. Del Pino, P. Felmer, Local mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. 4 (1996) 121–137. [8] J.M. Bezerra do Ó, U. Severo, Quasilinear Schrödinger quations involving concave and convex nonlinrearities, Commun. Pure Appl. Anal. 8 (2009) 621–644. [9] J.M. Bezerra do Ó, Olímpio H. Miyagaki, Sérgio H.M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations 248 (2010) 722–744. [10] X.D. Fang, A. Szulkin, Multiple solutions for a quasilinear Schrödinger equation, J. Differential Equations 254 (2013) 2015–2032. [11] J. Liu, Y. Wang, Z. Wang, Soliton solutions for quasilinear Schrödinger equations II, J. Differential Equations 187 (2003) 473–493. [12] J. Liu, Y. Wang, Z. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations 29 (2004) 879–901. [13] X. Liu, J. Liu, Z.Q. Wang, Ground states for quasilinear Schrödinger equations with critical growth, Calc. Var. 46 (2013) 641–669. [14] A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in RN , J. Differential Equations 229 (2006) 570–587. [15] 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. [16] D. Ruiz, G. Siciliano, Existence of ground states for a nonlinear Schrödinger equation, Nonlinearity 23 (2010) 1221–1233.

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