Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity

J. Differential Equations 254 (2013) 1977–1991

Contents lists available at SciVerse ScienceDirect

Journal of Differential Equations www.elsevier.com/locate/jde

Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity Claudianor O. Alves ∗,1 , Marco A.S. Souto 2 Universidade Federal de Campina Grande, Unidade Acadêmica de Matemática e Estatística, CEP 58429-900, Campina Grande, PB, Brazil

a r t i c l e

i n f o

Article history: Received 16 May 2012 Revised 1 November 2012 Available online 8 December 2012 MSC: 35J20 35J65 Keywords: Superlinear problem Positive solution Variational methods

a b s t r a c t In this paper we investigate the existence of positive ground state solution for the following class of elliptic equations

−u + V (x)u = K (x) f (u ) in R N , where N  3, V , K are nonnegative continuous functions and f is a continuous function with a quasicritical growth. Here, we prove a Hardy-type inequality and use it together with variational method to get a ground state solution. © 2012 Elsevier Inc. All rights reserved.

1. Introduction In recent years, many authors have considered the existence of solution for the following class of elliptic equations

⎧ ⎨ −u + V (x)u = K (x) f (u ) in R N , u (x) > 0  in R N , ⎩ 1,2 N R , u∈D

(P )

for N  3 and assuming that V , K : R N → R and f : R → R are continuous functions with V , K being nonnegative functions and f having a quasicritical growth.

* 1 2

Corresponding author. E-mail addresses: [email protected] (C.O. Alves), [email protected] (M.A.S. Souto). C.O. Alves was partially supported by INCT-MAT, PROCAD, CNPq/Brazil 620150/2008-4 and 303080/2009-4. M.A.S. Souto was supported by INCT-MAT, PROCAD, and 302650/2008-3 from CNPq/Brazil.

0022-0396/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jde.2012.11.013

1978

C.O. Alves, M.A.S. Souto / J. Differential Equations 254 (2013) 1977–1991

Problem ( P ) appears in a lot of problems, for example, the solutions of this class of problems are related to the existence of standing wave solutions for nonlinear Schrödinger equation like

ih

∂Ψ = −h2 Ψ + W (x)Ψ − f (x, Ψ ) for all x ∈ Ω, ∂t

(NLS)

where h > 0 and Ω is a domain in R N . Eq. (NLS) is one of the main objects of the quantum physics, because it appears in problems involving nonlinear optics, plasma physics and condensed matter physics. An important class of problems associated with ( P ) is the zero mass case, which occurs with the potential V vanishing at infinity, that is,

lim

|x|→+∞



V (x) = 0



shortly V (∞) = 0 .

(V ∞ )

In [5], Ambrosetti, Felli and Malchiodi studied the zero mass case with

f (s) = s p

with 2 < p < ( N + 2)/( N − 2)

and V , K satisfying the following assumptions: V , K : R N → R are smooth functions and there exist

a1 1 + |x|τ and

τ , ξ, a1 , a2 , a3 > 0 such that

 V (x)  a2 and 0 < K (x) 

a3 1 + |x|ξ

∀x ∈ R N

,

(VK)

τ , ξ verifying N +2 N −2

−

4ξ

τ ( N − 2)

if 0 < ξ < τ ,

< p,

or

1 < p,

when ξ  τ .

The condition (VK ) is interesting, because in Opic and Kufner [14] was showed that it can be used to prove that the space E given by

 E= u∈D

1,2



R

N





 2

V (x)u dx < +∞

: RN

endowed with the norm

 u 2 =



 |∇ u |2 + V (x)u 2 dx

RN

is compactly embedded into the weighted Lebesgue space

   R N = u : R N → R: u is measurable and K (x)|u | p +1 dx < ∞ .

p +1 

LK

RN

In [4], Ambrosetti and Wang have considered also the condition (VK ), however the inequality on V is assumed only outside of a ball centered at origin. In [9], Bonheure and Van Schaftingen have introduced a new set of hypotheses on K , by using the Marcinkiewicz spaces L r ,∞ (R N ) for r > 1, which permit to show continuous and compact embeddings

C.O. Alves, M.A.S. Souto / J. Differential Equations 254 (2013) 1977–1991

1979

q

from E into L K (R N ) for some q > 1. The space L r ,∞ (R N ) is formed by measurable functions h : R N → R that verify



1

hr ,∞ = sup

|h| dx < +∞.

1

| D |1 − r

D ⊂R N

D

An important subspace of L r ,∞ (R N ) is L 0 (R N ), which is the closure of L ∞ (R N ) ∩ L 1 (R N ) in L r ,∞ (R N ). In that paper, it is proved that the embedding r ,∞

p

E → L K R N



(1.1)

is continuous for all p ∈ [2, 2∗ ] if K ∈ L r ,∞ (R N ). If K ∈ L 0 (R N ), the embedding (1.1) is compact for all p ∈ [2, 2∗ ). In the above papers, the main tool used is the variational method, where the authors obtain critical points of the energy functional associated with ( P ), given by r ,∞

J (u ) =

1 2

 u 2 −

K (x) F (u ) dx

∀u ∈ E

RN

t

where F (t ) = 0 f (s) ds. The main point is that the above embeddings permit to use variational techniques very well, such as, the Mountain Pass Theorem. Motivated by the above references, more precisely by papers [4] and [5], we study problem ( P ) for a general class of nonlinearity, which includes the nonlinearity used in [4] and [5]. Moreover, a more general condition involving the functions V and K is assumed, such that (VK ) can be seen as a particular case. Hereafter, we say that ( V , K ) ∈ K if the following conditions hold: (I) V (x), K (x) > 0 ∀x ∈ R N and K ∈ L ∞ (R N ). (II) If { A n } ⊂ R N is a sequence of Borel sets such that | A n |  R, for all n and some R > 0, then

 lim

r →+∞ A n ∩ B rc (0)

K (x) dx = 0,

uniformly in n ∈ N.

(K 1 )

(III) One of the below conditions occurs:

K V

  ∈ L ∞ RN

(K 2 )

→ 0 as |x| → +∞.

(K 3 )

or there is p ∈ (2, 2∗ ) such that

K (x) 2∗ − p

[ V (x)] 2∗ −2

At this moment, it is very important to observe that ( K 1 ) is weaker than any one of the below conditions: (a) There are r  1 and ρ  0 such that K ∈ L r (R N \ B ρ (0)); (b) K (x) → 0 as |x| → ∞; (c) K = H 1 + H 2 , with H 1 and H 2 verifying (a) and (b) respectively.

1980

C.O. Alves, M.A.S. Souto / J. Differential Equations 254 (2013) 1977–1991

A simple computation shows that if V , K verify (VK ), then K ∈ K. However, there are functions V , K such that ( V , K ) belongs to K, but V , K don’t satisfy (VK ). To see this, let { B n } be a disjoint sequence of open balls in R N centered in ξn = (n, 0, 0, . . . , 0) and consider a nonnegative function H 3 such that

H 3 ≡ 0 in R N \

∞ 



H 3 (x) dx = 2−n .

H 3 (ξn ) = 1 and

Bn,

n

Bn

A direct computation shows that the pairs ( V , K ) given by

K (x) = V (x) = H 3 (x) +

1 ln(2 + |x|)

and

K (x) = H 3 (x) +



1

V (x) = H 3 (x) +

and

ln(2 + |x|)

1

2∗∗ −2

2 −p

ln(2 + |x|)

for some p ∈ (2, 2∗ ), belong to K, but the functions V , K don’t satisfy the condition (VK ). Related to the function f , we assume the following conditions:

lim sup

( f1)

f (s) s

s→0

= 0 if ( K 2 ) holds

or

lim sup s→0+

| f (s)| s p −1

< +∞ if ( K 3 ) holds.

( f 2 ) f has a quasicritical growth, that is, lim sup s→+∞

f (s)

= 0. ∗ s 2 −1

( f 3 ) s−1 f (s) is a non-decreasing function in (0, +∞) and its primitive F is superquadratic at infinity, that is,

lim

F (s)

s→+∞

s2

= +∞.

(SQ )

Related to condition ( f 3 ), we would like to mention that (SQ ) was first used in the papers of Liu and Wang [12] and Liu, Wang and Zhang [13] and that it is weaker than the well-known Ambrosetti– Rabinowitz condition (see [3]):

( f 3 ) There exists θ > 2 such that 0 < θ F (s)  sf (s)

∀ s > 0.

The below functions are typical examples of functions that verify ( f 1 )–( f 3 ):



f (s) = s+

(i) and

p

where s+ = max{s, 0}

C.O. Alves, M.A.S. Souto / J. Differential Equations 254 (2013) 1977–1991

 f (s) =

(ii)

log 2(s+ ) p , log(1 + s)s,

1981

s  1, s ∈ (1, +∞),

for some p ∈ (2, 2∗ ). Note that in the example (ii), the function f does not satisfy condition ( f 3 ). Now, we are able to state our main theorem. Theorem 1.1. Suppose that ( V , K ) ∈ K and ( f 1 )–( f 3 ) hold. Then, problem ( P ) possesses a positive ground state solution, that is, a positive solution u ∈ E with energy equal to the mountain pass level associated with J . In the proof of Theorem 1.1, we use a Hardy-type inequality involving V and K , see Proposition 2.1, together with a version of Mountain Pass Theorem with Cerami condition, see Section 2. It is very important to observe that Theorem 1.1 improved the results of existence of solutions proved in [4] and [5], once that our hypotheses are more general. Before to conclude this introduction, we would like to cite the papers by Berestycki and Lions [8], Badiale, Pisani and Rolando [6], Benci, Grisanti and Micheletti [7], Ghiment and Micheletti [11], Alves and Souto [1] and Alves, Montenegro and Souto [2] and references therein, where problems related to the zero mass case were also considered. The paper is organized as follows. In the next section we present some preliminary results. In Section 3, we prove Theorem 1.1 and in Section 4 we show that our arguments can be used to study a class of problems where potential V can be null at some set of R N . 2. Preliminary results The Euler–Lagrange functional associated with ( P ) is given by

J (u ) =

1 2

 u 2 −

K (x) F (u ) dx,

∀u ∈ E .

RN

From the conditions on f , the functional J ∈ C 1 ( E , R) and its Gateaux derivative is given by

J (u ) v =





 ∇ u ∇ v + V (x)uv dx −

RN

 K (x) f (u ) v dx,

∀u , v ∈ E .

RN

It is easy to check that critical points of J are weak solutions of ( P ). Once that we intend to find positive solutions, we will assume that

f (s) = 0 ∀s ∈ (−∞, 0].

(2.2)

From ( f 1 ) and ( f 2 ), it follows that J satisfies the geometry of the mountain pass (see [3]). Hence, there is a sequence (un ) ⊂ E such that

J (u n ) → c

and







1 + un   J (un ) → 0,

where c is the mountain pass level given by

c = inf max J γ ∈Γ t ∈[0,1]





γ (t )

with

      Γ = γ ∈ C [0, 1], H 01 (Ω) /γ (0) = 0 and J γ (1)  0 .

(2.3)

1982

C.O. Alves, M.A.S. Souto / J. Differential Equations 254 (2013) 1977–1991

The above sequence is the so-called Cerami sequence for J at level c, see [10]. From (2.2), a direct computation shows that, without loss of generality, we can assume that (un ) is formed by nonnegative functions. Our first result in this section is the following Hardy-type inequality. q

Proposition 2.1 (Hardy-type inequality). Assume that ( V , K ) ∈ K. Then, E is compactly embedded in L K (R N ) p for all q ∈ (2, 2∗ ) if ( K 2 ) holds. If ( K 3 ) holds, E is compactly embedded in L K (R N ). Proof. The proof will be made into two parts, firstly we consider the condition ( K 2 ), and after ( K 3 ). By assuming that ( K 2 ) is true, fixed q ∈ (2, 2∗ ) and given ε > 0, there are 0 < s0 < s1 and C > 0 such that



K (x)|s|q  ε C V (x)|s|2 + |s|2

∗



  ∗ + C K (x)χ[s0 ,s1 ] |s| |s|2

∀s ∈ R.

(2.4)

Hence,



 K (x)|u |q  ε C Q (u ) + C

K (x) dx

∀u ∈ E

(2.5)

A ∩ B rc (0)

B rc (0)

where



 V (x)|u |2 dx +

Q (u ) = RN

∗

|u |2 dx,

RN

and









A = x ∈ R N : s0  u (x)  s1 . If ( v n ) is a sequence such that v n  v in E, there is M 1 > 0 such that





 |∇ v n |2 + V (x)| v n |2 dx  M 1 and

RN



∗

| v n |2 dx  M 1 ∀n ∈ N, RN

implying that ( Q ( v n )) is bounded. On the other hand, setting









A n = x ∈ R N : s0   v n (x)  s1 , the last inequality implies that ∗



∗

| v n |2 dx  M 1 ∀n ∈ N,

s0 2 | A n |  An

showing that supn∈N | A n | < +∞. Therefore, from ( K 1 ), there is an r > 0 such that

 K (x) dx < A n ∩ B rc (0)

ε s21

∗

for all n ∈ N.

(2.6)

C.O. Alves, M.A.S. Souto / J. Differential Equations 254 (2013) 1977–1991

1983

Now, (2.5) and (2.6) lead to

 K (x)| v n |q dx  ε C M 1 + s21



∗

K (x) dx < (C M 1 + 1)ε

∀n ∈ N.

(2.7)

F n ∩ B rc (0)

B rc (0)

Once that q ∈ (2, 2∗ ) and K is a continuous function, it follows from Sobolev embeddings



 K (x)| v n |q dx =

lim

n→+∞ B r ( 0)

K (x)| v |q dx.

(2.8)

B r ( 0)

Combining (2.7) and (2.8),



 K (x)| v n |q dx =

lim

n→+∞

RN

K (x)| v |q dx,

(2.9)

RN

which yields q



in L K R N

vn → v



  ∀q ∈ 2, 2∗ .

Now, we will suppose that ( K 3 ) holds. First of all, it is important to observe that for each x ∈ R N fixed, the function ∗ g (s) = V (x)s2− p + s2 − p ,

∀s > 0

2∗ − p

has C p V (x) 2∗ −2 as its minimum value, where

Cp =

2∗ − 2



2∗ − 2

2∗− p

p−2

2 −2

2∗ − 2

.

Hence, ∗

2 −p ∗ C p V (x) 2∗ −2  V (x)s2− p + s2 − p ,

Combining the last inequality with ( K 3 ), given

∀x ∈ R N and s > 0.

ε ∈ (0, C p ), there is r > 0 large enough, such that



K (x)|s| p  ε V (x)|s|2 + |s|2

∗



∀s ∈ R and |x|  r ,

leading to



 K (x)|u | p dx  ε

B rc (0)



V (x)|u |2 + |u |2

∗



∀u ∈ E .

B rc (0)

If ( v n ) is a sequence such that v n  v in E, there is M 1 > 0 such that



 V (x)| v n |2 dx  M 1

RN

∗

| v n |2 dx  M 1 ∀n ∈ N,

and RN

1984

C.O. Alves, M.A.S. Souto / J. Differential Equations 254 (2013) 1977–1991

and so,

 K (x)| v n | p dx  2ε M 1

∀n ∈ N.

(2.10)

B rc (0)

Once that p ∈ (2, 2∗ ) and K is a continuous function, it follows from Sobolev embeddings

 lim

n→+∞ B r ( 0)

 K (x)| v n | p dx =

K (x)| v | p dx.

(2.11)

B r ( 0)

From (2.10) and (2.11),



 K (x)| v n | p dx =

lim

n→+∞

RN

K (x)| v | p dx,

RN

implying that

vn → v finishing the proof of the proposition.

p



in L K R N ,

2

Next lemma is an important step to prove that the Cerami sequence obtained in (2.3) is bounded. Lemma 2.2. Suppose that f satisfies ( f 1 )–( f 2 ) and ( V , K ) ∈ K. Let ( v n ) be a sequence such that v n  v in E. Then



 K (x) F ( v n ) dx =

lim

n→+∞

RN

K (x) F ( v ) dx

RN

and



 K (x) f ( v n ) v n dx =

lim

n→+∞

RN

K (x) f ( v ) v dx.

RN

Proof. We will begin the proof by assuming that ( K 2 ) occurs. From ( f 1 )–( f 3 ), fixed q ∈ (2, 2∗ ) and given ε > 0, there is C > 0 such that

     K (x) F (s)  ε C V (x)|s|2 + |s|2∗ + K (x)|s|q for all s ∈ R.

(2.12)

From Proposition 2.1,



 K (x)| v n |q dx →

RN

K (x)| v |q dx,

RN

then there is r > 0 such that

 K (x)| v n |q dx < ε B rc (0)

∀n ∈ N.

(2.13)

C.O. Alves, M.A.S. Souto / J. Differential Equations 254 (2013) 1977–1991

1985

Since ( v n ) is bounded in E, there is M 1 > 0 such that



 V (x)| v n |2  M 1

∗

V (x)| v n |2  M 1 .

and

RN

RN

Combining the last inequalities with (2.12) and (2.13),

    

  K (x) F ( v n ) dx < (2C M 1 + 1)ε ∀n ∈ N.

(2.14)

B rc (0)

Now, if ( K 3 ) holds, repeating the same arguments explored in the proof of Proposition 2.1, given small enough, there is r > 0 large enough such that

  ∗ ∀s ∈ R and |x|  r . K (x)  ε V (x)|s|2− p + |s|2 − p Hence,

      ∗   ∀s ∈ R and |x|  r . K (x) F (s)  ε V (x) F (s)|s|2− p +  F (s)|s|2 − p From ( f 1 ) and ( f 2 ), there are C , s0 , s1 > 0 verifying







K (x) F (s)  ε C V (x)|s|2 + |s|2

∗



∀s ∈ I and |x|  r

where I = {s ∈ R: |s| < s0 or |s| > s1 }. Thereby, for any u ∈ E, we have the following estimate



 K (x) F (u )  ε C Q (u ) + C

K (x) dx A ∩ B rc (0)

B rc (0)

with

 Q (u ) =

 V (x)|u |2 dx +

RN

∗

|u |2 dx,

RN

and









A = x ∈ R N : s0  u (x)  s1 . Once that ( v n ) is bounded in E, there is M 1 > 0 such that



 V (x)| v n |2  M 1

RN

∗

V (x)| v n |2  M 1 .

and RN

Thus,



 K (x) F ( v n ) dx  2M 1 ε + C

B rc (0)

K (x) dx A n ∩ B rc (0)

ε>0

1986

C.O. Alves, M.A.S. Souto / J. Differential Equations 254 (2013) 1977–1991

where









A n = x ∈ R N : s0   v n (x)  s1 . Repeating the same arguments used in the proof of Proposition 2.1, it follows that

 K (x) dx → 0 as |x| → +∞, A n ∩ B rc (0)

and so, for n large enough

    

 

K (x) F ( v n ) dx  (2M 1 + 1)ε .

B rc (0)

To conclude the proof, we need to show that

 lim

n→+∞ B r ( 0)

 K (x) F ( v n ) dx =

K (x) F ( v ) dx. B r ( 0)

However, this limit follows by using the compactness lemma of Strauss [8, Theorem A.I, p. 338], in the following way: B r (0) is a bounded domain, | v n | L 2∗ ( B r (0)) is bounded and limit ( f 2 ) together with the convergence almost everywhere imply the limits as required. 2 Now, we are able to prove that the Cerami sequence given in (2.3) is bounded in E. Lemma 2.3. The Cerami sequence (un ) ⊂ E given in (2.3) is bounded. Proof. Let tn ∈ [0, 1] be such that J (tn un ) = maxt 0 J (tun ). We claim that ( J (tn un )) is bounded from above. In fact, if either tn = 0 or tn = 1, we are done. Thereby, we can assume tn ∈ (0, 1), and so, J (tn un )un = 0. From this,

2 J (tn un ) = 2 J (tn un ) − J (tn un )tn un =

 K (x) H (tn un ) dx

RN

where

H (s) = sf (s) − 2F (s)

∀s ∈ R.

Recalling that un  0, the condition ( f 3 ) gives that H is non-decreasing for each x. Therefore,

 2 J (tn un ) 

K (x) H (un ) dx = 2 J (un ) − J (un )un = 2 J (un ) + on (1).

RN

Since ( J (un )) converges, it follows that ( J (tn un )) is bounded from above.

C.O. Alves, M.A.S. Souto / J. Differential Equations 254 (2013) 1977–1991

1987

Now, we are able to prove that (un ) is bounded. To this end, suppose by contradiction that un  → ∞ and set w n = uunn  . Since  w n  = 1, there exists w ∈ E such that w n  w in E. Next, we will show that w = 0. First of all, notice that

on (1) +

1 2





K (x) F (un )

=

un 2

RN

The condition ( f 3 ) implies that for each

K (x) F (un )

dx =

| u n |2

RN

w n2 dx.

τ > 0, there is ξ > 0 such that

F (s)

τ

| s |2

for |s|  ξ.

Hence,

on (1) +

1 2



K (x) F (un )



| u n |2

Ω∩{|un |ξ }

 | w n |2 dx  τ

K (x)| w n |2 dx, Ω∩{|un |ξ }

where Ω = {x ∈ R N : w (x) = 0}. By Fatou’s Lemma



1

K (x) w 2 dx

τ

2

∀ τ > 0.

Ω

Therefore,

 K (x) w 2 dx = 0. Ω

Once that K (x) > 0 for all x ∈ R N , we have that |Ω| = 0, showing that w = 0. Notice that for each B > 0, one has uB  ∈ [0, 1] for n sufficiently large. Thus n

J (tn un )  J

B

u n 

= J (B wn) =

un

B2 2

 −

K (x) F ( B w n ).

RN

From Lemma 2.2

 lim

n→∞

K (x) F ( B w n ) = 0,

(2.15)

RN

from where it follows that

lim inf J (tn un )  n→∞

B2 2

for every B > 0,

which is a contradiction with Lemma 2.3, once that ( J (tn un )) is bounded from above.

2

1988

C.O. Alves, M.A.S. Souto / J. Differential Equations 254 (2013) 1977–1991

3. Proof of Theorem 1.1 Hereafter, (un ) denotes the Cerami sequence given in (2.3), that is,

J (u n ) → c



and





1 + un   J (un ) → 0.

By Lemma 2.3, it follows that (un ) is bounded, up to some subsequence, we can assume that there is u ∈ E such that

un  u

in E .

Recalling that J (un )un = on (1), we derive

 lim un 2 = lim

n→+∞

K (x) f (un )un dx.

n→+∞

(3.16)

RN

By Lemma 2.2,



 K (x) f (un )u dx =

lim

n→+∞

RN

K (x) f (u )u dx

RN

then,

 lim un 2 =

n→+∞

K (x) f (u )u dx.

RN

Since J (un )u = on (1),

 u 2 =

K (x) f (u )u dx.

(3.17)

RN

Thereby, from (3.16) and (3.17),

lim un 2 = u 2

n→+∞

showing that

un → u

in E .

Consequently

J (u ) = c

and

J ( u ) = 0,

implying that u is a ground state solution for J . Since un  0, we have that u  0 and u = 0. The positivity of u follows by using maximum principle.

C.O. Alves, M.A.S. Souto / J. Differential Equations 254 (2013) 1977–1991

1989

4. Final comments The main goal of this section is to show that we can apply, with few modifications, the same arguments used in the proof of Theorem 1.1 to study a large class of problems. Here, we assume that V , K are nonnegative functions, not necessarily strictly positive, and f verifies the following conditions:

( f 4 ) lim sup|s|→0 |s|2∗sf+(Vx,(sx))|s|2 = 0, uniformly in x, where 2∗ = 2N /( N − 2). ( f 5 ) f has a quasicritical growth, that is, lim sup |s|→+∞

sf (x, s)

| s |2

∗

= 0,

uniformly in x.

( f 6 ) s−1 f (x, s) is a non-decreasing function in (0, +∞) and non-increasing in (−∞, 0), for all x ∈ R N and its primitive F is superquadratic at infinity, that is, there is a positive function G : R N → R such that

lim

F (x, s)

|s|→+∞ G (x)|s|2

= +∞ (uniformly at x).

Before to state the main result this section, we have to fix some notations. In what follows, let us denote by Ω ⊂ R N the following set

  Ω = x ∈ R N : V (x) > 0 and by W : Ω → [0, +∞) the function given by

K (x)

W (x) =

for some

1

[ V (x)] α

α  1. Moreover, we say that K ∈ Kˆ , if ( K 1 ) holds, and the following conditions are verified:

(i) K (x)  0 ∀x ∈ R N and there is p ∈ (2, 2∗ ) such that K ∈ L ηq (R N \ Ω) for

(2, p ). Moreover, K ∈ L r ,∞ (R N ) for some r  1 verifying

p

1 2

−

1 N

+

1 r

ηq =

2∗ 2∗ −q

for all q ∈

=1

and

  sf (x, s)  K (x)|s|2∗

for all x ∈ R N and s ∈ R.

(K 0 )

(ii) W ∈ L β (Ω) for β  1 and β1 + α1 = 1. The main result of this section is the following:

ˆ and that ( f 4 )–( f 6 ) are satisfied. Theorem 4.1. Suppose that V is a continuous nonnegative potential, K ∈ K Then, problem



−u + V (x)u = f (x, u ) in R N , u ∈ D 1,2 R N

possesses a nontrivial ground state solution.

( P 1)

1990

C.O. Alves, M.A.S. Souto / J. Differential Equations 254 (2013) 1977–1991

Using the same idea explored in Section 3, we look for critical points of the functional

J 1 (u ) =

1 2

 u 2 −

∀u ∈ E .

F (x, u ) dx,

RN

For this class of problems, we will use the following version of Proposition 2.1:

ˆ . Then E is compactly embedded in Proposition 4.2 (Hardy-type inequality). Assume that K belongs to K q L K (R N ) for all q ∈ (2, p ). Proof. From the definition of W , given that





 α1

K (x)|s|q  ε W (x) V (x)

ε > 0, there are 0 < s0 < s1 such that for all x ∈ Ω , we have

   ∗ 2 |s| α + K (x)|s| p + C K (x)χ[s0 ,s1 ] |s| |s|2

∀s ∈ R,

ˆ -(iii) implies that the embedding for some positive constant C . By a result found in [9], the condition K p

E → L K R N



is continuous. Thereby, combining this information with the fact that W ∈ L β (Ω), the same arguments used in the proof of Proposition 2.1 can be used to show that if ( v n ) is a sequence such that v n  v in E, there is M 1 > 0 such that

 K (x)| v n |q dx  M 1 ε

∀n ∈ N.

(4.18)

Ω∩ B rc (0)

Since p ∈ (2, 2∗ ), it follows from Sobolev embeddings



 K (x)| v n |q dx =

lim

n→+∞

Ω∩ B r (0)

K (x)| v |q dx.

(4.19)

Ω∩ B r (0)

Now, (4.18) combined with (4.19) leads to



 K (x)| v n |q dx =

lim

n→+∞

Ω

K (x)| v |q dx.

(4.20)

Ω

Recalling that K ∈ L ηq (R N \ Ω), the Lebesgue Theorem gives

 lim

n→+∞

R N \Ω

 K (x)| v n |q dx =

K (x)| v |q dx.

(4.21)

R N \Ω

Now, the proposition follows combining (4.20) and (4.21).

2

An immediate consequence of the last result is the corollary below.

ˆ hold for all p ∈ (2, 2∗ ), then Corollary 4.3. On the hypotheses of Proposition 4.2, if (i) and (iii) in the class K q E is compactly embedded in L K (R N ) for all q ∈ (2, 2∗ ).

C.O. Alves, M.A.S. Souto / J. Differential Equations 254 (2013) 1977–1991

1991

Next lemma is crucial in the study of the boundedness of Cerami sequence for J 1 and its proof follows the same arguments used in Section 2. Lemma 4.4. Suppose that f satisfies ( f 4 )–( f 6 ). Let ( v n ) be a sequence such that v n  v in E. Then



 F (x, v n ) dx =

lim

n→+∞

RN

F (x, v ) dx

RN

and

 lim

n→+∞

RN

 f (x, v n ) v n dx =

f (x, v ) v dx.

RN

Lemma 4.5. The Cerami sequence (un ) ⊂ E given in (2.3) is bounded. Proof. The proof follows, with few modifications, using the same type of arguments used in the proof of Lemma 2.3. 2 Proof of Theorem 4.1. The proof follows repeating the ideas explored in the proof of Theorem 1.1.

2

Acknowledgment The authors are grateful to the referee for his/her comments for improvement of the article. References [1] C.O. Alves, M.A.S. Souto, Existence of solutions for a class of elliptic equations in R N with vanishing potentials, J. Differential Equations 252 (2012) 5555–5568. [2] C.O. Alves, M. Montenegro, M.A.S. Souto, Existence of solution for three classes of elliptic problems in R N with zero mass, J. Differential Equations 252 (2012) 5735–5750. [3] A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381. [4] A. Ambrosetti, Z.-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differential Integral Equations 18 (2005) 1321–1332. [5] A. Ambrosetti, V. Felli, A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc. (JEMS) 7 (2005) 117–144. [6] M. Badiale, L. Pisani, S. Rolando, Sum of weighted Lebesgue spaces and nonlinear elliptic equations, NoDEA Nonlinear Differential Equations Appl. 18 (2011) 369–405. [7] V. Benci, C.R. Grisanti, A.M. Micheletti, Existence of solutions for the nonlinear Schrödinger equation with V (∞) = 0, in: Progr. Nonlinear Differential Equations Appl., vol. 66, 2005, pp. 53–65. [8] H. Berestycki, P.L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983) 313–346. [9] D. Bonheure, J. Van Schaftingen, Groundstates for nonlinear Schrödinger equation with potential vanishing at infinity, Ann. Mat. Pura Appl. 189 (2010) 273–301. [10] G. Cerami, Un criterio de esistenza per i punti critici su varieta ilimitate, Istit. Lombardo Accad. Sci. Lett. Rend. 112 (1978) 332–336. [11] M. Ghiment, A.M. Micheletti, Existence of minimal nodal solutions for the nonlinear Schrödinger equation with V (∞) = 0, preprint. [12] Z. Liu, Z.-Q. Wang, On the Ambrosetti–Rabinowitz superlinear condition, Adv. Nonlinear Stud. 4 (4) (2004) 563–574. [13] Y. Liu, Z.-Q. Wang, J. Zhang, Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (6) (2006) 829–837. [14] B. Opic, A. Kufner, Hardy-Type Inequalities, Pitman Res. Notes Math. Ser., vol. 219, Longman Scientific and Technical, Harlow, 1990.