Multiplicity of solutions for a scalar field equation involving a fractional p-Laplacian with general nonlinearity
aa r X i v : . [ m a t h . A P ] F e b MULTIPLICITY OF SOLUTIONS FOR A SCALAR FIELDEQUATION INVOLVING A FRACTIONAL p -LAPLACIAN WITHGENERAL NONLINEARITY H. P. BUENO, O. H. MIYAGAKI, AND A. L. VIEIRA
Abstract.
We investigate the existence of infinitely many radially symmetricsolutions for the following nonlinear scalar field equation involving a fractional p -Laplacian, ( − ∆ p ) s u = g ( u ) in R N , where s ∈ (0 , ≤ p < ∞ , 2 ≤ N = sp , N ∈ N and g ∈ C ( R , R ) is an oddfunction with both exponential and polynomial growth. The argument of theproof is based on the case N = 2 of the symmetric mountain pass approachdeveloped by Hirata, Ikoma and Tanaka in [14]. Introduction
In this paper we deal with the following nonlinear scalar field equation( − ∆ p ) s u = g ( u ) in R N , (1)where s ∈ (0 , ≤ p < ∞ , 2 ≤ N = sp , N ∈ N and ( − ∆ p ) s is the fractional p -Laplacian defined by( − ∆ p ) s u ( x ) := 2 lim ε → + Z R N \ B ε ( x ) | u ( x ) − u ( y ) | p − ( u ( x ) − u ( y )) | x − y | N + sp d y (2)(see also Section 2).In recent years great attention has been devoted to the study of elliptic equa-tions involving the fractional p -Laplacian operator. Mainly when p = 2, it appearsin many models arising from concrete applications in Biology (e.g., population dy-namics), Physics (e.g., continuum mechanics, phase transition phenomena), GameTheory and Financial Mathematics, see [5, 6]. But the operator also draws attentionfrom a purely mathematical point of view, because of the challenging difficulties dueto its both nonlocal and nonlinear character [19].The problem (1) has already evolved into an elaborate theory whose literature istoo broad to attempt any comprehensive synthesis on a single paper. We refer theinterested reader to the papers [5, 6, 13, 19] and references therein. Mathematics Subject Classification.
Key words and phrases.
Fractional p -Laplacian; symmetric criticality; Moser-Trudinger in-equality; exponential and polynomial growth.First author takes part in the project 422806/2018-8 by CNPq/Brazil.Second author was supported by Grant 2019/24901-3 by S˜ao Paulo Research Foundation(FAPESP) and Grant 307061/2018-3 by CNPq/Brazil. In the paper [14], Hirata, Ikoma and Tanaka extend slightly the results of Beresty-cki, Gallou¨et and Kavian [7], but unified the treatment developed before for the cases N = 2 and N ≥
3. They studied existence and multiplicity of nontrivial radiallysymmetric solutions to (1) in the limit case s = 1 and p = 2, that is, − ∆ u = g ( u ) in R N , u ∈ H ( R N ) . (3)To obtain infinitely many solutions to problem (3), the authors applied simul-taneously the mountain pass theorem and its symmetric version (see [21, chapter9]) as follows. After a truncation of the nonlinearity g , they obtained a function h which satisfies the general Ambrosetti-Rabinowitz condition ([14, Cor. 2.2]).Then they introduced a comparison functional J ( u ), which satisfies the Palais-Smale compactness condition to obtain an unbounded sequence of minimax valuesfor the energy functional I ( u ) associated to (3) ([14, Lemma 3.2]). By introducinga suitable auxiliary functional ˜ I ( θ, u ) and combining with a Pohozaev identity anda Trudinger-Moser type inequality [1], the authors proved that the unbounded se-quence of minimax values was, in fact, an unbounded sequence of critical values to I ( u ). In this proof, the Pohozaev identity played a crucial role when proving thecompactness condition for the Palais-Smale sequence.When N = 2, we observe that the problem (3) can be seen as the particular limitcase s = 1 and p = 2 of the problem (1). This motivate us to ask whether if it ispossible to study other cases in which N = sp of the problem (1) in the case of anonlinearity g satisfying polynomial and exponential growth.However, the method used by the authors in [14] does not apply immediatelyto problem (1), since a Pohozaev identity for the fractionary p -Laplacian is stillunknown.Therefore, inspired by Bueno, Caqui and Miyagaki [10] and by Bueno, Caqui,Miyagaki and Pereira [11] and references therein, we consider in (1) a nonlinearity g with both exponential and polynomial growth satisfying( g ,sp ) g ( t ) ∈ C ( R , R ), g ( t ) is odd and G ( t ) = R t g ( τ ) dτ ≥ , ∀ t ∈ R ;( g ,sp ) lim | t |→∞ | g ( t ) | e α | t | NN − s = 0 , ∀ α > α > α > , in the critical case );( g ,sp ) lim | t |→ | g ( t ) || t | p − t = 0;( g ,sp ) lim | t |→∞ G ( t ) | t | p = + ∞ . An example of nonlinearity g satisfying the hypothesis ( g ,sp ) − ( g ,sp ) is givenby g ( t ) = | t | p − t log(1 + | t | ) . Observe that we require that G ( t ) ≥ t ∈ R , a condition that will beuseful when applying Fatou’s Lemma.As in [8, 14], we consider the problem (1) in the closed subspace of radial functionsand apply the Principle of Symmetric Criticality due to Palais [18]. ULTIPLICITY OF SOLUTIONS FOR A SCALAR FIELD EQUATION 3
We point out that, for nonlinearities with exponential growth in the case ofa problem like (1) with N = sp , Ozawa [17] proved a version of the Trudinger-Moser inequality for fractional Sobolev spaces, see also Theorem 2.1 in the sequel.Moreover, for N ≥ s = 1 and p = N , a Trudinger-Moser inequality for theSobolev space W ,p ( R p ) was proved by Adachi and Tanaka[1] (see also Theorem2.2 in the sequence). The Trudinger-Moser inequality plays an important role inproblem like (1) because it can be used to prove both the geometry of the symmetriccase and also a compactness condition for the Palais-Smale sequence.By simple arguments of real analysis, we obtain a version of a result by Berestyckiand Lions [8, Theorem 10] for the problem (1) on the fractional Sobolev subspaceof radial functions. That is, we prove that, for any n ∈ N , there exists an oddcontinuous map π n from the unit sphere of R N into the fractional Sobolev subspaceof radial functions, such that R R N G ( π n ( σ ))d x ≥ σ in the unit sphere, seeTheorem 3.1. As a consequence, we obtain the necessary coerciveness hypothesisused to prove the geometry of the symmetric mountain pass theorem in a finitedimensional subspace.Our main result is the following. Theorem 1.1.
Let s ∈ (0 , , p ∈ [2 , + ∞ ) , sp = N ≥ , N ∈ N and g be a functionsatisfying ( g ,sp ) − ( g ,sp ) . Then (1) possesses infinitely many radially symmetricsolutions ( u n ) n ∈ N such that I ( u n ) → ∞ as n → ∞ .The same result is valid in the limit case s = 1 . Additionally supposing that, for all non-trivial weak solution v we have Z R N G ( v )d x > Theorem 1.2.
There exists a non-negative solution u to the problem (1) such that I ( u ) ≤ I ( v ) and for any non-trivial solution v to (1) . In fact, hypothesis (4) implies that the sequence of critical values obtained in theproof of Theorem 1.1 is constant (see Lemma 3.5). Details can be found in [5].2.
Preliminaries
We recall some useful results on fractional Sobolev space. For details see e.g.,[12, 13, 15].We consider the fractional Sobolev space W s,p ( R N ) := { u ∈ L p ( R N ) : [ u ] ps,p < ∞} , where [ u ] ps,p := Z Z R N | u ( x ) − u ( y ) | p | x − y | N + sp d x d y is the Gagliardo seminorm of u . This space, endowed with the natural norm k u k s,p =([ u ] ps,p + k u k pp ) p (where k u k p = k u k L p ( R N ) ) is a reflexive Banach space (see [12]). H. P. BUENO, O. H. MIYAGAKI, AND A. L. VIEIRA
Due to Principle of Symmetric Criticality, it is sufficient to find solutions to theproblem (1) in the closed subspace of radial functions, that is, in the space W s,pr ( R N ) := { u ∈ W s,p ( R N ); u ( | x | ) = u ( x ) } . We recall that W s,pr ( R N ) is compactly embedded in L q ( R N ) for any q ∈ ( p, + ∞ ), s ∈ (0 , p ∈ [1 , + ∞ ), N ≥ sp = N . (See [16, Theorem II.1].)A weak solution of (1) satisfies, for any v ∈ W s,pr ( R N ), h ( − ∆ p ) s u, v i = Z R N g ( u ) v d x, where h ( − ∆ p ) s u, v i = Z Z R N | u ( x ) − u ( y ) | p − ( u ( x ) − u ( y ))( v ( x ) − v ( y )) | x − y | N + sp d x d y. As proved in [15], we know that is continuous the linear functional η
7→ h ( − ∆ p ) s u, η i . For the reader’s convenience, we recall the main results about Trudinger-Moserinequalities for this class of fractional Sobolev spaces. We start stating a simplifiedversion of Theorem 2.4 in [4].
Theorem 2.1.
Let s = Np ∈ (0 , and < p < ∞ satisfy s + p = 1 . Then thereexist positive constants γ and c ( γ ) such that, for all u ∈ W s,pr ( R N ) with k u k s,p ≤ Z R N e γ | u ( x ) | pp − − X ≤ j
such that Z R p Φ p α (cid:18) | u ( x ) |k∇ u k p (cid:19) pp − ! d x ≤ C α k u k pp k∇ u k pp , u ∈ W ,pr ( R p ) \{ } , (6) where Φ p ( η ) = e η − p − X j =0 j ! η j . We consider the quotient space X r = { u ∈ W s,pr ( R N ) : u ≡ C ⇔ C = 0 } ≡ W s,pr ( R N ) / ∼ C, endowed with the natural norm k · k s,p . It is known that ( X r ; k · k s,p ) is a reflexiveBanach space. Moreover, we have the compact embedding X r ֒ → ֒ → L q ( R N ) ULTIPLICITY OF SOLUTIONS FOR A SCALAR FIELD EQUATION 5 for any q > p , the embedding being continuous in the case q ≥ p . (See [13, Theorem6.9] and [16, Theorem II.1].) Remark 2.3.
As a consequence the subset L = { [ u ] s,p : u ∈ X r , k u k p = 1 } is pre-compact in L q ( R N ) . See [13, Theorem 7.1] . We denote by λ r = inf L . It is well-known that λ r is attained and positive. Moreover, k u k p ≤ λ − p [ u ] s,p , forany u ∈ X r . Furthermore, [ · ] s,p defines a norm equivalent to k · k s,p on X r , whichwill be used from now on. 3. Proof of Theorem 1.1
Our first result simply adapts the arguments of Theorem 10 of Berestycki-Lions [8] for the space X r . Let S n − stand for the unit sphere in R N . Theorem 3.1.
For all n ∈ N , there exists an odd continuous map π n : S n − → X r such that ( i ) π n ( σ ) is radially symmetric in S n − ; ( ii ) 0 π n ( S n − ) ; ( iii ) Z R N G ( π n ( σ ))d x ≥ , for all σ ∈ S n − . This result follows by considering the subset V ⊂ X r and the functional J : X r → R given by V = (cid:26) w ∈ X r : Z R N G ( w ) dx = 1 (cid:27) , J ( w ) = Z R N G ( w )d x. It is easy to verify that J satisfies the intermediate value theorem in a suitableconvex set. Thus, V = ∅ and Theorem 3.1 is a consequence of that.Since the derivative of the “energy” functional I ( u ) = 1 p [ u ] ps,p − Z R N G ( u )d x, is given by I ′ ( u ) · η = Z Z R N | u ( x ) − u ( y ) | p − ( u ( x ) − u ( y ))( η ( x ) − η ( y )) | x − y | N + sp d x d y − Z R N g ( u ) η d x, we see that critical points of I ( u ) are weak solutions to (1). Lemma 3.2.
The functional I satisfies the geometry of the symmetric mountainpass theorem, that is, ( i ) There are β, ρ > such that: I ( u ) ≥ β > for k u k s,p = ρ and I ( u ) ≥ for k u k s,p ≤ ρ ;( ii ) For any n ∈ N , there exists an odd continuous mapping γ n : S n − → X r such that I ( γ n ( σ )) < , ∀ σ ∈ S n − . H. P. BUENO, O. H. MIYAGAKI, AND A. L. VIEIRA
Proof.
Since g is odd and continuous, it follows I (0) = 0. Fixed any θ > p , it followsfrom ( g ,sp ) that there exists C > | G ( t ) | ≤ C | t | θ e α | t | NN − s , ∀ t ∈ R . Therefore, I ( u ) = 1 p [ u ] ps,p − Z R N G ( u )d x ≥ p [ u ] ps,p − C Z R N e α | u | NN − s | u | θ d x. Thus, for any fixed r > p , we can take α ∈ (0 , α ) such that α r ∈ (0 , α ). Byapplying H¨older’s inequality with r ′ = r/ ( r − I ( u ) ≥ p [ u ] ps,p − C (cid:18)Z R N e αr | u | NN − s dx (cid:19) r (cid:18)Z R N | u | θr ′ dx (cid:19) r ′ . Suppose now that k u k s,p ≤
1. It follows from the Trudinger-Moser inequality(Theorem 2.1) that I ( u ) ≥ p [ u ] ps,p − C ′ [ u ] θs,p = [ u ] ps,p (cid:18) p − C ′ [ u ] θ − ps,p (cid:19) . Since θ − p >
0, by taking [ u ] s,p small enough, we conclude the proof of ( i ).The fact that π n is continuous guarantees the existence of M > k π n ( σ ) k s,p ≤ M, for all σ ∈ S n − . Let us define, for t ≥ φ tn ( σ )( x ) = π n ( σ ) (cid:16) xt (cid:17) . It follows I ( φ tn ) = t N − sp p [ π n ( σ )] ps,p − t N Z R N G ( π n ( σ ))d x ≤ M p p − t N and we conclude that I ( φ tn ) → −∞ as t → ∞ . Therefore we can chose ¯ t > I ( φ ¯ tn ) < σ ∈ S n − . Thus, γ n ( σ )( x ) := φ ¯ tn ( σ )( x ) (7)satisfies the required properties, concluding the proof of ( ii ). (cid:3) Adapting some ideas of Bueno, Caqui and Miyagaki [10], we obtain.
Lemma 3.3.
Any ( P S ) c -sequence ( u j ) is bounded in X r .Proof. Suppose that k u j k s,p → + ∞ as j → ∞ . We set v j = u j k u j k s,p , j ∈ N . Hence, v j ⇀ v ∈ X r , v j → v in L q ( R N ) for all q > p and v j ( x ) → v ( x ) a.e. in R N .Let us suppose initially that v = 0. Then, Θ = { x ∈ R N ; v ( x ) = 0 } has positiveLebesgue measure and | u j ( x ) | = | v j ( x ) |k u j ( x ) k s,p → ∞ , ∀ x ∈ Θ. Since G ( u j ) k u j k ps,p = G ( u j ) | v j | p | u j | p , ULTIPLICITY OF SOLUTIONS FOR A SCALAR FIELD EQUATION 7 it follows from ( g ,sp ) that lim inf j →∞ G ( u j ) k u j k ps,p = + ∞ . Our hypothesis ( g ,sp ) then implieslim sup j →∞ Z v =0 G ( u j ) k u j k ps,p d x ≥ Z v =0 lim inf j →∞ G ( u j ) | v j | p | u j | p d x = ∞ . But c + o (1) = I ( u j ) = 1 p [ u j ] ps,p − Z R N G ( u j )d x = 1 p k u j k ps,p − Z R N (cid:18) | u j | p p + G ( u j ) (cid:19) d x and therefore1 p k u j k ps,p − ( c + o (1)) = Z R N | u j | p p + G ( u j )d x ≥ Z R N G ( u j )d x. Hence, 1 p − ( c + o (1)) k u j k ps,p ≥ Z R N G ( u j ) k u j k ps,p d x ≥ Z v =0 G ( u j ) | v j | p | u j | p d x. Thus, lim sup j →∞ G ( u j ) | v j | p | u j | p d x ≤ p and we have reached a contradiction.Suppose now that v = 0. Since I ′ ( u j ) →
0, we have I ′ ( u j ) · φ = o (1) k φ k s,p forany φ ∈ X r . Let A : X r → ( X r ) ∗ be the operator defined by h A ( u ) , v i = Z R N g ( u ) v d x, ∀ u, v ∈ X r . (8)We have that, h A ( v j ) , φ i − Z R N g ( u j ) φ k u j k p − s,p d x = o (1) k φ k s,p k u j k p − s,p , ∀ φ ∈ X r (9)Since v j ⇀ v and the functional φ
7→ h A ( u ) , φ i is continuous, it follows that h A ( v j ) , φ i → h A ( v ) , φ i = 0for any φ ∈ X r . Hence, passing to the limit in (9) as j → ∞ , it follows that Z R N g ( u j ) φ k u j k p − s,p d x → , ∀ φ ∈ X r . Thus, there exists a constant
C > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R N g ( u j ) k u j k p − s,p φ k φ k s,p d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C, ∀ j ∈ N and ∀ φ ∈ X r and this yields (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R N g ( u j ) k u j k p − s,p φ d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C | φ k s,p , ∀ φ ∈ X r . (10) H. P. BUENO, O. H. MIYAGAKI, AND A. L. VIEIRA
For every j ∈ N , set T j ( φ ) = Z R N g ( u j ) φ k u j k p − s,p d x, φ ∈ X r . It follows that { T j } is a family of bounded linear functionals and sup j ∈ N | T j ( φ ) | ≤ C for all φ ∈ X r . Hence, sup j ∈ N k T j k < ∞ . Taking into account the embedding X r ֒ → L q ( R N ) for q ∈ [ p, + ∞ ), the Hahn-Banach Theorem guarantees the existence of a continuous linear functional ˜ T j de-fined in L q ( R N ) such that ˜ T j ( φ ) = T j ( φ ) and k ˜ T j k ( L q ( R N )) ∗ = k T j k , for all φ ∈ X r .Therefore, there exist functions h j ∈ L q ′ ( R N ) such that˜ T j ( φ ) = Z R N h j φ d x, and k ˜ T j k ( L q ( R N )) ∗ = k h j k L q ′ ( R N ) . Hence, for all φ ∈ L q ( R N ) we have Z R N h j φ d x − Z R N g ( u j ) φ k u j k p − s,p d x = 0 . It follows from ( g ,sp ) that Z R N | g ( u j ) |k u j k p − s,p ! q ′ d x ≤ C q ′ k u j k q ′ ( p − s,p Z R N e αq ′ | u j | NN − s d x. Since e αq ′ | u j | NN − s ∈ L ( R N ), it follows that g ( u j ) k u j k p − s,p ∈ L q ′ ( R N ) . Thus, h j − g ( u j ) k u j k p − s,p ∈ L q ′ ( R N ) ⊂ L loc ( R N ) and h j ( x ) = g ( u j ( x )) k u j k p − s,p a.e. in R N . Therefore, k h j k q ′ = k ˜ T j k q = k T j k ≤ K for some K >
0. Taking φ = v j yields (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R N g ( u j ) k u j k p − s,p v j d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) g ( u j ) k u j k p − s,p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) q ′ k v j k q = k h j k q ′ k v j k q ≤ K k v j k q . But v j → L q ( R N ) implies R R N g ( u j ) v j k u j k p − s,p d x →
0. Therefore, k v j k ps,p = I ′ ( u j ) v j k u j k p − s,p + Z R N g ( u j ) k u j k p − s,p v j d x → v j → X r . This a contradiction, since k v j k s,p = 1. (cid:3) Lemma 3.4.
Passing to a subsequence if necessary, the ( P S ) c -sequence ( u j ) con-verges strongly in X r . ULTIPLICITY OF SOLUTIONS FOR A SCALAR FIELD EQUATION 9
Proof.
Since Lemma 3.3 guarantees that ( u j ) is bounded, we have u j ⇀ u ∈ X r , u j → u in L q ( R N ) for any q ∈ ( p, + ∞ ) and u j ( x ) → u ( x ), a.e. in R N . Therefore,( u j − u ) is a bounded sequence in X r . Since I ′ ( u j ) → X r ,we have I ′ ( u j )( u j − u ) →
0. Hence, the operator A defined in (8) satisfies h A ( u j ) , u j − u i = I ′ ( u j )( u j − u ) + Z R N g ( u j )( u j − u )d x = Z R N g ( u j )( u j − u )d x + o (1)and it follows from ( g ,sp ) that h A ( u j ) , u j − u i ≤ C Z R N e α | u j | NN − s | u j − u | dx + o (1) . If 0 < α < α , we can take r > < αr < α and applying H¨older’sinequality we obtain h A ( u j ) , u j − u i ≤ C (cid:18)Z R N e r ′ α | u j | NN − s d x (cid:19) r ′ (cid:18)Z R N | u j − u | r d x (cid:19) r + o (1)and it follows from the Trudinger-Moser inequality that h A ( u j ) , u j − u i ≤ C k u j − u k r → , as j → ∞ . Thus, h A ( u j ) , u j − u i → , as j → ∞ . Hence, k u j − u k ps,p = h A ( u j ) , u j − u i − I ′ ( u ) · ( u j − u ) − Z R p g ( u )( u j − u )d x + Z R N g ( u j )( u j − u )d x → . We are done. (cid:3)
For every n ∈ N we set b n = inf γ ∈ Γ n max σ ∈ D n I ( γ ( σ )) , (11)where Γ n = { γ ∈ C ( D n , X r ) : γ is odd and γ = γ n in ∂ D n } , D n denotes the uni-tary disc in R n , with ∂ D n = S n − and γ n is defined in (7).We set ˜ γ n ( σ ) = (cid:26) | σ | γ n ( σ | σ | ) , σ ∈ D n \{ } , , σ = 0 . Since ˜ γ n ∈ Γ n , we have Γ n = ∅ for all n ∈ N .Considering a ( P S ) b n -sequence ( u j ), it is easy see that, { u ∈ X r ; k u k s,p = ρ } ∩ γ ( D n ) = ∅ , ∀ γ ∈ Γ n . Thus, it follows from Lemma 3.2 that 0 < β ≤ b n for any n ∈ N , with β as inLemma 3.2. Lemma 3.5.
The functional I ( u ) possesses an unbounded sequence of critical val-ues. In fact, ( i ) b n is a critical value for I ( u ) for any n ∈ N ; ( ii ) b n → ∞ as n → ∞ . Proof.
As a consequence of the symmetric mountain pass theorem, b n is criticalvalue for any n ∈ N . As in [21, Chapter 9], we define˜Γ n = n h ( D m \ Z ); h ∈ Γ n , m ≥ n, Z ∈ E m and gen ( Z ) ≤ m − n o , where E m is a family of closed subsets A ⊂ R m \{ } such that A = − A and gen ( A )is the Krasnoselski genus of A . We consider the sequence ( d n ) of minimax valuesfor the I ( u ) given by d n = inf A ∈ ˜Γ n max u ∈ A I ( u ) . It follows that d n ≤ b n and d n ≤ d n +1 , ∀ n ∈ N . Since I ( u ) satisfies the ( P S )-condition, it also follows that d n → ∞ as n → ∞ . Since d n ≤ b n , ∀ n ∈ N , we havethat b n → ∞ as n → ∞ and the proof of the case s ∈ (0 ,
1) is complete.In the limit case s = 1 the adequate Sobolev space is W ,p ( R p ) and resultsequivalent to those presented in the Section 2 are still valid (see, e.g., in [3, 9]).Hence, we can adapt the arguments exposed above to obtain the result. (cid:3) References [1] S. Adachi and K. Tanaka:
Trudinger type inequalities in R N and their best exponentes , Proc.Amer. Math. Soc. 128 (2000), no. 7, 2051-2057.[2] R. A. Adams: Sobolev spaces, Academic Press, New York-London, 1975.[3] M. Badiale and E. Serra: Semilinear elliptic equations for beginners. Existence results via thevariational approach, Springer, London, 2011.[4] A. Bahrouni: Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity , Commun. Pure Appl. Anal. (2017),no. 1, 243-252.[5] V. Ambrosio: Mountain pass solutions for the fractional Berestycki-Lions problem , Adv.Differential Equations (2018), no. 5-6, 455-488.[6] P. Belchior, H. Bueno, O.H. Miyagaki and G.A. Pereira: Remarks about a fractional Choquardequation: ground state, regularity and polynomial decay , Nonlinear Anal. (2017), 38-53.[7] H. Berestycki, T. Gallou¨et and O. Kavian, ´Equations de champs scalaires euclidiens nonlin´eaires dans le plan , C. R. Acad. Sci. Paris S´er. I Math. 297 (1983), no. 5, 307-310.[8] H. Berestycki and P.L. Lions:
Nonlinear scalar field equations, II, Existence of infinitelymany solutions , Arch. Rational Mech. Anal. (1983), no. 4, 347-375.[9] H. Brezis : Functional analysis, Sobolev spaces and partial differential equations, Springer,New York, 2011.[10] H. P. Bueno, E. H. Caqui and O. H. Miyagaki: Critical fractional elliptic equations with ex-ponential growth without Ambrosetti-Rabinowitz type condition , arXiv:2004.02578[math.AP].[11] H. P. Bueno, E. H. Caqui, O. H. Miyagaki and F. R. Pereira:
Critical concave convexAmbrosetti-Prodi type problems for fractional p-Laplacian , Adv. Nonlinear Stud. (2020),no. 4, 847-865.[12] F. Demengel and G. Demengel: Functional spaces for the theory of elliptic partial differentialequations, Springer, London, 2012.[13] E. Di Nezza, G. Palatucci and E. Valdinoci: Hitchhikers’s guide to the fractional Sobolevspaces , Bull. Sci. Math., (2012), no. 5, 521-573.[14] J. Hirata, N. Ikoma and K. Tanaka:
Nonlinear scalar field equations in R N : mountain passand symmetric mountain pass approaches , Topol. Methods Nonlinear Anal. (2010), no. 2,253-276.[15] A. Iannizzotto and M. Squassina: Weyl-type laws for fractional p -eigenvalue problems ,Asymptot. Anal. (2014), no. 4, 233-245. ULTIPLICITY OF SOLUTIONS FOR A SCALAR FIELD EQUATION 11 [16] P.L. Lions:
Sym´etrie et compacit´e dans les espaces de Sobolev , J. Functional Analysis (1982), no. 3, 315-334.[17] T. Ozawa: On critical cases of Sobolev’s inequalities , J. Funct. Anal. (1995), 259-269.[18] R. S. Palais:
The principle of symmetric criticality , Comm. Math. Phys. (1979), no. 1,19-30.[19] G. Palatucci: The Dirichlet problem for the p -fractional Laplace equation , Nonlinear Anal. (2018), part B , 699-732.[20] P. Pucci, M. Xiang and B. Zhang: Multiple solutions for nonhomogeneous Schr¨odinger-Kirchhoff type equations involving the fractional p-Laplacian in R N , Calc. Var. Partial Dif-ferential Equations , (2015), no. 3, 2785-2806.[21] P. H. Rabinowitz: Minimax methods in critical point theory with applications to differentialequations , CBMS Regional Conference Series in Mathematics , the American Math. Society,Providence, RI, 1986.[22] M. Willem: Minimax theorems, Progress in Nonlinear Differential Equations and their Ap-plications, 24. Birkh¨auser Boston, 1996.(H. P. Bueno) Department of Matematics, Universidade Federal de Minas Gerais,31270-901 - Belo Horizonte-MG, Brazil
Email address : [email protected] (O. H. Miyagaki) Department of Mathematics, Universidade Federal de S˜ao Carlos,13565-905 - S˜ao Carlos-SP, Brazil
Email address : [email protected], [email protected] (A. L. Vieira) Department of Matematics, Universidade Federal de Minas Gerais,31270-901 - Belo Horizonte-MG, Brazil and UFVJM, 39100-000 - Mucuri, Brazil
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