Multi-bump solutions for a class of quasilinear equations on $R$

This paper is concerned with the existence of multi-bump solutions 
to a class of quasilinear Schrodinger equations in $R$. 
The proof relies on variational methods and combines some arguments 
given by del Pino and Felmer, Ding and Tanaka, and Sere.


Introduction.
Recently, there has been growing interested in the quasilinear Schrödinger equation where i 2 = −1, k, x, t ∈ R, z = ∂z ∂x and W : R → R is a continuous function.

Knowledge of the solutions of
where w ∈ R, has a great importance for studying standing wave solutions for (1), that is, solutions of the form z(x, t) = e −iwt v(x). As observed in [18], the quasilinear equation (1) with k > 0 has a great relevance because it appears in mathematical models associated with several physical phenomena such as in the study of plasma physics, see for example [23,27], in the theory of superfluid film and in dissipative quantum mechanics, see for example [15,16,20].
In [5], Ambrosetti and Wang proved the existence of positive solutions for a large class of quasilinear elliptic equations, which includes the following problem provided that the positive parameter λ is sufficiently small, q ≥ 3, V is a real function satisfying certain hypotheses and the problem has a unique solution for λ = 0. The discussion of existence of solutions for equations of the type in terms of the parameter λ has been extensively investigated in several papers. We refer the reader to [1,2,3,6,7,8,11,13,14], and references therein.
(V ) In [13] the existence of 2 m − 1 solutions is proved assuming that λ is large. Moreover, it is stated that for non-empty subset Γ ⊂ {1, . . . , m} there exists a multibump solution, that is, a positive solution u λ of (3) which converges, as λ → +∞, to a least energy solution of where Ω Γ = ∪ j∈Γ Ω j . Existence of multi-bump solutions for problems involving the p-Laplacian operator and nonlinearities with critical growth for N ≥ 2 has been dealt with in [1], [2] and [3].
Motivated by [5] and [13], we are interested in finding multi-bump solutions to (2) assuming that W (x) = λV (x) + 1 + w and k = 1 2 . More precisely, we consider the following quasilinear elliptic problem where q > 3, the function V verifies the condition (V ) and λ is a positive parameter large enough. The main difficulty in proving our result is the presence of the quasilinear term R |v| 2 |v | 2 in the energy functional associated with the problem. Since this term is homogeneous of order 4 and non-convex, we have serious difficulties to show the Palais-Smale condition. Moreover, it is not clear that the characterization of the mountain pass values used in [13] can be used here, because the operator and the nonlinearity do not have the same degree of homogeneity. Thus, we modify the sets that appear in the minimax arguments explored in [13].
Our main result is the following Theorem 1.1. Suppose that q > 3 and (V ) holds. Then, for any non-empty subset Γ ⊂ {1, . . . , m}, there exists λ * > 0 such that, for λ ≥ λ * , the equation (5) has a family {u λ } of solutions such that, for any sequence λ n → +∞, there exists a subsequence λ ni such that u λn i converges strongly in H 1 (R) to a function u which satisfies u(x) = 0 for x ∈ Ω Γ and the restriction u| Ωj , for j ∈ Γ, is a least energy solution of Corollary 1.1. Under the assumptions of Theorem 1.1, there exists λ * > 0 such that (5) has at least 2 m − 1 solutions for all λ ≥ λ * .
The organization of this paper is as follows: Section 2 sets up the variational framework and a modified functional is introduced which satisfies the Palais-Smale conditions. In Section 3 some technical lemmas and propositions related to existence of multi-bump solutions are proved. Finally, Section 4 offers a proof of the main theorem. Notation: In this paper we use the following notations: • B R (z) denotes the open interval with center at z and radius R.
• In all the integrals we omit the symbol "dx".

2.
Preliminaries. In this section, we introduce the variational framework used to prove our results. Let A direct argument works to show that for each λ ≥ 0, (E, · λ ) is a Hilbert space and that the imbedding E → H 1 (R) is continuous. The positive solutions of (5) are critical points of the functional I : E → R given by For an open set Θ ⊂ R, we also write Since V is nonnegative, we have The next result is a consequence of the previous considerations: For any open set Θ ⊂ R, there exist δ 0 > 0 and ν 0 ∈ (0, 1 2 ) such that for all u ∈ E(Θ) and λ > 0.
2.1. Modified functional. In this subsection, based on the arguments used in [13], we introduce a modified functional which satisfies the Palais-Smale conditions. Since we intend to find positive solutions, we define f (t) = t q for every t ∈ [0, +∞) and f (t) = 0 for every t ∈ (−∞, 0).
Taking ν 0 > 0 given by Lemma 2.1, let a be a positive real such that f (a) a = ν 0 and define the functionf Now, we fix a non-empty subset Γ ⊂ {1, . . . , m} and take the sets where Ω j is a bounded open set verifying Moreover, we consider the function where χ Γ denotes the characteristic function of the set Ω Γ . The function g is Caractheódory and, for x ∈ R, t → g(x, t) is a C 1 function satisfying and every critical point of Φ λ is a weak solution of An important point that we would like to stress is the fact that if u is a positive weak solution of (7) with u(x) ≤ a in R \ Ω Γ , then u is a positive weak solution of (5).

(PS) condition.
In this section we make use of the modification on the nonlinearity f in order to show that Φ λ satisfies the Palais-Smale condition ((PS) for short).
In the following, we state a result whose proof follows the same arguments developed in [22,Lemma 2]. This result will be essential to study the behavior of the Palais-Smale sequences.
Lemma 2.2. For any T > 0, the functional N : E → R given by is weakly sequentially lower semicontinuous, that is, Proof. From (8), where n → 0 as n → +∞. From the definition of g and condition (g 1 ), ν 0 s 2 , ∀x ∈ Ω Γ and s ∈ R and consequently which together with Lemma 2.1 yields Then there exists K > 0 such that u n λ ≤ K for all n ∈ N.
Proposition 2.1. The functional Φ λ satisfies the Palais-Smale condition at the level c, for every c ∈ R, that is, any sequence satisfying (8) possesses a convergent subsequence in E.
Proof. Let (u n ) ⊂ E be a Palais-Smale sequence at the level c ∈ R. By Lemma 2.3, (u n ) is bounded in E. Hence there exist u ∈ E and a subsequence, still denoted by (u n ), such that (u n ) is weakly convergent to u in E.
Choose R > 0 such that Ω Γ ⊂ B R (0) and let Ψ R be a cut-off function such that for some constant C > 0. From (9)-(11) and Lemma 2.1, there exists C > 0 such that Observing that Ω Γ ⊂ B R (0), it follows from definition of g that Thereby, this inequality and (12) then show By (13) and (14), we get On the other hand, since u n u weakly in E, u n − u 2 λ = u n , u n − u n , u + o n (1), where · , · denotes the inner product in E.

It follows that
From (15), Since E is continuously embedded in L p (R) for 1 ≤ p ≤ +∞, for each δ > 0 fixed, there exists T > 0 such that we deduce that where the o n (1) terms were obtained using the converge u n → u in C(−T, T ) and the boundedness of sequences (u n ) in L ∞ (−T, T ) and (u n ) in L 2 (−T, T ). Therefore, we can conclude that This completes the proof of Proposition 2.1.
In the following, we use a version of the notion of Palais-Smale sequence introduced in [13]. We say that (u n ) is a (P S) sequence if        u n ∈ E, for every n, (Φ λn (u n )) is bounded, Φ λn (u n ) * λn → 0, as n → +∞, λ n → +∞, as n → +∞.
(P S) Proposition 2.2. Let (u n ) be a (P S) sequence. Then, up to subsequence, there exists u ∈ H 1 (R) such that (u n ) weakly converges to u in H 1 (R). Moreover, i) u ≡ 0 on R \ Ω Γ and u is a nonnegative solution of ii) u n − u λn → 0, as n → +∞, which implies u n → u in H 1 (R). iii) u n also satisfies Proof. As in the proof of Lemma 2.3, it follows that there exists K > 0 such that u n 2 λn ≤ K ∀n ∈ N. In particular, (u n ) is bounded in H 1 (R). Thus, we can assume that u n u weakly in H 1 (R) and u n (x) → u(x) a.e. in R for some u ∈ H 1 (R).
In order to prove i), given k ∈ N, define the set Combining this inequality with Fatou Lemma, we have leading to u = 0 on A k for every k ∈ N. Observing that we can conclude that u = 0 on R \ Ω. Hence u ∈ H 1 0 (Ω j ) for every j ∈ {1, . . . , m} (recall that Ω = Ω 1 ∪ . . . . ∪ Ω m ).
Arguing as in the proof of Proposition 2.1, for any R > 0 sufficiently large, Since u n u weakly in H 1 (R) and u = 0 on R \ Ω, we have and R g(x, u n )u n = R g(x, u n )u + o n (1) = R g(x, u)u + o n (1).
Combining (16)- (18), Repeating the same arguments explored in the proof of Proposition 2.1, the last equality implies that u n − u 2 λn → 0 as n → +∞. In particular, u n → u in H 1 (R), which gives that ii) holds.
In order to complete the proof of i), for any j ∈ Γ and φ ∈ C ∞ 0 (Ω j ), we have Φ λn (u n )φ = o n (1), which implies where we have used that V (x) = 0 on Ω j and the definition of g. By using the convergence u n → u in H 1 (R), the fact that u ∈ H 1 0 (Ω j ), and the density of Choosing φ = u − = min{u, 0} as a test function in (19), we find which yields u − = 0 in H 1 0 (Ω j ) for all j ∈ Γ, that is, u(x) ≥ 0 for all x ∈ Ω j and j ∈ Γ. Thus, u is a nonnegative weak solution of For iii), observing that u = 0 on R \ Ω and V (x) = 0 on Ω, it follows that 3. The existence of multi-bump solutions. We start this section with the definition of the functionals I j : H 1 0 (Ω j ) → R and Φ λ,j : H 1 (Ω j ) → R given by and whose critical points are respectively associated with the weak solutions of the problems and Note the Neumann boundary condition in the latter boundary value problem. Since q > 3, a familiar argument shows that these functionals satisfy the hypotheses of the mountain pass theorem, hence there are values c j and c λ,j given by (1)) < 0}. Moreover, since I j and Φ λ,j satisfy the Palais-Smale condition, there exist nonnegative functions w j ∈ H 1 0 (Ω j ) and w λ,j ∈ H 1 (Ω j ) such that I j (w j ) = c j , I j (w j ) = 0 (24) and Φ λ,j (w λ,j ) = c λ,j , Φ λ,j (w λ,j ) = 0.
Since Ω j ⊂ Ω j , it is easy to verify that c λ,j ≤ c j . We also observe that there exists η > 0, independent of λ and j such that Effectively, this follows from Φ λ,j (w λ,j )(w λ,j ) = 0 and the Sobolev imbedding Lemma 3.1. Let (λ n ) be a real sequence such that λ n → +∞ as n → +∞. For any j ∈ Γ, the corresponding critical levels c λn,j satisfy lim n→+∞ c λn,j = c j .

3.1.
Special minimax values of Φ λ . By continuity of I j , there exists R > 1 such that for all j ∈ Γ. By using the definition of c j , we can verify that Recalling that Γ ⊂ {1, .., m}, it can be assumed that Γ = {1, . . . , s} for some s ≤ m. Define We observe that γ 0 ∈ Γ * , thus Γ * = ∅ and b λ,Γ is well defined.

Conclusion.
In this section we establish the proof of Theorem 1.1. The goal is to find a positive solution u λ which is close to a least energy solution in each Ω j , j ∈ Γ, provided λ is sufficiently large. To this end, the key ingredients are the following propositions. Hereafter, Perhaps it is appropriate at this point to note that the use of the ball in the definition of the set A λ µ permits us to adapt the arguments exploit by Ding and Tanaka to prove Proposition 4.2 in [13]. We recall that we modify the sets that appear in the minimax arguments employed in [13]. We We have the following uniform estimate of Φ λ (u) * λ on the annulus (A λ 2µ \ A λ µ ) ∩ Φ cΓ λ : Proposition 4.1. There exist σ 0 > 0 and λ * > 0 such that Φ λ (u) * λ ≥ σ 0 for all λ > λ * and u ∈ (A λ 2µ \ A λ µ ) ∩ Φ cΓ λ . Proof. Suppose by contradiction that there exist λ n → +∞ and u n ∈ (A λn 2µ \ A λn µ ) ∩ Φ cΓ λn such that Φ λn (u n ) * λn → 0. Since u n ∈ A λn 2µ , Φ λn (u n ) is bounded, and then we can assume Φ λn (u n ) → c ∈ (−∞, c Γ ]. Hence (u n ) is a (P S) sequence, and by Proposition 2.2, we can assume that and u n 2 λn,R\ΩΓ → 0.
We claim that both of them do not occur. In fact, if (i) holds, from (35)-(37) u n ∈ A λn µ for large n, which contradicts u n ∈ (A λn 2µ \ A λn µ ). Now, if (ii) occurs, from (35)-(37) again, which is a contradiction with the fact that u n ∈ (A λn 2µ \ A λn µ ). This completes the proof of Proposition 4.1.
In the following proposition, we have to suppose a stronger condition on µ than previously, for instance, µ ≤ min{min j cj 4 , 1}.