MULTI-BUMP SOLUTIONS FOR SCHRÖDINGER EQUATION INVOLVING CRITICAL GROWTH AND POTENTIAL WELLS

In this paper, we consider the following Schrödinger equation with critical growth −∆u+ (λa(x)− δ)u = |u| −2u in R , where N ≥ 5, 2∗ is the critical Sobolev exponent, δ > 0 is a constant, a(x) ≥ 0 and its zero set is not empty. We will show that if the zero set of a(x) has several isolated connected components Ω1, · · · ,Ωk such that the interior of Ωi(i = 1, 2, ..., k) is not empty and ∂Ωi(i = 1, 2, ..., k) is smooth, then for any non-empty subset J ⊂ {1, 2, · · · , k} and λ sufficiently large, the equation admits a solution which is trapped in a neighborhood of ⋃ j∈J Ωj . Our strategy to obtain the main results is as follows: By using local mountain pass method combining with penalization of the nonlinearities, we first prove the existence of single-bump solutions which are trapped in the neighborhood of only one isolated component of zero set. Then we construct the multi-bump solution by summing these one-bump solutions as the first approximation solution. The real solution will be obtained by delicate estimates of the error term, this last step is done by using Contraction Image Principle.

1. Introduction and main results.We consider the following Schrödinger equation: ) where N ≥ 5, λ > 0 is a parameter, δ > 0 is a constant, 2 * is the critical Sobolev exponent.Solutions of this type equation are related to the existence of the standing wave solutions of the following Schrödinger equation: where V (x) is a given potential, λ > 0 is a parameter.
The first author is supported by NSFC grants NSFC(11171171,11331010) and the second author is supported by SRF for ROCS, SEM and NSFC(11171028).

YUXIA GUO AND ZHONGWEI TANG
In mathematical literature, in recent years, much attention has been devoted to the study of the existence of one-bump or multi-bump bound states for the following Schrödinger equation: with w(x) = h − 2 p−2 u(x) and h −2 = λ.For subcritical cases, i.e. 2 < p < 2 * , there are enormous investigations on problem (1.3) under various assumptions on the potential function.For instance, under the assumption that V (x) is a bounded function having a non-degenerate critical point, for sufficiently small h > 0, Floer and Weinstein [16] established the existence of a family standing wave solutions u h for (1.3).Moreover they showed that the solutions u h concentrate near the non-degenerate critical point of V (x) as h tends to 0. Their results were later generalized, by Oh [19,20], to the higher-dimensional case and the existence of multi-bump solutions concentrating near several nondegenerate critical points of V (x) as h tends to 0 was obtained.For more related results, we refer the readers to Ambrosetti, Badiale and Cingolani [1], Ambrosetti, Malchiodi and Secchi [2], Bartsch and Tang [3], Bartsch and Wang [4], Byeon and Wang [6], [7], Cingolani and Lazzo [11], Cingolani and Nolasco [12], Del.Pino and Felmer [13], [14] and the references therein.
As far as the critical case is concerned, due to the lack of the compactness, the problem get more challenges.There are some results under stronger assumptions on V (x).We firstly refer to the work of Benci and Cerami [5], they considered the following problem: as |x| → ∞.
(1.4) For = 1, under the assumption that V (x) ≡ 0 and V (x) is sufficiently small, they proved the existence of bound states for (1.4);For > 0 small, under the assumption that the interior part of zero sets Ω := intV −1 (0) of V (x) is nonempty and V (x) is sufficiently small, Chabrowski and Yang [9] proved that the problem (1.4) admits cat{Ω} many solutions; For V (x) is strictly positive and admits a local mimima at some point, using local mountain pass method combining penalization of the nonlinearity, Zhang, Chen and Zou [23] obtained the existence of bound states of (1.4) when small.Recently, Tang [22] considered the problem (1.1) with critical exponents and indefinite potential function V (x) = λa(x) − δ, where a(x) ≥ 0, λ > 0 is a parameter and δ > 0 is a constant which can be arbitrary large such that the operator −∆ + λa(x) − δ is indefinite.Under some suitable assumptions on a(x) and δ, the author proved the existence of least energy solution which localized near the potential well inta −1 (0) for λ large.
When the zero set of a(x) admits more than one isolated connected components, it is natural to ask whether (1.1) has a family of solutions u λ which converges, as λ → ∞, to the least energy solution in some selected isolated zero sets of a(x) and to 0 elsewhere?In this paper, we aim to answer this question and the answer is affirmative.More precisely, we will construct solutions such that these solutions have several bumps in R N .Moreover, we will show that these solutions converge to a limit solution u such that the restriction of u| ∂Ωi ( see the follows for the definition of Ω i ) is exactly the least energy solution of the "limit problem" (see problem (D Ωi )).
Our strategy in proving the existence of multiple bump solutions is first to prove the existence of one-bump solutions which are trapped on one non-empty zero set Ω i (i = 1, 2, ...k).Then we try to glue these one-bump solutions together and prove that the sum of these one-bump solutions, up to an error term, will be the real solution which is multi-bump type.To finish this, instead of using Lypunov-Schmit reduction method, here we use Contraction Image Principle, we will show the problem can be reduced to find a fixed point to a related operator in a small ball and we can prove that for a proper small ball, the operator is contractible.
Remark 1.By the assumption (A 3 ), there is a positive number ρ > 0, such that Remark 2. From the assumptions (A 1 ) − (A 3 ) , we can see that the zero set Ω of a(x) is a bounded domain in R N and thus the operator and we can denote its eigenvalues by 0 By assumption (A 4 ), it is easy to see that the operator −∆ + λa(x) − δ is positive definite in H 1 (R N ) for λ ≥ 0 properly large.Before the statement of the main theorem, we introduce some notations first.Set with the induced norm: for λ properly large.Moreover, there is a positive number ν 0 > 0 such that for λ ≥ 0 large enough, Now we define the variational functional by: then the critical points of J λ correspond to the solutions of (1.1).Let Then it is easy to check that any nontrivial critical points of J λ belongs to N λ .And its least energy set lies on the level: For any i ∈ {1, 2, • • • , k}, the following problem is somehow the "limit" problem of (1.1): and its least energy lies on the level where and is the corresponding variational functional of the "limit" problem (D Ωi ).Our first result is about the existence of single-bump solutions and it is: Then for any i ∈ {1, 2, • • • , k} and δ > 0 sufficiently small, there exits Λ 0 > 0 such that for λ ≥ Λ 0 , (1.1) admits a solution W i λ such that, for any sequences λ n → +∞, {W i λn } has a subsequence converging to w such that w ≡ 0 for x ∈ R N \ Ω i and w i δ := w| Ωi is a least energy solution of (D Ωi ).That is w i δ solves (D Ωi ) and I i (w i δ ) = c i .Moreover, there are two constants C, c > 0 such that (1.9) Let us denote by R i (x) the Robin function of domain Ω i , i.e.R i (x) = H i (x, x), where H i (x, y) is the regular part of the Green function G i (x, y), namely, To obtain the existence of multiple-bump type solutions, we need the following further assumption on the domain of Ω i : (A 5 ) For any i ∈ {1, 2, • • • , k}, all the critical points of the Robin function R i (x) of domain Ω i are non-degenerate.Our second result is concerned about the existence of multiple-bump solutions and it is: Theorem 1.2.Assume (A 1 )−(A 5 ) hold.Then for any J ⊂ {1, 2, • • • , k} and δ > 0 sufficiently small, there exits Λ 0 > 0 such that for λ ≥ Λ 0 , (1.1) admits a solution u λ such that, for any sequences λ n → +∞, {u λn } has a subsequence converging to u such that u ≡ 0 for x ∈ R N \ (∪ j∈J Ω j ) and u| Ωj (j ∈ J) is a least energy solution of (D Ωj ).Namely for j ∈ J, u| Ωj solves (D Ωj ) and I j (u) = c j .
The paper is organized as follows: In section 2, we concerned with the existence of one-bump solution, moreover we will show that the one-bump solution is exponential decay outside of its trapped domain.Section 3 is devoted to the existence of the multi-bump solution, to prove this, we will first show the non-degenerate result for the linearized operator around the approximation solution.The existence of the desired multi-bump solution is obtained by using the Contraction Image Principle.
2. One-bump solutions.In this section, we consider the existence of one-bump solutions of (1.1) for λ large.Suppose a(x) and δ satisfy the conditions (A 1 ) − (A 4 ).We will show that for any 1 ≤ i ≤ k, as λ large enough, (1.1) has a solution u λ which is trapped nearby the isolated zero set Ω i of a(x).Furthermore, we will study the asymptotically behavior outside of the domain Ω i and give the proof of Theorem 1.1.
To proceed, we will divide the proof of Theorem 1.1 into three parts.In the first part, we show the existence of one-bump solution and in the second part, we show the asymptotic behavior of the one-bump solution.In the third part, we give the exponential decay of the one-bump solution.And as a consequence, the proof of Theorem 1.3 follows immediately.
2.1.Penalization of the nonlinearity.As we know that the variational functional J λ of (1.1) is non-compact because of the critical growth of the nonlinearity and the unboundedness of the domain.To overcome these difficulties, we modify the functional J λ by penalizing the nonlinearity term of the equation, then we show that, under some energy level sets, the modified functional satisfies the Palais-Smale ( P.S. for shortness ) condition.For any small constant we define a function f (t) by: Let us denote and where χ Ω (x) is the Characteristic function of Ω, u + = max{u, 0} and F (s) = t 0 f (s)ds.We define the modified functional by: Then one can check that a critical point of J λ,i corresponds to a solution of the following equation By the definition of g i (x, t), we see that 2.2.Compactness of the modified functional.In this subsection, we will show that the functional J λ,i satisfies the Palais-Smale condition under certain energy level.
Lemma 2.1.Suppose that {u n } is a (P.S.) c sequence of the modified functionalJ λ,i , that is a sequence satisfying Then there exists a positive constant Λ 0 > 0 such that for any λ ≥ Λ 0 , {u n } is bounded.That is there exists a constant C which is independent of λ and n such that lim Proof.Since {u n } is a (P.S.) c sequence, we have which, by the definition of γ 0 in (2.1), implies that the estimate (2.5) holds for some constant C > 0 independent of λ ≥ 0.
Proposition 1. Suppose that {u n } is a (P.S.) c sequence for J λ,i with where S is the best Sobolev constant.Then there exists a subsequence of {u n } which converge strongly in E λ to a critical point u of J λ,i such that J λ,i (u) = c.
Proof.By Lemma 2.1, we know that {u n } is bounded.Thus there exists a subsequence of {u n } ( still denoted by {u n } ) such that Then by standard arguments, we can see that J λ,i (u) = 0 and J λ,i (u) ≥ 0. Next we show that u n → u strongly in E λ .Let v n = u n − u, it follows from the Brezis-Lieb's Lemma that {v n } is also a Palais-Smale sequence of J λ,i satisfying J λ,i (v n ) → 0 and Hence it is sufficient to prove that v n → 0 strongly in E λ .We show this by contradiction.Without loss of generality, up to a subsequence, we assume on the contrary that lim n→∞ v n On the other hand By the definition of f (t) and F (t), we have (2.9) Note that v n → 0 in L 2 (B R (0)) for any fixed R > 0. Take Λ 0 > 0 properly large such that Λ 0 a 0 ≥ δ, where a 0 := inf |x|≥R a(x) > 0. Then we have for λ (2.10) We obtain that b ≥ S N 2 , which contradicts with (2.9) and hence v n → 0 strongly in E λ .
Now we prove the existence of the critical point of the modified functional J λ,i (u).
By Proposition 1, we know that the functional J λ,i (u) satisfies the (P.S.) c con- To prove the existence of the critical point of J λ,i (u).It is sufficient to check that J λ,i has a (P.S.) c sequence with c < 1 N S N 2 .To do this, we use the standard Mountain Pass Lemma.
Step 1: We show that there exist ρ 0 > 0 and β 0 > 0 such that where It follows that Step 2: We show that there is a e i ∈ E λ such that J λ,i (e i ) < 0. Indeed, for Thus there exists e i ∈ H 1 0 (Ω i ) such that J λ,i (e i ) = I i (e i ) < 0.
Step 3: We define the following two minimax values. and where It is easy to see that c i 0 is indeed the least energy of the functional I i (Ω i ) and hence c i 0 = c i , where c i is defined in (1.8).By the definition of c i λ and the arguments of Step 1 and Step 2, it is easy to see that 0 < β 0 < c i λ ≤ c i 0 .It follows from the results of Capozzi, Fortunato and Palmieri [8], we know that By the above three steps arguments, using the standard Mountain Pass Lemma, we indeed have proved that Lemma 2.2.There is a (P.S.
Combining Proposition 1 and Lemma 2.2, we have proved the following existence result which is the main gredient of this subsection.
Asymptotic behavior of the one-bump solutions.In this subsection, we study the asymptotic behavior of one-bump solutions W i λ (1 ≤ i ≤ k) obtained in Proposition 2 as λ large.We have the following result.
where ω i δ is a least energy solution of (D Ωi ) such that I i (ω i δ ) = c i .Proof.For any sequence {λ n } ∞ n=1 with λ n → ∞ as n → ∞, let us denote W i λn is the corresponding critical points of J λn,i obtained in Proposition 2. To finish the proof of Proposition 3, we only need to show that up to a subsequence, W i λn → ω i δ , as n → ∞.
Indeed, since J λn,i (W i λn ) = c i λn ≤ c i and J λn,i (W i λn ) = 0 for all n ≥ 1, it is easy to see that there is a constant C > 0 such that for all n ≥ 1 Thus up to a subsequence we may assume that

We have
Cm Next we show that w = 0 in Ω j for j = i.Indeed for any ψ ∈ H 1 0 (Ω j ), since J λn,i (W i λn ) • ψ = 0. Take a limit as n → ∞, we have w satisfies the following problem: By the definition of f (w) we have f (w) ≤ γ 0 w.On the other hand, it follows from the choice of γ 0 that the operator −∆ − δ − γ 0 is positively definite.Hence (2.13) implies that w ≡ 0 in Ω j for j = i.We proved that the support of w supp w ⊂ Ω i .
Similarly, for any ψ ∈ H 1 0 (Ω i ), since J λn,i (W i λn ) • ψ = 0, we obtain that w is a solution of the following problem: In the following, we will show that w is indeed a least energy solution of (2.14).
It is sufficient to show that W i λn → w strongly in which implies that w is a nontrivial solution of (2.14).By the definition of c i , one can see that I i (w) ≥ c i .Moreover which indicates that I i (w) = c i and thus w i δ := w is a least energy solution of (2.14).At last, we come to show that, W i λn → w (as n → ∞) strongly in H 1 (R N ).We show this by a contradiction argument.Let us denote w n := W i λn − w, taking into account of (1.5), it is sufficient to prove that w n λn → 0 as n → ∞.Suppose on the contrary that, up to a subsequence, lim inf

Exponential decay of one-bump solution.
In this subsection, we will show that the critical point W i λ of J λ,i obtained in Proposition 2 is indeed a solution of the original problem (1.1).More precisely, we have the following proposition.Proposition 4.There are two constants C, c > 0 such that which implies that for λ large enough, we have, for any and thus W i λ (x) is also a solution of the original problem (1.1).Before giving the proof of Proposition 4, we firstly present an L ∞ estimate for the solutions W i λ outside of Ω i with W i λ λ ≤ M .More precisely, we have Lemma 2.3.Suppose W i λ are the critical points of J λ,i such that W i λ λ ≤ M , where M is a constant independent of λ.Then we have for some constant C 0 > 0 independent of λ such that for λ large, Proof.We prove Lemma 2.3 by Moser's iteration.The similar arguments can be found in the paper by Ni, Pan and Takagi [18], for the completeness, we give the details of the proof.Note that W i λ → w i δ strongly, as λ → ∞, where w i δ is the least energy solution of "limit" problem (D(Ω i )).By the work of Rey [21], we know that for δ > 0 small enough, w i δ concentrates at the point x 0 ∈ Ω i , which is a critical point of Robin function R i (x) of the domain Ω i .Let us denote Ω i,t0 := {x ∈ Ω i : dist{x, ∂Ω i } > t 0 }, where t 0 is a small positive constant and we may choose t 0 small enough such that x 0 ∈ Ω i,t0 .Hence for a small number η 0 > 0 (which we will be specified later), there is a δ 0 > 0, such that for δ < δ 0 , we have Now we are ready to use Moser's iteration argument to obtain the desired estimates.Let ψ denote a smooth cut-off function and β > 1 an arbitrary number, both of them will be specified later.Multiply (2.3) by ψ 2 (W i λ ) β , we have By a direct computation, we have On the other hand, by Hölder's inequality and Young's inequality, we have 2 Note that for any x ∈ R N and u ≥ 0, we have g i (x, u) ≤ u 2 * −1 .Thus the inequality (2.21) leads to By Sobolev imbedding theorem, we have Combine with (2.23) and (2.22), we get (2.24) Now for y ∈ R N \ Ω i and 0 < r < t0 4 , we specify the cut-off function ψ by . By Höder's inequality, we have

.26)
In the following, we will use above estimates combining with Moser's iteration argument to prove (2.19) , where β > 1 will be chosen later, then (2.24) becomes where ψ is a cut-off function supported in B 2r (y) with y ∈ R N \ Ω i and r will be specified later in each step of the iteration process.Again by Höder's inequality, the first term in (2.27) can be estimated by , where q = N 2 N −2 > N and 2 < 2q q−2 < 2 * .Thus, for any ε > 0, where σ = N −2 2 .It follows from (2.27), we have where, using the fact that y ∈ R N \ Ω i and also (2.26), (2.20), C 1 can be estimated as follows we obtain from (2.28) that where C 2 is a constant independent of β.Now for r ≤ r 2 < r 1 ≤ 2r, we choose ψ such that ψ ≡ 1 in B r2 (y) and ψ ≡ 0 in R N \ B r1 (y).Then by a direct computation, we deduce from (2.29) that i.e where h = 1 + β, R = max{r, 1} and C 3 is a constant independent of r, β.Set .
Then we can rewrite (2.31) in terms of N (•, •): where k where Since the above estimate is independent of y ∈ R N \ Ω i , we indeed have proved (2.19) with A direct result of the arguments in the proof of Lemma 2.3 is the following exterior Harnack-type inequality: Lemma 2.4.Suppose all the assumptions of Lemma 2.3 are satisfied.Then for any 0 < r < 1  4 min{ρ, t 0 }, there is a constant C > 0 such that for any y ∈ R N \ Ω i , it holds sup x∈Br(y) . (2.36) Now we are ready to present the proof of Proposition 4.
Proof of Proposition 4. By the proof of Proposition 2, one can find a constant M > 0 independent of λ such that for λ ≥ Λ 0 , we have (2.37) On the other hand, by the assumption on a(x), there is a positive number a 0 > 0 such that a(x) ≥ a 0 for all x ∈ R N \ Ω ρ , where Ω = inta −1 (0) is the interior of the zero set a(x).Thus for λ large enough, it holds that λa(x) − δ ≥ λ 2 a 0 , for all x ∈ R N \ Ω ρ .As a consequence of (2.37), we have which implies that And we may assume that 2M a0 ≥ 1, otherwise we can take M is properly large.Note that for any 0 < r < 1  4 min{ρ, t 0 } and q > 2 * fixed, by the interpolation inequality, we have for any y ∈ R N \ Ω, where α ∈ (0, 1) is such that 2 * = 2α + (1 − α)q i.e., α = q−2 * q−2 .We may choose q > 2 * such that α = 1 2 .By Lemma 2.3, we have where ω N is the volume of the unit ball B 1 (0).Combining (2.38)-(2.40),we obtain that, for any y ∈ R N \ Ω 2ρ B2r(y) B2r(y) where . As a consequence of Lemma 2.4, we have for any which implies that (2.42) At this point, we indeed have shown that the L ∞ norm of W i λ confined in R N \ Ω 2ρ goes to zero as λ goes to ∞, which implies that W i λ is also a solution of the original problem (1.1) for λ large.However, as stated in Proposition 4, we are going to prove a more strong estimate than (2.42) for the L ∞ norm of W i λ confined in R N \ Ω 2ρ .More precisely, we will show that W i λ (x) L ∞ (R N \Ω 2ρ ) can be exponentially decay as λ large.
Firstly, by Lemma 2.3, we have for λ large, sup Thus by the definition of g i (x, t), for λ large, it holds that We may choose a positive number b such that 0 < b < a 0 = inf x∈R N \Ω ρ a(x).Then by (2.43), there is a Λ (2.44) Let us denote y λ = (W i λ ) 2 , then a direct calculation combining with estimate (2.44) yields

Now we consider the following problem:
Then by a similar argument as in Byeon and Wang [6], there is a constant C > 0 such that U (x) ≤ C e −bλ dist(x,∂Ω ρ ) .Since y λ is bounded on ∂Ω ρ , by the Maximum principle, there is a constant M 0 > 0 such that for all x ∈ R N \ Ω ρ y λ (x) ≤ CU (x) ≤ M 0 e −λbdist(x,∂Ω ρ ) , which is equivalent to To finish the proof of Proposition 4, we need to show that for all x ∈ Ω 2ρ \ Ω 2ρ i , we still have the above estimate (2.45).In fact, since a(x) = 0 in Ω \ Ω i , to obtain the above estimate for x ∈ Ω 2ρ \ Ω 2ρ i , The Maximum principle can not be applied directly, we will use the property of the first eigenfunction of −∆ − δ − γ 0 on Ω 3ρ j .More precisely, let Φ 1 j be the first eigenfunction and ν j 1 be the first eigenvalue of the following problem: x ∈ ∂Ω 3ρ j . (2.46) By the assumptions on δ, (A 4 ) and γ for the choosing of γ 0 ), we know that ν j 1 > 0 and Φ 1 j (x) > 0 in Ω 3ρ j and we may assume that Φ(x) 1 j ≥ 1 for x ∈ ∂Ω 2ρ j .On the other hand, note that for we have This contradicts with (3.5).We complete the proof of Proposition 5.

3.2.
Existence of multi-bump solutions.In this subsection, we are going to prove the existence of solutions of the type (3.2) for (1.1) and our main results are: Theorem 3.1.Under the assumption (A 1 ) − (A 5 ), there exists Λ 2 ≥ Λ 1 such that for λ ≥ Λ 2 and 0 < δ < δ 0 , (1.1) has a solution Remark 4. Indeed, if u λ = W λ + ψ * λ is a solution of (1.1) such that (3.6) hold.Then by a similar arguments as we did in Section 2 for the existence of one-bump solution W i λ , we know that u λ → j∈J ω j δ as λ → ∞, and there are two constants C, c > 0 such that for x ∈ R N \ Ω 2ρ J , |u λ (x)| ≤ Ce −c λ 2 , where Ω J = ∪ j∈J Ω j .Moreover, by the strong maximum principle, it is easy to see that u λ > 0 in R N .Hence u λ is a |J|−bump positive solution of (1.1) which, for λ large, concentrates at Ω J , here |J| denote the number of the elements of J. Now we give the proof of Theorem 3.1.
Proof.Indeed, u λ = W λ + ψ λ is a solution of (1.1) for some ψ λ ∈ E λ is equivalent to ψ λ is a solution of the following problem: which is equivalent to In the following, we will show that the operator N λ : E λ → E λ has a fixed point in a small Ball B ρ λ (0) ⊂ E λ , where ρ λ is a small number depend on λ and will be specified later.
From Proposition 5, we know that for λ ≥ Λ 1 , the operator L λ (W λ ) is invertible and there is a constant M > 0 which is independent of λ such that L −1 λ (W λ ) ≤ M. On the other hand, by the exponential decay of W i λ outside of Ω ρ i , it is not difficult to verify that there are two constants C 1 , c 1 > 0 which is also independent of λ such that S(x) E * λ ≤ C 1 e −c 1 λ 2 .