MULTIPLE SOLUTIONS FOR SINGULARLY PERTURBED SEMILINEAR ELLIPTIC EQUATIONS IN BOUNDED DOMAINS

We are concerned with the multiplicity of solutions of the following singularly perturbed semilinear elliptic equations in bounded domains Ω:−ε2∆u+ a(·)u= u|u|p−2 in Ω, u > 0 in Ω, u= 0 on ∂Ω. The main purpose of this paper is to discuss the relationship between the multiplicity of solutions and the profile of a(·) from the variational point of view. It is shown that if a has a “peak” in Ω, then (P) has at least three solutions for sufficiently small ε.


Main theorem
We are concerned with the multiplicity of solutions for the following singularly perturbed semilinear elliptic equations: (P) ε −ε 2 ∆u + a(•)u = u|u| p−2 in Ω, u > 0 in Ω, u = 0 on ∂Ω, (1.1) where ε ∈ R + , Ω ⊂ R N (N ≥ 1) is a bounded domain, p ∈ (2,2 * ) (2 * denotes the critical exponent of the Sobolev embedding H 1 (Ω) L p (Ω) given by 2 * = 2N/(N − 2) if N ≥ 3 and 2 * = +∞ if N = 1,2).The main purpose of this paper is to discuss the relationship between the multiplicity of solutions of (P) ε and the shape of the profile of a(x) when ε is small.In order to characterize the topological feature of a(x), we introduce the following condition (A) K,r,c,c,δ,ρ for positive numbers r, c, c, δ, ρ, and a closed subset K of Ω.
Roughly speaking, the condition above implies that a(x) has a "peak" in K (condition (iii)), the value of a(x) on ∂K is uniformly less than the level of the peak (condition (iv)), and ∂K forms a set which surrounds the peak and is homotopically equivalent to S N−1 (condition (i)).
Then our main result reads as follows.
Theorem 1.1.For any positive numbers ρ, c, c with c < c, there exists a (sufficiently small) positive number δ depending on ρ, c, c, and Ω such that if a satisfies (A) K,r,c,c,δ,ρ for some r > 0 and a closed subset K of Ω, then there exists a positive number ε 0 so that (P) ε admits at least three solutions for all ε ∈ (0,ε 0 ]. We give some examples of the function a(•) satisfying the assumption of Theorem 1.1.
Example 1.2.Let Ω be a bounded domain which contains the closure of B(0,R).Let a ∈ C(Ω) and assume that there exist some positive numbers ρ, c, c with c < c and a closed subset L of B(0,R) such that ( Take R 1 (< R) and r > 0 such that Then a(•) satisfies (A) K,r,c,c,δ,ρ for K = B(0,R 1 ), δ ∈ (0,c − c), and r, c, c, ρ above.

Michinori Ishiwata 187
Remark 1.4.(1) It would be a routine work to prove that (P) ε admits at least one solution u 0 (the "ground state solution") for all ε ∈ (0,∞) with the aid of the well-known Mountain Pass lemma and the compactness of the Sobolev embedding H 1 L p .However, one cannot expect in general the existence of multiple solutions.Indeed, for example, when a(x) ≡ 1 and Ω = ball, the uniqueness result for sufficiently small ε is known (Dancer [6]).Theorem 1.1 says that immediately after a(x) is perturbed to have a "peak", other solutions u 1 , u 2 should appear even if the perturbation is very small (in the radial case, u 0 and one of the u 1 and u 2 , say u 1 , may be geometrically equivalent to each other, i.e., they may coincide via rotation in R N , so one gets at least two geometrically distinct solutions, u 0 ∼ u 1 and u 2 ).This "generation of higher energy solution" is a consequence of the change of topology of some level sets of the functional associated to (P) ε caused by the nontrivial shape of a(x).It is the purpose of this paper to discuss the effect of this change of topology on the multiplicity of solutions.
(2) It is already known that if a ∈ C(Ω) and the global minimum set of a(x), a min = {x ∈ Ω;a(x) = min y∈Ω a(y)}, is homotopically equivalent to S N−1 , then there exist at least cata min = catS N−1 = 2 solutions for small ε (here cat means the Ljsternik-Schnirelman category, see Definition 3.3 below).It should be noted that our assumption (A) K,r,c,c,δ does not require that a(x) should have a global minimum set in Ω as is stated in Example 1.3, but requires that "nearly" global minimum set of a(x) should contain the set ∂K which is homotopically equivalent to S N−1 .
(3) Another type of multiplicity result for −∆u + u = a(x)u + f (x) in R N , based on an argument similar to ours, is discussed in Adachi and Tanaka [1].

Known results and notation
2.1.Known results.The interest in (P) ε arises from several physical and mathematical backgrounds.
In the physical context, (P) ε can be regarded as a (reduced) nonlinear Schrödinger equation and small parameter ε corresponds to the Dirac constant .
It is well known that when can be well-approximated by 0 (this approximation is called "semiclassical approximation"), quantum mechanical equation may have a solution corresponding to a "semiclassical" state, concentrating around a classical mechanical equilibrium.It is also well known that the classical equilibrium is often given as the point which minimizes the potential energy.
So it is reasonable to expect that for small ε, (P) ε has a semiclassical solution concentrating around a point which attains the minimum of the energy potential a(x).Hence the structure of a min = {x ∈ Ω | a(x) = min y∈Ω a(y)}, the minimum set of a(x), may play a significant role for the existence and the multiplicity of solutions of (P) ε .
In the mathematical context, (P) ε can be regarded as a typical model exemplifying the following feature.In many semilinear elliptic problems including small parameters (e.g., semilinear elliptic equations involving the critical exponent [10], stationary Cahn-Hilliard equation [2], Ginzburg-Landau equation [3]), it is commonly observed that if the parameter is small enough, then the existence and multiplicity of solutions are controlled by the finite-dimensional object.As for singularly perturbed equations, del-Pino and Felmer [7,8] and Cingolani and Lazzo [5] obtain the following result.Proposition 2.1 (effect of weight function, del Pino and Felmer [7]).Assume that a(x) is a locally Hölder continuous function and Λ is a bounded set compactly contained in Ω.Also assume that there exists a positive constant α such that inf x∈Ω a(x) ≥ α and, min ∂Λ a(x) > inf Λ a(x).Then for sufficiently small ε, (P) ε admits a solution u ε , which concentrates to a point in Λ where the minimum of a(x) is attained as ε → 0. Proposition 2.2 (effect of weight function, del Pino and Felmer [8]).Assume that a(x) ∈ C 1 (Ω) and there exists a positive constant α such that inf x∈Ω a(x) ≥ α.Let x 0 ∈ Ω be a "topologically nontrivial critical point" of a(x) (this class of critical points includes the local minimum, the local maximum, and the saddle point of a(x).For the precise definition, see [8, page 249]).Then for sufficiently small ε, (P) ε admits a solution u ε , which concentrates to x 0 as ε → 0. Proposition 2.3 (effect of the topology of a min , Cingolani and Lazzo [5]).Assume that a(x) ∈ C(R N ) and lim |x|→∞ a(x) = a 0 > min x∈R N a(x) > 0. Then for sufficiently small ε, (P) ε admits at least cata min solutions.Here cata min denotes the Ljsternik-Schnirelman category of a min (see Definition 3.3

below).
The finite-dimensional objects referred to above in Propositions 2.1, 2.2, and 2.3 are the local minimum set (point) of a(x), the "topologically nontrivial" critical set (point) of a(x), and the global minimum set of a(x), respectively.
Our problem (P) ε also bears some interesting aspect in the context of the so-called "variational problem with lack of compactness".As stated in Section 1, for problem (P) ε with bounded Ω, one can easily find that there exists at least one solution of (P) ε , the ground state solution, with the aid of the compactness of the Sobolev embedding H 1 (Ω) L p (Ω).On the other hand, in the case of unbounded Ω, the situation changes drastically.That is, (P) ε may not have a ground state solution.From the point of view of the variational analysis, this nonexistence is caused by the breakdown of the Palais-Smale condition for the functional associated with (P) ε due to the fact that the Sobolev embedding H 1 (Ω) L p (Ω) is no longer compact for unbounded Ω.
Even though we are concerned with (P) ε in bounded domains (the original problem), the analysis of (P) ε in R N with some weight function determined by a(x) (the limiting problem) plays a crucial role in investigating the multiple existence of solutions of (P) ε .That is, the lack of compactness of the variational problem associated with (P) ε in unbounded domains with (suitably chosen) weight functions causes the multiplicity of solutions of (P) ε in bounded domains.In other words, for small ε, (P) ε can be treated as a problem on "almost unbounded domains" with Palais-Smale condition.
Applying propositions above to our problem, we find that the following facts hold true.
(2) Suppose that a(x) has a global maximum point in Ω and a(x) ∈ C 1 (Ω).Then Proposition 2.2 implies that there exists at least one solution of (P) ε for small ε, which concentrates to the global maximum point of a(x) as ε → 0.
As is pointed out in Section 1 (Example 1.3 and Remark 1.4), in our Theorem 1.1, we need not assume a min = ∂K nor a(x) ∈ C 1 (Ω).

Michinori Ishiwata 189
Moreover, our argument here relies on the comparison of variational structures of the original and the limiting problem with nontrivial weight function and, seems somewhat different from those in [5,7,8].

Notation.
We here fix the notation frequently used in this paper.
Let ω be a domain of R N , and we use the following notation.
(2.1) (v) When a(x) ≡ α > 0, we denote by v ε,α,ω the minimizer of S p (ε,α,ω) which is radially symmetric with respect to the origin, and ) stands for a cut-off function such that ϕ r is radially symmetric with respect to the origin and We also denote ϕ ε,r (x) := ϕ r (εx).
For any y ε ∈ ∂K/ε, we put and Φ ε,α (y ε ) := v ε,α,yε for all y ε ∈ ∂K/ε.Here K and r are a compact set and a positive constant which appear in the condition (A) K,r,c,c,δ,ρ in Section 1.
We occasionally suppress the subscript α when no confusion occurs.(vii) We denote a ε (x) := a(εx).(viii) Let X be a Banach manifold and a ∈ R. Then for I ∈ C 1 (X;R), we put where (dI) u represents the Fréchet derivative of I at u ∈ X.

Variational tools and preliminary facts
3.1.Variational tools.Our main tool relies on the variational approach.We here prepare some terminology frequently used later on.Definition 3.1 (Palais-Smale condition).Let M be a C 1 Banach-Finsler manifold and (b) J is said to satisfy the (PS) c -condition if (PS) c every (PS) c -sequence of J contains a strongly convergent subsequence.
(In the above T u M denotes the tangent space of M at u.) Our approach is based on the following fundamental principle.
Fundamental principle in Morse theory.Suppose that M is a Banach-Finsler manifold and I ∈ C 1 (M) satisfies the following assumptions: (1) In order to compare the topology of sets, various kinds of topological invariants are known.We will here use the notion of the "category" of sets.We use the following notation.
Definition 3.2.Let M be a topological space, and let A and x be a closed subset and a point of M, respectively.
Denote " x for all x ∈ A, and η(1,x) = x for all x ∈ A.

Definition 3.3 (notion of category). Let X be a topological space and let M, A be two closed subsets of X with
We simply denote cat Ω [Ω] by catΩ.In terms of this notion, Ljusternik-Schnirelman theorem (category version) reads as follows.

Michinori Ishiwata 191
In this paper, we use the variational method on the constraint manifold.In order to guarantee that the critical point on the manifold gives the critical point in the original space, we need the following version of Lagrange multiplier rule.Proposition 3.5 (Lagrange multiplier rule, [12, Proposition 5.12]).Let X be a Banach space, ψ ∈ C 2 (X;R), and In the proof of Theorem 1.1, we have to compare the category of two sets.For this purpose we use the following comparison theorem of category.Proposition 3.6 (comparison theorem for category).Let a and b be closed subsets of topological spaces A and B, respectively.Suppose that there exist Φ ∈ C(a;b) and Let a j = Φ −1 (b j ) ⊂ a. Then it is easy to see that a = m j=1 a j and a j is closed in A. Set Then it is easy to see that lim holds for all x ∈ a j .Hence g j ∈ C([0,1] × a j ;A).It is also obvious that for all x ∈ a j , g j (0,x Therefore it holds that a = m j=1 a j , a j is a closed subset of A, and a j {β(u j )} by g j in A. Hence, by the definition of the category cat In order to prove the existence of the critical point which has the higher energy, we use the following version of minimax principle.Proposition 3.7 (minimax principle).Let M be a C 1,1 Banach-Finsler manifold and let A be a metric space.Suppose A 0 ⊂ A is a compact subset, ϕ ∈ C(∂A 0 ;M), and If I satisfies (PS) c and sup y∈∂A0 I • ϕ(y) < c, then c gives a critical value of I.

Preliminary facts.
Setting v(x) = u(εx), (the weak form of) problem (P) ε can be rewritten as As for (P ) ε , the following fact is well known.For the convenience, we briefly give the sketch of proof.Proposition 3.8 (variational formulation of (P ) ε ).To find nontrivial solutions of (P ) ε is equivalent to (V) find critical points of I 1,aε,Ω/ε on M p (Ω/ε).
Proof.Sufficiency of (V).Assume that (V) has a solution, that is, there exists Therefore, by Proposition 3.5, there exists C ∈ R such that Michinori Ishiwata 193 Testing (3.11) with h = u, we get Then, by virtue of the fact that u = 0 and a ε > 0, we have Testing also (3.11) with h = u − = min(0,u), we obtain whence follows u − = 0 and u = u + ≥ 0.
Necessity of (V) also follows from arguments similar to those above.
For S p (ε,α,ω), it is well known that the following result holds.
Proposition 3.9 (existence and uniqueness for ground state in R N [9]).For any ε > 0 and α > 0, there exists a positive minimizer v ε,α,R N for S p (ε,α,R N ) which is unique (up to translation) and radially symmetric with respect to the origin.Especially, the map α → S p (ε,α,R N ) is continuous.
As we will see, the nontriviality of the topology of some level sets of I 1,aε,Ω/ε is the consequence of the nontriviality of that of ∂K.In order to discuss this relationship between the level set of I 1,aε,Ω/ε (in function space) and ∂K/ε (in R N ), we use the "truncated barycenter" β R (u) and a family of comparison function v ε,α,yε where y ε ∈ ∂K/ε (see Section 2.2 for definitions).
It is obvious Namely, the following holds.
Proof.(a) Take any α > 0, ε > 0, y ∈ ∂K and set y ε = y/ε.Since we will consider the limit ε → 0, without loss of generality we can assume that Then it follows that (3.16) As for (A), we find that, in view of (3.15), Then, since ϕ ε,r v α L p (R N ) → v α L p (R N ) as ε → 0, for suitable positive constant C 1 and C 2 , we have In view of the principle of Morse theory, in order to establish the existence of critical points of I 1,aε,Ω/ε , it is enough to verify the existence of a pair of level sets of I 1,aε,Ω/ε which have a difference in topology.The existence of such a pair of level sets of I 1,aε,Ω/ε is the consequence of the existence of that of I 1,bc,c,ρ,R N , the "limiting functional" associated to Note that by the assumption (ii) of (A) K,r,c,c,δ,ρ and the definition above, for any ε ∈ (0,1), (3.20) We next investigate the topology of the level set of I 1,bc,c,ρ,R N near its infimum level.

Limiting problem.
Hereafter we fix positive constants c, c, ρ.Let b c,c,ρ (x) be the function satisfying (B) c,c,ρ in the previous subsection.Throughout this subsection, we denote the limiting functional I 1,bc,c,ρ,R N as I c,∞ .In view of Proposition 3.9, we have the following.
Lemma 3.11 (inf is not achieved in the limiting problem).(a Proof.Suppose that the following claim holds true. Claim 3.12.Let b ∈ L N/2 (R N ) satisfy the following condition for some c > 0: Then we have the following: Then it is easy to see that Lemma 3.11 follows from the claim above with b(x) = b c,c,ρ (x).
Proof of Claim 3.12.(a) It is clear that for all u ∈ M p (R N ), we have To get the converse inequality, we will use some special sequence (v n ).Let v c ∈ M p (R N ) be a positive minimizer of S p (1,c,R N ) whose existence is guaranteed by Proposition 3.9.
Let (y n ) ⊂ R N be any sequence which satisfies Moreover, by virtue of . (3.26) it is also easy to see that, for R 1 and R 2 above, holds for large n.
Then we have (3.28) Using (3.25) and (3.27), we have Michinori Ishiwata 197 we have As in (a), let v c be a positive minimizer for S p (1,c,R N ).Suppose that the claim is false, that is, there exists w ∈ M p (R N ) such that . Then from Proposition 3.9, we find that w > 0 in R N .Therefore, in view of b(x) ≥ c and b(x) ≡ c, we have which leads to a contradiction.
This result implies that all the minimizing sequences possess no convergent subsequence.Combining this fact with the compactness of embedding H 1 L p loc , we find the following.

Lemma 3.13 (behavior of minimizing sequences of the limiting problem [4, Lemma 2.2]).
For any minimizing sequence Proposition 3.14 (concentration lemma at infinity for the limiting functional).For all s ∈ (0,R) and positive numbers ρ, c , c with c < c, there exists α > 0 which satisfies the following.
Proof.Suppose that the claim is not true.Then we find that for some ρ > 0, s ∈ (0,R), c, c with c < c, there exist Also it is easy to see that (3.36) yields the boundedness of (v n ) ⊂ H 1 (R N ).Therefore, it follows that as n → ∞.Furthermore, by Proposition 3.9 and (3.38), it holds that Therefore, by virtue of (3.36), (3.40), and (3.41), we find that v n ⊂ M p R N is a minimizing sequence of I c,∞ .Hence, together with Lemma 3.13, we have as n → ∞.But this is impossible in view of (3.37).
This proposition says that [I c,∞ ≤ S p + α] (⊂ M p (R N )), the infinite-dimensional object, can be compared with the B(0,s) c (⊂ R N ), the finite-dimensional object, with the aid of β R (v).We will see in the next subsection that this correspondence between the finitedimensional object and the infinite-dimensional object is also observed in the "original" problem.
Michinori Ishiwata 199 3.4.Original problem.Now we proceed to the analysis of the original problem.Throughout this subsection, we assume that a(x) is a function satisfying the condition (A) K,r,c,c,δ,ρ for some positive numbers ρ, c, c, δ with c < c and δ ∈ (0,c − c).We also assume that b c+δ,c (x) is a function defined by the condition (B) c+δ,c,ρ in the last part of Section 3.2.
As in the previous subsection, we denote the limiting functional I 1,bc,c,ρ,R N as I c,∞ and the original functional I 1,aε,Ω/ε as I ε .
We regard the original functional I ε (v) = Ω/ε |∇v| 2 + a(εx)|v| 2 as a perturbed functional of the limiting functional I c+δ,∞ for suitable δ > 0. Furthermore with the aid of Φ ε,c+δ (see Section 2.2 for definition), we can embed the topology of ∂K/ε into the level set of I ε near its infimum level when ε is small enough.Proposition 3.15 (construction of an embedding mapping from R N to the function space).For any δ > 0, Proof.First we are going to show the following fact.
uniformly in y ∈ R N (see Section 2.2 for the definition of v ε,α,yε ).
We next prove that under the condition (A) K,r,c,c,δ,ρ , the relation between the level set of functional I c,∞ and B(0,s) c described in Proposition 3.14 still holds for the perturbed functional I ε .Proposition 3.17 (concentration lemma at infinity for original functional).For any positive numbers ρ, s, c, and c with c < c, there exist δ > 0, η > 0 such that for any a(x) satisfying (A) K,r,c,c,δ,ρ with some r > 0 and some closed subset K of Ω, the following holds: there exists Michinori Ishiwata 201 ε ∈ (0,1) such that for all ε ∈ (0,ε], (3.53) Take any function a(x) which satisfies the condition (A) K,r,c,c,δ,ρ for some r > 0.

Proof of Theorem 1.1
Now we are in the position to give a proof of Theorem 1.1.In this section, we denote the original functional I 1,aε,Ω/ by I ε .Take any positive numbers ρ, c, c with c < c and R, s with R > s.Then Proposition 3.17 implies that there exist δ > 0, η > 0 such that for any a(x) satisfying (A) K,r,c,c,δ,ρ with some r > 0 and some closed subset K of Ω, the following holds: there exists ε 1 ∈ (0,1) such that for all ε ∈ (0,ε 1 ], Take any a(x) which satisfies (A) K,r,c,c,δ,ρ .Then, by Proposition 3.15 (with δ, η above) and Lemma 3.11(a), we see that there exists ε 2 > 0 such that for all ε ∈ (0,ε 2 ], for all y ε ∈ ∂K/ε.Note that since s < R, we can take ρ 0 so small that Then, by Lemma 3.10(a), we find that there exists ε 3 > 0 such that for all ε ∈ (0,ε 3 ] and y ε ∈ ∂K/ε.

Claim 4.1.
There exist c 1 , c 2 such that (a) c 1 is not a critical value of I ε and where m 1 is the number of critical points in [ Proof of Claim 4.1.(a) Without loss of generality, we can assume that there exists c 1 which satisfies and c 1 is not a critical value of I ε , otherwise we already have infinitely many critical values and the proof is finished.Then, by (4.3), Then it is easy to see that Moreover, because of (4.4), (4.5), and (4.6), we have g([0,1], y ε ) ⊂ B(0,s) c .Hence g is a homotopy between β R • Φ ε,c+δ and the natural injection from ∂K/ε into B(0,s) c , that is, c+δ , and β = β R .Then (4.2), (4.9), and (I) assure that all the hypothesis of Proposition 3.6 is fulfilled.Hence

Mathematical Problems in Engineering
Special Issue on Time-Dependent Billiards

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: [I ≤ a] M and [I ≤ b] M have a "difference in topology."Then there exists a critical value c ∈ [a,b].

Lemma 3 .
13 says that, for anyv ∈ M p (R N ) such that I c,∞ (v) is very close to its infimum S p (1,b c,c,ρ (•),R N ), vis almost concentrated at infinity.So by Lemma 3.10(b), we find that |β R (v)| R. Thus we get the following first key result stating the nontriviality of some level set of I c,∞ near its minimum level.