NODAL BUBBLE-TOWER SOLUTIONS FOR A SEMILINEAR ELLIPTIC PROBLEM WITH COMPETING POWERS

. In this paper, we consider the following semilinear elliptic problem where NN − 2 < q < p < p ∗ or q > p > p ∗ , p ∗ = N +2 N − 2 , N ≥ 3. We show that if q is ﬁxed and p is close enough to N +2 N − 2 , the above problem has radial nodal bubble tower solutions, which behave like a superposition of bubbles with diﬀerent orders and blow up at the origin.

For the supercritical case p > p * , there are few results up to now. The Pohozaev argument shows that there are no finite energy solutions of (1.1) provided that p > p * , q = 1. When q > p > p * , Kwong, McLeod, Peletier and Troy [19] proved that there exists a unique positive radial solution with fast decay O(|x| 2−N ), as |x| → ∞. Troy [32] showed that for any integer k ≥ 1, there is a radially symmetric solution with exactly k positive zeros and decay to zero at the rate O(|x| 2−N ).
The motivation for studying problem (1.1) comes from the study of the following Dirichlet problem where q > p > p * , B 1 (0) is the unit ball in R N . Merle and Peletier [25,26] have investigated the asymptotic behavior of positive solutions of (1.2) as ε → 0. However, their results show that the decay of the radial solutions must satisfy the fast decay, i.e. u(|x|) = O(|x| 2−N ), as |x| → ∞. Moreover, Dancer, Santra [12] and Dancer, Santra and Wei [13] considered the following singular perturbed problem where 1 < q < p < p * , Ω is a smooth bounded domain in R N . They studied the asymptotic behavior of the least energy solutions as ε goes to zero, which is also closely related to problem (1.1). From [18,19,20], we know that there is a unique positive radial solution for problem (1.1). It is natural to ask if there are some other sign-changing solutions of (1.1)? Our purpose in this paper is to give a positive answer to this question. Actually, we construct nodal bubble tower solutions for (1.1) which behave like a superposition of bubbles. It seems that this is the first existence results of nodal bubble tower solutions for (1.1).
More precisely, we consider the subcritical case where ε > 0, q ∈ ( N N −2 , p * ) is fixed. Our main result concerning problem (1.4) is the following. Theorem 1.1. Assume that N ≥ 3, then for any integer k ≥ 1, there exist ε k > 0 such that for ε ∈ (0, ε k ) problem (1.4) has a radial nodal bubble tower solution u ε , which has the form , M 1 , · · · , M k are positive constants depending only on N and k, and o(1) → 0 uniformly on compact subsets of R N as ε → 0.
For the supercritical case, we consider where ε > 0, q > p * is fixed. Then we have Theorem 1.2. Assume that N ≥ 3, then for any integer k ≥ 1, there exist ε k > 0 such that for ε ∈ (0, ε k ) problem (1.5) has a radial nodal bubble tower solutionû ε , which has the form ,M 1 , · · · ,M k are positive constants depending only on N and k, and o(1) → 0 uniformly on compact subsets of R N as ε → 0.  (1.5). See [28] for more general relation between tower of positive or sign-changing bubble solutions and the sign of ε.
Remark 1.4. The basic elements to construct nodal bubble tower solutions are the radial functions U µ (|x|), defined by The main idea of the paper is motivated by [8,15]. More precisely, we will use the Lyapunov-Schmidt reduction argument to prove Theorem 1.1 and 1.2, which reduces the construction of the solutions to a finite-dimensional variational problem. It is worthwhile pointing out that bubble tower concentration phenomena for nonlinear elliptic problems with critical Sobolev exponent has been observed in [8,10,15,14,28,29]. However, as far as we know, there are few results for problem (1.1).
As a final remark, our results seem to be connected with the work of Campos [8]. More precisely, Campos considered where ε > 0, q ∈ ( N N −2 , p * ), or ε < 0, q > p * . He constructed positive radial solutions, which look like a superposition of bubbles. Our results in this paper can been seen somewhat dual counterparts in [8].
This paper is organized as follows. In Section 2, we will give some preliminaries needed in the later sections. The finite dimensional reduction argument is carried out in Section 3. We will prove the main results in Sections 4, 5.
2. Some preliminaries. In this section, we will give some basic estimates, which will be used in the later sections. We are interested in finding radially symmetric solutions for problem (1.4), and then it can reduce the following problem −u − N −1 r u = |u| p * −1−ε u − |u| q−1 u, in R, u (0) = 0, u(r) → 0, as r → ∞.
The energy functional corresponding to problem (2.3) is It is easy to see that W (y) is the unique solution of the problem For given Λ i > 0, i = 1, 2, · · · , k, set We will look for a solution of problem (2.3) of the following form where φ is small. Set First, we give some asymptotic estimates of W i and V in the following lemma.
Next, we give the asymptotic expansions of the energy functional I ε (V ), which will turn out to be very important in looking for critical points of I ε (V + φ).
Proof. Recall that Note that It follows from Lemma 2.1, we find where It is easy to check that Thus, from Lemma 2.1, we find where Thus, we find Finally, it is easy to check that Note that Therefore, we can obtain (2.7) immediately and the proof of Proposition 2.2 is complete.
3. The finite dimensional reduction. In this section, we will perform the finite dimensional procedure, which reduces problem (2.3) to a finite-dimensional problem on R.

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Proof. We will first show that lim n→∞ φ n L ∞ = 0.
Arguing by contradiction, we may assume that φ n L ∞ = 1. Multiplying (3.1) by Z n and integrating by parts, we find This defines an almost diagonal system in the c n i 's as n → ∞. Since q > N N −2 and Z n (y) is a solution of Note that Z n (y) = O(e −|y−ξ n | ), by the dominated convergence Theorem, we know that lim n→∞ c n i = 0. Assume that y n ∈ R is such that |φ n (y n )| = 1, it follows from (3.1) and the elliptic regular theory that we can assume that there is an and a fixed M > 0, such that |ξ n − y n | ≤ M for n large enough. Setφ n (y) = φ n (y + ξ n ). From (3.1), we see that up to a subsequence, there isφ such thatφ n →φ uniformly over compact sets of R andφ is a nontrivial bounded solution of Thus, by nondegeneracy in [30],φ = cW , c = 0. However, which is a contradiction. Thus, lim n→∞ φ n L ∞ = 0. Next we shall establish that lim n→∞ φ n * → 0. Now we see that (3.1) possesses the following form where If 0 < σ < min{p * − 1, 2q − p * − 1, 1}, we find e −σ|y−ξ n i | with θ n → 0.
Choosing C > 0 large enough, we see that is a supersolution of (3.2), and −ϕ n (y) will be a subsolution of (3.2). Thus, and the result follows.
The following proposition is a direct consequence of proposition 1 in [15] combining with Lemma 3.1.
Proposition 3.2. There exist positive numbers ε 0 , δ 0 , R 0 , such that if ξ 1 , · · · , ξ k satisfy then for ε ∈ (0, ε 0 ) and h ∈ C * , problem (3.1) has a unique solution φ = T ε (h). Moreover, there exists C > 0 such that For later purposes, we need to study properties of the differentiability of the operator T ε on the variables ξ i and the space L(C * ) of the linear operator in C * . For simplicity, we will use the notation ξ = (ξ 1 , ξ 2 , · · · , ξ k ). We also consider numbers ε 0 , δ 0 and R 0 given by Proposition 3.2, and let for 0 < ε < ε 0 , We now define the following map We have the following results.
Proposition 3.3. For each h ∈ C * , the map ξ → S ε (ξ, h) is of class C 1 . Besides, there is a constant C > 0 such that Proof. Fix h ∈ C * , and let φ = T ε (h). Recall that φ satisfies and plus orthogonality conditions for some constants c i . We define the constants b j satisfying

Now we consider the following intermediate problem
(3.4) In order to solve problem (3.4), we rewrite it as Let us fix a large number M > 0, ξ satisfies the following conditions In order to prove that (3.5) is uniquely solvable with respect to φ * , we need to estimate R ε , N ε (φ) and their derivative correspondents in the · * -norm, where where and First, we consider p * > 2 and 2q − p * > 2. Thus, The case for p * ≤ 2 is similar. D φ N ε (φ) can be estimated similarly and the proof of the lemma is completed.
Lemma 3.5. Assume that (3.6) holds, then there exists C > 0 such that
The next proposition enables us to reduce the problem of finding a solution for (2.3) to a finite dimensional problem.
Proposition 3.6. Suppose that conditions (3.6) hold. Then there exists a positive constant C such that, for ε > 0 small enough, problem (3.5) admits a unique solution φ = φ(ξ), which satisfies Moreover, φ(ξ) is of class C 1 on ξ with the · * -norm, and where τ > 0 is a small constant.
Proof. Let us consider the operator then we know that problem (3.5) is equivalent to the fixed point problem φ = A ε (φ). We will use the contraction mapping theorem to solve it. Set 2 }, where ρ > 0 will be fixed later.
We will show that A ε is a contraction map from E ρ to E ρ . In fact, for ε > 0 small enough, we find provided ρ is chosen large enough, but independent of ε. Thus, A ε maps E ρ into itself. Moreover, Hence, Thus, there is a unique φ ∈ E ρ , such that φ = A ε (φ). Now consider the differentiability of ξ → φ(ξ).
Thus, the linear operator D φ B(ε, φ) is invertible in C * with uniformly bounded inverse depending continuously on its parameters. Differentiating with respect to ξ, we deduce where all these expressions depend continuously on their parameters.

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By the implicit function theorem, wee see that φ(ξ) is of class C 1 and Thus, The proof of Proposition 3.6 is concluded.
4. Proof of Theorem 1.1. In this section, we will prove Theorem 1.1. To do so, we will choose ξ such that V + φ is a solution of (2.3), where φ is the map obtained in Proposition 3.6.
Recall that It is well-known that if ξ is a critical point of K ε (ξ), then V + φ is a solution of (2.3). Next, we will prove that K ε (ξ) has a critical point. To this end, we need the next lemma, which is important in finding the critical points of K ε .
Lemma 4.1. The following expansion holds where O(ε 1+τ ) is uniformly in the C 1 -sense on the vectors ξ satisfying (2.5).
Proof. Note that DI ε (V + φ)[φ] = 0, we have Thus, . Differentiating with respect to ξ , we get that In a similar way, we find that Thus, the result follows.

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For the Λ ε obtained above, let is a solution of (2.3).
Note that where Thus, using the transformation (2.2), we see (1 + o(1)), and the proof of Theorem 1.1 is concluded.

5.
Proof of Theorem 1.2. In this section, we will give the proof of Theorem 1.2.