A CONCENTRATION PHENOMENON OF THE LEAST ENERGY SOLUTION TO NON-AUTONOMOUS ELLIPTIC PROBLEMS WITH A TOTALLY DEGENERATE POTENTIAL

. In this paper we study the following non-autonomous singularly perturbed Dirichlet problem: for a totally degenerate potential K . Here ε > 0 is a small parameter, Ω ⊂ R N is a bounded domain with a smooth boundary, and f is an appropriate super- linear subcritical function. In particular, f satisﬁes 0 < liminf t → 0+ f ( t ) /t q ≤ limsup t → 0+ f ( t ) /t q < + ∞ for some 1 < q < + ∞ . We show that the least energy solutions concentrate at the maximal point of the modiﬁed distance function D ( x ) = min { ( q + 1) d ( x,∂A ) , 2 d ( x,∂ Ω) } , where A = { x ∈ ¯Ω | K ( x ) = max y ∈ ¯Ω K ( y ) } is assumed to be a totally degenerate set satisfying A ◦ (cid:54) = ∅ .

The problem (1) arises in various mathematical models deriving from biological population theory, chemical reactor theory, etc. For example, the problem (1) with the Neumann boundary condition arises in the following Keller-Segel system: where u 1 (x, t) is the population of amoebae at place x and time t and u 2 (x, t) is the concentration of the chemical. This model was proposed by Keller and Segel in [11]. If we take φ(u 2 ) = log u 2 and k(x, u 1 , u 2 ) = −au 2 + K(x)u 1 with K(x) > 0 onΩ, and consider steady states of the above system, then the system for steady states is reduced to (1) with the Neumann boundary problem and f (u) = u p for some p > 0. We refer to Lin, Ni and Takagi [15] for the details for the derivation.
In this paper, we consider the problem with the Dirichlet boundary condition. The autonomous case of (1), that is, K(x) ≡ 1, has been widely studied for a long time. In [17], Ni and Wei showed that the least energy solution concentrates at the most centered point of Ω when f satisfies a uniqueness-nondegeneracy assumption on the limiting equation. Furthermore, in [7], del Pino and Felmer showed that the same results holds by a simpler approach without a uniqueness-nondegeneracy assumption for f . Also, in [4,5], Byeon studied the optimal conditions for f for the existence of the concentrating solution at the most centered point of Ω. On the other hand, there are many studies on the existence and the properties of higher energy solutions. Especially, many researchers showed there exist positive solutions with single and multiple peaks (see [6,21,14,8] and the references therein).
The non-autonomous case of (1), that is, K(x) ≡ const., also has been studied by many researchers when f (u) = u p . In [19], Ren showed that the least energy solutions concentrates at the maximal point of K(x) onΩ. In [18], Qiao and Wang studied the multiplicity of positive solutions to (1). In [22], Zhao studied on the number of interior peaks of solutions to (1) with the Neumann boundary condition.
We define the energy functional associated to (1) as and the least energy associated to (1) as e ε,Ω (K) := inf Moreover, we call the solution u ε of (1) the least energy solution if it holds that Similarly, we define the energy functional associated to the following problem: as and the least energy associated to (3) as First, we state the existence and the basic properties of the least energy solution to (1) as follows.
Proposition 1. We assume that a function K satisfies conditions (K0)-(K1) and a function f satisfies conditions (f0)-(f4). Then, for any ε > 0 there exists a positive least energy solution u ε ∈ C 2 (Ω) to (1). Furthermore, the following statements hold: (iii): There exist constants C 1 and C 2 independent of ε such that (iv): Passing to a subsequence, we have where v ε (y) := u ε (εy + x ε ) and w is a least energy solution to (3).
When f (u) = u p , Proposition 1 has been proved by Ren ([19]). We can prove Proposition 1 by almost the same argument as in [19] for general nonlinearities f (u) satisfying (f0)-(f4). So, we omit the details. For example, refer to [13,Appendix].
For the precise asymptotic location of the maximum point x ε and the precise asymptotic expansion of the least energy e ε,Ω (K), we show the following theorem, which is the main result of this paper. We will use the next notation: Theorem 1.1. We assume that a function K satisfies conditions (K0)-(K2) and a function f satisfies conditions (f0)-(f5). Let u ε be a least energy solution of (1) and x ε be a maximum point of u ε . Then the following statements hold: (iii): Passing to a subsequence, we have

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Our proof of Theorem 1.1 is based on the modified argument of [7] employing the rearrangement technique and the precise asymptotic behavior of solutions to several auxiliary problems (lemma 3.2 and 3.3). Especially, we do not need to assume the uniqueness-nondegeneracy condition for the nonlinearity f (u).
Another similar problem to (1) is the nonlinear Schrödinger equation, that is, for a potential V which satisfies B • = ∅, where In [16], Lu and Wei showed that the least energy solution of the problem (5) concentrates at the maximal point of d(x, ∂B) on B when f (u) = u p and V satisfies the assumptions that B • is connected and V (x) = O(d(x, ∂B) 2 ) near ∂B. Recently, in [13], the author extended the results to the more general f (u) and V (x) by using the modified argument of [7] employing the rearrangement technique. This paper is organized as follows: In section 2, we prepare some facts, that is, the properties of the least energy solutions to (3), the modified Bessel functions, and the Schwarz symmetrization. In section 3, we show that the asymptotic expansion of the least energy in the special case that Ω is a ball and a function K is radially symmetric. In section 4, we show the main result, that is, Theorem 1.1. In section 5, we give some concrete examples to illustrate the effect of the function D(x). We always assume that B(x; r) denotes an open ball with center x, radius r > 0, P • denotes the interior of P , P c denotes the complement of P and d(x, P ) denotes the distance from x to P . We also use the notation B r (x) = B(x; r) and B r = B r (0).

Preliminaries.
2.1. The basic properties of the least energy solutions to (3). First, we remark the next fact on the problem (3), which is well known, for example, see [2,3,9]. Proposition 2. We assume that a function f satisfies conditions (f0)-(f2), (f4). Then there exists a solution w ∈ C 2 (R N ) ∩ H 1 (R N ) of (3) and w satisfies that (i): w is decreasing and radially symmetric,

2.2.
Modified Bessel function. We will need the asymptotic properties of the modified Bessel functions to show the asymptotic expansion of the least energy. The next definition is taken from [20].
Definition 2.1. We define the modified Bessel function of first kind I ν by Also, we define the modified Bessel function of second kind K ν by and K n (x) := lim ν→n K ν (x) for n ∈ Z, x > 0.
The next results are basic properties of the modified Bessel functions I ν (x) and K ν (x). For the proof, see [20, p.77-80, p.202-203].
Proposition 3. For all ν ∈ R, x > 0, the following statements hold: (ii): Both I ν and K ν are solutions of the equation ] denotes the Wronskian of I ν (x) and K ν (x). In particular, I ν and K ν are linearly independent. (iv): As x → +∞, we have Proposition 3 yields the next results.
2.3. Schwarz symmetrization. The Schwarz symmetrization plays important roles in the proof of Theorem 1.1. We just recall the definition and basic properties.
Moreover, we define the Schwarz symmetrization u * : where E * denotes the open ball centered at the origin and having the same measure as E, i.e. |E * | = |E|.
The Schwarz symmetrization has the next basic properties.
Proposition 4 ([12, p.14]). Let E ⊂ R N be a bounded measurable set and u : E → R be a measurable function. Then, the following statements hold: (i): u and u * are equimeasurable.
The next result is very important for the proof of Theorem 1.1. For the proof, see [12,Theorem 1.2.2].
3. In the case A is a ball. In this section, we prove the expansion of e ε,Ω (K) in the special case. Let r > 1. We assume that a Borel measurable function K :B(0; r) → R satisfies the following: . For ρ > 0, we define the energy functional as Moreover, we define the least energy as In this section, it is our goal to show the next Theorem.

3.1.
Upper bound of C ρ,r (K). The next two Lemmas are crucial in the proof of the upper bound of C ρ,r (K).
We define w ρ as

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Then, w ρ satisfy By the uniqueness of the solution of (10), there exist unique α ρ and β ρ such that Note that it holds that In particular, Claim 1. (1)) as ρ → +∞.
where a ρ ∈ R and lim ρ→+∞ a ρ = 1. Then, it holds that Proof. We will use the same argument as in the proof of Lemma 3.2. Let ν := Then, z ρ satisfies that where z ρ (+∞) = 0 follows by the exponential decay of v ρ . In fact, by using the comparison principle for v ρ (s) and w ρ (s) := exp( √ a ρ (ρ − 1)) exp(− √ a ρ s), we have v ρ (s) ≤ w ρ (s) for ρ − 1 ≤ s < +∞. By the uniqueness of the solution of (16), there exists a unique α ρ such that Note that it holds that By the elementary calculation, we have We will calculate the upper bound of z ρ ( √ a ρ (ρ − 1)). By (17) and (18), we have The above equality and (19) Now, we shall show the upper bound of C ρ,r (K).
Proof of Proposition 7. Assume that r > 1. Let w ∈ H 1 (R N ) ∩ C 2 (R N ) be a least energy solution to (3). Moreover, let w ρ ∈ H 1 (B rρ \ B ρ−1 ) ∩ C 2 (B rρ \ B ρ−1 ) be a unique radially symmetric solution of the equation where Note that w ρ (s) > 0 for ρ − 1 < s < rρ by the strong maximum principle. Define w ρ as C ρ,r (K) = inf where t ρ is a unique positive constant such that the last equality holds.
Proof of Claim 1. Assume that there exists a subsequence {ρ j } ⊂ {ρ} such that where t j := t ρj . Then, by the definition of t j , we may estimate Integrating by parts, we obtain where the first inequality follows from w ρj (y) ≤ w(ρ j − 1) and the second inequality follows from w(ρ j − 1) ≤ C exp(−ρ j + 1) and Lemma 3.2. Hence, letting j → ∞ in (23), we have This is impossible since w is a solution to (3).
We return to the proof of the upper bound of C ρ,r (K). By (21), we may decompose J ρ,r (K) into the following three parts:

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First, we shall estimate A. By Proposition 2 (ii), (iii), we may estimate Next, we shall estimate B. By integration by parts, we may estimate By sup x∈R N |∇w(x)| < +∞ and w(r) ≤ Ce −r , we obtain that (1)) .
Finally, we shall estimate D. By integration by parts, we obtain that Since w ρ is the solution to (20), we may estimate where the first inequality follows from w ρ (y) ≤ w(ρ − 1) and the second inequality follows from (f5) and Proposition 2 (iii). By Lemma 3.2, we have (1)) . (1)) .
The next corollary is convenient when we show Theorem 1.1.

3.2.
Lower bound of C ρ,r (K). First, we note the following result which gives the information of the least energy solution when K and Ω are radially symmetric.
We can show Proposition 8 by combining the well known argument with the rearrangement technique. So we omit the proof of Proposition 8. For example, see [13].
Proof of Proposition 9. By Proposition 8, we obtain a positive radially symmetric least energy solution v ρ ∈ H 1 0 (B rρ ) to the equation For any 1/2 ≤ t ≤ 2, we may estimate that Proof. Assume that there exist a subsequence {ρ j } ⊂ {ρ} and δ > 0 such that where t j := t ρj . Assume that (32) holds for infinitely many j. By the definition of t j , where the inequality follows from (32). By v ρj → w in H 1 (R N ), letting j → ∞, we obtain that This is impossible since w is a solution of (3). Similarly, (33) is possible only for finitely many j. (1)) .
By Propositions 4-6, we obtain that u * ε ∈ H 1 0 (Ω) \ {0} and for any t > 0, By Corollary 2, passing to a subsequence, we obtain Consequently, (35), (38) and (39) yield that (1) We show (iii) of Theorem 1.1 by using the argument of [7]. Specifically. we reduce the energy e ε,Ω (K) to the case Ω and A are balls, and derive the upper and lower bound of e ε,Ω (K) by using the inequalities which are proved in section 3. But, since the problem which is considered in [7] has not the potential K, we have to modify the argument of [7].

4.2.
Proof of the lower bound of e ε,Ω (K). Our main idea to prove the lower bound of e ε,Ω (K) is to derive two inequalities. One represents effectiveness of the potential K and the other represents effectiveness of the Dirichlet boundary condition. We prove the former in Proposition 10 and the latter in Proposition 11. After that, we prove the lower bound of e ε,Ω (K) by using them. While we can show Proposition 10 to use almost the same as the argument of [7], we need to modify the argument of [7] to prove Proposition 11. First, we remark some important Lemmas to prove Propositions 10 and 11.
Proof of Lemma 4.3. Choose a smooth function η such that where C is a constant which is independent of z. Since u is a weak solution of −∆u = g in B(z; 2), taking η 2 u as a test function, we have B(z; 2) ∇u(x) · ∇(η(x) 2 u(x)) dx = B(z; 2) g(x)η(x) 2 u(x) dx.