Local uniqueness problem for a nonlinear elliptic equation

In this paper, we consider the following nonlinear Schrodinger equation \begin{document}$ \begin{eqnarray*} - \varepsilon^{2}\Delta u_{ \varepsilon}+u_{ \varepsilon} = K(x)u_{ \varepsilon}^{p-1} & & {\rm{in\;}}\mathbb{R}^{N}, \end{eqnarray*} $\end{document} where \begin{document}$ N\ge3 $\end{document} and \begin{document}$ 2 . Under mild assumptions on the function \begin{document}$ K $\end{document} and using the local Pohozaev identity method developed by Deng, Lin and Yan [ 10 ], we show that multi-peak solutions to the above equation are unique for \begin{document}$ \varepsilon>0 $\end{document} sufficiently small.


Introduction and main result.
1.1. Introduction. In this paper, we consider the nonlinear elliptic problem (1.1) where > 0 is a parameter, N ≥ 3 and 2 < p < 2N/(N − 2), and K is a bounded positive continuous function in R N .
Problem (1.1) and its variants arise in many applications such as chemotaxis, population genetics, chemical reactor theory. It is also a typical case of the more general problem which, for instance, stems from the study of standing waves with the type ψ(x, t) = e −iEt/ u(x) to the nonlinear Schrödinger equation In physics concerning e.g. nonlinear optics, plasma, condensed matter, the function V (x) represents the potential acting on the particle and K(x) is a particleinteraction term, which avoids spreading of the wave packets in the time-dependent version of the above equation. Furthermore, to describe the transition from quantum to classical mechanics, we let → 0 and thus the existence and uniqueness of solutions ψ for small has an important physics interest. Following Oh [21] and many others in the literature, we call solutions of (1.2) semiclassical states hereafter. Beginning from the pioneering paper by A. Floer and A. Weinstein [13], where (1.2) is considered with N = 1, p = 4 and K ≡ 1, a great deal of work has been devoted to the study of existence of problem (1.2). It would be impossible to list all the significant contributions since then, we mainly mention e.g. [1,2,3,5,9,11,12,16,20,21,22,23] and refer the interested readers to the references therein. Since we are concerned with the case K ≡ 1, we mention e.g. [1,2,3,5,9,20]. In the work [5], Bartch and Peng considered (1.2) in the case when both V and K are radial functions. They proved the existence of positive radially symmetric solutions concentrating simultaneously on multiple spheres. In the works [1,2,3,9], the authors constructed multipeak semiclassical solutions to problem (1.2) under quite general assumptions on V and K. In particular, V and K are allowed to be unbounded or vanishing at infinity. In the work [20], Noussair and Yan considered problem (1.1). Under the assumption that (i) K is a bounded continuous positive function in R N and K has a strict local minima x 0 , (ii) K is Hölder continuous in a neighborhood of x 0 , Noussair and Yan [20] constructed k-peak solutions concentrating at x 0 for every k ≥ 1.
In this paper, we are concerned with local uniqueness of positive concentrating solutions to problem (1.1). Here, by local uniqueness, it means that if u 1 , u 2 are two semiclassical solutions of equation (1.1) concentrating at the same family of concentration points, then u 1 ≡ u 2 for sufficiently small. As to the definition of concentrating solutions, it is standard and we refer to e.g. Cao and Heinz [6].
Let us now review some known results in the respect of local uniqueness. It seems that the first result in this respect is the uniqueness of solutions concentrating at one point for Dirichlet problems with critical nonlinearity on bounded domains given by Glangetas [14]. By calculating the number of single-bump(single-peak) solutions to (1.2) (with K ≡ 1), Grossi [15] proved that there is one solution concentrating at any nondegenerate critical point of V (x). We remark that the uniqueness of singlebump(single-peak) solutions concentrating at some degenerate critical point of V (x) is true in [15] as well. Later, Cao and Heinz [6] proved the uniqueness of multi-bump solutions to (1.2) (with K ≡ 1) which concentrate at the nondegenerate critical points of V . The results in [6,14] are obtained by using the topological degree. Recently, Deng, Lin and Yan [10] proved the local uniqueness and periodicity for the solutions with infinitely many bumps of the prescribed scalar curvature problem which involves the critical Sobolev exponent by the technique of Pohozaev identity. Using the idea of Deng et al. [10], Cao, Li and Luo [7] established local uniqueness of multi-peak solutions to (1.2) (with K ≡ 1). In particular, in the work [7], they do not need to assume V is nondegenerate at concentration points. The same idea has also been used by Guo, Peng and Yan [17] to study uniqueness of solutions concerning polyharmonic operators.
We conclude from the above review that great progress on local uniqueness of semiclassical solutions to problem (1.2) in the case K ≡ 1 have been obtained. However, in the case K ≡ 1, even though there have been many works on the existence of semiclassical solutions as aforementioned, there seems to have no results for problem (1.2) in the respect of local uniqueness. This is the direct motivation of this work. However, to avoid too much involved in the arguments, we will restrict ourselves to the typical case V ≡ 1 and K ≡ 1 in problem (1.2). That is, we will only consider problem (1.1) and leave the more general case to interested readers.

1.2.
Main result and strategy of proof. To state our main results, let us introduce some notations first. Denote It is well known (see e.g. Cao and Peng [9]) U j can be represented by scaling the unique (Kwong [18]) positive radial solution U of the problem However, to simplify the notations, we keep using the notation U j instead of a scaling of U .
Then, following the scheme of Cao and Peng [9], one can construct solutions to equation (1.1) in the form x − a j, + ω for some numbers α(a j, ), β(a j, ) → 1 as → 0, where a j, ∈ R N satisfying a j, → a j as → 0, (1.5) and for A = (a 1, , . . . , a k, ). However, it is noted that by combining the improved Lyapunov-Schimdt reduction (see e.g. Li et al. [19]), one can in fact construct a family of solutions concentrating at {a 1 , . . . , a k } ⊂ R N for equation (1.1) in the more brief form where a j, and ω are defined as in the above. Thus, a natural question is whether solutions constructed in this way is unique. To answer this question, we shall assume (K1) There exist m > 1, k i,α = 0 and δ > 0 such that for all 1 ≤ i ≤ k and 1 ≤ α ≤ N . Then, our main result reads as Theorem 1.1. Assume that K is a bounded positive continuous function satisfying the assumption (K1). Assume that u (i) , i = 1, 2, are two positive solutions of problem (1.1) concentrating at k different points {a 1 , · · · , a k } in R N . Then, u (1) ≡ u (2) for sufficiently small. Moreover, where α(a j, ), β(a j, ) → 1 as → 0, a j, ∈ R N and w ∈ H 1 (R N ) satisfying, as → 0, Since in the work of Cao, Li and Luo [7], they have proved this type of results for equation (1.2) in the case K ≡ 1, we will follow the line of Cao et al. [7]. To control the size of this work, we now explain the strategy of proving Theorem 1.1 and devote the rest sections to the proof of the main steps (see Theorem 1.2).
As the first step of proving Theorem 1.1, one can prove by a standard argument (see e.g. Cao, Li and Luo [7] or Cao and Heinz [6]) that if u is a positive solution of problem (1.1) concentrating at k different points {a 1 , · · · , a k } in R N , then u can be written in the form (1.8), where α, β converges to 1, and ω ∈ ∩ k j=1 E ,j,aj, , where Then, as the second step, one notices that it is sufficient to assume that α(a j, ) = β(a j, ) = 1 by the arguments as that of Cao, Li and Luo [7, Proposition 2.5], and assume that ω belongs to the simpler set ∩E ,aj, , where E ,aj, is defined as in (1.4), and ω satisfies for some constant ρ > 0. Indeed, solutions satisfying α(a j, ) = β(a j, ) = 1 and ω ∈ ∩E ,aj, can be constructed directly using the Lyapunov-Schmidt reduction (see e.g. Li et al. [19]). Furthermore, (1.9) can be proved using the same argument as that of Cao, Li and Luo [7] (see the section "Analysis of w and v " in their Appendix) or using the arguments of Cao et. al. [8]. In the work of Li et al. [19], estimates of type (1.9) has also been proved directly. We remark that in this step, the important well known nondegeneracy result (Kwong [18]) is used: With the help of the above explanation, we are reduced to prove the following theorem. . Moreover, assume that ω ∈ ∩E ,aj, , where E j,aj, is defined as in (1.4), and ω satisfies (1.9). Then, u (1) ≡ u (2) for sufficiently small. Moreover, let Then as → 0, there hold Before closing the introduction, let us briefly explain our proof for the above simplified theorem. To prove Theorem 1.2, we use a contradiction argument as that of Cao, Li and Luo [7]. More precisely, if u (i) , i = 1, 2, are two distinct solutions of problem (1.1) as stated in Theorem 1.2, then it is clear that the function is nonzero and ξ L ∞ (R N ) = 1. However, we will use the equations satisfied by ξ to show that ξ L ∞ (R 3 ) → 0 as → 0. This gives a contradiction, and thus the uniqueness result is obtained. To deduce the contradiction, delicate estimates on the asymptotic behaviors of solutions and the concentrating point a j, will be derived by a local Pohozaev type identity following the idea of Deng, Lin and Yan [10]. Throughout this paper, we use B R (x) (andB R (x)) to denote open (and close) balls in R N centered at x with radius R. Unless otherwise stated, we write u = R N u(x)dx to denote Lebesgue integral of an integrable function u over R N , and denote by u s the L s -norm of a function u in L s (R N ) for any 1 ≤ s ≤ ∞. We will use the same C to denote various generic positive constants, and use O(t) and o(t) to mean |O(t)| ≤ C|t| and o(t)/t → 0 as t → 0 respectively. We also use o(1) or o (1) to denote quantities that tend to 0 as → 0.
2. Preliminary estimates. In this section we prove the following estimates.
is a solution of equation (1.1) with a j, and ω satisfying the assumptions of Theorem (1.2). Then, For simplicity of notations and calculations, we hereafter assume without loss of generality that k = 2. The general cases follow easily. We first prove (2.2).
Proof of (2.2). Recall that for the linear operator L defined by there exist , δ > 0 sufficiently small, and ρ > 0 such that On the other hand, by the equation (1.1) of u and the equation (1.3 Hence, By the same argument as that of Cao and Peng [9, Lemma 3.1] and by the assumption (1.6), we have Since a 1 = a 2 and U i decays exponentially, we have for any given γ > 0. To estimate the last term, we use the assumption (K1). Note that In the last step, we used Hölder's inequality and the assumption (1.6). By the boundedness of K and the exponential decay of U i , we have for any given γ > 0. Hence, taking γ > N/2 + m, we derive Now, combining (2.4)-(2.7) and using -Young's inequality ab ≤ a 2 + C b 2 for any a, b > 0, we deduce By using a Pohozaev type identity, the assumption (K1), and (2.8), we will prove in below that (2.1) holds. This in turn implies that (2.2), and thus finishes the proof of (2.2).
Next we prove (2.1). We will use the following Pohozaev type identity.
Multiplying both sides of equation (1.1) by ∂ xα u for each 1 ≤ α ≤ N and then integrating by parts, Proposition 2 can be proved. We omit the details, see Cao, Li and Luo [7, Proposition 2.3].
The following type of Sobolev inequality will be used repeatedly: For any 2 ≤ q ≤ 6 there exists a constant C > 0 depending only on n, V , a and q, but independent of , such that holds for all ϕ ∈ H . For a proof, see e.g. (3.6) of Li et al. [19].
By an elementary inequality, we have Since U i , i = 1, 2, decay exponentially at infinity and |a 1 − a 2 | > δ > d, we have for any given γ > 0. Thus, by the choice (2.11) of d , we have Similarly, we derive where we have used (2.11) and (2.10). Combining the above estimate and (2.12), we obtain for any given γ > 0.
To estimate the left hand side of the above equation, we apply the assumption (K1) to get Using a scaling argument, we obtain By the exponential decay of U i and the fact that |a 1, − a 2, | ≥ d, we obtain From (2.10), we have Hence, Combining this estimate together with (2.13) and (2.14), we deduce | z α + a 1, ,α − a 1,α | m−2 ( z α + a 1, ,α − a 1,α ) u p ( z + a 1, ).

Using the elementary inequality
for any a, b ∈ R, where m * = min{m, 2} and C is positive constant depending only on m, and using the same argument as that of Cao, Li and Luo [7, Lemma 2.1], we conclude that and Since |a 1, ,α − a 1,α | = o (1) by assumption, using Young's inequality, we deduce To conclude, recall (2.8). We obtain To further prove (2.1), we assume, on the contrast, that there exist k → 0 such that |a 1, − a 1 |/ → A = 0. Then, from (2.15) we deduce However, since U 1 is radially symmetric-decreasing, this is impossible unless A = 0. We reach a contradiction. The proof of (2.1) is complete. Thus complete the proof of Proposition 1.
3. Proof of Theorem 1.2. In this section, we prove Theorem 1.2. We use a contradiction argument. As in Cao, Li and Luo [7], we assume that are two distinct solutions of equation (1.1). Set .
It is clear that ξ L ∞ (R N ) = 1. To obtain a contradiction, in the rest of this section, we prove Then, Theorem 1.2 follows from the above proposition. To prove (3.1), we will first prove that ξ = o(1) holds locally, and then prove that it holds at infinity. The proof is lengthy. We split it into several lemmas and propositions. Note that ξ satisfies equation Our first estimate reads as Proof. By (3.2), we have Since K is bounded, we have Direct computation gives where we have used (2.10) and (1.6). Hence, The desired result follows from the above directly.
We study the asymptotic behavior ofξ .
Next we prove Lemma 3.2. Let d α , 1 ≤ α ≤ N be the coefficients given in Lemma 3. Then, Proof. We will combine Proposition 1 and the identity (2.9) to prove that Once we prove (3.7), then d α = 0 since U 1 = U 1 (|x|) is a radially symmetric decreasing function, which implies that x α U xα is positive except at the origin. We prove (3.7) as follows. By (2.2) and Lemma 3.1, we can choose d > 0 such that Applying the Pohozaev type identity (2.9) to u (i) , i = 1, 2, in Ω = B d a 1, , we deduce (3.10) where 1 ≤ α ≤ N and Using the same computation as in the proof of (2.1), (3.9) and (3.8), we have As to A (x), we have for any given γ > 0 since U i , i = 1, 2, decay exponentially at infinity. Hence we can deduce as in the above that by taking γ sufficiently large, where we have used (2.10) again. Hence, by the above estimates, there holds the right hand side of (3.10) = O( N +m ).
This gives, Now we estimate the left hand side of (3.11). By the assumption (K1), we have Note that Thus, we have where we have used (2.1) in the second equality and the exponential decay of U i (i = 1, 2), (2.2) and (3.1) in the last equality. Hence, combining the estimate of l 2 and (3.11), together with the assumption k 1,α = 0, we have Equivalently, we obtain By the similar arguments as in (3.3) and note that |a 1, ,α − a 1,α | = o( ) by (2.1), we deduce from the above estimate that Since U 1 is radially symmetric, we infer from the above that (3.7) holds. Thus, d α = 0 for all 1 ≤ α ≤ N . The proof of Lemma (3.2) holds. Now the following result is natural.
Proof. By Lemma 3 and Lemma 3.2, we deduce thatξ → 0 in C 1 loc (R N ). Hence, for any fixed R > 0, there holds That is, as → 0.
i, ). We need the following estimate. Then Proof. We divide the proof into several steps.
Combining Claim 2 and Claim 3 finishes the proof.
Now we can prove the following estimate.
We may take R sufficiently large and sufficiently small such that i, ) for sufficiently small. Hence, applying the maximum principle to ξ in the above problem, we obtain the desired result. The result is complete. Now we can prove Theorem 1.2.
Proof of Theorem 1.2. By Proposition 4 and Proposition 5, we conclude that 3.1 holds. However, this contradicts ξ ∞ = 1. The proof of Theorem 1.2 is complete.