Asymptotic Behavior of Ground States and Local Uniqueness for Fractional Schrödinger Equations with Nearly Critical Growth

We study quantitative aspects and concentration phenomena for ground states of the following nonlocal Schrödinger equation (−Δ)su+V(x)u=u2s∗−1−𝜖inℝN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(-{\Delta })^{s} u+V(x)u= u^{2_{s}^{*}-1-\epsilon } \ \ \text {in}\ \ \mathbb {R}^{N},$\end{document} where 𝜖 > 0, s ∈ (0,1), 2s∗:=2NN−2s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2^{*}_{s}:=\frac {2N}{N-2s}$\end{document} and N > 4s, as we deal with finite energy solutions. We show that the ground state u𝜖 blows up and precisely with the following rate ∥u𝜖∥L∞(ℝN)∼𝜖−N−2s4s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\|u_{\epsilon }\|_{L^{\infty }(\mathbb {R}^{N})}\sim \epsilon ^{-\frac {N-2s}{4s}}$\end{document}, as 𝜖→0+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\epsilon \rightarrow 0^{+}$\end{document}. We also localize the concentration points and, in the case of radial potentials V, we prove local uniqueness of sequences of ground states which exhibit a concentrating behavior.


Introduction
In this paper we consider the following class of nonlocal equations where → 0 + , s ∈ (0, 1), 2 * s := 2N N−2s , N > 4s, V : R N → R is a potential function and (− s u(x) = c N,s PV R n u(x) − u(y) |x − y| N+2s dy is the fractional Laplacian. Here, c N,s is a normalizing constant, PV stands for the Cauchy principal value. As we are going to see, the restriction on the dimension is due to the fact that we look for finite L 2 -energy solutions. For fixed ∈ (0, 2 * s − 2), under suitable conditions on V (x), it is known that equation (1.1) admits a positive ground state u , see for instance [2,[21][22][23]. Moreover, if V (x) = 1, then u is spherically symmetric, see [12,18]. However, when = 0 it follows from a Pohozaev type identity that (1.1) has no solutions in H s (R N x · ∇V (x) ≥ 0 (and ≡ 0), see Theorem 9 or [10] in the special case V (x) ≡ 1. Therefore, it is natural to wonder what happens to the ground state u as → 0 + . The main motivation of this paper is to achieve a better understanding of this phenomenon. This type of problems for semilinear equations, with the so-called nearly critical growth, were first studied in the unit ball of R 3 by Atkinson and Peletier in [3] and then extended to spherical domains by Brezis and Peletier in [6] and non-spherical domains by Han in [25]. Indeed, they proved the solution u blows up in the sense that u L ∞ (R N ) ∼ − 1 2 as → 0 + . More recently, their results were extended to nonlocal problems in bounded domains in [13]. Precisely, the authors in [13]  where b n,s is a normalizing constant and x 0 ∈ is a critical point of the Robin function τ (x). Besides, in [27] the authors study the asymptotic behavior of solutions to the nonlocal nonlinear problem where p * s = Np N−ps , N > ps, p > 1. They prove that ground state solutions concentrate at a single point in and analyze the asymptotic behavior for sequences of solutions at higher energy levels as → 0. In particular, in the semi-linear case p = 2, they prove that for smooth domains the concentration point cannot lie on the boundary, and identify its location in the case of annular domains. Regarding the nonlocal problem (1.3) for p = 2, we also refer to [29] for a profile decomposition approach and to [30] for -convergence methods.
The purpose of this paper is twofold: on one side, under suitable conditions on V (x), we give a complete description of the blow up behavior of the ground states of (1.1); on the other side, we identify the location of the concentration points and then we establish local uniqueness of ground states.
Before stating our main results, let us make a few assumptions on V (x). Throughout this paper, we assume that V (x) satisfies the following conditions: The function x · ∇V (x) stays bounded in R N .
We consider here the fractional Sobolev space By Lions' concentration compactness, minimizers for S 2 * s − always exist and one may assume they do not change sign, [18,23]. Moreover, they are radially symmetric, see [24]. Here, we will consider only positive minimizers.
Recall also the Sobolev constant where Our main results are the following: (V 2 ), N > 4s and that u is a ground state of (1.1), namely satisfying (1.7), which has a maximum point x such that x → x 0 as → 0 + . Then, In particular we have Corollary 1 Assume N > 4s and that u is a minimizer for (1.4). Then, we have: Notice that in Theorem 1, we assume that the maximum point x does converge. However, under conditions (V 1 ) and (V 2 ), one of the main difficulties is that x may actually escape to infinity as → 0 + . In what follows, we prove that if N > 6s, then the maximum point x must be bounded, and therefore converging, up to a subsequence, to a global minimum point of V (x) provided inf x∈R N V (x) < V ∞ . More precisely, we have the following and that u is a ground state of (1.1) (in the sense of (1.7)) which has a maximum point x . Then, there exists a subsequence {x j } of {x } such that: What stated in Theorem 3 opens a natural question: is there more than one blow-up ground state sequence such that the maxima concentrate at the same point?
We do not have a full answer, however let us consider a special case. Assume V (x) is radial and that there exist two radial ground state sequences u 1 j and u 2 j of (1.1) such that We have the following local uniqueness result.
is radial, N > 4s and there exist two radial ground state sequences u 1 j and u 2 j of (1.1) satisfying u i j ∞ = u i j (0), i = 1, 2, (1), (2) and (3) of Theorem 3. Then, there exists 0 > 0 such that for any j ∈ (0 0 ), we have More precisely, up to rescaling, we have the following local uniqueness result Remark 1 In Theorems 4 we assume that

Overview
The asymptotic behavior of ground states to nonlocal problems has attracted remarkable attention in recent years. In [22], the authors studied the singularly perturbed fractional Schrödinger equation They proved that concentration points turn out to be critical points for V . Moreover, they proved that if the potential V is coercive and has a unique global minimum, then ground states concentrate at that minimum point as → 0. In [16], by means of a Lyapunov-Schmidt reduction method, the authors proved the existence of various type of concentrating solutions, such as multiple spikes and clusters, such that each of the local maxima converge to a critical point of V as → 0, see also [1,20]. In [5], the authors considered the nonlocal scalar field equation where 2 < p < q. For small, they proved the existence and qualitative properties of positive solutions when p is subcritical, supercritical or critical Sobolev exponent. For the existence of positive solutions of nonlocal equations with a small parameter see also [7,19]. Loosely speaking, all the results mentioned above were concerned with the characterization of concentration of ground states. The purpose of this paper is quite different as we focus on quantitative aspects of concentrating solutions . Let us emphasize that Theorem 3 can be seen as a nonlocal analog of the results in [31,40]. In [31], the authors studied the behavior of the ground states of equation Under some geometric assumptions on K(x), they proved the existence of ground states u . Moreover, the maximum point x of u ε is bounded and u L ∞ (R N ) ∼ − N−2 4 as → 0 + . In [40], the author further identified the location of the blow-up point. In the present paper, though conditions (V 1 ) and (V 2 ) guarantee the existence of the ground state solution u , it is not true in general that the maximum point x of u stays bounded as → 0 + and this yields a major difficulty.
The paper is organized as follows: existence of minimizers, local boundedness estimates of solutions and a Pohozaev type identity are established in the preliminary Section 2. In Section 3, we study the asymptotic behavior of ground states, including a uniform bound up to rescaling. Section 4 is devoted to identify the location of blow-up points, whence in Section 5 we prove the local uniqueness of ground states.
Throughout this paper, C will denote a positive constant which may vary from line to line.

Preliminaries
Here for the convenience of the reader we prove some auxiliary results. Consider first the following constrained minimization: In the special case V (x) = 1, minimizers for S 2 * s − always exist and do not change sign, see e.g. [18,23], Moreover, they are radially symmetric, see [24].
Then minimizer u for S ∞ 2 * s − exists and does not change sign, see e.g. [18,23]. Without loss of generality, we assume u is positive. Using this u as a test function we can show that if and v n → 0 in L 2 loc (R N ) as n → +∞ and by Bresiz-Lieb lemma, we have On the one hand we have On the other hand, by (V 1 ) and v n → 0 in L 2 loc (R N ) as n → +∞, we have Thus, we have lim and by (2.2), we get Therefore, by (2.3), (2.4), (2.5) and (2.6), we have Thus, from (2.8), we deduce that = 0 or = 1. If = 0, then from (2.7), we get s − , which is a contradiction. Thus, = 1, that is, w 2 * s − = 1 and thus w is a minimizer of S V 2 * s − .

Remark 2
Notice that in the proof of Theorem 6, condition S V By the Lagrange multiplier rule, there exists some λ > 0 such that w is a solution of the following equation By the maximum principle w > 0. In fact, w ≥ 0, and if there exists some x 0 such that w (x 0 ) = 0, then thus a contradiction.

Remark 3
If V (x) is radial, by means of symmetric rearrangement techniques, we may assume that w n is radially symmetric (cf. [32]). Thus, the minimizer w is radial.
Next we proof a Pohozaev type identity for the nonlocal equation The argument is similar to [4,33,34], where the Pohozaev identity for autonomous nonlocal equations was established, hence we just stress the differences.
Theorem 5 (Pohozaev identity) Let u ∈ H s (R N )∩L ∞ (R N ) be a positive solution to (2.11) and Proof Let u be a bounded weak nontrivial solution. Suppose that w is the harmonic extension of u, see e.g. [9]. Then, w satisfies (2.13) For r > 0, let Then, multiplying (2.13) by ((x, y) · ∇w)ϕ R and integrating in R N+1 + , we have, Q 2r div(y 1−2s ∇w)[((x, y) · ∇w)ϕ R ]dxdy = 0. (2.14) From (2.14), by integrating by parts, we get Q 2r (2.15) For the second integral in the last equality in (2.15), we have (2.18) Thanks to (2.17) and (2.18), we have Multiply (3.26) by wϕ R and integrate by parts to get Q 2r Proceed now as above to get Combining (2.18) and (2.20), we deduce that Finally, we prove a crucial local estimate. This type of estimate has been studied in Proposition 3.1 and Proposition 2.4 in [38,39]. Their methods relies on a localization method introduced by Caffarelli and Silvestre in [9,36] and the standard Moser iteration. However, these estimates contain the extension local domain Q R , which has no clear interpretation in terms of the original problem in R N that is our context. We now give another version of this estimate based on a more direct test function method and Moser's iteration.
where the constant C > 0 depends only on N , s, R, t and a(x) L t loc (R N ) .
Proof For β > 1 and T > 0, define the function Notice that ϕ(t) is a convex and differentiable function and thus where 0 < r < R has to be determined. For simplicity, in the following, we denote by ϕ := ϕ(u(x)) and ϕ := ϕ u (x).
Choose as test function φ(x) = η 2 ϕϕ to obtain (2.28) We obtain (2.29) and The estimate of I 3 is similar to the one for I 1 . Finally,

Asymptotic Behavior of Ground States
Let w be a positive minimizer for S V 2 * s − obtained in Theorem 6. Then, by the Lagrange multiplier rule, there exists λ > 0 such that w is a solution to the equation By multiplying both sides of equation (3.1) by w and then integrating, we get λ = S V 2 * s − . The energy associated with equation (3.1) is given by Thus, on the one hand we have On the other hand, if v is a nontrivial solution of (3.1), then it satisfies v 2 , by means of the mountain-pass theorem, (1.1) admits a positive ground state (see e.g. Theorem 1.4 in [23]). However, we don't know whether the mountain-pass solution and the minimal solution u obtained above do agree since uniqueness is not known. Anyway, in what follows, we will focus on the minimal solution u . We remark that in the special case V (x) = 1, the ground state is unique and radially symmetric, see [24].

Lemma 1 For any fixed
, and the result follows.
We next need the following result proved in [14].
s − } is uniformly bounded with respect to . Next, we further prove that lim By Hölder's inequality we have Thanks to (3.8) and (3.10), we get On the other hand, by (1.5), we have Thanks to (3.11) and (3.12), we have Next we prove that lim sup Once (3.14) is proved, the result follows from (3.13).
Recalling that u is a solution to (1.1) and that u attains S V So, we have These facts together with Lemma 13 imply the following Now let us prove that u ∞ blows up as → 0 + , namely Proof Suppose by contradiction the claim does not hold true. Then, there exists a sequence j → 0 + such that u j ∞ stays bounded. Let x j be a maximum point of u j . Define w j (x) = u j (x + x j ), then w j ∞ is bounded as well and we have that (− s w j ∞ is uniformly bounded with respect to j . As a consequence of this fact and of standard regularity results (see e.g. Lemma 4.4 in [8]), we deduce that w j C 2,α is uniformly bounded with respect to j , for some α ∈ (0, 1). By (3.22), [w j ] s = [u j ] s and w j 2 u j 2 are bounded. Thus, {w j } is bounded in H s V (R N ). Up to extracting a subsequence, which we still denote by {w j }, one has w j w 0 in H s V (R N ), w j → w 0 a.e. in R N and w j → w 0 in C 2,α loc (R N ). Moreover, by (3.6) one > 0. Let us now distinguish two cases: Case 1. {x j } j is bounded. Up to a subsequence, we may assume that x j → x 0 . Then, w 0 is a nonnegative classical solution of (3.23) It follows from the maximum principle that w 0 > 0. Thus, by Lemma 14 we have which is a contradiction. Case 2. {x j } j is unbounded. Up to a subsequence, we may assume that x j → ∞. Then, by (3.20) and the dominated convergence theorem, we have and similarly to the proof of (3.24), we get a contradiction.
2 ), we have that {u } is uniformly bounded with respect to , as established in the following Lemma 5 There exists K > 0, which does not depend on , such that any solutions u of (1.1) satisfies u ∞ ≤ K as → 0 .
Proof The claim can be achieved via Moser's iteration. Indeed, let w be the harmonic extension of u , see e.g. [9]. Then, w satisfies and k s = 2 1−2s 1−s) . Following Corollary 2.1 in [1], for each L > 0, we set and ψ = w 2(β−1) w , where β > 1 to be determined later on. By testing with ψ , we (3.28) Thus, Thus, from (3.29), Sobolev imbedding (see e.g. (2.9) in [13]) and Hölder's inequality, As u ∈ L 22 * s β (R N ), by using the fact that w ≤ w , we get Let L → +∞ and apply Fatou's lemma to get The claim now follows by iteration: Passing to the limit as m → +∞ in (3.35), we have u ∞ ≤ C u 2 * s . which concludes the proof.
Let x be the global maximum point of u and let μ > 0 be such that Then 0 < v (x) ≤ 1, v (0) = 1 and v satisfies the following (3.39) We have that (− s v ∞ is uniformly bounded with respect to . As a consequence of this fact and regularity results, we deduce that also v C 2,α is uniformly bounded with respect to , for some α ∈ (0, 1). Similarly to the proof of Lemma 15, there exists a sequence , still denoted by v , such that v → U in C 2,α loc (R N ), where U is the positive solution of equation and U(0) U ∞ = 1. From Theorem 1.2 in [11], (3.42) Finally, form Lemma 14 and (3.42), we obtain the following convergences , and μ → 1 as → 0 + . (3.43) Notice that up to now we do not know wether the global maximum point x turns out to be bounded or unbounded.

Lemma 8 Suppose that {x } is bounded. Then,
Proof Assume by contradiction that Eq. 3.44 does not hold. Then, there exist two sequences j → 0 and R j → +∞ such that for some δ > 0 and j = 1, 2, · · · . We distinguish two cases: Case 1. ε 0 > 0. By (3.22), {u } is bounded in H s V (R N ), passing to a subsequence {u j } if necessary, we may assume u j 0 in H s V (R N ). On the other hand, from Lemma 16, we know that u ε j ∞ is bounded and by regularity we deduce that u j C 2,α is uniformly bounded with respect to j , for some α ∈ (0, 1). Up to extracting again a subsequence, still denoted by {u j }, we have u j → u 0 in C 2,α loc (R N ). Thus, u 0 is a classical nonnegative solution of the equation which is a contradiction. Thus, u 0 (x) > 0 for all x ∈ R N . Now, by (3.21) and Lemma 17, observe that (3.47) Therefore, we get lim Similarly to the proof of Lemma 17, from (3.48) we get u j s,V u 0 s,V as j → +∞ and hence u j → u 0 in L 2 * s (R N ) as j → +∞. This contradicts (3.45). Case 2. ε 0 = 0. Thanks to Lemma 18, we obtain a contradiction from (3.45). Indeed, we have The proof is now complete.

Lemma 9
Suppose that {x } is unbounded. Then, for any fixed R > 0, lim sup Proof The proof is similar to Lemma 20. Suppose that the claim is not true. Then there exist a sequence j → 0 such that for some δ > 0 and j = 1, 2, · · · . The proof of the case 0 > 0 is similar to Lemma 20. For 0 = 0, we have The following lemmas will play an important role in our analysis.
where C is independent of . Thus, we conclude from (3.44) and (3.54) that This fact together with Lemma C.2 in [24] imply (3.56) Actually, we first fix 0 to applying Lemma C.2 in [24], and then we take the supremum with respect to . Finally, from Lemmas 17 and 14, we get (3.56). See also [22].

Lemma 11
Suppose that {x } is unbounded. Then there exists a constant C > 0 independent of such that for small 0, x. Then w (x) enjoys the following Furthermore, by condition (V 1 ), if we choose δ > 0 sufficiently small and R 1 > 0 large enough, we have for small 0, |x| ≥ R 1 and |V − 1 2s 0 x − x | ≥ R. Borrowing some results from [23], we also have where K is the Bessel kernel and which enjoys the following properties: and (3.63) From (3.60) and (3.61) we have Since {x } is unbounded, then there exists 0 1 0 such that |x | ≥ R + R 1 for 0 1 . Thus, for |y| ≤ R 1 , we get |y − x | ≥ |x | − |y| ≥ R. So, from (3.59) and (3.60), we obtain (3.65) By (3.62) and (3.63), we have x − x | > R 2 and small 0, we have That is, Since R is arbitrary, as well as R 2 is arbitrary, the proof is complete.
Remark 4 By using standard comparison arguments as in [23], we can prove results similar to (3.44) and (3.50). However, the constant C obtained there may depend on .

Lemma 12 There exists a positive constant C independent of , such that
(3.68) Proof Note that we do not assume that {x } is bounded or unbounded. From the definition of v and U , v (0) = U(0) = 1, and since v (x) ∈ C 2,α , by choosing some large C, (3.68) holds in a neighborhood of zero. Therefore, it is enough to establish (3.68) ifor |x| bounded away from zero. For this purpose, let (x) be the Kelvin transform of v , namely (3.69) Then, satisfies Now, we aim at proving that { } is uniformly bounded with respect to in a neighborhood of zero, and this will imply (3.68) by (3.69 Claim: a(x) ∈ L t loc (R N ) with some t > N 2s .
Assume for the moment the claim holds true and let us complete the proof. By Theorem 10, for any compact set K, we have (3.72) The last inequality follows from the facts μ → 1 and u 2 * s ≤ C u s,V (x) → CS N 4s as → 0 + . Thus, it remains to prove the claim. On the one hand, for r > 0 we get Proof By Pohozaev identity (2.12), we have (3.76) Since N > 4s, by Lemma 25 and the Lebesgue dominated convergence theorem, we get (3.77) By direct calculations, we deduce that (3.78) Finally, combine (3.76)-(3.78) to have as → 0 +o(1).  Proof of Theorem 1. The conclusions (1) and (2) in Theorem 1 follow from Propositions 13 and 26. Clearly, Corollary 2 is a particular case of Theorem 1.

Localizing Blow up Points
We next recall for convenience of the reader a few basic facts on fractional Sobolev spaces. Let β > 0 and p ∈ [1, ∞), We refer to [23] for the following results.

Proposition 4
The following properties hold true: For p ∈ [1, +∞) and β > 0, consider the Bessel potential space Then, L β,p (R N ) = W β,p (R N ), see Theorem 3.1 in [23]. On the other hand, from Theorem 5 in Chapter V of [37], for p ∈ [2, ∞) and 0 < β < 1, one has W β,p (R N ) ⊂ W β,p (R N ), where W β,p (R N ) is the usual fractional Sobolev space defined by W β,p (R N ) = u ∈ L p (R N ) : Our next target is to identify the location of the blow up points. For this purpose we adapt the method developed in [28], where the basic idea is to get an asymptotic expansion of the ground state energy and then to compare it with an upper bound of S V 2 * s − . This method has been used to deal with the localization of blow-up points of ground states to semilinear problems in [40].
Let us begin with establishing an upper bound for S V 2 * s − .

Theorem 7
Assume N > 4s and that u j is a ground state of (1.1) satisfying (1.6) which has a maximum point x j which enjoys x j → x 0 as j → ∞. Then

1) wherex 0 is a global minimum point of V (x) and
Then by inspection and by dominated convergence, we have By (3.79), we also get Thus, by using Taylor's formula, we get This concludes the proof.
For simplicity, set μ j := μ j , Define the operator L as follows: Then (4.8) can be rewritten as In order to prove Proposition 30, we need the following result from [17] Lemma 13 (Nondegeneracy) The solution U is nondegenerate in the sense that all bounded solutions of equation Clearly, X ⊂ L p (R N ) with N N−2s < p < +∞. For 1 < r < N 2s , define Y r := u ∈ L r (R N ) : Then L r (R N ) = X ⊕ Y q , where N N−2s < r < N 2s .

Lemma 14
Suppose N > 4s. Then for any 1 < q < N 4s , there exists a constant C > 0 such that u W 2s,r ≤ C( Lu r Lu q ), (4.10) Proof It is enough to prove u r ≤ C( Lu r Lu q ).
In fact, by Assume that u = 0. Otherwise, we are done. By homogeneity, we can replace u by u max u C 2 , u r } in (4.10). Thus, assume that there exists a sequence {u n } ⊂ Y r ∩ W 2s,r (R N ) ∩ C 2 (R N ) such that either u n C 2 = 1, u n r < 1, or u n C 2 < 1, u n r = 1, (4.11) and Lu n q Lu n r → 0. (4.12) Then, there exists u ∞ ∈ C 2 (R N ) such that after passing to a subsequence if necessary, u n → u ∞ in C 2 loc (R N ) and in particular, u n → u ∞ in L t loc (R N ), r ≤ t < 2 * s . Let I = (− −s the Riesz potentials defined by See Chapter V in [37]. Then, we have By Hardy-Littlewood-Sobolev inequality [26,37], we have I * Lu n r ≤ C Lu n q → 0 and Hölder's inequality yields where 1 q − 2s N = 1 r . Thus, {I * [(2 * s − 1)U 2 * s −2 u n } is a Cauchy sequence in L r (R N ) and then {u n } is a Cauchy sequence in L r (R N ). So, u ∞ ∈ L r (R N ), u ∞ ∈ Y r and (4.14) By (4.11), u ∞ ∈ L ∞ (R N ) and u ∞ ∈ X by (4.14). But since u ∞ ∈ Y r , we get u ∞ ≡ 0, which is a contradiction from (4.11).
Since z jk ∈ Y r , we get z ∈ Y r . Thus, (4.33) has at most one such solution, and z j in W 2s,r . Moreover, since (− s is invariant with respect to the action of the orthogonal group O(n) on R N (see [15]), if T denotes a rotation in R N , since (4.33) is invariant under rotation, then z(T x) − z(x) ∈ X. Consequently, z(T x) = z(x). This proves that z is radial.  Since ∇z(0) = 0, we get the result.
Proof It sufficient to prove z j → z in L ∞ (R N ) as j → ∞. In fact, by consider L(z j − z), the proof is analogous to the proof of Lemma 33.

Theorem 8
Assume N > 6s, u j is a ground state of (1.1) satisfying (1.6) which has a maximum point x j satisfying x j → x 0 as j → ∞. Then (4.38) By (4.15), we get Thus, Thus, we have  The proof is complete. Proof of Theorem 3. It follows from Theorems 1, 29, 37 and 38.

Local Uniqueness: Proof of Theorem 4
Let us argue by contradiction. Suppose that there exists a sequence j → 0 and two ground states far apart, namely u 1 j := u 1 j and u 2 j := u 2 j . Set Then v i j → U in C 2,β loc (R N ) for i = 1, 2 as j → ∞.
The proof of Theorem 4 is now complete.
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