Local Uniqueness and Refined Spike Profiles of Ground States for Two-Dimensional Attractive Bose-Einstein Condensates

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1 Local Uniqueness and Refined Spie Profiles of Ground States for Two-Dimensional Attractive Bose-Einstein Condensates Yujin Guo, Changshou Lin, and Juncheng Wei May 1, 17 Abstract We consider ground states of two-dimensional Bose-Einstein condensates in a trap with attractive interactions, which can be described equivalently by positive minimizers of the L critical constraint Gross-Pitaevsii energy functional. It is nown that ground states exist if and only if a < a := w, where a denotes the interaction strength and w is the unique positive solution of w w + w 3 = in. In this paper, we prove the local uniqueness and refined spie profiles of ground states as a a, provided that the trapping potential hx is homogeneous and Hy = hx + yw xdx admits a unique and non-degenerate critical point. Keywords: Bose-Einstein condensation; spie profiles; local uniqueness; Pohozaev identity. 1 Introduction The phenomenon of Bose-Einstein condensation BEC has been investigated intensively since its first realization in cold atomic gases, see 1, 5 and references therein. In these experiments, a large number of bosonic atoms are confined to a trap and cooled to very low temperatures. Condensation of a large fraction of particles into the same one-particle state is observed below a critical temperature. These Bose-Einstein condensates display various interesting quantum phenomena, such as the critical-mass collapse, the superfluidity and the appearance of quantized vortices in rotating traps e.g.5. Specially, if the force between the atoms in the condensates is attractive, the system collapses as soon as the particle number increases beyond a critical value, see, e.g., 3 or 5, Sec. III.B. Bose-Einstein condensates BEC of a dilute gas with attractive interactions in can be described, 5, 1 by the following Gross-Pitaevsii GP energy functional E a u := u + V x u dx a u 4 dx, 1.1 Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O. Box 711, Wuhan 4371, P. R. China. yjguo@wipm.ac.cn. Y. J. Guo is partially supported by NSFC grants and Taida Institute of Mathematical Sciences, National Taiwan University, Taipei 1617, Taiwan. cslin@math.ntu.edu.tw. Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z, Canada. jcwei@math.ubc.ca. J.C. Wei is partially supported by NSERC of Canada. 1

2 where a > describes the strength of the attractive interactions, and V x denotes the trapping potential satisfying lim x V x =. As addressed recently in 1, 11, ground states of attractive BEC in can be described by the constraint minimizers of the GP energy ea := inf E a u, 1. {u H, u =1} where the space H is defined by { } H := u H 1 : V x ux dx <. 1.3 The minimization problem ea was analyzed recently in, 1, 11, 1, 6 and references therein. Existing results show that ea is an L critical constraint variational problem. Actually, it was shown in, 1 that ea admits minimizers if and only if a < a := w, where w = w x is the unique up to translations radial positive solution cf. 7, 19, 14 of the following nonlinear scalar field equation w w + w 3 = in, where w H It turns out that the existence and nonexistence of minimizers for ea are well connected with the following Gagliardo-Nirenberg inequality ux 4 dx w ux dx ux dx, u H 1, 1.5 where the equality is attained at w cf. 5. Since E a u E a u for any u H, any minimizer u a of ea must be either non-negative or non-positive, and it satisfies the Euler-Lagrange equation u a + V xu a = µ a u a + au 3 a in, 1.6 where µ a R is a suitable Lagrange multiplier. Thus, by applying the maximum principle to the equation 1.6, any minimizer u a of ea is further either negative or positive. Therefore, without loss of generality one can restrict the minimizations of ea to positive functions. In this paper positive minimizers of ea are called ground states of attractive BEC. Applying energy estimates and blow-up analysis, the spie profiles of positive minimizers for ea as a a were recently discussed in 1, 11, 1 under different types of potentials V x, see our Proposition.1 for some related results. In spite of these facts, it remains open to discuss the refined spie profiles of positive minimizers. On the other hand, the local uniqueness of positive minimizers for ea as a.e. a a was also proved 11 by the ODE argument, for the case where V r = V x is radially symmetric and satisfies V r, see Corollary 1.1 in 11 for details. Here the locality of uniqueness means that a is near a. It is therefore natural to as whether such local uniqueness still holds for the case where V x is not radially symmetric. We should remar that all these results mentioned above were obtained mainly by analyzing the variational structures of the minimization problem ea, instead of discussing the PDE properties of the associated elliptic equation 1.6. By investigating thoroughly the associated equation 1.6, the main purpose of this paper is to derive the refined spie profiles of positive minimizers for ea as a a, and extend the above local uniqueness to the cases of non-symmetric potentials V x as well. Throughout the whole paper, we shall consider the trapping potential V x satisfying lim x V x = in the class of homogeneous functions, for which we define

3 Definition 1.1. hx in is homogeneous of degree p R + about the origin, if there exists some p > such that htx = t p hx in for any t >. 1.7 Following 9, Remar 3., the above definition implies that the homogeneous function hx C of degree p > satisfies hx C x p in, 1.8 where C > denotes the maximum of hx on B 1. Moreover, since we assume that lim x hx =, x = is the unique minimum point of hx. Additionally, we often need to assume that V x = hx C satisfies y is the unique critical point of Hy = hx + yw xdx. 1.9 The following example shows that for some non-symmetric potentials hx, Hy admits a unique critical point y, where y satisfies y and is non-degenerate in the sense that Hy det, where i, j = 1,. 1.1 x i x j Example 1.1. Suppose that the potential hx satisfies hx = x p 1 + δh θ, where p and δ R, 1.11 where h θ C, π satisfies π π h θ cos θdθ + h θ sin θdθ >. 1.1 One can chec from 1.1 that if δ is small enough, then Hy admits a unique critical point y = δŷ, where ŷ satisfies ŷ π π C 1 h θ cos θdθ, C h θ sin θdθ, as δ 1.13 for some positive constants C 1 and C depending only on w and p. Furthermore, if δ is small enough, then det Hy x i x j >, which implies that the unique critical point y of Hy is non-degenerate. Our first main result is concerned with the following local uniqueness as a a, which holds for some non-symmetric homogeneous potentials hx in view of Example 1.1. Theorem 1.1. Suppose V x = hx C is homogeneous of degree p, where lim x hx =, and satisfies y is the unique and non-degenerate critical point of Hy = hx + yw xdx. Then there exists a unique positive minimizer for ea as a a

4 The local uniqueness of Theorem 1.1 means that positive minimizers of ea must be unique as a is near a. It is possible to extend Theorem 1.1 to more general potentials V x = gxhx for a class of functions gx, which is however beyond the discussion ranges of the present paper. We also remar that the proof of Theorem 1.1 is more involved for the case where y occurs in Our proof of such local uniqueness is motivated by 3, 6, 9. Roughly speaing, as derived in Proposition.1 we shall first obtain some fundamental estimates on the spie behavior of positive minimizers. Under the non-degeneracy assumption of 1.14, the local uniqueness is then proved in Subsection.1 by establishing various types of local Pohozaev identities. The proof of Theorem 1.1 shows that if one considers the local uniqueness of Theorem 1.1 in other dimensional cases, where is replaced by R d and u 4 is replaced by u + 4 d for d, the fundamental estimates of Proposition.1 are not enough. Therefore, in the following we address the refined spie behavior of positive minimizers under the assumption To introduce our second main result, for convenience we next denote λ = where y is given by 1.14, and 1 p hx + y w +p xdx, 1.15 ψx = φx C wx + x wx, where φx C L is the unique solution of φ = and + 1 3w φx = w3 R w 4 hx + y w p R hx + y w in R, 1.16 and the nonzero constant C is given by C = wψ p φ with ψ 3 C L being the unique solution of 3.9. Using above notations, we shall derive the following theorem. Theorem 1.. Suppose V x = hx C is homogeneous of degree p, where lim x hx =, and satisfies 1.14 for some y. If u a is a positive minimizer of ea as a a, then we have u a x = λ w { 1 a a 1 +p λ x x a w a a 1 +p +a a 3+p +p ϕ λ x x a a a 1 +p + a a 1+p λ x x +p a ψ } + o a a 3+p +p a a 1 +p as a a 1.17 uniformly in for some function ϕ C L, where x a is the unique maximum point of u a satisfying λx a a a 1 +p y = a ao y as a a 1.18 for some y. 4

5 Theorem 1. is derived directly from Theorem 1.1 and Theorem 3.6 in Section 3 with more details, where ϕ C L is given explicitly. In Section 4 we shall extend the refined spie behavior of Theorem 1. to more general potentials V x = gxhx, where h x = hx is homogeneous and satisfies 1.14 and C gx 1 C holds in, see Theorem 4.4 for details. To establish Theorem 1. and Theorem 4.4, our Proposition.1 shows that the arguments of 1, 11, 1 give the leading expansion terms of the minimizer u a and the associated Lagrange multiplier µ a satisfying 1.6 as well. In order to get 1.17 for the rest terms of u a, the difficulty is to obtain the more precise estimate of µ a, which is overcome by the very delicate analysis of the associated equation 1.6, together with the constraint condition of u a. This paper is organized as follows: In Section we shall prove Theorem 1.1 on the local uniqueness of positive minimizers. Section 3 is concerned with proving Theorem 1. on the refined spie profiles of positive minimizers for ea as a a. The main aim of Section 4 is to derive Theorem 4.4, which extends the refined spie behavior of Theorem 1. to more general potentials V x = gxhx. We shall leave the proof of Lemma 3.4 to Appendix A. Local Uniqueness of Positive Minimizers This section is devoted to the proof of Theorem 1.1 on the local uniqueness of positive minimizers. Towards this purpose, we need some estimates of positive minimizers for ea as a a, which hold essentially for more general potential V x C satisfying V x = gxhx, where < C gx 1 C in R and hx is homogeneous of degree p..1 For convenience, we always denote {u } to be a positive minimizer sequence of ea with a a as, and define 1 pg λ = hx + y w +p xdx,. where V x = gxhx is assumed to satisfy.1 with p and y is given by 1.9. Recall from 1.4 that w x satisfies w dx = w dx = 1 w 4 dx,.3 see also Lemma 8.1. in 4. Moreover, it follows from 7, Prop. 4.1 that w admits the following exponential decay wx, wx = O x 1 e x as x..4 Proposition.1. Suppose V x = gxhx C satisfies lim x V x = and.1, and assume 1.9 holds for some y. Then there exist a subsequence, still denoted by {a }, of {a } and {x } such that I. The subsequence {u } satisfies a a 1 +p u x + xa a 1 +p λwλx as.5 w 5

6 uniformly in, and x is the unique maximum point of u satisfying lim λx a a 1 +p = y,.6 where y is the same as that of 1.9. Moreover, u satisfies a a 1 +p u x + xa a 1 +p Ce λ x in,.7 where the constant C > is independent of. II. The energy ea satisfies lim ea λ p + a = a p/+p a p..8 Proof. Since the proof of Proposition.1 is similar to those in 1, 11, 1, which handle 1.1 with different potentials V x, we shall briefly setch the structure of the proof. If V x C satisfies.1 with p, we note that hx satisfies 1.8. Tae the test function τ u τ x = A τ φxwτx, w where the nonnegative cut-off function φ C R satisfies φx 1 in, and A τ > is chosen so that R u τ x dx = 1. The same proof of Lemma 3 in 1 then yields that ea Ca a p p+ for a < a,.9 where the constant C > is independent of a. By.9, we can follow Lemma 4 in 1 to derive that there exists a positive constant K, independent of a, such that u a x 4 dx 1 K a a p+ for a < a,.1 where u a > is any minimizer of ea. Applying.9 and.1, a proof similar to that of Theorem.1 in 1 then gives that there exist two positive constants m < M, independent of a, such that ma a p p+ ea Ma a p p+ for a < a..11 Based on.11, similar to Theorems 1. and 1.3 in 1, one can further deduce that there exist a subsequence still denoted by {a } of {a } and {x }, where a a as, such that.7 and.8 hold, and a a 1 +p u x + xa a 1 +p λwλx strongly in H 1.1 w as, where x is the unique maximum point of u. Finally, since w decays exponentially, the standard elliptic regularity theory applied to.1 yields that.5 holds uniformly in e.g. Lemma 4.9 in 18 for similar arguments. We finally follow 1.9 and.5 to derive the estimate.6. Following.5, we define a ū x := ε ε u λ λ x + x, where ε := a a 1 +p >, 6

7 so that ū x wx uniformly in as. We then derive from 1.5 that ea = E a u = λ a ε ū x dx 1 ū 4 xdx + λ ε p a ū 4 xdx + 1 a R V ε λ x + x ū xdx.13 λ ε p a ū 4 xdx + 1 ε p a g ε λ λ x + x h x + λx ū ε xdx, which then implies from.5 that λx ε is bounded uniformly in. Therefore, there exist a subsequence still denoted by { λx ε Note that lim inf lim inf } of { λx ε } and y such that λx ε y as. g ε λ x + x h x + λx ε ū xdx B 1 ε = g hx + y w xdx. g ε λ x + x h x + λx ε ū xdx.14 Since u gives the least energy of ea and the assumption 1.9 implies that y is essentially the unique global minimum point of Hy = hx+yw xdx, we conclude from.13 and.14 that y = y, which thus implies that.6 holds, and the proof is therefore complete..1 Proof of local uniqueness Following Proposition.1, this subsection is focussed on the proof of Theorem 1.1, and in the whole subsection we always assume that V x = hx C is homogeneous of degree p and satisfies 1.14 and lim x hx =. Our proof is stimulated by 3, 6, 9. We first define the linearized operator L by L := + 1 3w in, where w = w x > is the unique positive solution of 1.4 and w satisfies the exponential decay.4. Recall from 14, that { erl = span, }..15 x 1 x For any positive minimizer u of ea, where a a as, one can note that u solves the Euler-Lagrange equation u x + V xu x = µ u x + a u 3 x in R,.16 where µ R is a suitable Lagrange multiplier and satisfies µ = ea a xdx u 4

8 Moreover, under the more general assumption.1, one can derive from.3 and.5 that u satisfies u 4 xdx = a a +p λ a + o1 as..18 It then follows from.3,.17 and.18 that µ satisfies where we denote Set µ ε λ 1 as +,.19 ε := a a 1 +p >. a ū x := ε ε u λ λ x + x, so that Proposition.1 gives ū x wx uniformly in as. Note from.16 that ū satisfies ε V ε ū x + λ λ x + x ū x = µ ε λ ūx + a a ū3 x in R.. Moreover, by the exponential decay.7, there exist C > and R > such that which then implies that ū x C e x for x > R,.1 ε V ε λ λ x + x ū x CC e x 4 for x > R, if V x satisfies.1 with p. Therefore, under the assumption.1, applying the local elliptic estimates see 3.15 in 8 to. yields that ū x Ce x 4 as x,. where the estimates.19 and.1 are also used. In the following, we shall follow Proposition.1 and. to derive Theorem 1.1 on the local uniqueness of positive minimizers as a a. Proof of Theorem 1.1. Suppose that there exist two different positive minimizers u 1, and u, of ea with a a as. Let x 1, and x, be the unique local maximum point of u 1, and u,, respectively. Following.16, u i, then solves the Euler-Lagrange equation u i, x + hxu i, x = µ i, u i, x + a u 3 i, x in R, i = 1,,.3 where V x = hx and µ i, R is a suitable Lagrange multiplier. Define ū i, x := a ε u i, x, where i = 1,..4 λ 8

9 Proposition.1 then implies that ū i, ελ x + x, wx uniformly in, and ū i, satisfies the equation ε ū i,x + ε hxū i,x = µ i, ε ūi,x + λ a a ū3 i, x in R, i = 1,..5 Because u 1, u,, we consider ξ x = Then ξ satisfies the equation where the coefficient C x satisfies u,x u 1, x u, u 1, L = ū,x ū 1, x. ū, ū 1, L ε ξ + C x ξ = ḡ x in,.6 C x := µ 1, ε λ a ū a, + ū, ū 1, + ū 1, + ε hx,.7 and the nonhomogeneous term ḡ x satisfies ḡ x := ε ū,µ, µ 1, ū, ū 1, L = λ4 a ū, ū4, ū4 1, a ε dx ū, ū 1, L.8 = λ4 a ū, ū a ε ξ, + ū 1, ū, + ū 1, dx, due to the relation.17. Motivated by 3, we first claim that for any x, there exists a small constant δ > such that B δ x ε ξ + λ ξ + ε hx ξ ds = Oε as..9 To prove the above claim, multiplying.6 by ξ and integrating over, we obtain that ε ξ µ i, ε ξ + ε hx ξ R = λ a ū a, + ū, ū 1, + ū 1, ξ λ4 a ū a ε ū, ξ ξ, + ū 1, ū, + ū 1, R λ a ū a, + ū, ū 1, + ū λ 4 a ū 1, + a ε ū,, + ū 1, ū, + ū 1, Cε as, since ξ and ū ελ i, x + x, are bounded uniformly in, and ελ ūi, x + x, decays exponentially as x, i = 1,. This implies that there exists a constant C 1 > such that I := ε R ξ + λ ξ + ε 9 hx ξ < C 1 ε as..3

10 Applying Lemma 4.5 in 3, we then conclude that for any x, there exist a small constant δ > and C > such that ε ξ + λ ξ + ε hx ξ ds C I C 1 C ε as, B δ x which therefore implies the claim.9. We next define ξ x = ξ ε λ x + x,, = 1,,,.31 and a ũ i, x := ε ε u i, λ λ x + x,, where i = 1,, so that ũ i, x wx uniformly in as in view of Proposition.1. Under the non-degeneracy assumption 1.14, we shall carry out the proof of Theorem 1.1 by deriving a contradiction through the following three steps. Step 1. There exist a subsequence {a } and some constants b, b 1 and b such that ξ x ξ x in C loc as, where Note that ξ satisfies ξ x = b w + x w + i=1 b i x i..3 where the coefficient C x satisfies ξ + C xξ = g x in,.33 C x := 1 ε+p a ũ, x + ũ,xũ 1, x + ũ 1, x.34 ε λ µ 1, + ε λ h ε x λ + x,, and the nonhomogeneous term g x satisfies g x := ũ, λ ε µ, µ 1, ũ, ũ 1, L = ũ, λ = a ũ, a a ε u4, u4 1, dx ũ, ũ 1, L.35 ũ, + ũ 1, dx. ξ ũ, + ũ 1, Here we have used.17 and.5. Since ξ L 1, the standard elliptic regularity then implies cf. 8 that ξ C 1,α C for some α, 1, where the constant loc R C > is independent of. Therefore, there exist a subsequence {a } and a function ξ = ξ x such that ξ x ξ x in C loc as. Applying Proposition.1, direct calculations yield from.17 and.18 that C x 1 3w x uniformly on as, and g x wx a w 3 ξ uniformly on as. 1

11 This implies from.33 that ξ solves Lξ = ξ + 1 3w ξ = a w 3 ξ w in..36 Since Lw + x w = w, we then conclude from.15 and.36 that.3 holds for some constants b, b 1 and b. Step. The constants b = b 1 = b = in.3. We first derive the following Pohozaev-type identity b hx + y x j x w b i i=1 hx + y x j x i w =, j = 1,..37 Multiplying.5 by ū i, x j, where i, j = 1,, and integrating over B δ x,, where δ > is small and given by.9, we calculate that ε = µ i, ε B δ x, = 1 µ i,ε B δ x, B δ x, B δ x, ū i, x j ū i, + ε ū i, ū i, + λ a x j a B δ x, ū i, ν jds + λ a 4a B δ x, hx ū i, x j ū i, B δ x, ū i, x j ū 3 i, ū 4 i, ν jds, where ν = ν 1, ν denotes the outward unit normal of B δ x,. Note that ε ū i, ū i, B δ x, x j = ε ū i, ū i, B δ x, x j ν ds + ε ū i, ū i, B δ x, x j = ε ū i, ū i, x j ν ds + 1 ε ū i, ν j ds, B δ x,.38 and ε B δ x, hx ū i, ū i, = ε hxū i, x j ν jds ε hx ū i, B δ x, B δ x, x. j We then derive from.38 that ε = ε +ε B δ x, λ a a B δ x, B δ x, hx ū i, x j B δ x, ū i, x j ū i, ν ds + ε hxū i, ν jds µ i, ε ū 4 i, ν jds. B δ x, B δ x, ū i, ν j ds ū i, ν jds.39 11

12 Following.39, we thus have ε hx ū, + ū 1, ξ dx B δ x, x j = ε ū, ξ B δ x, x j ν + ξ ū 1, ds x j ν +ε ξ ū, + ū 1, νj ds B δ x, +ε hx ū, + ū 1, ξ ν j ds µ 1, ε B δ x, λ a ū a, + ū 1, ū, + ū 1, ξ ν j ds B δ x, µ, µ 1, ε ū, ū, ū 1, ν jds. L B δ x, B δ x, ū, + ū 1, ξ ν j ds.4 We now estimate the right hand side of.4 as follows. Applying.9, if δ > is small, we then deduce that ε ū, ξ ds B δ x, x j ν ε ū, 1 ds ε ξ 1 ds Cε Cδ ε e as, B δ x, x j B δ x, ν.41 due to the fact that ū ελ, x+x, satisfies the exponential decay., where C > is independent of. Similarly, we have ε ξ ū 1, ds Cε Cδ ε e as, x j ν and ε B δ x, B δ x, ξ ū, + ū 1, νj ds Cε Cδ ε e as, On the other hand, since both ξ and µ, µ 1, ε are bounded uniformly in, we also get from. that ε hx ū, + ū 1, ξ ν j ds µ 1, ε ū, + ū 1, ξ ν j ds B δ x, B δ x, λ a ū a, + ū 1, ū, + ū 1, ξ ν j ds B δ x,.4 µ, µ 1, ε ū, ū, ū 1, ν jds L = oe Cδ ε as, B δ x, due to the fact that.8 gives µ, µ 1, ε λ4 a ū ū, ū 1, L a ε, + ū 1, ū, + ū 1, ξ M,.43 1

13 where the constants M > is independent of. Because hx is homogeneous of degree p, it then follows from.4 that for small δ >, oe Cδ ε = ε = ε3 λ = εp+3 B δ x, B λδ ε hx ū, x + ū 1, x ξ xdx x j ε ξ λ y + x, h ε y j λ y + x, ε ū, λ y + x ε, + ū1, λ y + x, dy λ B p+1 h y + λx, ε ξ λδ y j ε λ y + x, ε ε ū, λ y + x ε, + ū1, λ y + x 1, dy + o1 as. Applying 1.14, we thus derive from.6,.3 and.44 that hx + y hx + y = w ξ = w x j x j hx + y = b x w x j where j = 1,, which thus implies.37. b w + x w + b i We next derive b =. Using the integration by parts, we note that x x, ū i, ūi, ε = ε = ε B δ x, B δ x, B δ x, ū i, x x, ū i, + ε ν ū i, ε x x, ū i, + ν B δ x, B δ x, i=1 i=1 b i x i.44 hx + y x j x i w, ū i, x x, ū i, x x, ν ū i,..45 Multiplying.5 by x x, ū i,, where i = 1,, and integrating over B δ x,, 13

14 where δ > is small as before, we deduce that for i = 1,, x x, ū i, ūi, ε B δ x, = ε + λ a = ε + ε B δ x, a λ a a = µ i, ε µi, hx ū i, x x, ū i, B δ x, ū i, B δ x, B δ x, B δ x, ū 3 i, x x, ū i, { µ i, hx } x x, hx ū i, µi, hx x x, νds ū 4 i, + λ a 4a ū i, + + p ε B δ x, hxū i, λ a a ū 4 i, x x,νds ū 4 i, + I i, where the lower order term I i satisfies I i = µ i, ε ū i, + p \B δ x, ε hxū i, \B δ x, + λ a a ū 4 i, 1 \B δ x, ε ū i, x, hx B δ x, + ε ū i, µi, hx x x, νds B δ x, + λ a 4a ū 4 i, x x,νds, i = 1,. B δ x, Since it follows from.17 that µ i, ε ū i, λ a a we reduce from that ū 4 i, = ε 4 a λ µ i, + a a ε 4 λ ea + p ε hxū i, = I i + ε ε B δ x, B δ x, u 4 i, = a ε 4 λ ea, ū i, x x, ū i, ν x x, ν ū i,, i = 1,, which implies that + p ε hx ū, + ū 1, ξ = T

15 Here the term T satisfies that for small δ >, I I 1 T = ε x x, ν ū, + ū 1, ξ ū, ū 1, L B δ x, { x } x, ū, ν ξ + ν ū1, x x, ξ +ε B δ x, I I 1 = + oe Cδ ε as, ū, ū 1, L.49 due to. and.9, where the second equality follows by applying the argument of estimating.41. Using the arguments of estimating.41 and.4, along with the exponential decay of ū i,, we also derive that for small δ >, I I 1 ū, ū 1, L = µ, ε \B δ x, + λ a ū a, + ū 1, ū, + ū 1, ξ \B δ x, µ, µ 1, ε + ū 1, ū, ū 1, 1 L \B δ x, ε + λ a 4a B δ x, ε ū, + ū 1, ξ hxx x, νds B δ x, ū, + ū 1, ξ + p ε hx ū, + ū 1, ξ \B δ x, B δ x, ū, + ū 1, ū, + ū 1, ξ x x, νds + µ,ε ū, + ū 1, ξ x x, νds B δ x, µ, µ 1, ε + ū 1, ū, ū 1, x x,νds L B δ x, µ, µ 1, ε = ū 1, ū, ū 1, + 1 L \B δ x, 1 ε B δ x, Note from.43 that µ, µ 1, ε ū, ū 1, L B δ x, x, hx ū, + ū 1, ξ ū 1, x x,νds x, hx ū, + ū 1, ξ + oe Cδ ε as. \B δ x, ū 1, + 1 B δ x,.5 ū 1, x x,νds = Oe Cδ ε.51 as, where the constant C > is independent of. Moreover, we follow from the 15

16 first identity of.44 that 1 ε = 1 ε B δ x, i=1 x i, x, hx ū, + ū 1, ξ B δ x, = oe Cδ ε as, hx ū, x + ū 1, x ξ xdx x i.5 where we denote x, = x 1,, x,. Therefore, we deduce from.49.5 that T = oε 4+p as. Further, we obtain from.48 that oε 4+p = + p ε hx ū, + ū 1, ξ = + p λ ε4 h ε λ x + x ε, ū, λ x + x ε, + ū1, λ x + x 1, ξ xdx + p λ ε4 = + p ε4+p λ+p h ε λ x + x, ε ū 1, λ x + x ε, ū1, λ x + x 1, ξ xdx h x + λx, ε ū, ε λ x + x, +Oε 4+p x, x 1, as. ε + ū1, λ x + x 1, ξ xdx Since x + y hx + y = phx + y, by Proposition.1 and Step 1, we thus obtain from 1.14 and above that = hx + y wξ = b hx + y w w + x w + b i hx + y i=1 x i = b hx + y w + 1 R hx + y x w { = b hx + y w 1 R w hx + y + x hx + y } = pb hx + y w + b w y hx + y = pb hx + y w, which therefore implies that b =. By the non-degeneracy assumption 1.14, setting b = into.37 then yields that b 1 = b =, and Step is therefore proved. Step 3. ξ cannot occur. Finally, let y be a point satisfying ξ y = ξ L = 1. By the same argument as employed in proving Lemma 3.1 in next section, applying the maximum principle to.33 yields that y C uniformly in. Therefore, we conclude that ξ ξ uniformly on, which however contradicts to the fact that ξ on. This completes the proof of Theorem

17 3 Refined Spie Profiles In the following two sections, we shall derive the refined spie profiles of positive minimizers u = u a for ea as a a. The purpose of this section is to prove Theorem 1.. Recall first that u satisfies the Euler-Lagrange equation.16. Under the assumptions of Proposition.1, for convenience, we denote ε = a a 1 +p >, α := ε +p > and β := 1 + µ ε λ, 3.1 where µ R is the Lagrange multiplier of the equation.16, so that α and β as, where.19 is used. In order to discuss the refined spie profiles of u as, the ey is thus to obtain the refined estimate of µ equivalently β in terms of ε. We next define a w x := ū x wx := ε ε u λ λ x + x wx, 3. where x is the unique maximum point of u, so that w x uniformly in by Proposition.1. By applying.16, direct calculations then give that ū satisfies ū x + ε λ V ε λ x + x ū x = µ ε λ ūx + a a ū3 x in R. Relating to the operator L := + 1 3w in, we also denote the linearized operator L := + 1 ū + ū w + w in, so that w satisfies 1 L w x = α a ū3 x + 1 λ +p g ε x λ + x λx h x + ū x ε +β ū x in, w =, 3.3 where V x = gxhx satisfies the assumptions of Proposition.1 and the coefficients α > and β > are as in 3.1. Define 1 L ψ 1, x = α λ +p g ε x λ + x λx h x + ū x ε + 1 a ū3 x in, ψ 1, =, 3.4 L ψ, x = β ū x in, ψ, =. Note that the right hand side of 3.4 is orthogonal to the ernel of L, which then implies that both ψ 1, and ψ, exist. One can get that the solution w x of 3.3 then satisfies w x := ψ 1, x + ψ, x in. 3.5 We first employ Proposition.1 to address the following estimates of w as. Lemma 3.1. Under the assumptions of Proposition.1, where V x = gxhx, we have 17

18 1. ψ 1, x satisfies ψ 1, x = α ψ 1 x + oα as, 3.6 where ψ 1 x C L solves uniquely ψ 1 =, where y is given by ψ, x satisfies where ψ x solves uniquely i.e., ψ x C L satisfies 3. w satisfies Lψ 1 x = 1 a w3 x g λ +p hx + y wx in, 3.7 ψ, x = β ψ x + oβ as, 3.8 ψ =, Lψ x = wx in, 3.9 ψ = 1 w + x w. 3.1 w x := α ψ 1 x + β ψ x + oα + β as Proof. 1. We first derive ψ 1, Cα in by contradiction. On the contrary, we assume that ψ 1, L lim =. 3.1 α Set ψ 1, = ψ 1, ψ 1, L so that ψ 1, L = 1. Following 3.4, ψ 1, then satisfies ψ 1, + 1 ū + ū w + w ψ1, = α 1 ψ 1, λ +p g ε x λ + x λx h x + ū x + 1 ε a ū3 x in Let y be the global maximum point of ψ 1, so that ψ ψ 1, y = max 1, y y ψ 1, = 1. L Since both ū and w decay exponentially in view of.7, using the maximum principle we derive from 3.13 that y C uniformly in. On the other hand, applying the usual elliptic regularity theory, there exists a subsequence, still denoted by { ψ 1, }, of { ψ 1, } such that ψ 1, ψ 1 wealy in H 1 and strongly in L q loc R for all q,. Here ψ 1 satisfies ψ 1 =, L ψ 1 x = in, which implies that ψ 1 = i=1 c i y i. Since ψ 1 =, we obtain that c 1 = c =. Thus, we have ψ 1 y in, which however contradicts to the fact that 1 = ψ 1, y ψ 1 ȳ for some ȳ by passing to a subsequence if necessary. Therefore, we have ψ 1, Cα in. We next set ϕ 1, x = ψ 1, x α ψ 1 x, where ψ 1 x C L is a solution of 3.7. Then either ϕ 1, x = Oα or ϕ 1, x = oα as, and ϕ 1, satisfies ϕ 1, =, ϕ 1, + 1 ū + ū w + w ϕ 1, = α f x in, 18

19 where f x satisfies f x = w ū ū w ψ 1 x + 1 a ū3 x w 3 x + 1 λ +p g ε x λ + x λx h x + ū x ghx + y wx. ε One can note that f x uniformly as. Therefore, applying the previous argument yields necessarily that ϕ 1, x = oα as, and the proof of 3.6 is then complete. Also, the property.15 gives the uniqueness of solutions for Since the proof of 3.8 is very similar to that of 3.6, we omit the details. Further, the property.15 gives the uniqueness of ψ. Also, one can chec directly that w + x w/ is a solution of 3.9, which therefore implies that 3.1 holds. 3. The expansion 3.11 now follows immediately from 3.5, 3.6 and 3.8, and the proof is therefore complete. 3.1 Proof of Theorem 1. The main aim of this subsection is to prove Theorem 1. on the refined spie behavior of positive minimizers. In this whole subsection, we assume that the potential V x = hx C satisfies lim x hx = and 1.14, where hx is homogeneous of degree p. Following 3.1, from now on we denote for simplicity that o α + β = o α + o α β + o β as, 3.14 where α and β are defined in 3.1. We first use Lemma 3.1 to establish the following lemmas. Lemma 3.. Suppose that V x = hx C satisfies lim x hx = and 1.14 for some y, where hx is homogeneous of degree p. Then there exists an x such that the unique maximum point x of u satisfies α λx ε y α β y = α O x + o α + β as Proof. Multiplying 3.7 and 3.9 by x 1 and then integrating over, respectively, we obtain from 1.14 and.15 that Lw = w = hx + y w =, 3.16 x 1 x 1 x 1 where y is given by the assumption Similarly, we derive from 3.3 and 3.11 that 1 L w = β ū α x 1 x 1 x 1 a ū3 + ū λ +p h x + λx ε 3.17 = α β ψ 1 + oα β + β x I 1, 1 where the identity x 1 ψ = is used, since x 1 ψ is odd in x 1 by the radial symmetry 19

20 of ψ. We obtain from 1.14 and 3.16 that 1 I 1 = α x 1 a ū3 + ū λ +p h x + λx ε = α = α a + α = 3α a λ +p { 1 ū3 x 1 a w λ +p h x + λx ū hx + y w ε w 3w + 3ww + w x 1 h x + λx ū hx + y w x 1 ε x 1 w ψ 1 + oα + α β + I, } 3.18 where we have used the identity x 1 w ψ =, since x 1 w ψ is odd in x 1 by the radial symmetry of ψ. Further, applying 3.11 and 3.16 yields that λ +p I = α = + = α { h x + λx ū w + h x + λx hx + y } w x 1 ε ε hx + y w + oα + β x 1 λx λx y hx + y w + o y x 1 ε ε + x 1 hx + y ψ 1 + β x 1 λx ε y hx + y hx + y ψ x 1 w + o α + λx y + β, ε where.6 is used for the second identity. We thus get that 3 I 1 = α a w ψ x 1 λ +p hx + y ψ 1 x 1 + α β λ +p hx + y ψ + α λx x 1 λ +p y hx + y w x 1 ε +o α λx ε y + α + β On the other hand, we obtain from 3.16 that L w = Lw + L L w x 1 R x 1 x 1 = w x 3w + w 1 = 3α wψ 1 6α β wψ 1 ψ + oα x 1 x + α β

21 Combining 3.17, 3.1 and 3., we now conclude from 1.14 and 3.11 that α λx λ +p y hx + y w x 1 ε = α β ψ wψ 1 ψ 1 x 1 x 1 λ +p hx + y ψ x α a w ψ x 1 λ +p hx + y ψ 1 3 wψ1 x 1 x 1 +oα + β. We claim that the coefficient I 3 of the term α β in 3. satisfies I 3 : = ψ wψ 1 ψ 1 x 1 x 1 λ +p hx + y ψ x 1 = 1 λ +p y hx + y. x 1 w 3.3 If 3.3 holds, we then derive from 3. that there exists some x = x 1, x such that 1 R λx y λ +p α y α β hx + y x j ε 3.4 = α O x j + oα + β, j = 1,. By the non-degeneracy assumption of 1.14, we further conclude from 3.4 that 3.15 holds for some x, and the lemma is therefore proved. To complete the proof of the lemma, the rest is to prove the claim 3.3. Indeed, using the integration by parts, we derive from 3.1 that A : = ψ wψ 1 ψ x 1 x 1 = ψ 1 3 w ψ 1 3 ψ 1 x w x 1 x 1 x 1 = ψ 1 3 w ψ 1 x 1 x w ψ 1 + x ψ 1 x 1 x 1 = ψ x ψ 1 + ψ 1 x. x 1 x 1 x 1 w Since x + y hx + y = phx + y, we obtain from 1.14, 3.1 and 3.16 that B : = 1 λ +p hx + y ψ x 1 = 1 λ +p hx + y w + x w x 1 = 1 λ +p w hx + y + x hx + y x 1 x 1 = 1 { x λ +p w hx + y } + hx + y x x 1 x 1 = 1 λ +p whx + y x + 1 x 1 λ +p w y hx + y. x 1 1

22 By above calculations, we then get from 3.3 that I 3 = A + B = ψ w x ψ 1 x 1 x w ψ 1 whx + y λ +p x x λ +p w y hx + y x 1 := I λ +p y hx + y. x 1 w 3.5 Applying the integration by parts, we derive from 3.7 that ψ w ψ 1 whx + y x 1 λ +p x x 1 = ψ w 3w 3 ψ 1 x 1 a + whx + y λ +p = ψ R ψ1 + ψ 1 x x 1 x 1 = ψ R ψ1 x 1 x 1 x 1 = 1 R ψ1 x 1 x ψ1, x 1 x 1 which then gives from 3.5 that I 4 = ψ 1 x + x 1 = ψ 1 x + x 1 x x 1 ψ1 + x ψ 1 x 1 w w 3 x ψ1 x 1 w x 1 x ψ To further simplify I 4, we next rewrite ψ 1 as ψ 1 x = ψ 1 r, θ, where x = rcos θ, sin θ

23 and r, θ is the polar coordinate in. We then follow from 3.7 and 3.6 that I 4 = π π + = + + = = π π π π π π { r ψ1 r ψ1 } r r w cos θ dθdr r r + 1 r θθ w + w cos θ r ψ 1 r r dθdr r ψ 1 r rw cos θdθdr r ψ 1 r ψ1 r ψ 1 r r w + w cos θdθdr r r θθ w cos θdθdr { rw r w + w } cos θdθdr r r ψ 1 w cos θdθdr ψ1 r w cos θdθdr π ψ 1 w cos θdθdr =, i.e., I 4 =, which therefore implies that the claim 3.3 holds by applying Remar 3.1. Whether the point x in Lemma 3. is the origin or not is determined completely by the fact that whether the coefficient I 5 of the term α in 3. is zero or not, where I 5 satisfies I 5 := 3 a w ψ x 1 λ +p hx + y ψ 1 3 x 1 x 1 wψ 1. If hx is not even in x, it however seems difficult to derive that whether I 5 = or not. Lemma 3.3. Suppose that V x = hx C satisfies lim x hx = and 1.14 for some y, where hx is homogeneous of degree p. Then we have w := α ψ 1 + β ψ + α ψ 3 + β ψ 4 + α β ψ 5 + oα + β as, 3.8 where ψ 1 x, ψ x C L are given in Lemma 3.1 with g = 1, and ψ i x C L, i = 3, 4, 5, solves uniquely ψ i = and Lψ i x = f i x in, i = 3, 4, 5, 3.9 and f i x satisfies for some y, 3w 3wψ 1 a + hx + y λ +p ψ 1 w y λ 1+p hx + y, if i = 3; 3wψ f i x = + ψ, if i = 4; 3w 6wψ 1 ψ + ψ 1 a + hx + y λ +p ψ w λ +p y hx + y, if i = 5; where y is given by Moreover, ψ 4 is radially symmetric

24 Proof. Following Lemma 3.13, set v = w α ψ 1 β ψ, so that L w = L v + α ψ 1 + β ψ = L v + α L Lψ 1 + β L Lψ + α Lψ 1 + β Lψ w 3 = L v w α ψ 1 + β ψ 3w + w α a + hx + y w λ +p + β w. Applying 3.3, we then have w 3 L v = L w + w α ψ 1 + β ψ 3w + w + α a + hx + y w λ +p β w = w α ψ 1 + β ψ 3w + w + β ū w { 1 α a ū3 w3 + 1 λ +p h x + λx } ū hx + y w ε = w α ψ 1 + β ψ 3w + w + β w I 6, where I 6 satisfies I 6 = α a w 3w + 3ww + w + α { λ +p hx + y ū w + h x + λx } hx + y ū ε = α a w 3w + 3ww + w + α λ +p hx + y w + α λx λ +p y hx + y ū + o α + β ε 3w = α w a + hx + y λ +p + α a w 3w + w + α λx λ +p y hx + y ū + o α + β, ε where Lemma 3. is used in the second equality. y such that By Lemma 3. again, there exists α λx ε y α β y α y = o α + β as. We thus obtain from above that 3w L v = w α ψ 1 + β ψ 3w + w + β w α w a + hx + y λ +p α λx λ +p y hx + y ū α ε a w 3w + w + oα + β { 3w = α 3wψ1 a + hx + y λ +p ψ 1 w } λ 1+p y hx + y 3w +α β {6wψ 1 ψ + ψ 1 a + hx + y λ +p ψ 1 } λ +p w y hx + y +β 3wψ + ψ + oα + β in

25 Following 3.33, the same argument of proving Lemma 3.1 then gives 3.8. Finally, since f 4 x is radially symmetric, there exists a radial solution ψ 4. Further, the property.15 gives the uniqueness of ψ 4. Therefore, ψ 4 must be radially symmetric, and the proof is complete. Lemma 3.4. Suppose that V x = hx C satisfies lim x hx = and 1.14 for some y, where hx is homogeneous of degree p. Then we have wψ 1 =, wψ =, 3.34 and However, we have I = wψ4 + ψ = II = wψ 5 + ψ 1 ψ = + p < Here ψ 1 x,, ψ 5 x C L are given in Lemma 3.1 with g = 1 and Lemma 3.3. Since the proof of Lemma 3.4 is very involved, we leave it to the appendix. Following above lemmas, we are now ready to derive the comparison relation between β and α. Proposition 3.5. Suppose that V x = hx C satisfies lim x hx = and 1.14 for some y, where hx is homogeneous of degree p. Then we have where the constant C satisfies C = wψ p Moreover, w satisfies β = C α as, 3.37 ψ w := ψ 1 + C ψ α + ψ 3 + C ψ 4 + C ψ 5 α + oα as, 3.39 Here ψ 1 x,, ψ 5 x C L are given in Lemma 3.1 with g = 1 and Lemma 3.3. Proof. Note from 3. that w satisfies w = ū =, w + w i.e. ww + w =

26 Applying 3.4, we then derive from Lemma 3.3 that = ww + w = wα ψ 1 + β ψ + α ψ 3 + β ψ 4 + α β ψ 5 R + α ψ 1 + β ψ + α ψ 3 + β ψ 4 + α β ψ 5 + oα + β 3.41 = α wψ 1 + β wψ + β wψ 4 + ψ R +α β wψ 5 + ψ 1 ψ + α wψ 3 + ψ1 + oα + β = + p α β + α wψ 3 + ψ1 + oα + β, where Lemma 3.4 is used in the last equality. It then follows from 3.41 that wψ 3 + ψ1, and moreover, + p β + α wψ 3 + ψ 1 =, i.e., β = C α, where C is as in Finally, the expansion 3.39 follows directly from 3.37 and Lemma 3.3, and we are done. We remar from 3.1 and Proposition 3.5 that the Lagrange multiplier µ R of the Euler-Lagrange equation.16 satisfies µ = λ ε + λ C ε p + oεp as, 3.4 where λ > is defined by. with g = 1, and C is given by Moreover, following above results we finally conclude the following refined spie profiles. Theorem 3.6. Suppose that V x = hx C satisfies lim x hx = and 1.14 for some y, where hx is homogeneous of degree p. If u a is a positive minimizer of ea for a < a. Then for any sequence {a } with a a as, there exist a subsequence, still denoted by {a }, of {a } and {x } such that the subsequence solution u = u a satisfies for ε := a a 1 +p, u x = λ { 1 λx x w w ε ε +ε 3+p + ε 1+p λx ψ 1 + C x ψ ε ψ 3 + C ψ 4 + C ψ 5 λx x ε } + oε 3+p as 3.43 uniformly in, where the unique maximum point x of u satisfies λx y = ε +p ε O y as 3.44 for some y, and C is given by Here ψ 1 x,, ψ 5 x C L are given in Lemma 3.1 with g = 1 and Lemma

27 Proof. The refined spie profile 3.43 follows immediately from 3. and Also, Lemma 3. and 3.37 yield that the estimate 3.44 holds. Proof of Theorem 1.. Since the local uniqueness of Theorem 1.1 implies that Theorem 3.6 holds for the whole sequence {a }, Theorem 1. is proved. 4 Refined Spie Profiles: V x = gxhx The main purpose of this section is to derive Theorem 4.4 which extends the refined spie behavior of Theorem 1. to more general potentials V x = gxhx C, where V x satisfies lim x V x = and V. h x = hx satisfies 1.14 and is homogeneous of degree p, gx C m for some m N {+ } satisfies < C gx 1 C in R and Gx := gx g, D α G = for all α m 1, and D α G for some α = m. Here it taes m = + if gx 1. Remar 4.1. The property h x = hx in the above assumption V implies that y = must occur in For the above type of potentials V x, suppose {u } is a positive minimizer sequence of ea with a a as, and let w be defined by 3., where x is the unique maximum point of u. Then Lemma 3.1 still holds in this case, where α > and β > are defined in 3.1. Similar to Lemma 3., we start with the following estimates. Lemma 4.1. Suppose V x = gxhx C satisfies lim x V x = and the assumption V for p and m N {+ }. Then the unique maximum point x of u satisfies the following estimates: 1. If m is even, then we have λα x ε = o α + β + α ε m as If m is odd, then we have λα x ε = Oα ε m x + o α + β + α ε m as, 4. where x satisfies g x hx w + 1 x 1 λ m α =m x α x 1 α! Dα g hxw =. 4.3 Proof. Recall that ψ 1 x and ψ x are given in Lemma 3.1. Since h x = hx, we have ψ i x = ψ i x for i = 1, and thus ψ 1 = wψ 1 = wψ 1 ψ =. 4.4 x 1 x 1 x 1 7

28 Since 1.14 holds with y = as shown in Remar 4.1, the same calculations of then yield that oα + α β = L w x 1 = oα β + β α 1 x 1 a ū3 + ū λ +p g ε x λ + x λx h x + ε = oα β + β α ū3 a x w α λ +p g ε x x 1 λ + x λx h x + ū ghxw ε = oα β + β α λ +p g ε x x 1 λ + x λx h x + ū ghxw ε = oα β + β I 1, where the first equality follows from 3.1 and 4.4. Similar to 3.19, we deduce from 1.14 with y = that λ +p I 1 = α + { gh x + λx ū w + g h x + λx } hx w x 1 ε ε g ε x x 1 λ + x g h x + λx ū ε = oα + λx ε ε m + λ + β + g α =m x 1 1 α! λx hx x 1 ε x + λx ε αd α g w h x + λx ε ū + oε m, which then implies that I 1 = α ε m λ +p λ + α g λ+p α =m 1 x + λx αd α g x 1 α! ε λx hx x 1 ε h x + λx ε ū w + oα + α β + λx ε + α ε m. Combining 4.5 and 4.6, we then conclude from the estimate 3.11 that α λx g hx w λ+p x 1 ε = α ε m x α λ +p λ x 1 α! Dα g hxw + o α + β + α ε m. α =m If m is even, one can note that x α x 1 α! Dα g hxw =, α =m and it then follows from 4.7 and 1.14 with y = that 4.1 holds. If m is odd, we then derive from 4.7 that both 4. and 4.3 hold. 8

29 Lemma 4.. Suppose V x = gxhx C satisfies lim x V x = and the assumption V for p and m N {+ }. Let ψ 1 x and ψ x be given in Lemma 3.1 with y =. Then w satisfies w : = α ψ 1 + β ψ + α ψ 3 + β ψ 4 +α ε m ϕ + α β ψ 5 + o α + β + α ε m 4.8 as, where ψ i x C L, i = 3, 4, 5, solves uniquely ψ i = and Lψ i x = g i x in, i = 3, 4, 5, 4.9 and g i x satisfies g i x = 3w 3wψ1 a + ghx λ +p ψ 1, if i = 3; 3wψ + ψ, if i = 4; 3w 6wψ 1 ψ + ψ 1 a + ghx λ +p ψ, if i = Here ϕ C L solves uniquely Lϕx = 1 { x λ +p hx gw + 1 x α } λ m α! Dα g hxw α =m in, and ϕ =, 4.11 where x = holds for the case where m is even, and x satisfies 4.3 for the case where m is odd. Proof. Following Lemma 3.13, we set Similar to 3.3, we then have v = w α ψ 1 β ψ. L v = w α ψ 1 + β ψ 3w + w + β w α a ū3 w3 α λ +p g ε x λ + x λx h x + ū ghxw ε = w α ψ 1 + β ψ 3w + w + β w α a w 3w + 3ww + w I, 4.1 9

30 where I satisfies I = α { λ +p g ε x λ + x g h x + λx ū ε +g h x + λx hx ε = α { ε m λ +p λ λx +g α =m ε = α w ghx λ +p + α λ +p λx + α ε m λ +p+m α =m ū + ghx ū w } 1 x + α! λx αd α g h x + λx ū ε ε } hx ū + ghxw ε hx gū 1 α! x + λx ε αd α g α x + o + α ε m ε h x + λx ε ū + o α ε m, 4.13 where Lemma 4.1 is used in the last equality. Applying Lemma 4.1 again, we then obtain from 4.1 and 4.13 that L v = w α ψ 1 + β ψ 3w + w + β w α a w 3w + w 3w α w a + ghx λ +p α λx λ +p hx gū ε α ε m 1 x + α! λx αd α g ε = α λ +p+m α =m 3wψ 1 3w a + ghx λ +p ψ 1 3w a + ghx λ +p ψ α ε m λ +p { x hx gw + 1 λ m h x + λx ε ū + o α ε m +α β 6wψ 1 ψ + ψ 1 x α } α! Dα g hxw α =m +β 3wψ + ψ + o α + β + α ε m in, 4.14 where x = holds for the case where m is even, and x satisfies 4.3 for the case where m is odd. Following 4.14, the same argument of proving Lemma 3.1 then gives 4.8, and the proof is therefore complete. Proposition 4.3. Suppose V x = gxhx C satisfies lim x V x = and the assumption V for p and m N {+ }. Let ψ 1 x,, ψ 5 x C L be given in Lemma 3.1 with y = and Lemma 4., and ϕ is given by If m > + p, then and w satisfies β = C α, 4.15 w := ψ 1 + C ψ α + ψ 3 + C ψ 4 + C ψ 5 α + oα as, 4.16 where the constant C satisfies C := wψ 3 + ψ p 3

31 . If 1 m + p and m is odd, then β = C α and w satisfies w : = ψ 1 + C ψ α + ϕ α ε m + ψ 3 + C ψ 4 + C ψ 5 α + oα ε m as, 4.18 where the constant C is given by If 1 m < + p and m is even, consider S = x α α! Dα g hxw α =m Then for the case where S =, we have β = C α and w satisfies 4.18, where the constant C is given by However, for the case where S =, we have β = C 1ε m, 4. and w satisfies w := C 1ψ ε m + ψ 1α + C 1 ψ 4 ε m + oε min{+p,m} as, 4.1 where the constant C1 satisfies C1 m + p = + pλ +p+m α =m x α α! Dα g hxw If m = + p is even, then and w satisfies β = C α, 4.3 w : = ψ 1 + C ψ α + ψ 3 + C ψ 4 + C ψ 5 + ϕ α + oα as, 4.4 where the constant C satisfies C = wψ 3 + ψ1 + wϕ p Proof. The same argument of proving Lemma 3.4 with y = yields that wψ 1 =, wψ = and I = wψ4 + ψ =, 4.6 and II = wψ 5 + ψ 1 ψ = + p <

32 It thus follows from 3.4 and Lemma 4. that = ww + w = w α ψ 1 + β ψ + α ψ 3 + β ψ 4 + α ε m ϕ + α β ψ 5 R + α ψ 1 + β ψ + α ψ 3 + β ψ 4 + α ε m ϕ + α β ψ 5 +o α + β + α ε m = α wψ 1 + β wψ + β wψ 4 + ψ R +α β wψ 5 + ψ 1 ψ + α wψ 3 + ψ1 R +α ε m wϕ + o α + β + α ε m = + p α β + α wψ 3 + ψ1 + α ε m wϕ +o α + β + α ε m, 4.8 where 4.6 and 4.7 are used in the last equality. Following 4.8, we next carry out the proof by considering separately the following four cases: Case 1. m > + p. In this case, it follows from 4.8 that the constant C defined in 4.17 is nonzero and + p β + α wψ 3 + ψ1 =, i.e., β = C α. Moreover, the expansion 4.16 follows directly from 4.15 and Lemma 4., and Case 1 is therefore proved. Case. 1 m + p and m is odd. In this case, since m is odd and h x = hx, we obtain from 3.1 and 4.11 that wϕ = ϕlψ = ψ Lϕ = 1 { x λ +p hx gw + 1 x α w λ m α! Dα g hxw} + x w =. α =m We then derive from 4.8 that 4.17 still holds and thus β = C α. Further, the expansion 4.18 follows directly from 4.8 and

33 Case 3. 1 m < + p and m is even. Since m is even, then x = holds in Further, since x α hx is homogeneous of degree m + p, we then obtain from 4.11 that wϕ = ϕlψ = ψ Lϕ 1 x α = λ +p+m α =m α! Dα g hxw w + x w 1 x α = λ +p+m α =m α! Dα g hxw 1 x α + λ +p+m α =m α! Dα g hx x w 1 x α 4.9 = λ +p+m α =m α! Dα g hxw 1 { x λ +p+m w α α =m α! Dα ghx x α } +x α! Dα ghx = m + p x α λ +p+m α! Dα g hxw := m + p S, λ+p+m α =m where S is as in Therefore, if S =, then we are in the same situation as that of above Case, which gives that β = C α and w satisfies 4.18, where the constant C is given by We next consider the case where S =. By applying 4.9, in this case we derive from 4.8 that + p α β + α ε m wϕ =, which implies that β = C1 εm, where the constant C 1 satisfies 4. in view of 4.9. Further, the expansion 4.1 follows directly from 4. and Lemma 4.. Case 4. m = + p is even. In this case, we derive from 4.8 that + p α β + α wψ 3 + ψ1 + wϕ =, which gives that β = C α, where the constant C satisfies 4.5. Further, the expansion 4.4 follows directly from 4.3 and Lemma 4.. Applying directly Lemmas 4.1 and 4. as well as Proposition 4.3, we now conclude the following main results of this section. Recall that λ > is defined by. with y =, ψ 1 x,, ψ 5 x C L are given in Lemma 3.1 with y = and Lemma 4., and ϕ is given by Theorem 4.4. Suppose V x = gxhx C satisfies lim x V x = and the assumption V for p and m N {+ }. Let u a be a positive minimizer of 1.1 for a < a. Then for any sequence {a } with a a as, there exists a subsequence, still denoted by {a }, of {a } such that u = u a has a unique maximum point x and satisfies for ε := a a 1 +p, 33

34 1. If m > + p, then we have u x = λ { 1 λx x w w ε ε +ε 3+p uniformly in, where x satisfies + ε 1+p λx ψ 1 + C x ψ ε ψ 3 + C ψ 4 + C ψ 5 λx x ε } + oε 3+p as 4.3 x ε = Oε m y + oε +p as 4.31 for some y, and the constant C is given by Further, if m is even, then x satisfies x ε 3+p = o1 as If 1 m + p and m is odd, then we have u x = λ { 1 λx x w + ε 1+p λx w ε ε ψ 1 + C x ψ ε +ε 3+p ψ3 + C ψ 4 + C λx x ψ 5 ε +ε 1+m+p λx x } ϕ ε uniformly in, where x satisfies x ε m+1 + oε 1+m+p as 4.33 = O y as for some y, and the constant C is given by If m = + p is even, then we have u x = λ { 1 λx x w + ε 1+p λx w ε ε ψ 1 + Cψ x ε +ε 3+p λx ψ 3 + C ψ 4 + Cψ x } 5 + ϕ + oε 3+p ε as 4.35 uniformly in, where x satisfies 4.3 and the constant C is defined by If 1 m < + p and m is even, let the constant S be defined in Then for the case where S =, u satisfies 4.33 and x satisfies x ε m+1 However, for the case where S, u satisfies u x = λ w { 1 ε w λx x ε = o1 as λx + ε m 1 C1ψ x ε λx +ε m 1 C1 x ψ 4 + ε 1+p λx x } ε ψ 1 ε +oε min{+p,m} 1 as

35 uniformly in, where x satisfies 4.36, and the constant C1 is defined by 4.. Proof. 1. If m > + p, then 4.3 follows directly from Proposition 4.31, and 4.31 follows from Lemma 4.1. Specially, if m is even, then Lemma 4.1 gives y =, and therefore 4.31 implies If 1 m + p and m is odd, then Proposition 4.3 gives Moreover, it yields from 4. that x satisfies x ε = Oε m y + oε m as, which then implies 4.34 for some y. 3. If m = + p is even, then Proposition 4.34 gives 4.35, and we reduce from 4.1 that x satisfies If 1 m < + p and m is even, it then follows from 4.1 that x always satisfies Moreover, Proposition 4.33 gives that if S =, then u satisfies 4.33; if S, then u satisfies A Appendix: The Proof of Lemma 3.4 In this appendix, we shall follow Lemmas 3.1 and 3.3 to address the proof of Lemma 3.4, i.e., The proof of Under the assumptions of Lemma 3.4, we first note that the equation 3.7 can be simplified as due to the fact that ψ 1 =, Lψ 1 = w3 R w 4 hx + y w p R hx + y w in R, A.1 a = w = 1 w 4. A. By 1.14, 3.1 and A.1, we then have wψ 1 = Lψ ψ 1 = ψ Lψ 1 R w 3 = R w 4 + hx + y w w p + x w R hx + y w = + p + R w 4 w 3 x w + p R hx + y w hx + y w x w = + p + 1 x w 4 1 R w 4 + p R hx + y w hx + y x w = + p 1 1 hx p R hx + y w + y + x hx + y = + p 1 p + p =, w since x + y hx + y = phx + y and R w y hx + y =. Also, we deduce from 3.1 that wψ = ww + x w = w 1 x w =, 35

36 which thus completes the proof of The proof of By Lemmas 3.1 and 3.3, we obtain that I = wψ4 + ψ = ψ + Lψ, ψ 4 R = ψ + ψ, Lψ 4 = ψ + ψ, 3wψ + ψ R = 3 ψ + 6 wψ, 3 which implies that I 3π = ψ + wψ 3 π Here we have A = = where A. is used, and B = rw + rw rw + rw = rww + rw 3 = rw = 1 rw = Therefore, we get from A.3 that I 3π = r 3 w To further simplify I, recall that r w 3 w + 3 rww + rw 3 := A B. r 3 w + r 3 w 1 r 3 w w w + r 4 ww 3 3 rw = w + rw rw 3, rw + rw 4, r 3 w w w + r 4 ww 3. r dw r 4 ww 3 r 3 w w w := C + D + E. A.3 A.4 A.5 by which we then have C = = = = r 3 w dw = wr 3 w w 3r w + r w + rw rw 3 w r w + r 3 w r 3 w 3 rw 36 r 3 w + r 3 w 4.

37 Similarly, we have D = 1 r 4 w dw = 1 = = Note from A.5 that r 3 w w = 3 = 4 3 = 3 We thus derive that D + E = = 3 w 4r 3 w + r 3 w w + rw rw 3 r 3 w w r 3 w w r 3 w dw 3 = 3 w 3 r w + 3 rw r 3 w w rw r 4 dw r 3 w r 3 w 6. r 4 dw 6 w 3 3r w + r w + rw rw 3 r 3 w 4 3 r 3 w 4 3 r 3 w 4, r 3 w r 3 w 6 r 3 w 6. r 3 w 6 by which we conclude from A. and A.4 that I 3π = C + D + E = 1 3 In the following, we note that w satisfies rw 3 rw = rw rw 3, r >. r 3 w + r 3 w 4. A.6 A.7 Multiplying A.7 by r 3 w and integrating on,, we get that Note also that r 3 w rw = = r 3 w rw = r 3 w rw rw 3 = 1 r 3 w + r 3 w + 1 r 3 w 4. By combining above two identities, it yields that r 4 dw 1 4 r 4 dw = r 4 dw 4 r 3 w. r 3 w = r 3 w r 3 w 4. A.8 37

38 On the other hand, multiplying A.7 by r w and integrating on,, we obtain that r 3 w which then implies that r 3 w 4 = = = = r 3 w = r wrw = r ww r ww rw rw We thus conclude from A.8 and A.9 that rw 3 r 3 w + r ww + w 3r w + r 3 w r 3 w r 3 w, r 3 w + r 3 w 4 =, r 3 w 4. which therefore implies that I = in view of A.6, i.e., 3.35 holds. The proof of Following Lemmas 3.1 and 3.3 again, we get that II = ψ 5 Lψ + ψ 1 ψ = ψ Lψ 5 + ψ 1 R = w + x w6wψ 1 ψ + ψ 1 1 3w a + hx + y w + x w λ +p + 1 λ +p w + x w y hx + y w := A + B. r 3 wdw A.9 A.1 Since x + y hx + y = phx + y holds in, we derive from 1.14 and A.1 that B = 1 3w a + hx + y w λ +p + wx w + x w + 1 λ +p w + x w y hx + y w R 3w = R w 4 + hx + y w p R hx + y w + wx w 1 3w a + hx + y x 1 w + λ +p λ +p w y hx + y x w = 3 1 p 3 R w 4 x w 4 1 p R hx + y w hx + y x w 1 3w a + hx + y x 1 w + λ +p λ +p w y hx + y x w := 3 1 p p + C = p C, p p 38

39 where the term C satisfies C = 1 3w a + hx + y x 1 w + λ +p λ +p w y hx + y x w = 1 a x wx w 3 1 λ +p hx + y x w x w + 1 λ +p w y hx + y x w = 1 a w 3 x w + x x w + 1 { λ +p w hx + y x w + x hx + y x w } +hx + y x x w + 1 λ +p w y hx + y x w = 1 w 3 a + whx + y x x w λ +p + 1 a w 3 x w + + p λ +p whx + y x w = 1 w 3 a + whx + y x x w λ +p 1 + R w 4 x w 4 + p + p R hx + y w hx + y x w = 1 w 3 a + whx + y x + p x w 1 λ +p, p in view of A.1. We thus have B = 1 w 3 a + whx + y x p + 4p + x w λ +p. A.11 p We next calculate the term A as follows. Observe that 1 ψ 1 x x w = 1 x w ψ1 + x ψ 1 R 1 = ψ 1 x w x ψ1 x w R 1 = ψ 1 x w + w x ψ 1 + x x ψ1 R = ψ 1 x w + w 1 x ψ 1 + wx x ψ 1, which implies that = 1 ψ 1 x w + R ψ 1 x x w 1 w x ψ 1 39 wx x ψ 1. A.1