INFINITELY MANY POSITIVE SOLUTIONS OF NONLINEAR SCHRÖDINGER EQUATIONS WITH NON-SYMMETRIC POTENTIALS

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1 INFINITELY MANY POSITIVE SOLUTIONS OF NONLINEAR SCHRÖDINGER EQUATIONS WITH NON-SYMMETRIC POTENTIALS MANUEL DEL PINO, JUNCHENG WEI, AND WEI YAO Abstract. We consider the standing-wave problem for a nonlinear Schrödinger equation, corresponding to the semilinear elliptic problem u + V (x)u = u p 1 u, u H 1 (R 2 ), where V (x) is a uniformly positive potential and p > 1. Assuming that V (x) = V + a ( x m + O 1 ) x m+σ, as x +, for instance if p > 2, m > 2 and σ > 1 we prove the existence of infinitely many positive solutions. If V (x) is radially symmetric, this result was proved in [43]. The proof without symmetries is much more difficult, and for that we develop a new intermediate Lyapunov-Schmidt reduction method, which is a compromise between the finite and infinite dimensional versions of it. Contents 1. Introduction and statement of the main result 2 2. Description of the construction 6 3. Preliminaries The Lyapunov-Schmidt reduction Linear analysis Nonlinear analysis A further reduction process Projections The invertibility of T Reduction to one dimension Proof of Theorem 1.1: variational reduction Generalizations and discussion More general nonlinearities Sign-changing solutions Remarks on condition (1.14) The anisotropic case Optimal condition on the decay Higher dimensions Higher dimensional concentration phenomena Appendix A: Circulant matrices and proof of Lemma Appendix B: Energy expansion 47 1

2 2 MANUEL DEL PINO, JUNCHENG WEI, AND WEI YAO References Introduction and statement of the main result In this paper we consider the problem of finding positive solutions of the classical semilinear elliptic problem u + V (x)u = u p 1 u in, (1.1) where = N and p > 1. 2 x 2 j stands for the Laplace operator in, V (x) is a non-negative potential, Equation (1.1) arises in various branches of applied mathematics and physics (cf. [12] and references therein). For instance, in condensed matter physics one simulates the interaction effect among many particles to obtain a focusing nonlinear Schrödinger equation of the form i ψ t = 2 ψ + W (x)ψ ψ p 1 ψ in [0, ), (1.2) where i is the imaginary unit, the Planck constant and W (x) a given potential. Standing wave solutions of (1.2) are those of the form ψ(t, x) = e iλt/ u( 1 x) where u(x) is a real-valued function. Then (1.2) reduces to equation (1.1) for u, where V (x) = W ( x) λ. In what follows, we shall only consider positive, finite energy solutions of (1.1). Namely, we are concerned with the problem: { u + V (x)u u p = 0 in, u > 0 in, u H 1 ( (1.3) ). Associated to (1.3) is the energy functional E(u) = 1 } { u 2 + V (x)u 2 dx 1 u p+1 + dx, (1.4) 2 R p + 1 N where u + = max{u, 0}. In all what follows, we make the following structure assumptions on V and p: V is locally Hölder continuous, V L ( ) and V 0 = inf x V (x) > 0 (1.5) 1 < p < for N = 2 and 1 < p < N+2 for N 3. (1.6) N 2 Under these hypotheses, it is standard that classical solutions of (1.3) correspond precisely to non-trivial critical points of E in H 1 ( ). Let us denote the set of solutions of problem (1.3) by S V. A natural question is whether or not S V. When V is radially symmetric, the answer is yes (cf. Theorem 4.6 in [24]). But the answer is no when the potential is increasing along a direction (cf. Theorem 1.1 in [15]). If we further assume that lim x V (x) = V > 0, (1.7)

3 INFINITELY MANY POSITIVE SOLUTIONS 3 the existence of a positive solution of (1.3) has been widely investigated. For example, if we further suppose that inf V (x) < V, (1.8) x then one can show that (1.3) has a least energy (ground state) solution by using the concentration compactness principle (cf. [31, 32, 24, 39]). But if (1.8) does not hold, problem (1.3) may not have a least energy solution and solutions have to be seeked for at higher energy levels. Results in this direction are contained in [6, 7, 9], where a positive solution has been found by variational methods under a suitable decay condition on V at infinity. The structure of the solution set S V may be quite rich and interesting. Let us consider for instance the semi-classical limit case: { ε 2 u + W (x)u u p = 0 in, u > 0 in, u H 1 ( (1.9) ), where ε > 0 is a small parameter. Naturally, problem (1.9) is equivalent to problem (1.3) for V (x) = W (εx). It is known that as ε goes to zero, highly concentrated solutions near critical points of the potential W can be found, see [1, 10, 11] [17]-[20], [25, 27, 37, 42], or near higher dimensional stationary sets of other auxiliary potentials [3, 21, 33, 41]. The number of solutions of (1.9) may depend on the number or type of the critical points of Ṽ (x). It is rather difficult task to understand the structure of S V for an arbitrary potential V. For instance, a conspicuously unanswered question is whether or not S V for any potential V satisfying (1.7). Summing up, the above-mentioned work concern the existence of positive solutions, i.e., S V. There is less work on the multiplicity of positive solutions of (1.3), namely on estimating #(S V ). A seminal result in this direction was given in Coti-Zelati and Rabinowitz [16] where V (x) is spatially periodic. In that situation they prove the existence of infinitely many positive solutions, distinct up to periodic translations, via variational methods. Recently, by assuming that V = V ( x ) is radially symmetric, the second author and Yan [43] proved that problem (1.3) has infinitely many positive non-radial solutions if there are constants V > 0, a > 0, m > 1, and σ > 0, such that V (r) = V + a ( 1 ) r + O, as r +. (1.10) m r m+σ An alternative proof through min-max methods was given by Devillanova and Solimini [23]. The proof in [43] uses in essential way the radial symmetry of the potential V. On the other hand, it is conjectured there that the result should remain true when the symmetry requirement is lifted: Conjecture [43]. Problem (1.3) has infinitely many positive solutions if there are constants V > 0, a > 0, m > 1, and σ > 0, such that the potential V (x) satisfies V (x) = V + a ( x + O 1 ), as x +. (1.11) m x m+σ

4 4 MANUEL DEL PINO, JUNCHENG WEI, AND WEI YAO Results in this direction with non-symmetric potentials, as far as we know, there are only perturbative results (cf. [14, 4]). For instance, if V (x) tends to V from above with a suitable rate: V (x) V > 0, lim x ( ) V (x) e η x V = +, for some η (0, V ), (1.12) and V satisfies a global condition: sup V (x) V L N/2 (B 1 (x)) < V, (1.13) x where V is a sufficiently small positive constant (with no explicit expression), Cerami, Passaseo and Solimini [14] proved that problem (1.3) has infinitely many positive solutions by purely variational methods. (In [4], Ao and Wei gave a new proof of this result, using localized energy method. The new techniques also allow them to deal with more general nonlinearity.) The main purpose of this paper is to prove the above conjecture under some additional assumptions. In [43], the fact that V is radially symmetry allows to build a k-bump solution for an arbitrary k 1 with a k-dyadic symmetry, reducing the problem to just adjusting one parameter representing the location of a single bump along a given ray. A finite dimensional Lyapunov-Schmidt reduction method is used. When V is non-symmetric, we cannot constrain the bump configuration to any symmetry class. We are thus forced to deal with a large number of bumps and therefore with a huge number of parameters which need to be adjusted. This poses a tremendous difficulty in the construction comparatively to [43]. In Lyapunov-Schmidt reduction for problems like (1.3) the situation of adjusting a finite number of points (finite dimensional Lyapunov-Schmidt reduction method), and that of adjusting a higher dimensional object such a geodesic in a suitable metric as limiting concentration sets (infinite dimensional Lyapunov-Schmidt reduction method) have been treated. In this work we develop an intermediate Lyapunov-Schmidt reduction method, which consists of the finite dimensional procedure for large number of reduced equations, which in the limit become an ODE system of limiting Jacobi-type operators (see (8.9) below). Treating the discrete problem needs a method, technically delicate, which we interpret as an intermediate procedure between the finite and the infinite dimensional one (see for instance [22], [21], [38] and references therein for the latter). The main difference between the intermediate and infinite dimensional reduction, is that in the latter procedure only the variations in the normal direction are needed so the usual Jacobi operator for a curve appears. In the former procedure we also need to take into account variations in the tangential direction of points, which in the limit may be interpreted as a reparametrization of the curve. This seems to be a new procedure, with potentially many interesting applications. Our main result removes the symmetry assumption on V when N = 2. Theorem 1.1. Let N = 2. Suppose that V (x) satisfies (1.5) and (1.11) for some constants V > 0, a > 0, and min { 1, p 1 } m > 2, σ > 2. (1.14) 2

5 INFINITELY MANY POSITIVE SOLUTIONS 5 Then problem (1.3) has infinitely many non-radial positive solutions, whose energy can be made arbitrarily large. If (1.11) holds in the C 1 sense, then σ > 2 in (1.14) can be improved to be σ > 1. The condition on p can be further relaxed if we assume more regularity of the condition (1.11) or if p is an integer. The results in [14, 4] make us think that condition (1.11) could be improved. In fact we believe that the optimal condition should be (1.12). We stress that our result does not require a global perturbative assumption such as (1.13) on V. In addition, it is worth pointing out that the results on the existence of positive solutions in [6, 7, 9] do not include the polynomial decay case (1.11). Finally, we remark that for N 3, Theorem 1.1 holds if we assume the following additional symmetry assumption on V : after suitably rotating the coordinate system, V (x) = V (x, x ) = V (x, x ), (1.15) where x = (x, x ) R 2 2. An open question is whether or not the same result holds when N 3 with no extra assumption made. Throughout the paper, we shall use the following notation and conventions: For quantities A K and B K, we write A K B K to denote that there exists a positive constant C such that 1/C A K /B K C for K sufficiently large; A K = O(B K ) means that A K /B K are uniformly bounded as K tends to infinity; A K = o(b K ) denotes that A K /B K 0 as K. For simplicity, the letter C denotes various generic constant which is independent of K. It is allowed to vary from line to line, and also within the same formula. We will use the same y = y 2 for the Euclidean norm in various Euclidean spaces when no confusion can arise and we always denote the inner product of a and b in by a b. For the index j {1, 2,..., K}, we shall always use the convention that j 1 = K if j = 1 and j + 1 = 1 if j = K. The cardinality of a finite set E will be denoted by #E; The Lebesgue measure of a set E will be denoted by E. The transpose of a matrix A will be denoted by A T. For each function w(x) defined in, if w is radially symmetric, then there is a real function w(r) such that w(x) = w( x ). With slight abuse of notation, we will simply write w(r) instead of w(r). In the next section, we will describe the procedure of our construction and give the main ideas of each step. Acknowledgments: The authors are grateful to Frank Pacard for sharing his ideas and for useful discussions. Manuel del Pino is supported by Fondecyt grant and Fondo Basal CMM. J. Wei is supported by a NSERC from Canada. The research of W. Yao is supported by Fondecyt Grant

6 6 MANUEL DEL PINO, JUNCHENG WEI, AND WEI YAO 2. Description of the construction We will prove the slightly more general version of the Theorem, for N 3 where we assume even symmetry in N 2 the remaining variables in the sense that where x = (x, x ) R 2 2. We assume this henceforth. V (x) = V (x, x ) = V (x, x ), (2.1) We shall briefly describe the solutions to be constructed later and will give the main ideas in the procedure of the construction. In particular, we shall introduce the intermediate reduction method, which we believe can be very useful in other contexts. Firstly, without loss of generality, we can assume that V = 1 by suitable scaling. As developed in [43], we will use the loss of compactness to build up solutions. More precisely, we will construct solutions with large number of spikes whose inter-distances and distances from the origin are sufficiently large. By the asymptotic behaviour of V at infinity, the basic building block is the ground state (radial) solution w of the limit problem at infinity: { w + w w p = 0, w > 0 in, w = w( x ), w H 1 ( (2.2) ). The solutions we construct will be small perturbations of the sum of copies of w, centered at some carefully chosen points on R 2 {0}, where 0 is the zero vector in 2. Let K N + be the number of spikes, whose locations are given by Q j, j = 1,..., K. We define K w Qj (x) = w(x Q j ) and U(x) = w Qj (x), for x. (2.3) A natural and central question is how to choose Q j s such that a small perturbation of U will be a genuine solution. Assuming that inf Q j and inf Q j Q l, 1 j K j l by the asymptotic behaviour of V at infinity and the property of w, one can get (at least formally) the following energy expansion E(U) = KI 0 + a 0 K Q j m 1 2 γ 0 w( Q j Q l ) +other terms, (2.4) j l } {{ } J(Q 1,...,Q K ) where I 0, a 0 and γ 0 are positive constants. Here E(U) is the energy functional defined at (1.4) and we denote the leading order expansion as J(Q 1,..., Q K ). Observe that for any rotation R θ around the origin in, there holds J(R θ Q 1,..., R θ Q K ) = J(Q 1,..., Q K ). Hence any critical point of J(Q 1,..., Q K ) is degenerate. Therefore, except in the symmetric class, it is not easy to find critical points of small perturbations of J(Q 1,..., Q K ). This means

7 INFINITELY MANY POSITIVE SOLUTIONS 7 that it is not easy to apply the localized energy method directly. However, this observation gives us some enlightenment in the non-symmetric setting. Actually, when we restrict Q j s on a plane, this suggests us to introduce one more parameter to deal with the degeneracy due to rotations as we will see in Section 5. Under the condition (1.15), there is no essential difference between N = 2 and N 3. Hence from now on we will restrict Q j s on the plane R 2 {0}. To describe further the configuration space of Q j s, we define where Q 0 j = (R cos θ j, R sin θ j, 0) R 2 {0}, for j = 1,..., K, θ j = α + (j 1) 2π K R. Here α is the parameter representing the degeneracy due to rotations, and R is a positive constant to be determined later. Observe that each point Q 0 j depends on α. Thus we write Q 0 j = Q 0 j(α). When α = 0, the Q 0 j s are the points used in [43]. If V (x) is radially symmetric, it is obvious that the parameter α plays no role in the construction in [43]. But it is very important in our construction as we will see in Section 6. For the constant R, we introduce the so-called balancing condition: where a 0 = a 2 a 0 mr m 1 = 2 sin π K Ψ( 2R sin π K ), (2.5) w 2 dx > 0, and Ψ is the interaction function defined by Ψ(s) = w(x s e) div ( w p (x) e ) dx. (2.6) Here e can be any unit vector in (cf. [34, 35]). The balancing condition (2.5) can either be understood as a consequence of a conservation law or can be seen as a condition such that the approximation U is very close to a genuine solution (cf. Appendix in [35]). Assuming that d = 2R sin π K +, as K +, we will see that (cf. Lemma 3.3) Q 0 j = R m K ln K, and min 2π j l { Q0 j Q 0 l } = d m ln K. Next we define a small neighbourhood of Q 0 = (Q 0 1,..., Q 0 K ) on (R2 {0}) K in a suitable norm to be made precise and introduce another parameter. Let f j, g j R, j = 1,..., K, we define where Q j = Q 0 j + f j n j + g j t j = (R + f j ) n j + g j t j, (2.7) n j = (cos θ j, sin θ j, 0), and t j = ( sin θ j, cos θ j, 0). Keep in mind that f j and g j measure the displacement in the normal and tangential directions respectively.

8 8 MANUEL DEL PINO, JUNCHENG WEI, AND WEI YAO Writing Q j = Q j (α), n j = n j (α) and t j = t j (α), we note the following trivial but important fact: Q j (α + 2π) = Q j (α), α R, and j = 1,..., K. (2.8) We can now introduce another parameter q and define a suitable norm. Denote q = (f 1,, f K, g 1,, g K ) T R 2K, that is, q j = f j and q K+j = g j for j = 1,..., K. We define where for j = 1,..., K, q = ( f 1,, f K, ġ 1,, ġ K ) T, and q = ( f 1,, f K, g 1,, g K ) T, f j = (f j+1 f j ) K 2π, fj = (f j+1 2f j + f j 1 ) K2 4π, 2 ġ j = (g j+1 g j ) K 2π, g j = (g j+1 2g j + g j 1 ) K2 4π 2, f K+1 = f 1, f 0 = f K, g K+1 = g 1, g 0 = g K. Observe that if f j = f(θ j ) for some 2π periodic smooth function f, then difference of f and f j is the 2nd order central difference of f. With these notation, we can define the configuration space of Q j s by { Λ K = (Q 1,..., Q K ) (R 2 {0}) } K Qj is defined by (2.7) and q 1, f j is the forward where q = q + q + q is a norm on R 2K. In the following, we assume that Q j is defined by (2.7), the parameter α R and the parameter q satisfy q = q + q + q 1. (2.9) For any (Q 1,..., Q K ) Λ K, an easy computation shows that for j = 1,..., K, and Define ρ = min j l { Q j Q l }, it follows that Q j = R + f j + O(R 1 ), Q j+1 Q j = d + 2(f j + ġ j ) π K + O(K 2 ). ρ = d + O(K 1 ), and We will prove Theorem 1.1 by showing the following result. min { Q j } = R + O(1). (2.10),...,K Theorem 2.1. Under the assumption of Theorem 1.1, there is a positive integer K 0 such that: for all integer K K 0, there exist α [0, 2π) and (Q 1,..., Q K ) Λ K such that problem (1.3) has two solutions of the form K u(x) = w(x Q j ) + φ(x), (2.11)

9 INFINITELY MANY POSITIVE SOLUTIONS 9 where φ H 1 ( ) + φ L ( ) 0 as K +. Moreover, the energy of u is given by ( 1 E(u) = K 2 1 ) w p+1 dx + o(1). (2.12) p + 1 Remark 1. It is worth pointing out that the solutions constructed in this paper are different from those found in [14, 4]. The reason is simply that the inter-distances and distances from the origin of the spikes of the solutions given in (2.11) tend to infinity uniformly as K goes to infinity, but those of the solutions found in [14, 4] do not. Remark 2. The fact that we can find at least two solutions of the form (2.11) is nontrivial. This is due to the fact that we need to choose the first starting point Q 0 1. It turns out that there are at least two such points to choose (see Section 6). Remark 3. As K +, (f j, g j ) is the discretization of two second order ordinary differential equations (8.9). To prove Theorem 2.1, it is sufficient to show that for K sufficiently large there are parameters α and q such that U + φ is a genuine solution for a small perturbation φ. To achieve this goal, we will adopt the techniques in the singularly perturbed problem. Unlike problem (1.9), there is no apparent parameter in (1.3). As stated in Theorem 2.1, we use the number of the spikes as the ε type parameter. This idea comes directly from [43] and goes back at least as far as to [30]. Before we sketch the procedure of our proof, we briefly introduce the abstract set-up of the Lyapunov-Schmidt reduction (although it is always used in a framework that occurs often in bifurcation theory). Let X, Y be Banach spaces and S(u) is a C 1 map from X to Y. To study the equation S(u) = 0, a natural way is to find approximations first and then to look for genuine solutions as (small) perturbations of approximations. Assume that U λ are the approximations, where λ Λ is the parameter (we think of Λ as the configuration space). Writing u = U λ + φ, then solving S(u) = 0 amounts to solve L[φ] + E + N(φ) = 0, (2.13) where L[φ] = S (U λ )[φ], E = S(U λ ), and N(φ) = S(U λ + φ) S(U λ ) S (U λ )[φ]. Here S (U λ ) is the Fréchet derivative of S at U λ, E denotes the error of approximation, and N(φ) denotes the nonlinear term. In order to solve (2.13), we try to invert the linear operator L so that we can rephrase the problem as a fixed point problem. That is, when L has a uniformly bounded inverse in a suitable space, one can rewrite the equation (2.13) as φ = L 1 [E + N(φ)] = A(φ). What is left is to use fixed point theorems such as contraction mapping theorem. The Lyapunov-Schmidt reduction deals with the situation when the linear operator L is Fredholm and its eigenfunction space associated to small eigenvalues has finite dimensional. Assuming that {Z 1,..., Z n } is a basis of the eigenfunction space associated to small eigenvalues of L, we can divide the procedure of solving (2.13) into two steps:

10 10 MANUEL DEL PINO, JUNCHENG WEI, AND WEI YAO (i) solving the projected problem for any λ Λ, L[φ] + E + N(φ) = n c j Z j, φ, Z j = 0, j = 1,..., n, where c j may be constant or function depending on the form of φ, Z j. (ii) solving the reduced problem c j (λ) = 0, j = 1,..., n, by adjusting λ. Let us now turn to our problem (1.3). In this case, S(u) = u + V (x)u u p +, L[φ] = φ + V (x)φ pu p 1 φ, E = U + V (x)u U p, N(φ) = (U + φ) p + + U p + pu p 1 φ. Observe that all of these quantities depend implicitly on α and q even though this is not apparent in the notation. By the Lyapunov-Schmidt reduction, the procedure of construction is made up of several steps which we explain next and postpone the proofs of major facts in later sections. Step 1: Solving the projected problem. Let α R and q satisfy (2.9). We look for a function φ and some multiplier β R 2K such that { L[φ] + E + N(φ) = β U, q (2.14) φ Z Qj dx = 0, j = 1,..., K, where the vector field Z Qj is defined by Z Qj (x) = w(x Q j ). (2.15) By direct computation, we have U q = ( Z Q1 n 1,, Z QK n K, Z Q1 t 1,, Z QK t K ) T. This is the first step in the Lyapunov-Schmidt reduction. It is done in Section 4 through some a priori estimates and contraction mapping theorem. A required element in this step is the non-degeneracy of w (cf. Lemma 3.1). It is worth pointing out that the function φ and the multiplier β found in Step 1 depend on the parameters α and q. Hence we write φ = φ(x; α, q) and β = β(α, q). Step 2: Solving the reduced problem By Step 1, it is known that β is small. But it is not easy to solve β(α, q) = 0 directly since the linear part of the expansion of β in q is degenerate (due to the invariance of J(Q 1,..., Q K ) under rotations).

11 More precisely, let us write INFINITELY MANY POSITIVE SOLUTIONS 11 β(α, q) = T q + Φ(α, q), where T q is the linear part and Φ(α, q) denotes the remaining term. As we will see in Section 5, T q does not depend on α and there is a unique vector (up to a scalar) q 0 = (0,..., 0, 1,..., 1) }{{}}{{} T R 2K K K such that T q 0 = 0. By the Lyapunov-Schmidt reduction again (called the secondary Lyapunov-Schmidt reduction), the step of solving the reduced problem β(α, q) = 0 can be divided into two steps. To write the projected problem of β = 0 in a proper form, note that K U α = R U + g j K ( f j U g j g j U f j where q = ( g, f) for q = ( f, g). Hence we define ) = (Rq 0 + q ) U q, β = β γ(rq 0 + q ), for every γ R. (2.16) Obviously the new multiplier β depends on the parameters α, q and γ. Thus we write β = β(α, q, γ). Step 2.A: Solving β(α, q, γ) = 0 by adjusting γ and q. In this step, for each α R, we are going to find parameters (γ, q) such that β(α, q, γ) = 0, and q q 0. (2.17) It can be seen as the step of solving the projected problem in the secondary Lyapunov-Schmidt reduction. To achieve it, we will use the condition (1.14). This step is done in Section 5 by using various integral estimates and contraction mapping theorem. A key element in this step is the invertibility of an 2K 2K matrix whose proof is given in Appendix A. When Step 2.A is done, we denote the unique solution of (2.17) by (γ(α), q(α)). Then the original problem (1.3) is reduced to the problem γ(α) = 0 of one dimension. Step 2.B: Solving γ(α) = 0 by choosing α. At the last step, we want to prove that there exists an α such that γ(α) = 0. As a result, the function u = U + φ is a genuine solution of (1.3). This step is the second step of solving the reduced problem in the secondary Lyapunov- Schmidt reduction. To achieve this step, by Step 2.A, the function φ = φ(x; α, q(α)) found in Step 1 solves the following problem: { L[φ] + E + N(φ) = γ(α) U, α (2.18) φ Z Qj dx = 0, j = 1,..., K, where all of the quantities depending implicitly on (α, q) are taken values at (α, q(α)). To solve γ(α) = 0, we first apply the so-called variational reduction (often used in the localized

12 12 MANUEL DEL PINO, JUNCHENG WEI, AND WEI YAO energy method) to show that equation γ(α) = 0 has a solution if the reduced energy function F (α) = E(U + φ) has a critical point. Secondly, by using (2.8), it is easy to check that F (α) is 2π periodic in α. Hence it has at least two critical points. More details of this step will be given in Section 6. Finally, this paper is organized as follows. Some preliminary facts and estimates are explained in Section 3. In Section 4 we apply the standard Lyapunov-Schmidt reduction for Step 1. Section 5 contains a further reduction process for Step 2.A which reduces the original problem to one dimension. In Section 6 we carry out Step 2.B and then complete the proof of Theorem 2.1. At the last, we discuss some possible extensions in Section Preliminaries In this section we present some preliminary facts and some useful estimates. First we recall some basic and useful properties of the standard spike solution w defined by (2.2) and those of the interaction function Ψ defined in (2.6). Lemma 3.1. If 1 < p < for N = 2 and 1 < p < N+2 for N 3, then every positive solution N 2 of the problem: { u + u u p = 0 in, u > 0 in, u H 1 ( (3.1) ), has the form w( Q) for some Q, where w(x) = w( x ) C ( ) is the unique positive radial solution which satisfies lim r N 1 2 e r w (r) w(r) = c N,p, lim r r w(r) = 1. (3.2) Here c N,p is a positive constant depending only on N and p. Furthermore, the Morse index of w is one and w is nondegenerate in the sense that Ker ( + 1 pw p 1) L ( ) = Span { x1 w,, xn w}. Proof. This result is well known. For the proof we refer the reader to [12] for the existence, [26] for the symmetry, [29] for the uniqueness, Appendix C in [36] for the nondegeneracy, and [8] for the Morse index. Lemma 3.2. For s sufficiently large, where c N,p > 0 is a constant depending only on N and p. Ψ(s) = c N,p s N 1 2 e s ( 1 + O(s 1 ) ). (3.3) Proof. This lemma follows from Taylor s theorem and the Lebesgue dominated convergence theorem. We omit it here and refer to [27, 34] for details. Next we study the balancing condition (2.5). Assuming that d = 2R sin π +, as K +, K

13 INFINITELY MANY POSITIVE SOLUTIONS 13 by the expansion (3.3), both positive numbers R and d are uniquely determined by K. Moreover, we have the following expansions. Lemma 3.3. For K sufficiently large, ( d = m ln K + m N 3 ) ln(m ln K) + O(1), (3.4) 2 R = m 2π K ln K + 1 ( m N 3 ) K ln(m ln K) + O(K). 2π 2 Proof. From the balancing conditon(2.5), the number d satisfies the equation By using (3.3), for K sufficiently large, equation (3.5) becomes d m+1 Ψ(d) = a 0 m ( 2 sin π K ) m. (3.5) from which we obtain N 1 m+1 c N,p d 2 e d (1 + O(d 1 )) = a 0 m(2π) m K m( 1 + O(K 2 ) ), N 1 m+1 d 2 e d K m. Let d = m ln K + d 1 with d 1 = o(ln K), we have It follows that N 1 m+1 c N,p (m ln K) 2 e ( d o(1) ) = a 0 m(2π) m (1 + O(K 2 )). e d N 1 1 m+1 (m ln K) 2. Therefore, ( d 1 = m N 3 ) ln(m ln K) + O(1), 2 from which we get the expansions of d and R. In the next section, we will apply the Lyapunov-Schmidt reduction. After refinements by many authors working on the subject or on closely related problems, this type of argument is rather standard now. However technical difficulties arise when the number of spikes goes to infinity or the number of spikes is infinity (cf. [34]). To deal with these difficulties, the following lemmas are useful. Lemma 3.4. There exists a constant C N depending only on N such that for any K N + and any Q = (Q 1,..., Q K ) ( ) K, { } # Q j lρ/2 Qj x < (l + 1)ρ/2 C N (l + 1) N 1, (3.6) for all x and all l N, where ρ = min j l { Q j Q l }. In particular, if Q = (Q 1,..., Q K ) (R 2 {0}) K, then for all x and all l N, { } # Q j lρ/2 Q j x < (l + 1)ρ/2 6(l + 1). (3.7)

14 14 MANUEL DEL PINO, JUNCHENG WEI, AND WEI YAO Proof. When ρ = 0, the result is trivial. It remains to consider the case ρ > 0. For l = 0, it suffices to take C N 1. For l 1, let Q jk s (k = 1,..., n) be the points satisfying By the triangle inequality, we have Hence for all k = 1,..., n, lρ/2 Q jk x < (l + 1)ρ/2. (l 1)ρ/2 y x < (l + 2)ρ/2, y B ρ/2 (Q jk ). B ρ/2 (Q jk ) B (l+2)ρ/2 (x) \ B (l 1)ρ/2 (x). Since B ρ/2 (Q jl ) B ρ/2 (Q jk ) = for l k, we conclude that n Bρ/2 (Q jk ) B(l+2)ρ/2 (x) \ B (l 1)ρ/2 (x). k=1 (l+2) Therefore, taking C N = sup N (l 1) N l N+, we have (l+1) N 1 n (l + 2) N (l 1) N C N (l + 1) N 1, which implies (3.6). If Q = (Q 1,..., Q K ) (R 2 {0}) K, the above argument implies that n Bρ/2 (Q jk ) R 2 {0} B(l+2)ρ/2 (x) \ B (l 1)ρ/2 (x) R 2 {0}, k=1 which implies that n (l + 2) 2 (l 1) 2 6(l + 1). Therefore, we get the estimate (3.7) if we restrict Q = (Q 1,..., Q K ) on (R 2 {0}) K. Given Q = (Q 1,..., Q K ) ( ) K with ρ = min j l { Q j Q l } > 0, for any l N, we divide into K + 1 parts: { Ω l j = x R } N x Q j = min x Q l lρ/2, j = 1,..., K, 1 l K and Ω l K+1 = RN \ K Ω l j. Then the interior of Ω l j Ω l l is an empty set for j l. Lemma 3.5. Suppose that Γ (r) is a positive decreasing function defined on [0, ) such that for some b R and η > 0, Γ (r) r b e ηr as r. (3.8) Then there exist positive constants ρ 0 and C (independent of K) such that (i) for all K, l N +, all (Q 1,..., Q K ) ( ) K with ρ ρ 0, and all x Ω l j 0 (j 0 = 1,..., K), we have K Γ ( x Q j ) Cl N 1 Γ ( x Q j0 ). (3.9)

15 INFINITELY MANY POSITIVE SOLUTIONS 15 In particular, if (Q 1,..., Q K ) (R 2 {0}) K, then K Γ ( x Q j ) ClΓ ( x Q j0 ). (ii) for all (Q 1,..., Q K ) ( ) K with ρ ρ 0 and all j 0 {1,..., K}, j j 0 Γ ( Q j0 Q j ) CΓ (ρ). (3.10) Remark 4. A similar result holds when Γ (r) has polynomial decay. For example, if for some integer n N +, Γ (r) r b as r +, where b < n, then there are positive constants ρ 0 and C (independent of K) such that for all K N +, all (Q 1,..., Q K ) (R n {0}) K with ρ ρ 0, and all j 0 {1,..., K}, j j 0 Γ ( Q j Q j0 ) CΓ (ρ). This kind of property is useful and important in the construction of infinitely many solutions of problem with critical growth. Proof. Given x Ω l j 0, by definition we have x Q j0 lρ/2 and x Q j0 x Q j, j = 1,..., K. Thus there is an integer 0 l 0 l such that l 0 ρ/2 x Q j0 < (l 0 + 1)ρ/2. By the property of Γ (r) and Lemma 3.4, for ρ sufficiently large, we have K Γ ( x Q j ) C N (l 0 + 1) N 1 Γ ( x Q j0 ) + C N + s=l 0 +1 (s + 1) N 1 Γ (sρ/2) C N (l 0 + 1) N 1 Γ ( x Q j0 ) + C(l 0 + 2) N 1 Γ ( (l 0 + 1) ρ/2 ) Cl N 1 Γ ( x Q j0 ), where in the second inequality we use the following inequality: + s=l 0 +1 (s + 1) N 1 Γ (sρ/2) (l 0 + 2) N 1 Γ ( (l 0 + 1)ρ/2 ) C. To prove it, for ρ sufficiently large, by (3.8), we have Γ (sρ/2) ( s ) be Γ ( (l 0 + 1)ρ/2 ) C η(s l 0 1)ρ/2. l 0 + 1

16 16 MANUEL DEL PINO, JUNCHENG WEI, AND WEI YAO Hence + s=l 0 +1 (s + 1) N 1 Γ (sρ/2) (l 0 + 2) N 1 Γ ( (l 0 + 1)ρ/2 ) C C C + ( s ) N 1+be η(s l 0 1)ρ/2 l s=l e η(s l 0 1)ρ/4 s=l e ηtρ/4 dt C. In particular, if (Q 1,..., Q K ) (R 2 {0}) K, then by (3.7), we can take N = 2 in the above arguments. To deduce (3.10) from (3.9), denote Q = (Q 1,..., Q j0 1, Q j0 +1,..., Q K ) ( ) K 1, and ρ = min { Q j Q l } j j0, l j 0 ρ. j l Take j 1 {1,..., K} such that and choose l N + satisfying Then by (3.9), we have ( Γ ( Q j0 Q j ) C j j 0 Q j0 Q j1 = min l j 0 { Qj0 Q l }, (l 1) ρ/2 < Q j0 Q j1 l ρ/ Q j 0 Q j1 ρ ) N 1Γ ( Qj0 Q j1 ) CΓ ( Q j0 Q j1 ) + CΓ ( ρ) CΓ (ρ), where in the second inequality we use the following inequality: Q j0 Q j1 N 1 Γ ( Q j0 Q j1 ) C ρ N 1 Γ ( ρ ), for Q j0 Q j1 ρ ρ 0. To prove it, we only need to apply (3.8). A simple corollary is the following result which is useful in our construction. Corollary 3.6. There are positive constants ρ 0 and C (independent of K) such that for all K, l N +, all (Q 1,..., Q K ) (R 2 {0}) K with ρ ρ 0, and all x Ω l j 0 (j 0 = 1,..., K), we have K w Qj (x) Clw Qj0 (x), (3.11) and j j 0 e η Qj Qj0 Ce ηρ. (3.12)

17 INFINITELY MANY POSITIVE SOLUTIONS 17 To analyze the interactions between spikes, we prove some estimates concerning convolution of functions with suitable exponential decays. Lemma 3.7. Given Γ 1, Γ 2 two positive continuous radial functions on with the following property: Γ 1 (r) r b 1 e η 1r, and Γ 2 (r) r b 2 e η 2r, as r, where b 1, b 2 R, η 1 > 0, η 2 > 0. Let ξ tends to infinity. Then, the following asymptotic estimates hold: (i) If η 1 < η 2, then Γ 1 (x ξ)γ 2 (x) dx ξ b 1 e η1 ξ. Clearly, if η 1 > η 2, a similar expression holds, by replacing b 1 and η 1 with b 2 and η 2. (ii) If η 1 = η 2, suppose that b 1 b 2 for simplicity. Then Γ 1 (x ξ)γ 2 (x) dx ξ b 1+b 2 + N+1 2 e η 1 ξ, if b 2 > N+1 2, ( ξ b 1 ln ξ ) e η 1 ξ, if b 2 = N+1 2, ξ b 1 e η 1 ξ, if b 2 < N+1 2. Proof. This result follows from the Lebesgue dominated convergence theorem. The argument is standard and is omitted here, we refer the reader to Lemma 3.7 in [2] for details. By the property of w, as a corollary of Lemma 3.7, we have the following integral estimates. Lemma 3.8. Suppose that Q j Q k is sufficiently large, then the following estimates hold: (i) for every p > 1, w Qj w p Q k dx = (γ 0 + o(1))w( Q j Q k ), where γ 0 = w p (x)e x 1 dx > 0 is a constant; (ii) w Qj w Qk dx = O ( e Q k Q j Q k Q j (N 3)/2) ; (iii) let Ω k = { x R N x Q k = min x Q j }, then 1 j K w p Q j w Qk dx = O ( e p+1 ) 2 Q j Q k Q j Q k N 3 2, Ω k and wq 2 j w p 1 Q k dx = O ( p+1 ) min{2, e 2 } Q j Q k Q j Q k N 3 2. Ω k Proof. Since the argument of proof is somewhat standard, we give only the main ideas of the proof. (i) It follows from Lemma 3.1 and Lebegue s dominated convergence theorem (see e.g. the arguments used in the proof of Lemma 2.5, in [30]). (ii) By Lemma 3.1 and a simple computation, we get the estimate from Lemma 3.7.

18 18 MANUEL DEL PINO, JUNCHENG WEI, AND WEI YAO (iii) By the definition, for all x Ω k, w Qj (x) w Qk (x) for every 1 j K. Lemma 3.1 and Lemma 3.7, we have w p Q j w Qk dx w p+1 2 Q j w p+1 2 Q k dx Ce p+1 2 Q j Q k Q j Q k N 3 2, Ω k Ω k and wq 2 j w p 1 Q k Ω k dx C w min{2, p+1 2 } p+1 min{2, 2 Q j w } Q k Ω k dx p+1 min{2, Ce 2 } Q j Q k Q j Q k N 3 2. Hence by Using above integral estimates, we can get the expansion of the energy of approximate solution. Lemma 3.9. For K sufficiently large, for any α R and q satisfies (2.9) we have E(U) = KI 0 + ( a 0 + o(1) ) K Q j m 1 ( γ0 + o(1) ) w( Q i Q j ) 2 + O(KR 2m ) + O ( p+1 min{2, Ke i j 2 }d d N 3 2 where γ 0 = w p (x)e x 1 dx is a positive constant given in Lemma 3.8, I 0 = ( p + 1) w p+1 dx, and a 0 = a w 2 dx. R 2 N Proof. The proof is delayed to Appendix B. 4. The Lyapunov-Schmidt reduction The aim of this section is to achieve Step 1 in the procedure of our construction described in Section 2. Before stating the main result, we first introduce some notation. Let η (0, 1) be a constant chosen later, we define the weighted norm: h = sup ( K e ) η x Q j 1, h(x) (4.1) x where Q j is defined in (2.7). In what follows, we assume that (Q 1,..., Q K ) Λ K, i.e., the parameter q satisfies (2.9). We first claim that h L ( ) C h and h L q ( ) CK h for 1 q <. (4.2) Indeed, the second inequality in (4.2) follows directly from h(x) h K ), e η x Q j, x.

19 INFINITELY MANY POSITIVE SOLUTIONS 19 To prove the first inequality in (4.2), it suffices to show that K e η x Qj C. Indeed, for any x, we can choose l N + and j 0 {1,..., K} such that x Ω l j 0 \ Ω l 1 j 0. Hence by Lemma 3.5, K 0 < e η x Qj Cl N 1 e η(l 1)ρ/2 C. (4.3) Denote B = { h L ( ) h < }. Then B is a Banach space with the norm h. To show the completeness, suppose that {h n } is a Cauchy sequence in B. By (4.2), {h n } is also a Cauchy sequence in L ( ). Hence h n converges to a function h in L ( ). By the definition of Cauchy sequence, for any ε > 0, there is n 0 N such that ( K 1 h n (x) h k (x) e j ) η x Q hn h k < ε, x, if n, k n 0. Letting k, we get ( K 1 h n (x) h (x) e j ) η x Q < ε, x, if n n 0, which implies that h n h 0 as n. Now we can state our main result in this section. Proposition 4.1. Suppose that V (x) satisfies (1.11) for constants V > 0, a R, m > 0 and σ > 0. If N 3, we further assume (1.15). Then there is a positive integer K 0 such that: for all K K 0, every α R, and q satisfies (2.9), there exists a unique function φ W 2,2 ( ) B K and a unique multiplier β R 2K such that { L[φ] + E + N(φ) = β U, q (4.4) φ Z Qj dx = 0, j = 1,..., K, where B K = } {φ L ( p η min{1, ) : φ C 0 K 2 }m (ln K) 1 2. Here C 0 is a positive constant independent of K. Moreover, (α, q) φ(x; α, q) is of class C 1, and R 1 φ α + φ q C ( K min{1, p η ) 2 }m (ln K) 1 2 min{p 1,1}. The proof of Proposition 4.1 is somewhat standard and can be divided into two steps: (i) study the invertibility of the linear operator; (ii) apply fixed point theorems.

20 20 MANUEL DEL PINO, JUNCHENG WEI, AND WEI YAO 4.1. Linear analysis. Let M denotes an 2K 2K matrix defined by U U M jk = dx, j, k = 1,..., 2K. (4.5) q j q k Lemma 4.2. For K sufficiently large, given any vector b R 2K, there exists a unique vector β R 2K such that M β = b. Moreover, for some constant C independent of K. β C b, (4.6) Proof. To prove the existence, it is sufficient to prove the a priori estimate (4.6). Suppose that β j = β, by the definition, we have K M jk βk = b j. (4.7) k=1 For the entries M jk, by Lemma 3.3 and Lemma 3.8, we get M jk Ce d d N 3 2 CK m (m ln K) m, k j, (4.8) and ( w ) 2 M jj = dx = c0 > 0, j = 1,..., 2K. (4.9) R x N 1 Hence by (4.7)-(4.9), for K sufficiently large, we have c 0 β c 0 β j k j from which the desired result follows. M jk β k + b j c 0 2 β + b, We can now formulate our main result in this subsection. Lemma 4.3. Under the assumption of Proposition 4.1, there is a positive integer K 0 such that: for all K K 0, every α R, and q satisfies (2.9), and for all h B, there exists a unique function φ W 2,2 ( ) B and a unique multiplier β R 2K such that { L[φ] = h + β U, q (4.10) φ Z Qj dx = 0, j = 1,..., K. Moreover, we have for some positive constant C independent of K. φ + β C h, (4.11) Proof. To solve (4.10), we first consider weak solutions. Define H = {u H 1 ( ) ( ) } u, ( + 1) 1 Z Qj = 0, j = 1,..., K.

21 INFINITELY MANY POSITIVE SOLUTIONS 21 Then H is a Hilbert space with the standard inner product: (u, v) = ( u v + uv) dx. Since the vector function Z Qj decays exponentially at infinity, by integration by parts, it is not hard to show that for φ H 1 ( ), φ H is equivalent to φ Z Qj dx = 0, j = 1,..., K. As usual, φ H is a weak solution of (4.10) if and only if it satisfies the following equation: { } φ ϕ + V (x)φϕ pu p 1 φϕ dx = hϕ dx, ϕ H. By the Riesz representation theorem, the last equation can be written as φ + K[φ] = ĥ, where ĥ is defined by duality and K is a linear compact operator due to the exponential decay of U and V (x) 1 C x m for x large. Using the Fredholm alternative, showing that equation (4.10) has a unique weak solution is equivalent to showing that it has a unique solution for h = 0. Moreover, by (4.2), h L q ( ) for all 1 < q <. By the standard elliptic regularity results, φ W 2,q ( ). Hence φ is a strong solution and φ L ( ) by the Sobolev imbedding theorem. Therefore, to prove Lemma 4.3, it is sufficient to prove the a priori estimate (4.11). To prove (4.11), we first multiply equation (4.10) by U and integrate over q RN to obtain M β = L[φ] U R q dx h U dx, (4.12) N R q N where M is an 2K 2K matrix defined in (4.5). By the integration by parts, L[φ] Z Qk dx = φ L[Z Qk ] dx. Observe that L[Z Qk ] = (V (x) 1) w Qk p ( ) U p 1 w p 1 Q wqk k. We claim that L[φ] Z Qk dx Cde min{1, p 2 }d φ. (4.13) Indeed, on one hand, by the assumption (1.11) and φ H, we have (V (x) 1) w Qk φ dx C(R m 1 ln K + R m σ ) φ. On the other hand, by mean value theorem and (3.11), for x Q k < 2m ln K, we have U p 1 w p 1 Q k Cw p 2 Q k w Qj. j k

22 22 MANUEL DEL PINO, JUNCHENG WEI, AND WEI YAO Thus by Lemma 3.8, p(u p 1 w p 1 Q k ) w Qk φ dx Cde min{1, p 2 }d φ. Combining the above estimates we get (4.13). Since w decays exponentially at infinity, we have hz Qk dx C h. (4.14) Combining the above estimates (4.13) and (4.14), by Lemma 4.2, we get ) β C (de min{1, p 2 }d φ + h. (4.15) Now we prove the a priori estimate (4.11). First we show that φ <. To prove it, by the maximum principle, we prove that there exist constants τ and C (all independent of K) such that for all x \ K B(Q j, τ), φ(x) C ( ) K L[φ] + sup φ L (B(Q j,τ)) e η x Qj. (4.16) 1 j K To prove the above pointwise estimate, we first show the independence of τ on K, for x \ K B(Q j, τ), by Lemma 3.4, we have U(x) Q j x <ρ/2 w(τ) + C w(x Q j ) + l=1 lρ/2 Q j x <(l+1)ρ/2 l N 1 e lρ/2 Cw(τ). l=1 Thus we can take τ sufficiently large but independent of K such that w(x Q j ) pu p 1 (x) (V 0 η 2 )/4, x \ K B(Q j, τ). (4.17) Now we claim that for τ sufficiently large (independent of K), in \ K B(Q j, τ), L[W ] c 0 W, and L[W + ] c 0 W + where W ± (x) = K e±η x Q j and c 0 > 0 is a constant independent of K. Indeed, for x \ K B(Q j, τ), L[W ± ] = K { V (x) η 2 N 1 } x Q j η pu p 1 e ±η x Qj V 0 η 2 W ±, 2 by the assumption (V 1) and inequality (4.17).

23 INFINITELY MANY POSITIVE SOLUTIONS 23 The remaining part in the proof of (4.16) is to apply the maximum principle for the linear operator L in \ K B(Q j, τ) to obtain φ(x) C ( ) K L[φ] + sup φ L (B(Q j,τ)) e η x Qj + δ 1 j K K e η x Q j for any δ > 0, where C is a constant independent of K and δ. Letting δ 0, we get the desired estimate (4.16). Hence ( ) φ C L[φ] + sup 1 j K φ L (B(Q j,τ)) <. (4.18) Now we can prove the a priori estimate (4.11). Arguing by contradiction, assume that there is a sequence of (φ (K), h (K) ) satisfying (4.10) such that φ (K) = 1, and h (K) = o(1), as K. (For simplicity, in the following we will drop (K) in the superscript) As a consequence of (4.15), β U ( ) q (x) C K de min{1, p 2 }d φ + h e η x Qj. Since φ C φ and h = o(1), we get L[φ] = o(1). Hence (4.18) implies that there exists a subsequence of Q j such that φ L (B(Q j,τ)) C > 0 (4.19) for some fixed constant C (independent of K). Since φ 1, by elliptic regularity estimates, we get φ C 1 ( ) C. Applying Ascoli-Arzela s theorem, one can find a subsequence of Q j such that φ(x + Q j ) converge (on compact sets) to φ. It is not hard to show that φ is a bounded (weak and then strong) solution (actually bound by e η x ) of φ + φ pw p 1 φ = 0. Furthermore, since φ satisfies the orthogonality condition φ Z Qj dx = 0, the limit function φ satisfies φ w = 0. By the non-degeneracy of w, one has φ 0, which is in contradiction with (4.19). This completes the proof of Lemma 4.3. Remark 5. If V (x) is a bounded measurable function such that there is no nontrivial solution of φ + V (x)φ = 0, φ(x) Ce η x in, (4.20) our arguments still work by adding 0 to the points Q j s. Remark 6. Since the Morse index of w is finite, using a similar argument in the proof of Lemma 4.3 (cf. [4]), one can show that φ H 1 ( ) C h L 2 ( ) (4.21)

24 24 MANUEL DEL PINO, JUNCHENG WEI, AND WEI YAO for some positive constant C independent of K. Indeed, since τ is independent of K, one can first prove that } C φ 2 H 1 ( ) { φ 2 + V (x)φ 2 pu p 1 φ 2 dx Nonlinear analysis. Summarizing, for any h B, by Lemma 4.3, there is a unique function φ H W 2,2 ( ) B satisfying (4.10). Hence we can define a linear operator from B to H W 2,2 ( ) B and denote it by L 1. Then the equation (4.4) is equivalent to φ = L 1[ E + N(φ) ]. Before we give the complete proof of Proposition 4.1, we first show the estimate of the error. Lemma 4.4. Given (Q 1,..., Q K ) Λ K, then for any fixed 0 < η < 1 and K sufficiently large, there is a constant C (independent of K) such that Proof. By the definition, we have E = p η min{1, E CK 2 }m (ln K) 1 2. (4.22) K {( K ) p K } (V (x) 1) w Qj w Qj w p Q j. } {{ } E 1 } {{ } E 2 Claim 1: There exists a constant C (independent of K) such that E 1 CR m CK m (ln K) m. (4.23) Claim 2: There exists a constant C (independent of K) such that E 2 Cd N 1 p η min{1, 2 e 2 }d p η min{1, CK 2 }m N 1 min{ (ln K) 2,m+1}. (4.24) If both Claim 1 and Claim 2 are true, the desired estimate (4.22) follows. Proof of Claim 1: Note that for x < R/3, by the triangle inequality, we have x Q j Q j x R/2. Hence for all x < R/3, by V L ( ) and Lemma 3.1, we get E 1 (x) C K w(x Q j ) Ce (1 η)r/2 K e η x Q j CK m 3 K e η x Q j. For x R/3, by the assumption (1.11), we have V (x) 1 CR m. Hence for all x R/3, K E 1 (x) CR m w(x Q j ) CR m Combining these estimates, Claim 1 follows. K e η x Q j.

25 INFINITELY MANY POSITIVE SOLUTIONS 25 Proof of Claim 2: For x Ω l K+1, where l N + is chosen later, we have Since ρ > m 2 Thus for all x Ω l K+1, K E 2 (x) K p 1 w p Q j (x) + CK p 1 K ln K, by choosing l > 4(p+m+2) m(p 1) K w p Q j (x) e p x Q j CK p 1 e (p η)lρ/2 K (independent of K), we have K p 1 e (p η)lρ/2 K p 1 K (p η)lm/4 CK m 3. K E 2 (x) CK m 3 e η x Qj. e η x Q j. By the definition of Ω l j, j = 1,..., K, x Q j x Q k for all x Ω l j and 1 k K. Thus x Q k ρ 2 for k j. Hence by mean value theorem and (3.11), for all x Ωl j, we have ( K ) p E 2 (x) w Qk w p ( K ) p 1 Q j + w p Q i p w Qk k=1 i j w Qk + k=1 k j k j Cl (N 1)(p 1) w p 1 Q j w Qk. Since l is independent of K, for all x Ω l j, by theorem 3.1 and (3.12) we have E 2 (x) Ce (p 1) x Q j k j ρ N 1 2 e x Q k k j Cρ N 1 2 e η x Q p η j min{1, e 2 } Q j Q k k j w p Q k Combining these estimates, Claim 2 follows. Cd N 1 p η min{1, 2 e 2 }d e η x Qj. Now we are in the position to give the proof of Proposition 4.1. Proof of Proposition 4.1. Let C 0 be a positive number to be determined later, we define } B K = {φ L ( p η min{1, ) : φ C 0 K 2 }m (ln K) 1 2. Then B K is a non-empty closed set in B. Now we define a map A : B K H W 2,2 ( ) B by A(φ) = L 1[ E + N(φ) ]. Now solving equation (4.4) is equivalent to finding a fixed point for the map A.

26 26 MANUEL DEL PINO, JUNCHENG WEI, AND WEI YAO Since φ is uniformly bounded for φ B K, by the mean value theorem, there is a positive constant C such that for all φ B K, and for all φ 1, φ 2 B K, Thus by (4.3), one has and we have that N(φ) C φ min{p,2}, N(φ 1 ) N(φ 2 ) C( φ 1 min{p 1,1} + φ 2 min{p 1,1} ) φ 1 φ 2. N(φ) C φ min{p,2}, N(φ 1 ) N(φ 2 ) C( φ 1 min{p 1,1} + φ 2 min{p 1,1} ) φ 1 φ 2. Hence by Lemma 4.3 and Lemma 4.4, for K sufficiently large and C 0 large we have and p η min{1, A(φ) C( E + N(φ) ) C 0 K 2 }m (ln K) 1 2, A(φ 1 ) A(φ 2 ) C N(φ 1 ) N(φ 2 ) 1 2 φ 1 φ 2, which shows that A is a contraction mapping on B K. Hence there is a unique φ B K such that (4.4) holds. Now we come to the differentiability of φ(x; α, q) of (α, q). Consider the following map T : R R 2K B R 2K B R 2K of class C 1 : ( + 1) 1 S(U + φ) β 1 U ( + 1) q T (α, q, φ, β) φ Z = Q1 dx., φ Z QK dx where B = W 2,2 ( ) B. Equation (4.4) is equivalent to T (α, q, φ, β) = 0. By the above argument, we know that, given α R and q satisfying (2.9), there is a unique local solution (φ(α, q), β(α, q)). For simplicity, in the following, we write (φ, β) = (φ(α, q), β(α, q)). We claim that the linear operator T (α, q, φ, β) (φ, β) : B R 2K B R 2K (α,q,φ, β)

27 INFINITELY MANY POSITIVE SOLUTIONS 27 is invertible for K large. Then the C 1 -regularity of (α, q) (φ, β) follows from the Implicit Function Theorem. Indeed we have T (α, q, φ, β) (φ, β) [ϕ, ζ] (α,q,φ, β) ( + 1) 1 S (U + φ)[ϕ] 1 U ζ ( + 1) q ϕ Z = Q1 dx.. ϕ Z QK dx Since φ C 0 K 2 }m (ln K) 1 2, by Lemma 4.2, the argument in the proof of Lemma 4.3 shows that T (α,q,φ, β) is invertible for K sufficiently large. This concludes the proof (φ, β) (α,q,φ, β) of Proposition 4.1. Next we study the dependence of φ on (α, q). Assume that we have two solutions corresponding to two sets of parameters. One of them denoted by min{1, p η L[φ] + E + N(φ) = β q U, corresponds to the parameters α and q; the other denoted by L[ φ] + E + N( φ) = β q U, corresponds to the parameters α and q. Observe that φ is L 2 -orthogonal to q U while φ is L 2 -orthogonal to q U. To compare φ with φ, we first choose a vector ω so that φ ω = φ + ω q U satisfies the same orthogonality condition as φ. Moreover, by the equation of φ, the function φ ω satisfies the equation L[ φ ω ] + ( L L)[ φ] ω L[ q U] + E + N( φ) + β ( q U q U) = β q U. Taking the difference with the equation satisfied by φ, we get L[ φ ω φ] = (L L)[ φ] + ω L[ q U] + (E E) + (N(φ) N( φ)) Note that by (2.7), for j = 1,..., K, we have β ( q U q U) + ( β β) q U. Q j Q j C(R α α + q q ). Assume that (R α α + q q ) 1/2, then we have (L L)[ φ] p η min{1, CK 2 }m (ln K) 1 2 (R α α + q q ), p η min{1, ω L[ q U] CK 2 }m (ln K) 1 2 ω,

28 28 MANUEL DEL PINO, JUNCHENG WEI, AND WEI YAO E E p η min{1, CK 2 }m (ln K) 1 2 (R α α + q q ), N(φ) N( φ) C( φ p 1 + φ p 1 ) φ φ + C φ min{p 1,1} (R α α + q q ) p η min{1, CK 2 }m(p 1) (ln K) p 1 2 φ φ + C ( p η ) min{1, K 2 }m (ln K) 1 2 min{p 1,1}(R α α + q q ), β ( q U q U) C β (R α α + q q ) Hence by Lemma 4.3, p η min{1, CK 2 }m (ln K) 1 2 (R α α + q q ). φ ω φ + β β C ( p η ) min{1, K 2 }m (ln K) 1 2 min{p 1,1}(R α α + q q ) p η min{1, + CK 2 }m (ln K) 1 2 ω p η min{1, + CK 2 }m(p 1) (ln K) p 1 2 φ φ. On the other hand, by the definition of φ ω, we have Hence ω C φ (R α α + q q ) p η min{1, CK 2 }m (ln K) 1 2 (R α α + q q ). φ φ + β β C ( p η ) min{1, K 2 }m (ln K) 1 2 min{p 1,1}(R α α + q q ). Therefore, we conclude that R 1 φ α + φ q C ( K min{1, p η ) 2 }m (ln K) 1 2 min{p 1,1}. 5. A further reduction process The main purpose of this section is to achieve Step 2.A. As explained in Section 2, we define Then equation (4.4) becomes β = β γ(rq 0 + q ), for every γ R. (5.1) L[φ] + E + N(φ) = β U q + γ U α. (5.2) Note that φ does not depend on γ, but β depends on the parameters α, q and γ and we write β = β(α, q, γ).