MULTIPLICITY OF SYMMETRIC SOLUTIONS FOR A NONLINEAR EIGENVALUE PROBLEM IN R n

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1 Electronic Journal of Differential Euations, Vol. 2005(2005, No. 05, pp ISSN: URL: or ftp ejde.math.txstate.edu (login: ftp MULTIPLICITY OF SYMMETRIC SOLUTIONS FOR A NONLINEAR EIGENVALUE PROBLEM IN R n DANIELA VISETTI Abstract. In this paper, we study the nonlinear eigenvalue field euation u + V ( x u + ε( pu + W (u = µu where u is a function from R n to R n+1 with n 3, ε is a positive parameter and p > n. We find a multiplicity of solutions, symmetric with respect to an action of the orthogonal group O(n: For any Z we prove the existence of finitely many pairs (u, µ solutions for ε sufficiently small, where u is symmetric and has topological charge. The multiplicity of our solutions can be as large as desired, provided that the singular point of W and ε are chosen accordingly. 1. Introduction In this paper, we find infinitely many solutions of the nonlinear eigenvalue field euation u + V ( x u + ε( p u + W (u = µu, (1.1 where u is a function from R n to R n+1 with n 3, ε is a positive parameter and p N with p > n. The choice of the nonlinear operator p + W is very important. The presence of the p-laplacian comes from a conjecture by Derrick (see [14]. He was looking for a model for elementary particles, which extended the features of the sine-gordon euation in higher dimension; he showed that euation u + W (u = 0 has no nontrivial stable localized solutions for any W C 1 on R n with n 2. He proposed then to consider a higher power of the derivatives in the Lagrangian function and this has been done for the first time in [6]. So the p-laplacian is responsible for the existence of nontrivial solutions. As concerns W, it denotes the gradient of a function W, which is singular in a point: this fact constitutes a sort of topological constraint and permits to characterize the solutions of (1.1 by a topological invariant, called topological charge (see [6]. The free problem u ε 6 u + W (u = Mathematics Subject Classification. 35Q55, 45C05. Key words and phrases. Nonlinear Schrödinger euations; nonlinear eigenvalue problems. c 2005 Texas State University - San Marcos. Submitted October 22, Published January 2, Supported by M.U.R.S.T., project Metodi variazionali e topologici nello studio di fenomeni non lineari. 1

2 2 DANIELA VISETTI EJDE-2005/05 has been studied in [6], while the concentration of the solutions has been considered in [1]. In [7] and [8] the authors have studied problem (1.1 respectively in a bounded domain and in R n. In [3] the authors have proved the existence of infinitely many solutions of the free problem, which are symmetric with respect to the action of the orthogonal group O(n. In this paper, we find a multiplicity of solutions, symmetric with respect to the action considered in [3], of problem (1.1 in R n : For any Z we prove the existence of finitely many pairs (u, µ solutions of problem (1.1 for ε sufficiently small, where u is symmetric and has topological charge. The multiplicity of the solutions can be as large as one wants, provided that the singular point ξ = (ξ 0, 0 (ξ 0 R, 0 R n of W and ε are chosen accordingly. The basic idea is to consider problem (1.1 as a perturbation of the linear problem when ɛ = 0. In terms of the associated energy functionals, one passes from the non-symmetric functional J ɛ (defined in (2.10 to the symmetric functional J 0. Non-symmetric perturbations of a symmetric problem, in order to preserve critical values, have been studied by several authors. We recall only [2], which seems to be the first work on the subject, and the papers [10] and [11]. In fact, the existence result is a result of preservation for the functional J ɛ of some critical values of the functional J 0, constrained on the unitary sphere of L 2 (R n, R n+1. Since the topological charge divides the domain Λ of the energy functional J ɛ into connected components Λ with Z, the solutions are found in each connected component and in two different ways: as minima and as min-max critical points of the energy functional constrained on the unitary sphere of L 2 (R n, R n+1. More precisely we can state: Given Z, for any ξ = (ξ 0, 0 (with ξ 0 > 0 and 0 R n and for any ε > 0, there exist µ 1 (ε and u 1 (ε respectively eigenvalue and eigenfunction of the problem (1.1, such that the topological charge of u 1 (ε is. Moreover, given Z \ {0} and k N, we consider ξ = (ξ 0, 0 with ξ 0 large enough and 0 R n. Let λ j be the eigenvalues of the linear problem (1.1 with ɛ = 0. Then for ε sufficiently small and for any j k with λ j 1 < λ j, there exist µ j (ε and u j (ε respectively eigenvalue and eigenfunction of the problem (1.1, such that the topological charge of u j (ε is. 2. Functional setting Statement of the problem. We consider from now on the field euation u + V ( x u + ε r ( p u + W (u = µu, (2.1 where u is a function from R n to R n+1 with n 3, ɛ is a positive parameter and p, r N with p > n and r > p n (for technical reasons we prefer to re-scale the parameter ɛ. The function V is real and we denote with W the gradient of a function W : R n+1 \ {ξ } R, where ξ is a point of R n+1, different from the origin, which for simplicity we choose on the first component: ξ = (ξ 0, 0, (2.2 with ξ 0 R, ξ 0 > 0 and 0 R n. Throughout the paper, we assume the following hypotheses on the function V : [0, + R:

3 EJDE-2005/05 MULTIPLICITY OF SYMMETRIC SOLUTIONS 3 (V1 lim V (r = + r + (V2 V ( x e x L p (R n, R (V3 ess inf r [0,+ V (r > 0 The assumptions on the function W : R n+1 \ {ξ } R are as follows: (W1 W C 1 (R n+1 \ {ξ }, R (W2 W (ξ 0 for all ξ R n+1 \ {ξ } and W (0 = 0 (W3 There exist two constants c 1, c 2 > 0 such that ξ R n+1, 0 < ξ < c 1 = W (ξ + ξ c 2 ξ np p n (W4 There exist two constants c 3, c 4 > 0 such that ξ R n+1, 0 ξ < c 3 = W (ξ c 4 ξ. (W5 For all ξ R n+1 \{ξ }, ξ = (ξ 1, ξ with ξ 1 R, ξ R n and for all g O(n, there holds W (ξ 1, g ξ = W (ξ 1, ξ. The space E. We define the following functional spaces: Γ(R n, R is the completion of C 0 (R n, R with respect to the norm z 2 Γ(R n,r = R n [V ( x z(x 2 + z(x 2 ]dx (2.3 Γ(R n, R n+1 is the completion of C 0 (R n, R n+1 with respect to the norm For s 1, we set u 2 Γ(R n,r n+1 = R n [V ( x u(x 2 + u(x 2 ]dx. (2.4 n+1 u s L R = u s dx = u i s s L s (R n,r n (2.5 n with u = (u 1, u 2,..., u n+1. The spaces Γ(R n, R and Γ(R n, R n+1 are Hilbert spaces, with scalar products (z 1, z 2 Γ(Rn,R = [V ( x z 1 z 2 + z 1 z 2 ] dx, (2.6 R n (u 1, u 2 Γ(R n,r n+1 = [V ( x u 1 u 2 + u 1 u 2 ] dx. (2.7 R n We recall a compact embedding theorem (see for example [4] into L 2. Theorem 2.1. The embedding of the space Γ(R n, R into the space L 2 (R n, R is compact. We define the Banach space E as the completion of the space C 0 (R n, R n+1 with respect to the norm i=1 u E = u Γ(R n,r n+1 + u L p. (2.8 The space E satisfies some useful properties which are listed in the next proposition. They follow from Sobolev embedding theorem and from [3, Proposition 8]. Proposition 2.2. The Banach space E has the following properties: (1 It is continuously embedded into L s (R n, R n+1 for 2 s + ;

4 4 DANIELA VISETTI EJDE-2005/05 (2 It is continuously embedded into W 1,p (R n, R n+1 ; (3 There exist two constants C 0, C 1 > 0 such that for every u E u L (R n,r n+1 C 0 u E, u(x u(y C 1 x y 1 n p u W 1,p (R n,r n+1 ; (4 If u E then lim x u(x = 0. The energy functional J ɛ. In the space E, by Proposition 2.2, it is possible to consider the open subset Λ = {u E : ξ u(r n }. (2.9 On Λ we consider the functional [1 J ɛ (u = R 2 u V ( x u 2 + ɛr p u p + ɛ r W (u ] dx, (2.10 n which is the energy functional associated to the problem (2.1. It is easy to verify the following lemma (see Lemma 2.3 of [8]. Lemma 2.3. The functional J ɛ is of class C 1 on the open set Λ of E. The topological charge. On the open set Λ a topological invariant can be defined. Let Σ be the sphere of center ξ and radius ξ 0 in R n+1. Let P be the projection of R n+1 \ {ξ } onto Σ: P (ξ = ξ + ξ ξ ξ ξ. (2.11 Definition 2.4. For any u Λ, u = (u 1,..., u n+1 the open and bounded set K u = {x R n : u 1 (x > ξ 0 } is called port of u. Then the topological charge of u is the number ch(u = deg(p u, K u, 2ξ. To use some properties of the topological charge, we need to recall the following result, whose proof can be found in [6]. Proposition 2.5. If a seuence {u m } Λ converges to u Λ uniformly on A R n, then also P u m converges to P u uniformly on A. This proposition permits to prove the continuity of the charge with respect to the uniform convergence: Theorem 2.6. For every u Λ there exists r = r(u > 0 such that, for every v Λ v u L (R n,r n+1 r = ch(v = ch(u. The connected components of Λ. The topological charge divides the open set Λ into the following sets, each of them associated to an integer number Z: Λ = {u Λ : ch(u = }. (2.12 By Theorem 2.6, we can conclude that the sets Λ are open in E. Moreover it is easy to see that Λ = Z Λ, Λ p Λ = if p and each Λ is a connected component of Λ.

5 EJDE-2005/05 MULTIPLICITY OF SYMMETRIC SOLUTIONS 5 3. Symmetry and compactness properties Action of O(n. We consider the following action of the orthogonal group O(n on the space of the continuous functions C(R n, R n+1 : T : O(n C(R n, R n+1 C(R n, R n+1 (g, u T g u (3.1 where T g u(x = (u 1 (gx, g 1 ũ(gx, (3.2 with u(x = (u 1 (x, ũ(x = (u 1 (x, u 2 (x,..., u n+1 (x. (3.3 In particular O(n acts on the space E and so one can prove the following result. Lemma 3.1. The open subset Λ E and the energy functional J ɛ are invariant with respect to the action ( Remark 3.2. More precisely every connected component Λ of Λ is invariant with respect to the action ( of the orthogonal group O(n. Moreover for any u E and for any g O(n T g u E = u E. Let F denote the subspace of the fixed points with respect to the action ( of O(n on E: F = {u E : g O(n T g u = u}. (3.4 Remark 3.3. The set F is a closed subspace. The set Λ F = Λ F is a natural constraint for the energy functional J ɛ. In fact, if u Λ F is a critical point for J Λ ɛ F, it is a global critical point (see [3]: Lemma 3.4. For every u Λ F and v E, we have being P the projection of E onto F. J ɛ(u(v = J ɛ(u(p v, We denote by Λ F the subset of the invariant functions of topological charge : Λ F = Λ F. Results of compactness. Next proposition provides a compact embedding for the subspace of the invariant functions of E into L s (R n, R n+1 : Proposition 3.5. The space F euipped with the norm E is compactly embedded into L s (R n, R n+1 for every s [2, 2n n 2. The proof is a conseuence of [3, Proposition 4] and of Theorem 2.1. We set S = {u E : u L 2 (R n,r n+1 = 1}. (3.5 To get some critical points of the functional J ɛ on the C 2 manifold Λ S we use the following version of Palais-Smale condition. For J ɛ C 1 (Λ, R, the norm of the derivative at u S of the restriction Ĵɛ = J ɛ Λ S is defined by Ĵ ɛ(u = min t R J ɛ(u tg (u E,

6 6 DANIELA VISETTI EJDE-2005/05 where g : E R is the function defined by g(u = R n u(x 2 dx. Definition 3.6. The functional J ɛ is said to satisfy the Palais-Smale condition in c R on Λ S (on Λ S, for Z if, for any seuence {u i } i N Λ S ({u i } i N Λ S such that J ɛ (u i c and Ĵ ɛ(u i 0, there exists a subseuence which converges to u Λ S (u Λ S. To obtain the Palais-Smale condition, we need a few technical lemmas (see [8] and [6]. Lemma 3.7. Let {u i } i N be a seuence in Λ (with Z such that the seuence {J ɛ (u i } i N is bounded. We consider the open bounded sets Then the set i N Z i R n is bounded. Z i = {x R n : u i (x > c 3 }. (3.6 Lemma 3.8. Let {u i } i N Λ be a seuence weakly converging to u and such that {J ɛ (u i } i N R is bounded, then u Λ. Lemma 3.9. For any a > 0, there exists d > 0 such that for every u Λ J ɛ (u a inf x R n u(x ξ d. Now it is possible to prove (see [8] that the functional J ɛ satisfies the Palais- Smale condition on Λ S for any c R and 0 < ɛ 1. As a conseuence the following proposition holds: Proposition The functional J ɛ satisfies the Palais-Smale condition on Λ F S (on Λ F S for Z for any c R and 0 < ɛ 1. Proof. Given a Palais-Smale seuence {u m } m N for J ɛ on Λ F S Λ S, it strongly converges to a function u Λ S by Proposition 2.1 of [8]. As the subspace F is closed (see Remark 3.3, u Λ F. 4. Eigenvalues of the Schrödinger operator Existence of the eigenvalues. We define the following subspace of invariant functions with respect to the action of O(n (see ( : Γ F (R n, R n+1 = {u Γ(R n, R n+1 : g O(n T g u = u}. (4.1 By Proposition 3.5 the identical embedding of Γ F (R n, R n+1 into L 2 (R n, R n+1 is continuous and compact. Then there exists a monotone increasing seuence { λ m } m N of eigenvalues with 0 < λ 1 λ 2 λ m m + λ m = inf max E m E m v E m, v 0 v 2 Γ F (R n,r n+1 v 2, L 2 (R n,r n+1 where E m is the family of all m-dimensional subspaces E m of Γ F (R n, R n+1. Also there exists a seuence {ϕ m } m N Γ F (R n, R n+1 of eigenfunctions, orthonormal in L 2 (R n, R n+1, such that (ϕ m, v ΓF (R n,r n+1 = λ m (ϕ m, v L2 (R n,r n+1, v Γ F (R n, R n+1, m N.

7 EJDE-2005/05 MULTIPLICITY OF SYMMETRIC SOLUTIONS 7 Regularity of the eigenfunctions. The eigenfunctions ϕ m have been found in the space Γ F (R n, R n+1. Nevertheless they possess some more regularity properties, as it can be shown using the following theorem: Theorem 4.1. If V (x + as x, then for any z H 1 (R n, R such that the following estimate holds: z + V (xz = λz where a > 0 is arbitrary and C a > 0 depends on a. For the proof of this theorem, see [9, p. 169]. z(x C a e a x, (4.2 Proposition 4.2. The eigenfunctions ϕ m Γ F (R n, R n+1 of the Schrödinger operator + V ( x belong to the Banach space E. Proof. We prove the result for the real-valued eigenfunctions e m so that the statement of the proposition follows immediately. By the regularity result of Agmon- Douglis-Nirenberg, if z Γ F (R n, R is such that z λz = V z and if V z L 2 (R n, R L p (R n, R, then z W 2,p (R n, R. So we only have to verify that V z L 2 (R n, R L p (R n, R. By Theorem 4.1 and ( V 2 we get V ( x z(x p dx C V ( x e x p < +. R n L p (R n,r Moreover, if R > 0 is such that for x R n \ B R n(0, R V ( x > 1, we have V ( x z(x 2 dx R n ( < C B R n (0,R V ( x 2 e p x dx + R n \B R n (0,R V ( x p e p x dx < +. Useful properties. We give here another variational characterization of the eigenvalues (see for example [13] and [16] and we introduce the subspaces spanned by the eigenfunctions. Definition 4.3. For m N we consider the following subspaces of Γ F (R n, R n+1 : F m = span[ϕ 1,..., ϕ m ], (4.3 F m = {u Γ F (R n, R n+1 : (u, ϕ i L 2 (R n,r n+1 = 0 for 1 i m}. (4.4 Lemma 4.4. The following properties hold: and Moreover, λ m = min u Γ F (R n,r n+1, u 0 (u,ϕ i L 2 (R n,r n+1 =0 i=1,...,m 1 u 2 Γ F (R n,r n+1 u 2 L 2 (R n,r n+1 (4.5 (ϕ i, ϕ j ΓF (R n,r n+1 = λ i δ ij i, j N. (4.6 u F m, u 0 = λ 1 u 2 Γ F (R n,r n+1 u 2 L 2 (R n,r n+1 λ m, (4.7

8 8 DANIELA VISETTI EJDE-2005/05 u F m, u 0 = u 2 Γ F (R n,r n+1 u 2 L 2 (R n,r n+1 5. Min-max values λ m+1. (4.8 The functions Φ ɛ. We introduce here a particular class of functions in E, which are invariant with respect to the action of the orthogonal group O(n. Let us consider the functions ϕ : R n R n+1 defined in the following way (see [12]: ( ϕ1 ( x ϕ 2 ( x x for x 0 x ϕ(x = ( (5.1 ϕ 1 (0 for x = 0 0 where ϕ i : [0, + R for i = 1, 2. In fact for any g O(n and x R n T g ϕ(x = ϕ(x. By Proposition 4.2, the set F m defined in (4.3 is a subset of E. Then, for any m N, let S(m denote the m-dimensional sphere: where S has been defined in (3.5. Fixed an integer k N, we introduce the number M k = S(m = F m S, (5.2 u L (R n,r n+1. (5.3 u S(k Then we choose the first coordinate ξ 0 of the point ξ = (ξ 0, 0 in such a way that ξ 0 > 2M k. (5.4 We can now introduce for any Z \ {0} the functions Φ a : R n R n+1 of type (5.1: ( Φ a,1 ( x Φ Φ a,2 a(x = ( x x for x 0 ( x Φ a,1 (0 (5.5 for x = 0 0 where with Φ a,1 ( x = { 2ξ 0 [cos(π x + 1] for R 1 x R 2 0 for 0 x R 1 or x R 2 Φ a,2 ( x = a x e x sin(±π x (i a > 0 (ii The sign in the argument of the sine in Φ a,2 is eual to the sign of, (iii R 1 is a constant depending on the parity of : { 0 if is odd, R 1 = R 1 ( = 1 if is even, (5.6

9 EJDE-2005/05 MULTIPLICITY OF SYMMETRIC SOLUTIONS 9 (iv R 2 is a positive constant depending on : { if is odd, R 2 = R 2 ( = + 1 if is even. Next lemma computes the topological charge of the functions just defined (see [3]. Lemma 5.1. For any Z \ {0}, the functions Φ a defined in (5.5, (5.6, with the hypotheses (i-(iv, belong to E and have topological charge ch (Φ a =. Proof. The functions Φ a belong to the space E. If we consider the components where 2 i n + 1, we have f 1 (x 1, x 2,..., x n = Φ a,1 ( x, f i (x 1, x 2,..., x n = Φ xi a,2 ( x x, x f 1 2 = Φ a,1 ( x 2 ( x f i 2 C Φ a,2 ( x 2 + Φ a,2 ( x 2 x 2 and conseuently n+1 ( f i 2 L 2 (R n,r n C Φ a,1 (r 2 + Φ a,2 (r 2 + Φ a,2 (r 2 r 2 r n 1 dr i=1 0 < + n+1 f i p L p (R n,r n C i=1 0 < + ( Φ a,1 (r p + Φ a,2 (r p + Φ a,2 (r p r p r n 1 dr Moreover the following ineualities hold: n V ( x f i (x 2 dx R n i=0 R2 C V (r(φ a,1 (r2 r n 1 dr + V ( x (Φ a,2 ( x 2 dx R 1 R n C + V ( x e x L p (R n,r a 2 x 2 e x L (R n,r < +, p p 1. where = The functions Φ a belong to the space Λ. In fact, if Φ a,2 ( x = 0, then x N {0} and hence Φ a,1 ( x {0, 4ξ 0}, so that Φ a(r n ξ. The functions Φ a have topological charge. Let P be the projection introduced in (2.11 of R n+1 onto the sphere Σ of center ξ and radius ξ 0 in R n+1 ; then P Φ a(x = (Φ Φ a,1 ( x ξ0 + ξ a,1 ( x ξ02 +(Φ 0 a,2 ( x 2 Φ a,2 ( x x (Φ a,1 ( x ξ02 +(Φ a,2 ( x 2 x

10 10 DANIELA VISETTI EJDE-2005/05 If K Φ a is the port of Φ a, we can consider on it the local coordinates obtained by the stereographic projection of the sphere Σ from the origin onto the plane Π = {ξ 1 = 2ξ 0 }: p : Σ ( Π (ξ 1, ξ 2,..., ξ n+1 2ξ ξ 2 0 ξ, ξ3 1 ξ,..., ξn+1 1 ξ. 1 Then the function Φ a in the new coordinates becomes Φ a(x = p P Φ a(x = f a( x x x, where fa( x Φ a,2 = ( x. (5.7 Φ a,1 ( x ξ 0 + ξ 0 (Φ a,1 ( x ξ (Φ a,2 ( x 2 The topological charge is therefore ch(φ a = deg ( Φ a, K Φ a, 0. Let δ be a positive parameter, δ < 3 4 and let i 1, i 2 N {0}, with i 1 = R 1, i 2 = max{i N {0} : 2i + 1 R 2 }. Then the sets K i = {x R n : 2i δ < x < 2i + δ}, for i N {0}, i 1 i i 2, are disjoint and their union satisfies the inclusion: i 2 i=i 1 K i K Φ a. Moreover this subset of K Φ a contains all the zeros of the function Φ a, that is: { } x K Φ a : Φ i 2 a = 0 K i. i=i 1 By the excision and the additive properties of the topological degree we can write deg(φ a, K Φ a, 0 = To conclude we want to prove that In fact consider the function deg(φ a, K i, 0 = i 2 i=i 1 deg(φ a, K i, 0. { sign( for i = 0, 2 sign( for i N. v 0 (x = f a(δ x, δ where fa( x is defined in (5.7. Since v 0 coincides with Φ a on the boundary of K 0, i.e. for any x K 0 Φ a(x = fa( x x x = v 0(x, the degrees of the two functions coincide too, so deg(φ a, K 0, 0 = deg(v 0, K 0, 0 = sign(.

11 EJDE-2005/05 MULTIPLICITY OF SYMMETRIC SOLUTIONS 11 Finally, for 1 i i 2, set K + i = {x R n : x < 2i + δ}, K i = {x R n : x < 2i δ} ; then the degrees satisfy deg ( Φ a, K i, 0 = deg ( Φ a, K + i, 0 deg ( Φ a, K i, 0. Analogously to the previous argument, we introduce the functions: As v ± i v + i (x = f a(2i + δ 2i + δ coincides with Φ a on the boundary of K ± i This completes the proof. deg ( Φ a, K + i, 0 = deg(v + i, K+ i deg ( Φ a, K i, 0 = deg(v i, K i The following corollary is now immediate. x, v i (x = f a(2i δ x. 2i δ, we conclude that, 0 = sign(,, 0 = sign(. Corollary 5.2. For all Z the connected component Λ F is not empty. Lemma 5.3. Fixed Z \ {0}, there exists â > 0 such that for every a â the functions Φ a have the following properties: (i The distance of Φ a from the point ξ is ξ 0, i.e. d(φ a, ξ = inf x R n Φ a(x ξ = ξ 0. (ii If we expand Φ a of a factor t 1, tφ a Λ F and d(tφ a, ξ = inf x R n tφ a(x ξ = ξ 0. Proof. (i We prove that there exists a sufficiently large such that Φ a(x ξ ξ 0 for all x R n. For x R n with 0 x R 1 or x R 2, it is immediate that Φ a(x ξ 2 = a 2 x 2 e 2 x sin 2 (π x + ξ 2 0 ξ 2 0. As for x R n with R 1 x R 2 there holds: Φ a(x ξ 2 = ξ 2 0[2 cos(π x + 1] 2 + a 2 x 2 e 2 x sin 2 (π x = ( 4ξ 2 0 a 2 x 2 e 2 x cos 2 (π x + 4ξ 2 0 cos(π x + ξ a 2 x 2 e 2 x. Let f a : [0, + R be the function f a (r = ( 4ξ 2 0 a 2 r 2 e 2r cos 2 (πr + 4ξ 2 0 cos(πr + a 2 r 2 e 2r. We consider the polynomial P (y = P α (y = ( 4ξ 2 0 α 2 y 2 + 4ξ 2 0y + α 2, where α = α a (r = are r, on the interval [ 1, +1]. Now, if α 2 = 4ξ0, 2 the only zero of P (y is y = 1 and therefore on [ 1, 1] P (y is nonnegative. On the contrary, if α 2 4ξ0, 2 the zeros of P (y are: y 1,2 = 2ξ2 0 ± ( { α 2 2ξ ξ0 2 = α2 2 α α 2 4ξ 2 0

12 12 DANIELA VISETTI EJDE-2005/05 For α 2 > 4ξ 2 0 we have y 1 = 1 and y 2 > 1, so P (y 0 on [ 1, 1]. For 2ξ 2 0 α 2 < 4ξ 2 0, we have y 2 1, so P (y 0 and conseuently a 2 r 2 e 2r 2ξ 2 0 = f a (r 0. If we consider 2ξ0 a (5.8 R 2 e R2 and R 1 = 1 (i.e. even, there always holds α 2 2ξ0. 2 If on the contrary is odd and so R 1 = 0, for a as in (5.8 (α a (r 2 < 2ξ0 2 for 0 r < r 1, where r 1 is such that (α a (r 1 2 = 2ξ 2 0. (5.9 We choose a sufficiently large to have r : then cos(πr (0, 1] for any r [0, r 1 and so min r [0,r 1 f a(r 0. (ii For any x R n with 0 x R 1 or x R 2, it is immediate that tφ a(x ξ 2 = t 2 a 2 x 2 e 2 x sin 2 (π x + ξ 2 0 ξ 2 0. On the contrary for x R n, R 1 x R 2, there holds: tφ a(x ξ 2 = ξ 2 0[2t cos(π x + 2t 1] 2 + t 2 a 2 x 2 e 2 x sin 2 (π x = t 2 ( 4ξ 2 0 a 2 x 2 e 2 x cos 2 (π x + 4t(2t 1ξ 2 0 cos(π x + ξ 2 0(2t t 2 a 2 x 2 e 2 x. As before we consider f a : [0, + R defined by f a (r = t 2 ( 4ξ 2 0 a 2 r 2 e 2r cos 2 (πr + 4t(2t 1ξ 2 0 cos(πr + 4t(t 1ξ t 2 a 2 r 2 e 2r. The polynomial P (y becomes P (y = P α (y = t 2 (4ξ 2 0 α 2 y 2 + 4t(2t 1ξ 2 0y + 4t(t 1ξ t 2 α 2. If α 2 = 4ξ0, 2 the only zero of P (y is y = 1 and so on [ 1, 1] P (y is nonnegative. If α 2 4ξ0, 2 the zeros of P (y are y 1,2 = (2 4tξ2 0 ± ( 2ξ0 2 tα 2 { 1 t(4ξ0 2 = α2 4(1 tξ 2 0 tα2 t(4ξ 2 0 α2 For α 2 > 4ξ 2 0 we have y 1 = 1 and y 2 > 1, then P (y 0 in [ 1, 1]. For 2ξ 2 0 α 2 < 4ξ 2 0, there holds y 2 1, so P (y 0 and conseuently a 2 r 2 e 2r 2ξ 2 0 = f a (r 0. Now, with the choice of a done in (i and R 1 = 1 ( even, α 2 2ξ0. 2 Finally, if R 1 = 0 and a is as in (i, α 2 < 2ξ0 2 for 0 r < r (where r 1 is as in (5.9, min f a (r 0. r [0,r 1 This completes the proof.

13 EJDE-2005/05 MULTIPLICITY OF SYMMETRIC SOLUTIONS 13 Definition 5.4. For any Z \ {0} and for â as in Lemma 5.3, we define the function Φ = Φ â. (5.10 Evidently for i = 1, 2 we pose (Φ i = Φ â.,i Moreover we introduce the rescaled functions Φ ɛ, with Z\{0} and 0 < ɛ 1: ( Φ ɛ(x = Φ x. (5.11 ɛ Remark 5.5. (1 The functions Φ ɛ belong to Λ F. (2 By definition of Φ ɛ and by Lemma 5.3 the image of Φ ɛ does not intersect the point ξ and the distance of the image from the point is ξ 0. (3 Even if we expand the functions Φ ɛ (0 < ɛ 1 of a factor t 1, their image is such that they do not meet the point ξ and the distance is still ξ 0. Hence tφ ɛ Λ F for all t 1 and ɛ (0, 1]. Remark 5.6. The norms of the functions Φ ɛ satisfy the following eualities depending on the parameter ɛ: Φ ɛ 2 L 2 (R n,r n+1 = ɛn Φ 2 L 2 (R n,r n+1, (5.12 Φ ɛ 2 L 2 = ɛn 2 Φ 2 L 2, (5.13 Φ ɛ p L p = 1 ɛ p n Φ p L p. (5.14 The functions Φ ɛ own some fundamental properties, which are presented in the following lemma. Lemma 5.7. Given Z\{0} and k N, we consider ξ = (ξ 0, 0 with ξ 0 > 2M k, where M k = u S(k u L (R n,r n+1, and 0 R n. There exist ˆρ > 0 and ɛ, with 0 < ɛ 1, such that for all 0 < ɛ ɛ we have (i Φ ɛ + ˆρ u L2 (R n,r n+1 1 for all u S(k, (ii Φ ɛ + ˆρ u L 2 (R n,r n+1 > 0, (iii (iv inf ɛ (0,ɛ ] u S(k inf x R n ɛ (0,ɛ ] u S(k Φ ɛ(x + ˆρ u(x Φ ɛ + ˆρ u L 2 (R n,r n+1 ξ > ξ 0 2, Φ ɛ + ˆρ u Φ Λ S for all u S(k. ɛ + ˆρ u L 2 (R n,r n+1 Proof. (i For any ρ > 0 and 0 < ɛ 1 we have Let ɛ be such that Φ ɛ + ρu L 2 (R n,r n+1 ɛ n 2 Φ L 2 (R n,r n+1 + ρ. ( 2 n 1 ɛ <, Φ L 2 (R n,r n+1 ɛ 1. Then there exists ˆρ > 0 such that Φ ɛ L2 (R n,r n+1 + ˆρ 1. (5.15

14 14 DANIELA VISETTI EJDE-2005/05 (ii As Φ ɛ + ˆρ u L2 (R n,r n+1 ˆρ Φ ɛ L2 (R n,r n+1, reducing if necessary ɛ, we get Φ ɛ + ˆρ u L 2 (R n,r n+1 > 0. (iii By (ii of Lemma 5.3 we deduce that for all u S(k inf x R n ɛ (0,ɛ ] Φ ɛ(x Φ ξ = ξ 0. ɛ + ˆρ u L2 (R n,r n+1 To get (iii it is sufficient to prove that, reducing if necessary ɛ, for all ɛ ɛ We observe that u S(k u S(k ˆρ u L (R n,r n+1 Φ ɛ + ˆρ u L2 (R n,r n+1 ˆρ u L (R n,r n+1 Φ ɛ + ˆρ u L2 (R n,r n+1 Since M k < ξ0 2, for ɛ sufficiently small we have (iii. (iv follows immediately from (iii. < ξ 0 2. ˆρ M k inf u S(k Φ ɛ + ˆρ u L2 (R n,r n+1 M k 1 ɛ n 2 ˆρ Φ L2 (R n,r n+1 The values c ɛ,j. Using the properties of the functions Φ ɛ seen in Lemma 5.7, it is possible to introduce the following subsets of Λ F S: Definition 5.8. Fixed k N, Z \ {0} and 0 < ɛ ɛ, where ɛ is defined in Lemma 5.7, we set M ɛ,j = { Φ ɛ + ˆρ u Φ : u S(j } (5.16 ɛ + ˆρ u L2 (R n,r n+1 with j k and ˆρ defined in Lemma 5.7. We pose by convention M ɛ,0 =. Remark 5.9. We outline the following properties of the sets M ɛ,j : (i M ɛ,j 1 M ɛ,j ; (ii M ɛ,j ΛF S; (iii M ɛ,j is a compact set; (iv M ɛ,j is a sub-manifold of ΛF for 0 < ɛ ɛ (see Lemma 5.7. Next definition introduces the min-max values c ɛ,j. Definition Fixed k N, for all Z \ {0}, j k and 0 < ɛ ɛ (ɛ is defined in Lemma 5.7, we define the following values: c ɛ,j = inf h H ɛ,j v M ɛ,j J ɛ ( h(v, (5.17 where H ɛ,j are the following sets of continuous transformations: H ɛ,j {h = : Λ F S Λ F S : h continuous, h M = id M ɛ,j 1 ɛ,j 1 We observe that H ɛ,j+1 H ɛ,j. Lemma Fixed k N, for all Z \ {0}, j < k and 0 < ɛ ɛ, we have (i c ɛ,j R, (ii c ɛ,j c ɛ,j+1.. }.

15 EJDE-2005/05 MULTIPLICITY OF SYMMETRIC SOLUTIONS Main results Minima. We recall now the Deformation Lemma: Lemma 6.1 (Deformation Lemma. Let J be a C 1 -functional defined on a C 2 - Finsler manifold E. Let c be a regular value for J. We assume that: (i J satisfies the Palais-Smale condition in c on M, (ii there exists k > 0 such that the sublevel J c+k is complete. Then there exist δ > 0 and a deformation η : [0, 1] E E such that: (a η(0, u = u for all u E, (b η(t, u = u for all t [0, 1] and u such that J(u c 2δ, (c J(η(t, u is non-increasing in t for any u E, (d η(1, J c+δ J c δ. To apply Lemma 6.1 on each connected component Λ F, with Z \ {0}, intersected with the unitary sphere S we need the completeness of the sub-levels of the functional J ɛ. It is simple to verify next: Lemma 6.2. For any Z, ɛ (0, 1] and c R, the subset Λ F S J c ɛ of the Banach space E is complete. Now we get easily the minimum values of the functional J ɛ on each set Λ F S: Theorem 6.3. Given Z, for any ξ = (ξ 0, 0 with ξ 0 > 0 and 0 R n and for any ɛ > 0, there exists a minimum for the functional J ɛ on the subset Λ F S of Λ S. Proof. For any t 1 we have that tφ Λ F (see (iii of Remark 5.5 and in particular the function is in Λ F S. This means that Λ F S is not Φ Φ L 2 (R n,r n+1 empty for all Z, since it is obvious that Λ F 0 S. The claim follows by the fact that Λ F S is not empty, the functional J ɛ is bounded from below and satisfies the Palais-Smale condition on Λ F S (see Proposition Remark 6.4. We point out that to have this result there is no need to reuire that the first coordinate ξ 0 of the point ξ is sufficiently large (see (5.4. In fact this assumption is necessary to have properties (iii and (iv of Lemma 5.7, while here we only have to show that Λ F S is not empty for all Z. Critical values. The next theorem is an existence and multiplicity result of solutions for the problem ( P ɛ. Theorem 6.5. Given Z \ {0} and k N, we consider ξ = (ξ 0, 0 with ξ 0 > 2M k, where M k = u S(k u L (R n,r n+1, and 0 R n. Then there exists ˆɛ (0, 1] such that for any ɛ (0, ˆɛ ] and for any 2 j k with λ j 1 < λ j, we get that c ɛ,j is a critical value for the functional J ɛ restricted to the manifold Λ F S. Moreover c ɛ,j 1 < c ɛ,j. The proof of this theorem is similar to the proof of Theorem 3.1 in [8], but for the convenience of the reader we summarize it here.

16 16 DANIELA VISETTI EJDE-2005/05 Proof. We begin with some notation: if u F we define the projections j P Fj u = (u, ϕ i ΓF (R n,r n+1 ϕ i, Q Fj u = u P Fj u. (6.1 i=1 It is immediate that (Q Fj u, ϕ i ΓF (R n,r n+1 = λ i (Q Fj u, ϕ i L2 (R n,r n+1 = 0 i = 1,..., j. (6.2 We divide the argument into five steps. Step 1 For any h H ɛ,j the intersection of the set h(m ɛ,j with the set {u F : (u, ϕ i ΓF (R n,r n+1 = 0 i = 1,..., j 1} is not empty: in fact there exists v M ɛ,j such that P Fj 1 h(v = 0. This is obtained by an argument of degree theory (for the proof see [7]. Step 2 We prove that J ɛ (v λ j + σ(ɛ (6.3 v M ɛ,j c ɛ,j λ j + σ(ɛ (6.4 where lim ɛ 0 σ(ɛ = 0. First of all we verify that J 0 (v λ j + v M ɛ,j u S(j Q Fj Φ ɛ 2 Γ F (R n,r n+1 P Fj Φ ɛ + ˆρ u 2 L 2 (R n,r n+1 + Q F j Φ ɛ 2. (6.5 L 2 (R n,r n+1 In fact by Definition 5.8, (6.1 and (6.2 we have: J 0 (v = v M ɛ,j u S(j Φ ɛ + ˆρ u Φ ɛ + ˆρ u L 2 (R n,r n+1 2 Γ F (R n,r n+1 P Fj Φ ɛ + ˆρ u 2 Γ = F (R n,r n+1 + Q F j Φ ɛ 2 Γ F (R n,r n+1 u S(j P Fj Φ ɛ + ˆρ u 2 L 2 (R n,r n+1 + Q F j Φ ɛ 2 L 2 (R n,r n+1 ( P Fj Φ ɛ + ˆρ u 2 Γ F (R n,r n+1 u S(j P Fj Φ ɛ + ˆρ u 2 L 2 (R n,r n+1 Q Fj Φ ɛ 2 Γ + F (R n,r n+1 P Fj Φ ɛ + ˆρ u 2 L 2 (R n,r n+1 + Q F j Φ ɛ 2 L 2 (R n,r n+1 λ j + u S(j Q Fj Φ ɛ 2 Γ F (R n,r n+1 P Fj Φ ɛ + ˆρ u 2 L 2 (R n,r n+1 + Q F j Φ ɛ 2. L 2 (R n,r n+1

17 EJDE-2005/05 MULTIPLICITY OF SYMMETRIC SOLUTIONS 17 Using the definition of J ɛ and (6.5, we prove the following ineualities: c ɛ,j = inf h H ɛ,j v M ɛ,j J ɛ (v v M ɛ,j v M ɛ,j λ j + u S(j J 0 (v + ɛ r J ɛ (h(v v M ɛ,j R n ( 1 p v p + W (v dx Q Fj Φ ɛ 2 Γ F (R n,r n+1 P Fj Φ ɛ + ˆρ u 2 L 2 (R n,r n+1 + Q F j Φ ɛ 2 L 2 (R n,r n+1 (6.6 + ɛr p (Φ ɛ + ˆρ u p L p u S(j Φ ɛ + ˆρ u p L 2 (R n,r n+1 ( + ɛ r Φ ɛ + ˆρ u u S(j Φ dx. ɛ + ˆρ u L2 (R n,r n+1 R n W At this point we note that lim ɛ 0 Q Fj Φ ɛ 2 Γ F (R n,r n+1 = 0, in fact Q Fj Φ ɛ 2 Γ F (R n,r n+1 Φ ɛ 2 Γ F (R n,r n+1 = Φ ɛ(x 2 dx + V ( x Φ ɛ(x 2 dx, R n R n where the right-hand side tends to zero for ɛ 0, because (5.13 holds and [ ( x 2 ( ] V ( x Φ ɛ(x 2 dx = V ( x (Φ 1 + x 2 R n R ɛ (Φ 2 dx ɛ n ɛr2 ( r c V (r (Φ 2 1 r n 1 dr ɛr 1 ɛ ( + V ( x e x L p (R n,r e x x 2 (Φ 2 ɛ L (R n,r ( c max V r [0,R (rrn 1 2 (R 2 R 1 ɛ n 2] ( V + ( x e x L p (R n,r x 2 e x L (R n,r ɛ n where denotes the dual exponent of p. Moreover by (ii of Lemma 5.7 we obtain in fact 0<ɛ ɛ u S(j 1 P Fj Φ ɛ + ˆρ u 2 L 2 (R n,r n+1 + Q F j Φ ɛ 2 < +, L 2 (R n,r n+1 P Fj Φ ɛ 2 L 2 (R n,r n+1 ɛn Φ 2 L 2 (R n,r n+1, Q Fj Φ ɛ 2 L 2 (R n,r n+1 ɛn Φ 2 L 2 (R n,r n+1. Therefore the second term of the last ineuality of (6.6 goes to zero when ɛ goes to zero. Now we observe that the following ineuality holds: ( ɛ r Φ ɛ + ˆρ u p L ɛ r (p n p p p Φ L p + ɛ r p ˆρ u L p.

18 18 DANIELA VISETTI EJDE-2005/05 Then by this ineuality and (ii of Lemma 5.7 (we recall that r > p n, we have that the third term of the last ineuality of (6.6 tends to zero when ɛ tends to zero. Regarding the last term, we verify that ( Φ ɛ + ˆρ u Φ dx ɛ + ˆρ u L2 (R n,r n+1 R n W is bounded uniformly with respect to ɛ (0, ɛ] and u S(k. In fact by definition of Φ ɛ and by the exponential decay of the eigenfunctions (see Theorem 4.1 there exists a ball B R n(0, R such that, if we write u = j m=1 a mϕ m with j m=1 a2 m = 1, for all x R n \ B R n(0, R the following ineualities hold Φ ɛ(x + ˆρ u(x Φ ɛ + ˆρ u = ˆρ u(x L2 (R n,r n+1 Φ ɛ + ˆρ u L2 (R n,r n+1 ( j C ˆρ m=1 a m e x Φ ɛ + ˆρ u L 2 (R n,r n+1 Me x < c 3 where the constant M does not depend on u S(j nor on ɛ for ɛ small enough (see the point (ii of Lemma 5.7. By ( W 4 we get ( W Φ ɛ(x + ˆρ u(x Φ Φ c ɛ(x + ˆρ u(x 2 4 ɛ + ˆρ u L2 (R n,r n+1 Φ ɛ + ˆρ u 2 L 2 (R n,r n+1 for any x R n \ B R n(0, R. Concluding we have ( Φ ɛ + ˆρ u W R Φ dx n ɛ + ˆρ u L 2 (R n,r n+1 ( c 4 + W Φ ɛ + ˆρ u Φ dx ɛ + ˆρ u L 2 (R n,r n+1 B R n (0,R where the integral on the right hand side is bounded by (iii of Lemma 5.7. So we have the claim. Step 3 We prove that c ɛ,j λ j. By Step 1 and by the positivity of W we get c ɛ,j inf h H ɛ,j v M ɛ,j inf h H ɛ,j v M ɛ,j P Fj 1 h(v=0 λ j h(v 2 Γ F (R n,r n+1 h(v 2 Γ F (R n,r n+1 In fact by Step 1 for all h H ɛ,j we have that the set h(m ɛ,j intersects the set {u F : (u, ϕ i = 0 i = 1,..., j 1} and so from (4.8 we get the claim. Step 4 If λ j 1 < λ j, then for ɛ small enough we have: By Step 2 and 3 we obtain for ɛ small enough c ɛ,j 1 < c ɛ,j, (6.7 J ɛ (v < c v M ɛ,j. (6.8 ɛ,j 1 c ɛ,j 1 λ j 1 + σ(ɛ < λ j c ɛ,j,

19 EJDE-2005/05 MULTIPLICITY OF SYMMETRIC SOLUTIONS 19 J ɛ (v λ j 1 + σ(ɛ < λ j c v M ɛ,j. ɛ,j 1 Step 5 If λ j 1 < λ j, then c ɛ,j is a critical value for the functional J ɛ on the manifold Λ F S. By contradiction we pose that c ɛ,j is a regular value for J ɛ on Λ F S. By Proposition 3.10 and Lemmas 6.1 and 6.2, there exist δ > 0 and a deformation η : [0, 1] Λ F S Λ F S such that By (6.8 we can pose η(t, u = u η(0, u = u u Λ F S, t [0, 1], u J c ɛ,j 2δ ɛ, η(1, J c ɛ,j +δ ɛ J c ɛ,j δ ɛ. J ɛ (v < c ɛ,j 2δ. (6.9 v M ɛ,j 1 Moreover by definition of c ɛ,j there exists a transformation ĥ H ɛ,j such that v M J (ĥ(v ɛ ɛ,j < c ɛ,j + δ. Now by the properties of the deformation η and by (6.9 we get η ( 1, ĥ(. H ɛ,j and v M J ɛ( ɛ,j η(1, ĥ(v < c ɛ,j δ and this is a contradiction. Remark 6.6. (1 In the assumptions of Theorem 6.5 min u Λ F S J ɛ (u = c ɛ,1. Nevertheless the critical point corresponding to the minimum, found in Theorem 6.3, is not attained in the framework of Theorem 6.5. In fact, to conclude that a value c ɛ,j is critical, j must be strictly greater than one. (2 Provided that we choose suitably ξ and ɛ, it is possible to find as many solutions of ( P ɛ as we want. In fact, let us pose that we want K N solutions, then, since λ j, there exists k N, k > K, such that among the first k eigenvalues λ there are K jumps λ j < λ j+1, so that Theorem 6.5 gives K critical values. (3 For all Z \ {0}, ɛ (0, 1] the critical values c ɛ,j tend to the eigenvalues λ j when ɛ tends to zero. Acknowledgements. This paper arises from the Ph.D. thesis (see [18] of the author, who heartily thanks her ervisors V. Benci and A. M. Micheletti. References [1] M. Badiale, V. Benci and T. D Aprile, Existence, multiplicity and concentration of bound states for a uasilinear elliptic field euation, Calculus of Variations and Partial Differential Euations 12 3 (2001, [2] A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications, Transactions of the American Mathematical Society (1981, [3] V. Benci, P. D Avenia, D. Fortunato and L. Pisani, Solitons in several space dimensions: Derrick s problem and infinitely many solutions, Archive for Rational Mechanics and Analysis (2000, [4] V. Benci and D. Fortunato, Discreteness conditions of the spectrum of Schrödinger operators, Journal of Mathematical Analysis and Applications 64 3 (1978,

20 20 DANIELA VISETTI EJDE-2005/05 [5] V. Benci, D. Fortunato, A. Masiello and L. Pisani, Solitons and the electromagnetic field, Mathematische Zeitschrift (1999, [6] V. Benci, D. Fortunato and L. Pisani, Soliton-like solution of a Lorentz invariant euation in dimension 3, Reviews in Mathematical Physics 10 3 (1998, [7] V. Benci, A.M. Micheletti and D. Visetti, An eigenvalue problem for a uasilinear elliptic field euation, Journal of Differential Euations (2002, [8] V. Benci, A.M. Micheletti and D. Visetti, An eigenvalue problem for a uasilinear elliptic field euation on R n, Topological Methods in Nonlinear Analysis 17 2 (2001, [9] F.A. Berezin and M.A. Shubin, The Schrödinger Euation, Kluwer Academic Publishers [10] P. Bolle, On the Bolza problem, Journal of Differential Euations (1999, [11] P. Bolle, N. Ghoussoub and H. Tehrani, The multiplicity of solutions in non-homogeneous boundary value problems, Manuscripta Mathematica (2000, [12] C. Cid and P.L. Felmer, A note on static solutions of a Lorentz invariant euation in dimension 3, Letters in Mathematical Physics 53 1 (2000, [13] R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. I, Interscience, New York. [14] C.H. Derrick, Comments on Nonlinear Wave Euations as Model for Elementary Particles, Journal of Mathematical Physics 5 (1964, [15] N. Ghoussoub, Duality and perturbation methods in critical point theory, Cambridge University Press [16] A. Manes and A.M. Micheletti, Un estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine, Boll. U.M.I. (4 7 (1973, [17] P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential euations, C.B.M.S. Reg. Conf. 65, American Mathematical Society, Providence [18] D. Visetti, An eigenvalue problem for a uasilinear elliptic field euation, Ph.D. Thesis, Università di Pisa Dipartimento di Matematica Applicata U. Dini, Università degli studi di Pisa, via Bonanno Pisano 25/B, Pisa, Italy address: visetti@mail.dm.unipi.it