In this paper we are concerned with the following nonlinear field equation:

Transcription

1 Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 17, 2001, AN EIGENVALUE PROBLEM FOR A QUASILINEAR ELLIPTIC FIELD EQUATION ON R n V. Benci A. M. Micheletti D. Visetti Abstract. We study the field equation u + V (x)u + ε r ( pu + W (u)) = µu on R n, with ε positive parameter. The function W is singular in a point and so the configurations are characterized by a topological invariant: the topological charge. By a min-max method, for ε sufficiently small, there exists a finite number of solutions (µ(ε), u(ε)) of the eigenvalue problem for any given charge q Z \ 0}. 1. Introduction In this paper we are concerned with the following nonlinear field equation: (P ε ) u + V (x)u + ε r ( p u + 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, r N with p > n and r > p n. Here u = ( u 1,..., u n+1 ), being u = (u 1,..., u n+1 ) and the classical Laplacian operator. Moreover, p u denotes the (n + 1)-vector, whose i-th component is given by ( p u) i = ( u i p 2 u i ) Mathematics Subject Classification. 34A34, 35Q55, 47S10. Key words and phrases. Nonlinear systems, nonlinear Schrödinger equations, nonlinear eigenvalue problems. This paper was ported by M.U.R.S.T. (40% and 60% funds). c 2001 Juliusz Schauder Center for Nonlinear Studies 191

2 192 V. Benci A. M. Micheletti D. Visetti Finally, V is a real function V : R n R and W is the gradient of a function W : R n+1 \ ξ } R, where ξ is a point of R n+1 which for simplicity we choose on the (n + 1)-th component: (1) ξ = (0, ξ), with 0 R n and ξ R, ξ > 0. The motivation for considering an eigenvalue problem relative to a nonlinear equation such as (P ε ) needs some explanations. Let us consider the nonlinear Schrödinger equation (2) iψ t = ψ + V (x)ψ + ε r N(ψ) where N(ψ) is a nonlinear differential operator. The standing waves ψ(x, t) = u(x)e iµt of equation (2) are determined by the solutions of the following nonlinear eigenvalue problem (3) u + V (x)u + ε r N(u) = µu provided that (4) N(u(x)e iµt ) = e iµt N(u(x)). The nonlinear operator (5) N(u) = p u + W (u) can be extended to the complex functions in such a way to verify (4). The choice of the operator (5) is due to the fact that in a paper of 1964 Derrick ([13]) pointed out by a simple rescaling argument that equation ϕ + 1 c 2 ϕ tt f (ϕ) = 0, where f is the gradient of a nonnegative C 1 real function f and the function ϕ has domain R n with n > 2, has no nontrivial static solutions: We are faced with the disconcerting fact that no equation of type ϕ 1 c 2 ϕ tt = 1 2 f (ϕ) has any time-independent solutions which could reasonably be interpreted as elementary particles. He presents some conjectures and the first one is to consider higher powers for the derivatives: in fact in [4] (see also [7]) the authors proved that equation (6) ϕ p ϕ + W (ϕ) = 0,

3 An Eigenvalue Problem for a Quasilinear Elliptic Field Equation on R n 193 (where ϕ : R 3 R 4 ), has a family ϕ q } q Z\0} of nontrivial solutions with the energy concentrated around the origin. These solutions are characterized by a topological invariant ch( ), called topological charge, which takes integer values (see (9)). More precisely, for every q Z \ 0}, there exists a solution ϕ q with ch(ϕ q ) = q. An interesting concentration problem has been studied in [2], where the authors consider some bound states of a field equation like (6) with the addition of a potential depending on a parameter. Here we study the eigenvalue problem relative to equation (6), with the addition of a potential V ; so we look for critical points of a suitable constrained functional and not only minima. Throughout the paper we always assume these hypotheses on the function V : R n R: (V 1 ) lim V (x) =, x (V 2 ) V (x)e x L p (R n, R), (V 3 ) ess inf V (x) > 0. x Rn We note that (V 2 ) is a technical hypothesis. We need it to prove the regularity of the eigenfunctions of the linear eigenvalue problem (see Lemma 2.8), but it may be weakened. The assumptions on the function W : R n+1 \ ξ } R are the following: (W 1 ) W C 1 (R n+1 \ ξ }, R), (W 2 ) W (ξ) 0 for all ξ R n+1 \ ξ } and W (0) = 0, (W 3 ) there exist two constants c 1, c 2 > 0 such that ξ R n+1, 0 < ξ < c 1 W (ξ + ξ) and ξ c 1 > 0, (W 4 ) there exist two constants c 3, c 4 > 0 such that c 2 ξ np/(p n) ξ R n+1, 0 ξ < c 3 W (ξ) c 4 ξ. The energy functional associated to the problem (P ε ) is: [ 1 (7) J ε (u) = 2 u ] 2 V (x) u 2 + εr p u p + ε r W (u) dx. R n In [8] the authors proved the existence of solutions for the eigenvalue problem (P ε ) on a bounded domain Ω. In this paper we consider a more complex case, namely when the domain is R n and the potential is coercive, i.e. V (x) for x. We state the following existence results (see Theorem 3.1 and Theorem 3.2): Given q Z \ 0} and k N, we consider ξ = (0, ξ) with 0 R n and ξ large enough. Then for ε sufficiently small and for any j k with λ j 1 < λ j, there

4 194 V. Benci A. M. Micheletti D. Visetti exist µ j (ε) and u j (ε) respectively eigenvalue and eigenfunction of the problem (P ε ), such that the topological charge of u j (ε) is q. Moreover, given q Z, for any ξ = (0, ξ) (with 0 R n and ξ > 0) and for any ε > 0, there exist µ 1 (ε) and u 1 (ε) respectively eigenvalue and eigenfunction of the problem (P ε ), such that the topological charge of u 1 (ε) is q. Here λ j (see Subsection 2.4) are the eigenvalues of the linear problem u+ V (x)u = λu, since we have the discreteness of the spectrum of the Schrödinger operator + V, with lim x V (x) =, by a compact embedding theorem (see e.g. [5] and Theorem 2.1). Our aim is to find critical values of the energy functional J ε in the intersection of any connected component, characterized by the topological charge, with the unitary sphere in L 2 (R n, R n+1 ). The idea is to consider the functional J ε as a perturbation of the symmetric functional 1 J 0 (u) = R 2 [ u 2 + V (x) u 2 ] dx. n Non-symmetric perturbations of a symmetric problem, in order to preserve critical values, have been studied by several authors. We omit for the sake of brevity a complete bibliography and we recall only [3], which seems to be the first work on the subject, and the recent papers [10] and [11]. In this paper we give a result of preservation for the functional J ε of some critical values λ j of the functional J 0 restricted on the unitary sphere of L 2 (R n, R n+1 ) in the space Γ(R n, R n+1 ) (see Subsection 2.1). The content of the paper is divided into the following sections. In Section 2 there is the description of the functional setting, the definition of a topological invariant, called topological charge, and some arguments of eigenvalues theory. The compactness, that we lose because of the unbounded domain R n, is recovered by the compact embedding of [5] (see Theorem 2.1). Then, by some technical devices, we obtain the Palais Smale condition for the functional J ε (defined in (7)). The addition of the potential V breaks the translation invariance, so that the technical lemmas require some care. Section 3 is devoted to the proof of our main results. In Theorem 3.1 we state the existence of some critical values of the functional J ε on every component of the unitary sphere, characterized by the value of the topological invariant topological charge (see (11), (8), (10)). These critical values c q ε,j (see (28)) of the functional J ε are of min-max type. The construction of some suitable functions G q ε of topological charge q (see (26)) and some suitable manifolds M q ε,j (see (27)) is crucial in finding the critical values c q ε,j. In Theorem 3.2 we state the existence of the minimum of the functional J ε on every component of the unitary sphere, characterized by the topological charge (see (10)).

5 An Eigenvalue Problem for a Quasilinear Elliptic Field Equation on R n 195 Notations. We fix the following notations: x is the Euclidean norm of x R n, if ξ R n+1 some times we will use the notation ξ = ( ξ, ξ), where ξ R n and ξ R, if x R n and ρ > 0, then B(x, ρ) is the open ball with centre in x and radius ρ. 2. Functional setting 2.1. The space E. We shall consider the following functional spaces: Γ(R n, R) the completion of C0 (R n, R) with respect to the norm z 2 Γ(R n,r) R = V (x) z(x) 2 dx + z(x) 2 dx; n R n the space Γ(R n, R) is then a Hilbert space, whose scalar product is denoted by (z 1, z 2 ) Γ(R n,r). Γ(R n, R n+1 ) the completion of C0 (R n, R n+1 ) with respect to the norm u 2 Γ(R n,r n+1 ) R = V (x) u(x) 2 dx + n R n u(x) 2 dx; where u 2 = n+1 i=1 u i 2 and u 2 = n n+1 i=1 j=1 u j/ x i 2 ; analogously the space Γ(R n, R n+1 ) is a Hilbert space, whose scalar product is denoted by (u 1, u 2 ) Γ(Rn,R n+1 ). It is clear that the spaces Γ(R n, R) and Γ(R n, R n+1 ) are continuously embedded respectively into the Sobolev spaces H 1 (R n, R) and H 1 (R n, R n+1 ). At this point we recall a compact embedding theorem of Benci and Fortunato (see [5]), which will be important in the sequel: Theorem 2.1. The embedding of the space Γ(R n, R) into the space L 2 (R n, R) is compact. We shall denote by: E the completion of C0 (R n, R n+1 ) with respect to the norm ( ) 2/p u 2 E = V (x) u(x) 2 dx + u(x) 2 dx + u(x) p dx. R n R n R n The main properties of the Banach space E are summarized in the following lemma and corollary: Lemma 2.1. The Banach space E is continuously embedded into the space L s (R n, R n+1 ) for 2 s. For the proof see [4].

6 196 V. Benci A. M. Micheletti D. Visetti Corollary 2.1. (i) The Banach space E is continuously embedded into the Sobolev space W 1,p (R n, R n+1 ). (ii) 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 (p n)/p u L p (R n,r n+1 ). (iii) If u E then lim x u(x) = Topological charge and connected components of Λ. In the space E we can consider the open subset (8) Λ = u E ξ u(r n )}. We recall now the definition of topological charge introduced by Benci, Fortunato and Pisani in [7] (we report here the definition given in [4]). We write the n + 1 components of a function u E in the following way: where ũ : R n R n and u : R n R. u(x) = (ũ(x), u(x)), Definition 1. Let u be a function in Λ E, then the port of u is the following set: K u = x R n u(x) > ξ}, where ξ is defined in (1). The topological charge of u is the following function: deg(ũ, Ku, 0) if K u, (9) ch (u) = 0 if K u =. As a consequence of the fact that u is continuous and lim x u(x) = 0 (see Corollary 2.1), K u is an open bounded subset of R n. Since u Λ, if x K u, we have u(x) = ξ and ũ(x) 0. Therefore the previous definition is well posed. Moreover, the topological charge is continuous with respect to the uniform convergence (see [7]): Lemma 2.2. 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 (u) = ch (v). The set Λ E is divided into connected components by the topological charge: Λ = q Z Λ q,

7 where An Eigenvalue Problem for a Quasilinear Elliptic Field Equation on R n 197 (10) Λ q = u Λ ch (u) = q} Palais Smale condition for the energy functional. First of all we verify that the functional J ε is well defined on the set Λ, that is: J ε (u) < for all u Λ. It is enough to check that W (u(x)) dx <. In fact by (W R n 2 ) and (W 4 ) we have that W (u(x)) dx c 4 u(x) 2 dx + W (u(x)) dx, R n B R n \B where B = x R n u(x) B(0, c 3 )}. The first integral is bounded because B u(x) 2 dx R n u(x) 2 dx <. The second integral is bounded because by Corollary 2.1 the domain R n \ B is bounded. Lemma 2.3. The energy functional J ε is of class C 1 on the open set Λ of E. Proof. The first part of the energy functional is clearly of class C 1. Then we consider G(u) = W (u(x)) dx. Now we want to prove the Gateaux differentiability, hence we show R n that lim t 0 R n [ W (u + tv) W (u) t ] W (u) v dx = 0 for all u Λ and for all v E. The integrand clearly tends to zero pointwise. By the Lagrange Theorem we have that W (u(x) + tv(x)) W (u(x)) = tw (u(x) + θtv(x)) v(x) for t R small enough, where θ = θ(x, t) [0, 1]. As lim x u(x) = 0, there exists R 1 > 0 such that u(x) c3 /2, x R n \ B(0, R 1 ) u(x) + θtv(x) c 3, for t t with t suitably small. Then by (W 4 ), we have the following inequalities W (u(x) + θtv(x)) v(x) c4 [ u(x) + t v(x) ] v(x) for all x R n \ B(0, R 1 ), const v(x) for all x B(0, R 1 ). There are analogous inequalities for W (u(x)) v(x). So we can apply the Lebesgue s dominated convergence theorem.

8 198 V. Benci A. M. Micheletti D. Visetti To have the Fréchet differentiability of the functional G it remains to show that the Gateaux derivative v G (u)(v) = W (u) v dx R n u Λ, v E is continuous with respect to u. Let u i } i N be a sequence in Λ strongly converging to u 0 Λ, then we have G (u i ) G (u 0 ) E = [W (u i ) W (u 0 )] v v E R n v E =1 [ W (u i ) W (u 0 ) 2 v E R n v E =1 [ ] 1/2 C W (u i ) W (u 0 ) 2, R n ] 1/2 v L2(R n,r n+1) where C is a constant. Obviously we have that for all x R n W (u i (x)) W (u 0 (x)) 0. Moreover, there exists R 2 > 0 such that u0 (x) c x R n 3 /2, \ B(0, R 2 ) u i (x) c 3, for i large enough; hence, for i large enough, we have c4 u W i (x) for all x R n \ B(0, R 2 ), (u i (x)) const for all x B(0, R 2 ), c4 u W 0 (x) for all x R n \ B(0, R 2 ), (u 0 (x)) const for all x B(0, R 2 ), and consequently W (u i (x)) W (u 0 (x)) 2 c 2 4 ( u i (x) + u 0 (x) ) 2 for all x R n \ B(0, R 2 ), const for all x B(0, R 2 ). We can now apply the generalized version of the Lebesgue s dominated convergence theorem and conclude that G (u i ) G (u 0 ) E 0. We put (11) S = u E u L 2 (R n,r n+1 ) = 1}. To get some critical points of the functional J ε on the C 2 manifold Λ S we use the following version of Palais Smale condition. For J ε C 1 (Λ, R), the norm of the derivative at u S of the restriction Ĵε = J ε Λ S is defined by Ĵ ε(u) = min t R J ε(u) tg (u) E, where g : E R is the function defined by g(u) = R n u(x) 2 dx.

9 An Eigenvalue Problem for a Quasilinear Elliptic Field Equation on R n 199 Definition 2. The functional J ε is said to satisfy the Palais Smale condition in c R on Λ S (on Λ q S, for q Z) if, for any sequence u i } i N Λ S (u i } i N Λ q S) such that J ε (u i ) c and Ĵ ε(u i ) 0, there exists a subsequence which converges to u Λ S (u Λ q S). To obtain the Palais Smale condition, we need a few technical lemmas. Lemma 2.4. Let u i } i N be a sequence in Λ q (with q Z) such that the sequence J ε (u i )} i N is bounded. We consider the open bounded sets (12) Z i = x R n u i (x) > c 3 }. Then the set i N Z i R n is bounded. Proof. By contradiction we pose that i N Z i is unbounded; then there exist a sequence of indices ν i for i and a sequence of points x νi } i N such that x νi Z νi and x νi. By (12) we have: (13) u νi (x νi ) > c 3 ; we consider the numbers R νi = R > 0 for all x B(x νi, R) u νi (x) > c 3 /2}. We claim that R νi 0 for i. In fact, if R νi 0, there exists M > 0 such that R νi > M for infinitely many indices. Then for such indices we have: V (x) u νi (x) 2 dx V (x) u νi (x) 2 dx R n B(x νi,r νi ) ( ) 2 c3 V (x) dx, 2 B(x νi,m) but B(x νi V (x) dx and this is a contradiction.,m) We choose now for every i N a point x νi B(x νi, R νi ), i.e. such that (14) u νi ( x νi ) = c 3 /2; it is clear that x νi x νi = R νi 0. As the functions u i are equiuniformly continuous, i.e. for all x, y R n and for all i N (see (ii) of Corollary 2.1) u i (x) u i (y) C 1 x y (p n)/p u i Lp (R n,r n+1 ) const x y (p n)/p, then u νi (x νi ) u νi ( x νi ) tends to zero for i. On the other hand, by (13) and (14), there holds: u νi (x νi ) u νi ( x νi ) u νi (x νi ) u νi ( x νi ) > c 3 /2. The next two lemmas are the Propositions 3.8 and 3.9 of [7]. The addition of the potential V in our equation leads to the loss of translation invariance. Hence we give a proof of Lemma 2.6. (see Proposition 3.9 in [7]), because the arguments of [7] partially fall.

10 200 V. Benci A. M. Micheletti D. Visetti Lemma 2.5. Let u i } i N Λ be a sequence weakly converging to u and such that J ε (u i )} i N R is bounded, then u Λ. Lemma 2.6. For any a > 0, there exists d > 0 such that for every u Λ J ε (u) a inf x R n u(x) ξ d. Proof. By contradiction we pose that there exist a > 0 and a sequence u i } i N Λ such that for any i N J ε (u i ) a and inf x R n u i (x) ξ 1/i. As we have u i E const, up to a subsequence u i weakly converges to u in E. In particular u i converges to u pointwise. Moreover, by Lemma 2.5, we know that u Λ. We denote by x i } i N a sequence of points in R n such that u i (x i ) ξ. We claim that x i } i N is bounded. By contradiction let x i tend to. We consider now R i = R 0 for all x B(x i, R), u i (x) B(ξ, c 1 )}, where c 1 is the constant defined in (W 3 ); proceeding in the same way as in the proof of Lemma 2.4, we obtain that R i 0. For every i N we choose a point x i on the boundary of B(x i, R i ), i.e. x i is such that u i ( x i ) ξ = c 1 and x i x i 0. Now by the equiuniform continuity we have u i ( x i ) u i (x i ) 0, but this is absurd because u i ( x i ) u i (x i ) = u i ( x i ) ξ + ξ u i (x i ) c 1 u i (x i ) ξ and c 1 u i (x i ) ξ c 1 > 0. Then x i } i N is bounded and up to a subsequence x i x 0. Since we have u i (x i ) u(x 0 ) u i (x i ) u i (x 0 ) + u i (x 0 ) u(x 0 ), by equiuniform continuity and by pointwise convergence we can conclude that u i (x i ) u(x 0 ) 0. This means that u(x 0 ) = ξ and this is in contradiction with the fact that u Λ. Proposition 2.1. The functional J ε satisfies the Palais Smale condition on Λ S (on Λ q S for q Z) for any c R and 0 < ε 1. Proof. It is immediate that every Palais Smale sequence u i } i N on Λ S is bounded in E. Hence we can choose a subsequence, which for simplicity we denote again u i } i N, converging to a function u weakly in E, strongly in L 2 (R n, R n+1 ) (by Theorem 2.1) and uniformly on every compact subset of R n. As we have min J ε(u i ) tg (u i ) E 0, t R

11 An Eigenvalue Problem for a Quasilinear Elliptic Field Equation on R n 201 there is a sequence η i > 0, with η i 0 for i and a sequence t i R such that for all v E (15) [ u i v + V (x)u i v + ε r u i p 2 u i v + ε r W (u i ) v] dx R n 2t i u i v dx R η i v E. n From the substitution v = u i in (15), we obtain (16) [ u i 2 + V (x) u i 2 + ε r u i p + ε r W (u i ) u i ] dx 2t i η i u i E. R n Since u i } i N is bounded in E, the first three terms are bounded. Lemma 2.4 and by (W 4 ), we observe that W (u i ) u i dx W R Z (u i ) u i dx + W n i R (u i ) u i dx n \Z i W (u i ) u i dx + c 4 u i 2 L 2 (R n,r n+1 ) K Now, by where Z i is defined in (12) and K is a compact subset of R n such that i=1 Z i K. Hence the fourth term of (16) is bounded too and so t i } i N is bounded. Substituting now v = u i u, we get: [ u i (u i u) + V (x)u i (u i u) + ε r u i p 2 u i (u i u) R n + ε r W (u i ) (u i u)] dx 2t i u i (u i u) dx R η i u i u E n and we write R n [ u i (u i u) + V (x) u i (u i u) + ε r u i p 2 u i (u i u)] dx = ε r R n [W (u i ) (u i u) + 2t i u i (u i u)] dx + o(1). As t i } i N is bounded and u i } i N converges to u in L 2 (R n, R n+1 ) we get that t i R n u i (u i u) dx tends to zero. Moreover by Lemma 2.4 and (W 4 ) W (u i ) (u i u) dx W R Z (u i ) (u i u) dx n i + c 4 u i L2 (R n,r n+1 ) u i u L2 (R n,r n+1 ) C u i u L (K,R n+1 ) + c 4 u i L 2 (R n,r n+1 ) u i u L 2 (R n,r n+1 ),

12 202 V. Benci A. M. Micheletti D. Visetti where Z i is defined in (12), C is a constant and K is a compact subset of R n such that i=1 Z i K; hence this term tends to zero. Concluding [ u i (u i u) + V (x) u i (u i u) + ε r u i p 2 u i (u i u)] dx 0 R n for i. At this point we recall that p is a monotone operator (see [16] and [4]), and there exists ν > 0 such that for all u 1, u 2 E R n [ (u 1 u 2 ) 2 + V (x) u 1 u u 1 p 2 u 1 (u 1 u 2 ) u 2 p 2 u 2 (u 1 u 2 )] dx u 1 u 2 2 Γ(R n,r n+1 ) + ν (u 1 u 2 ) p L p. Hence we get our claim Eigenvalues of the Schrödinger operator. By the compactness result cited in Theorem 2.1 we obtain the discreteness of the spectrum of the Schrödinger operator +V (x) on L 2 (R n, R) (which is the self-adjoint extension of the operator +V (x) on C 0 (R n, R)). That is the spectrum of the operator + V (x) consists of a countable set of eigenvalues of finite multiplicity. The following sequence denotes the eigenvalues counted with their multiplicity: λ 1... λ k... We denote by e i } i N the sequence of the corresponding eigenfunctions, with (e i, e j ) L 2 (R n,r) = δ ij. We consider now the sequence of the eigenvalues of the problem λ 1... λ m... (17) u + V (x)u = λu with u Γ(R n, R n+1 ). If u = (u 1,..., u n+1 ), then (17) is equivalent to u i + V (x)u i = λu i with i = 1,..., n + 1. It is trivial that λ 1 = λ 1 =... = λ n+1 λ n+2, in fact if λ is an eigenvalue of multiplicity ν of the problem z + V (x)z = λz with z Γ(R n, R), then it is an eigenvalue of (17) of multiplicity (n + 1)ν. Moreover, if λ k < λ k+1, then λ (n+1)k < λ (n+1)k+1. If we set ẽ j = (e j, 0,..., 0), ẽ j+1 = (0, e j,..., 0),..., ẽ j+n = (0, 0,..., e j ), it is clear what we mean by the sequence of the eigenvectors ϕ i } i N corresponding to the sequence λ i } i N, which is an orthonormal set in L 2 (R n, R n+1 ).

13 An Eigenvalue Problem for a Quasilinear Elliptic Field Equation on R n 203 The main properties of the eigenvalues λ i } i N and λ i } i N are summarized in the following lemma: Lemma 2.7. The following properties hold: and λ i = λ i = min w Γ(R n,r) (w,e j ) L 2 (R n,r) =0 j=1,...,i 1 min u Γ(R n,r n+1 ) (u,ϕ j ) L 2 (R n,r n+1 ) =0 j=1,...,i 1 w 2 Γ(R n,r) w 2 L 2 (R n,r) u 2 Γ(R n,r n+1 ) u 2, L 2 (R n,r n+1 ) (e i, e j ) Γ(Rn,R) = λ i δ ij for all i, j N, (ϕ i, ϕ j ) Γ(Rn,R n+1 ) = λ i δ ij for all i, j N. If we set E m = span [e 1,..., e m ] and E m = w Γ(R n, R) (w, e i ) L2 (R n,r) = 0 for i = 1,..., m}, we get, w E m λ 1 w 2 Γ(R n,r) w 2 λ m, L 2 (R n,r) w Em w 2 Γ(R n,r) w 2 λ m+1. L 2 (R n,r) If we set, respectively, F m = span [ϕ 1,..., ϕ m ] and F m = u Γ(R n, R n+1 ) (u, ϕ i ) L2 (R n,r n+1 ) = 0 for i = 1,..., m}, we get (18) (19) u F m λ 1 u 2 Γ(R n,r n+1 ) u 2 λ m, L 2 (R n,r n+1 ) u Fm u 2 Γ(R n,r n+1 ) u 2 λ m+1. L 2 (R n,r n+1 ) The proof is a direct consequence of classical argumentations of spectral theory. Now we recall the following estimate about the eigenfunctions of the Schrödinger operator (see [9, p. 169]): Remark 1. If z Γ(R n, R) is such that z + V (x)z = λz, then for any a > 0 there exists a constant c a such that (20) z(x) c a e a x. By this result and the regularity theorems we get the following lemma.

14 204 V. Benci A. M. Micheletti D. Visetti Lemma 2.8. The eigenfunctions ϕ i Γ(R n, R n+1 ) of the Schrödinger operator + V (x) belong to the Banach space E. Proof. By a regularity result, if z Γ(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). By this fact the statement follows immediately. Now we verify that V z L 2 (R n, R) L p (R n, R). By Remark 1 and (V 2 ) we get V (x)z(x) p dx const V (x)e x p L p (R n,r) <. R n Moreover, if R > 0 is such that for x R n \ B(0, R) V (x) > 1, we have V (x)z(x) 2 dx R n ( ) const V (x) 2 e p x dx + V (x) p e p x dx <. B(0,R) R n \B(0,R) 3. Critical values of the energy functional on every manifold Λ q S 3.1. The functions G q ε. Fixed an integer k N, we define (21) M k = u L (R n,r n+1 ) u S(k) where (22) S(k) = F k S for all k N. At this point we choose the (n + 1)-th coordinate ξ of the point ξ defined in (1) in such a way that (23) ξ > 2M k. First of all we construct a function G ρ depending on a parameter ρ > 0. We consider two functions ϕ ρ, ψ ρ : R + [0, 1] of class C such that (24) ϕ ρ (r) = 1 for 0 r ρ 2, 0 for r 4ρ 2, ψ ρ (r) = 1 for 0 r 9ρ 2, 0 for r 16ρ 2. Moreover, ϕ ρ and ψ ρ take values between 0 and 1 for ρ 2 r 4ρ 2 and 9ρ 2 r 16ρ 2, respectively. We define: (25) G ρ : B(0, 5ρ) R n (R n R) \ ξ }, ( ) ξ x ψ ρ ( x 2 ) ρ x, 2ξϕ ρ( x 2 ). It is important to observe that the distance of the image of G ρ from the point ξ is ξ. We can now introduce for any q Z \ 0} the functions G q ε.

15 An Eigenvalue Problem for a Quasilinear Elliptic Field Equation on R n 205 Definition 3. If q Z \ 0} and 0 < ε 1, we set Gρi (γ q (x x i )/ε) for x B( x i, 5ερ i ) and i = 1,..., q, (26) G q ε(x) = 0 for x R n \ q i=1 B( x i, 5ερ i ), where G ρ is defined in (25), γ q is the following function from R n to R n (x1, x 2,..., x n ) for q > 0, γ q (x 1, x 2,..., x n ) = ( x 1, x 2,..., x n ) for q < 0, and the points x i and the radiuses ρ i are such that 1. B( x i, ρ i ) B( x j, ρ j ) = for all i j, i, j = 1,..., q, 2. G q 1 L 2 (R n,r n+1 ) < 1. Finally, we define G q = G q 1. Remark 2. We note that by construction the image of G q ε does not intersect the point ξ and the distance of the image from the point is ξ. Moreover, even if we expand the functions G q ε (0 < ε 1) of a factor t 1, their image is such that they do not meet the point ξ and the distance is still ξ. Hence tg q ε Λ q for all t 1 and ε (0, 1]. Remark 3. By the definition of the functions G q ε and by Remark 2 we can conclude that for any q Z we have that Λ q S. The following lemma presents some useful properties of the functions G q ε which will be crucial in the sequel: Lemma 3.1. There exist ρ > 0 and ε, with 0 < ε 1, such that for all 0 < ε ε we have (i) G q ε + ρu L2 (R n,r n+1 ) 1 for all u S(k), (ii) G q ε + ρu L2 (R n,r n+1 ) > 0, (iii) (iv) inf ε (0,ε] u S(k) inf x R n ε (0,ε] u S(k) G q ε(x) + ρu(x) G q ξ > ξ ε + ρu L 2 (R n,r n+1 ) 2, G q ε + ρu G q Λ q S for all u S(k). ε + ρu L2 (R n,r n+1 ) For the proof see [8] The critical values c q ε,j of the energy functional on Λ q S. Now we can introduce some definitions which we will use to study multiplicity of solutions. Definition 4. Fixed k N, q Z \ 0} and 0 < ε ε, where ε is defined in Lemma 3.1, we set (27) M q ε,j = G q } ε + ρu G q ε + ρu u S(j) L 2 (R n,r n+1 )

16 206 V. Benci A. M. Micheletti D. Visetti with j k and ρ defined in Lemma 3.1. Remark 4. It is trivial that for j k we have M q ε,j 1 Mq ε,j, where M q ε,0 =. By Lemma 3.1 we can claim that Mq ε,j Λ q S. Moreover, M q ε,j is a submanifold of Λ q S for ε sufficiently small. Definition 5. Fixed k N, for all q Z \ 0}, j k and 0 < ε ε (ε is defined in Lemma 3.1), we introduce the following values: (28) c q ε,j = inf h H q ε,j v M q ε,j J ε (h(v)), where H q ε,j are the following sets of continuous transformations: H q ε,j = h : Λ q S Λ q S h continuous, h M q ε,j 1 = id M q ε,j 1 }. We observe that H q ε,j+1 Hq ε,j. Lemma 3.2. Fixed k N, for all q Z \ 0}, j < k and 0 < ε ε, we have (i) c q ε,j cq ε,j+1, (ii) c q ε,j R. In the following we will use the version of the deformation lemma on a C 2 manifold which we now recall (see for example [14], [18] and [19]). Lemma 3.3 (Deformation Lemma). Let J be a C 1 -functional defined on a C 2 -Finsler manifold M. 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] M M such that: η(0, u) = u for all u M, η(t, u) = u for all t [0, 1] and all u J c 2δ, η(1, J c+δ ) J c δ. Lemma 3.4. For any q Z, ε (0, 1] and a R, the subset Λ q S Jε a the Banach space E is complete. of We give some notations: if u E we set j (29) P Fj u = (u, ϕ i ) Γ(Rn,R n+1 )ϕ i and Q Fj u = u P Fj u. i=1 It is immediate that (30) (Q Fj u, ϕ i ) Γ(R n,r n+1 ) = λ i (Q Fj u, ϕ i ) L 2 (R n,r n+1 ) = 0 for all i = 1,..., j. We can now prove the main result:

17 An Eigenvalue Problem for a Quasilinear Elliptic Field Equation on R n 207 Theorem 3.1. Given q Z\0} and k N, we consider ξ = (0, ξ) R n+1 with ξ > 2M k, where M k = u S(k) u L (R n,r n+1 ). Then there exists ε (0, 1] such that for any ε (0, ε] and for any j k with λ j 1 < λ j, we get that c q ε,j is a critical value for the functional J ε restricted to the manifold Λ q S. Moreover, c q ε,j 1 < cq ε,j and cq ε,j λ j for ε 0. Proof. In the following proof we will denote by L q and Γ the norms respectively in L q (R n, R n+1 ) and in Γ(R n, R n+1 ). We divide the argument into three steps. Step 1. We prove that (31) (32) J ε (v) λ j + σ(ε), v M q ε,j c q ε,j λ j + σ(ε), where lim ε 0 σ(ε) = 0. First of all we verify that (33) J 0 (v) λ Q Fj G q j + ε 2 Γ v M q u S(j) P Fj G q ε + ρu 2 ε,j L + Q 2 Fj G q ε 2. L 2 In fact by Definition 4, (29) and (30) we have: J 0 (v) = v M q ε,j u S(j) u S(j) G q 2 ε + ρu G q ε + ρu L 2 Γ ( PFj G q ε + ρu 2 Γ P Fj G q ε + ρu 2 + L 2 = u S(j) P Fj G q ε + ρu 2 Γ + Q F j G q ε 2 Γ P Fj G q ε + ρu 2 L + Q 2 Fj G q ε 2 L 2 Q Fj G q ε 2 ) Γ P Fj G q ε + ρu 2 L 2 + Q Fj G q ε 2 L 2 λ Q Fj G q j + ε 2 Γ u S(j) P Fj G q ε + ρu 2 L + Q 2 Fj G q ε 2. L 2 Now, by definition of J ε and (33), we prove the following inequalities: (34) c q ε,j = inf h H q ε,j v M q ε,j v M q ε,j J 0 (v) + ε r J ε (h(v)) v M q ε,j v M q ε,j R n J ε (v) ( ) 1 p v p + W (v) dx λ j + u S(j) + εr p u S(j) + ε r u S(j) R n W Q Fj G q ε 2 Γ P Fj G q ε + ρu 2 L + Q 2 Fj G q ε 2 L 2 (G q R n ε + ρu) p dx G q ε + ρu p L ( 2 ) G q ε + ρu dx. G q ε + ρu L 2

18 208 V. Benci A. M. Micheletti D. Visetti At this point we note that lim ε 0 Q Fj G q ε 2 Γ = 0; in fact by (29) and (30), by the fact that the port of G q ε is contained in the port of G q for all ε < 1 and by the fact that V L 2 loc (Rn, R), we have Q Fj G q ε 2 Γ G q ε 2 Γ G q ε 2 dx + V L2 (Ω,R) G q ε 2 L 4 R n = ε n 2 G q 2 dx + ε n 2 V L2 (Ω,R) G q 2 L 4 R n where Ω R n is the port of G q. Moreover, by (ii) of Lemma 3.1 we obtain 0<ε ε u S(j) 1 P Fj G q ε + ρu 2 L 2 + Q Fj G q ε 2 L 2 <, in fact P Fj G q ε 2 L ε n G q 2 2 L and Q 2 Fj G q ε 2 L ε n G q 2 2 L. Therefore the 2 second term of the last inequality of (34) goes to zero when ε goes to zero. Now we observe that the following inequality holds: ) (G q ε + ρu) p dx const (ε n p G q p dx + ρ p u p dx. R n R n R n Then by this inequality and (ii) of Lemma 3.1 (we recall that r > p n), we have that the third term of the last inequality of (34) tends to zero when ε tends to zero. As regards the last term, we verify that W ((G q R n ε + ρu)/ G q ε + ρu L 2) dx is bounded. In fact by definition of G q ε and by the exponential decay of the eigenfunctions (see Remark 1) there exists a ball B(0, R) such that, if we write u = j i=1 a iϕ i with j i=1 a2 i = 1, for all x Rn \ B(0, R) the following inequalities hold G q ε(x) + ρu(x) G q ε + ρu L 2 = ρ u(x) G q ε + ρu L 2 const ρ( j i=1 a i )e x G q Me x < c 3 ε + ρu L 2 where the constant M does not depend on u S(j) nor on ε for ε small enough (see the point (ii) of Lemma 3.1). By (W 4 ) we get ( ) G q W ε (x) + ρu(x) G q G q c ε(x) + ρu(x) 2 4 ε + ρu L 2 G q ε + ρu 2 L 2 for any x R n \ B(0, R). Concluding we have ( ) G q ε + ρu G q dx ε + ρu c 4 + L 2 R n W B(0,R) ( ) G q W ε + ρu G q dx ε + ρu L 2 where the integral on the right hand side is bounded by (iii) of Lemma 3.1. So we have the claim. Step 2. We prove that c q ε,j λ j.

19 An Eigenvalue Problem for a Quasilinear Elliptic Field Equation on R n 209 By positivity of W the following inequalities hold c q ε,j inf h H q ε,j v M q ε,j h(v) 2 Γ inf h H q ε,j v M q ε,j P Fj 1 h(v)=0 h(v) 2 Γ. By an argument of degree theory we get that for any h H q ε,j the intersection of the set h(m q ε,j ) with the set u E (u, ϕ i) Γ = 0 for all i = 1,..., j 1} is not empty, that is there exists v M q ε,j such that P F j 1 h(v) = 0 (for the proof see [8]). Now by (19) in Lemma 2.7 we obtain c q ε,j λ j. Step 3. If λ j 1 < λ j, then c q ε,j is a critical value for the functional J ε on the manifold Λ q S and c q ε,j 1 < cq ε,j for ε small enough. We begin by noting that (35) (36) v M q ε,j 1 c q ε,j 1 < cq ε,j, J ε (v) < c q ε,j ; in fact, by Steps 1 and 2, we obtain for ε sufficiently small, c q ε,j 1 λ j 1 + σ(ε) < λ j c q ε,j, J ε (v) λ j 1 + σ(ε) < λ j c q v M q ε,j. ε,j 1 Now we pose by contradiction that c q ε,j is a regular value for J ε on Λ q S. By Proposition 2.1 and Lemmas 3.3, 3.4 there exist δ > 0 and a deformation η : [0, 1] Λ q S Λ q S such that η(0, u) = u for all u Λ q S, η(t, u) = u η(1, J cq ε,j +δ ε ) J cq ε,j δ ε. By (36) we can pose for all t [0, 1] and all u J cq ε,j 2δ ε, (37) J ε (v) < c q ε,j 2δ. v M q ε,j 1 Moreover, by definition of c q ε,j there exists a transformation ĥ Hq ε,j such that v M q Jε(ĥ(v)) < cq ε,j ε,j + δ. Now by the properties of the deformation η and by (37) we get η(1, ĥ( )) Hq ε,j and v M q J ε(η(1, ĥ(v))) < cq ε,j ε,j δ and this is a contradiction Minima of the energy functional on Λ q S. Finally we can get the minimum values of the functional J ε on each manifold Λ q S, with q Z, for any ε > 0 and for any ξ = (0, ξ).

20 210 V. Benci A. M. Micheletti D. Visetti Theorem 3.2. Given q Z, for any ξ = (0, ξ) with 0 R n and ξ > 0 and for any ε > 0, there exists a minimum for the functional J ε on the submanifold Λ q S of Λ S. Proof. The claim follows by the fact that Λ q S is not empty (see Remark 3) and the functional J ε is bounded from below and satisfies the Palais Smale condition on Λ q S (see Proposition 2.1). Remark 5. The minimum critical value of J ε on Λ q S is not obtained by Theorem 3.1 and coincides by definition with c q ε,1 (Definition 5). Moreover, the minimum critical value c q ε,1 tends to λ 1 for ε that tends to 0. References [1] A. Abbondandolo and V. Benci, Solitary waves and Bohmian mechanics, Proc. Nat. Acad. Sci. U.S.A. (to appear). [2] M. Badiale, V. Benci and T. D Aprile, Existence, multiplicity and concentration of bound states for a quasilinear elliptic field equation, Calc. Var. Partial Differential Equations (to appear). [3] A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc. 267 (1981), [4] V. Benci, P. D Avenia, D. Fortunato and L. Pisani, Solitons in several space dimensions: Derrick s problem and infinitely many solutions, Arch. Rational Mech. Anal. 154 (2000), [5] V. Benci and D. Fortunato, Discreteness conditions of the spectrum of Schrödinger operators, J. Math. Anal. Appl. 64 (1978). [6] V. Benci, D. Fortunato, A. Masiello and L. Pisani, Solitons and the electromagnetic field, Math. Z. 232 (1999), [7] V. Benci, D. Fortunato and L. Pisani, Soliton-like solution of a Lorentz invariant equation in dimension 3, Rev. Math. Phys. 10 (1998), [8] V. Benci, A. M. Micheletti and D. Visetti, An eigenvalue problem for a quasilinear elliptic field equation, J. Differential Equations (to appear). [9] F. A. Berezin and M. A. Shubin, The Schrödinger Equation, Kluwer Academic Publishers, [10] P. Bolle, On the Bolza problem, J. Differential Equations 152 (1999), [11] P. Bolle, N. Ghoussoub and H. Tehrani, The multiplicity of solutions in nonhomogeneous boundary value problems, Manuscripta Math. 101 (2000), [12] R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. I, Interscience, New York. [13] C. H. Derrick, Comments on nonlinear wave equations as model for elementary particles, J. Math. Phys. 5 (1964), [14] N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory, Cambridge University Press, [15] T. Kato, Perturbation Theory for Linear Operators, Springer Verlag, Berlin, [16] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod-Gauthier Villar, Paris, 1969.

21 An Eigenvalue Problem for a Quasilinear Elliptic Field Equation on R n 211 [17] A. Manes and A. M. Micheletti, Un estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital. A (7) 4 (1973), [18] R. S. Palais, Lusternik Schnirelman theory on Banach manifolds, Topology 5 (1966), [19] P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, C.B.M.S. Reg. Conf. 65, Amer. Math. Soc., Providence, Manuscript received February 16, 2001 V. Benci, A. M. Micheletti and D. Visetti Dipartimento di Matematica Applicata U. Dini Università degli studi di Pisa via Bonanno 25/b Pisa, ITALY address: benci@dma.unipi.it, a.micheletti@dma.unipi.it, visetti@dm.unipi.it TMNA : Volume N o 2