Martin Schechter 1, and Wenming Zou 2, 1. Introduction WEAK LINKING THEOREMS AND SCHRÖDINGER EQUATIONS WITH CRITICAL SOBOLEV EXPONENT

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1 ESAIM: Control, Optimisation and Calculus of Variations August 23, Vol. 9, DOI: 1.151/cocv:2329 WEAK LINKING THEOREMS AND SCHRÖDINGER EQUATIONS WITH CRITICAL SOBOLEV EXPONENT Martin Schechter 1, and Wenming Zou 2, Abstract. In this paper we establish a variant and generalized weak linking theorem, which contains more delicate result and insures the existence of bounded Palais Smale sequences of a strongly indefinite functional. The abstract result will be used to study the semilinear Schrödinger equation u + V (x)u = K(x) u 2 2 u + g(x, u),u W 1,2 ( ), where N 4; V,K,g areperiodicinx j for 1 j N and is in a gap of the spectrum of +V ; K>. If <g(x, u)u c u 2 for an appropriate constant c, we show that this equation has a nontrivial solution. Mathematics Subject Classification. 35B33, 35J65, 35Q55. Received October 28, Introduction In this article, the aim is to study the following semilinear Schrödinger equation with critical Sobolev exponent and periodic potential: u + V (x)u = K(x) u 2 2 u + g(x, u), u W 1,2 ( ), (S) where N 4; 2 := 2N/(N 2) is the critical Sobolev exponent and g is of subcritical growth. First of all, we recall that the equation u + λu = u 2 2 u, λ, (1.1) has only the trivial solution u =inw 1,2 ( )(cf. [4]). Therefore, the existence of nontrivial solution of (S) is an interesting problem. Before we state the main result, we introduce the following conditions: (S 1 ) V,K C(, R),g C( R, R),k := inf x K(x) > ; V,K,g are 1-periodic in x j for j = 1,..., N; (S 2 ) σ( +V )andσ( +V ) (, ), where σ denotes the spectrum in L 2 ( ); (S 3 ) K(x ):=max x K(x) andk(x) K(x )=o( x x 2 )asx x and V (x ) < ; Keywords and phrases. Linking, Schrödinger equations, critical Sobolev exponent. 1 Department of Mathematics, University of California, Irvine, CA , USA; mschecht@math.uci.edu Supported in part by a NSF grant. 2 Department of Mathematical Sciences, Tsinghua University, Beijing 184, China; wzou@math.tsinghua.edu.cn Supported by NSFC grant (GP1119). This work was finished when W. Zou was visiting the University of California at Irvine during He thanks all the members of the Math Department of UCI for offering this position. c EDP Sciences, SMAI 23

2 62 M. SCHECHTER AND W. ZOU (S 4 ) g(x, u) c (1+ u p 1 ) for all (x, u) R, wherec > andp (2, 2 ). Moreover, g(x, u)/ u 2 1 asu uniformly for x ; (S 5 ) g(x, u)u > for all x and u. The main result is the following: Theorem 1.1. Assume that (S 1 S 5 ) hold. If k N 2 g(x, u)u, where m g := max, (1.2) m g 2 x,u R\{} u 2 then equation (S) has a solution u.particularly,ifk(x) k >, (S 3 ) can be deleted and the same result holds. An equivalent form of Theorem 1.1 is the following: Corrolary 1.1. Assume that (S 1 S 5 ) hold. Then the following Schrödinger equation u + V (x)u = K(x) u 2 2 u + βg(x, u), u W 1,2 ( ), has a nontrivial solution for all β (, 2k /(m g (N 2))]. If K(x) k >, condition (S 3 ) can be omitted and the same result holds. Remark 1.1. It is an open problem whether or not the results of the present paper remain true for the case of N = 3. This problem is also raised by Y.Y. Li in private communications. Now we make some comments on this problem and the main results. Under the hypotheses on V the spectrum of +V in L 2 ( ) is purely continuous and bounded below and is the union of disjoint closed intervals (cf. Th. XIII. 1 of [17] and Th of [13]), which makes the problem difficult to be dealt with. Recently, equation (S) was studied in [6], which also generalized the early results obtained in [7]. In [6], the assumption γg(x, u) ug(x, u) on R, (1.3) where γ =2;G(x, u) := u g(x, s)ds, was imposed in order to prove the boundedness of the Palais Smale sequence. Obviously, this condition contains the case of g. Condition (1.3) has three disadvantages: the first is that one has to compute the primitive function G of g; the second is that one has to check the second inequality of (1.3); the third is that (1.3) does not contain the sublinear (at infinity) case and some asymptotically linear (at infinity) case. But sometimes, it is either impossible to compute G so that (1.3) can be checked or the second inequality of (1.3) does not hold. These cases happen on the following three examples: { c u (i) g(x, u) := 2 ue sin2 u u 1 c u 2/3 ue sin2 u (1 + ln u ) u 1, { c u 2 u u 1 (ii) g(x, u) := c u 2/3 u u 1, (sublinear at infinity) { c u 2 u u 1 (iii) g(x, u) := c 2 (u + u 2/3 u) u 1, (asymptotically linear at infinity). However, we emphasize that the above examples satisfy the hypotheses of Theorem 1.1 of the present paper for appropriate c>. Moreover, conditions (S 4 )and(s 5 ) permit the nonlinearity g to be superlinear, asymptotically linear or sublinear. Evidently, if we set g(x, u)u m g (r) = max, x, u r or u 1/r u 2

3 WEAK LINKING THEOREMS AND SCHRÖDINGER EQUATIONS WITH CRITICAL SOBOLEV EXPONENT 63 then k / m g (r) as r. It is an open problem whether or not assumption (1.2) can be concealed or equivalently, Corollary 1.1 holds for all β>. On the other hand, it should be mentioned that (1.2) is the price to pay for relaxing (1.3). Equation (S) with K(x), i.e., the nonlinear term is of subcritical growth, has been studied by several authors (for example, cf. [1 3, 5, 8, 1 12, 24, 26] and the references cited therein). In those papers, the Ambrosetti Rabinowitz condition (1.3) with γ>2was needed. In [23], the authors considered the asymptotically linear case. In [25] (see also [3]), zero is an end point of σ( +V ). In [14], the author studied a special case u = Ku 5 in R 3 (see also [15] for higher dimension case on S N ). Very little is known for (S) with critical Sobolev exponent and periodic potential. Without (1.3) with γ 2, the problem becomes more complicated. The main obstacle is how to get a bounded Palais Smale sequence. To get over this road block, we establish a variant and generalized weak linking theorem. Roughly speaking, let E be a Hilbert space, let N E be a separable subspace, and let Q N be a bounded open convex set, with p Q. Let F be a weak continuous map of E onto N such that F Q = id and that F (u v) (F (u) F (v)) is contained in a fixed finite-dimensional subspace of E for all u, v E. Then under suitable hypotheses, Q links F 1 (p ) with respect to a suitable restricted class of deformations of Q. We will define a family of C 1 -functional {H λ } λ [1,2] which is related to problem (S). Since the spectrum of +V in L 2 ( ) is purely continuous, both positive and negative subspaces of the functional H λ are infinite-dimensional. Moreover, H λ is unbounded from both below and above, the so-called strongly indefinite functional. Furthermore, because of the weaker assumptions (particularly, without (1.3)), the usual minimax techniques (e.g. [1 7, 1, 11, 14, 15]), can not be used here. However, by using the new weak linking theorem, we show that {H λ } has a bounded Palais Smale sequence for almost every λ [1, 2]. The main idea is the Monotonicity Trick due to [1, 11] (see also [21] for an earlier application). Other applications of this trick can be found in [23, 27, 28, 3 32]. We also give the estimates of the energy, i.e., the energy lies in [ inf H λ, sup H 1 ]. By this way, there is no need to impose some strong conditions for proving the boundedness F 1 (p ) Q of Palais Smale sequences. In other words, we permit much more freedom for the nonlinearity. The paper is organized as follows: in Section 2, we establish a variant weak linking theorem. In Section 3, equation (S) will be studied. In Section 4, an Appendix will be given. 2. A variant weak linking theorem Let E be a Hilbert space with norm and inner product, and have an orthogonal decomposition E = N N, where N E is a closed and separable subspace. Since N is separable, we can define a new norm v w satisfying v w v, v N and such that the topology induced by this norm is equivalent to the weak topology of N on bounded subset of N (see Appendix of Sect. 4). For u = v + w E = N N with v N,w N, we define u 2 w = v 2 w + w 2,then u w u, u E. w Particularly, if (u n = v n + w n )is w -bounded and u n u, thenvn vweakly in N, w n w strongly in N, u n v+ w weakly in E (cf. [9]). Let Q N be a -bounded open convex subset, p Q be a fixed point. Let F be a w -continuous map from E onto N satisfying F Q = id; F maps bounded sets to bounded sets; there exists a fixed finite-dimensional subspace E of E such that F (u v) (F (u) F (v)) E, v, u E; F maps finite-dimensional subspaces of E to finite-dimensional subspaces of E. We use the letter c to denote various positive constants. where Q denotes the -boundary of Q. A := Q, B := F 1 (p ),

4 64 M. SCHECHTER AND W. ZOU Remark 2.1. There are many examples: (i) let N = E, N = E +, then E = E E + and let Q := {u E : u <R}, p = Q. For any u = u u + E, define F : E N by Fu := u, then A := Q, B := F 1 (p )=E + satisfy the above conditions; (ii) let E = E E +, z E + with z =1. For any u E, we write u = u sz w + with u E,s R, w + (E Rz ) := E + 1. Let N := E Rz. For R>, let Q := {u := u + sz : s R +,u E, u <R},p = s z Q, s >. Let F : E N be defined by Fu := u + sz + w + z, then F, Q, p satisfy the above conditions with B = F 1 (s z )={u := sz + w + : s,w + E + 1, sz + w + = s } In fact, according to the definition, F Q = id and F maps bounded sets to bounded sets. On the other hand, for any u, v E, wewriteu = u + sz + w +,v = v + tz + w + 1, then F (u) =u + sz + w + z, F(v) =v + tz + w + 1 z, therefore, F (u v) (F (u) F (v)) = F (u v) =u v + (s t)z + w + w + 1 z, ( (s t)z + w + w + 1 sz + w + + tz + w + 1 ) z Rz := E (an 1-dimensional subspace). For H C 1 (E,R), we define { Γ:= h :[, 1] Q E,h is w -continuous. For any (s,u ) [, 1] Q, there is a w neighborhood U (s,u ) such that {u h(t, u) :(t, u) U (s,u ) ([, 1] Q)} E fin, h(,u)=u, H(h(s, u)) H(u), u Q }, then Γ since id Γ. Here and then, we use E fin to denote various finite-dimensional subspaces of E whose exact dimensions are irrelevant and depend on (s,u ). The variant weak linking theorem is: Theorem 2.1. The family of C 1 -functional (H λ ) has the form H λ (u) :=I(u) λj(u), λ [1, 2]. Assume (a) J(u), u E; H 1 := H; (b) I(u) or J(u) as u ; (c) H λ is w -upper semicontinuous; H λ boundedsetstoboundedsets; (d) sup H λ < inf H λ, λ [1, 2]. A B Then for almost all λ [1, 2], there exists a sequence (u n ) such that is weakly sequentially continuous on E. Moreover, H λ maps sup u n <, H λ(u n ), H λ (u n ) C λ ; n

5 WEAK LINKING THEOREMS AND SCHRÖDINGER EQUATIONS WITH CRITICAL SOBOLEV EXPONENT 65 where C λ := inf sup H λ (h(1,u)) [inf H λ, sup H]. h Γ u Q B Q Before proving this theorem, let us make some remarks. Remark 2.2. Similar weak linking was developed in [18 2, 29]. In [18 2], conditions F N id and F (v w) =v Fw for all v N,w E were stated but not needed. All that was used was F Q id and F (v w) =v Fw for all v Q, w E. This was noted in [29]. Particularly, we emphasize that because the monotonicity trick was not used in [18 2,29], the boundedness of Palais Smale sequence was not a consequence of the Theorems. Therefore, some compactness conditions were introduced and played an important role. The results of [18 2, 29] can not be used to deal with equation (S). Remark 2.3. In [12] (see also [26]), some theorems were given which contained only a particular linking and the boundedness of Palais Smale sequence is also remained unknown. Therefore, in applications, Ambrosetti Rabinowitz type condition (1.3) with γ > 2 is needed. In [12,26], a τ-topology is specially constructed to accommodate the splitting of E into subspace and by this, a new degree of Leray Schauder type is established. The new degree is also applied in [23, 25, 27, 28]. Proof of Theorem 2.1. Step 1. We prove that C λ [inf H λ, sup H]. Evidently, by the definition of C λ, B Q C λ sup u Q H λ (u) sup u Q H 1 (u) sup H(u) <. u Q To show C λ inf B H λ for all λ [1, 2], we have to prove that h(1, Q) B for all h Γ. By hypothesis, the map Fh :[, 1] Q N is w -continuous. Let K := [, 1] Q. Then K is w -compact. In fact, since K is bounded with respect to both norms w and, for any (t n,v n ) K, we may assume that v n v weakly in E and that t n t [, 1]. Then v Q since Q is convex. Since on the bounded set Q N, the w w -topology is equivalent to the weak topology, then u n v.so,kis w -compact. By the definition of Γ, for any (s,u ) K, there is a w -neighborhood U (s,u ) such that {u h(t, u) :(t, u) U (s,u ) K} E fin, here and then, we use E fin to denote various finite-dimensional subspaces of E whose exact dimensions are irrelevant. Now, K (s,u) K U (s,u). Since K is w -compact, K j i=1 U (s i,u i), (s i,u i ) K. Consequently, Hence, by the basic assumptions on F, {u h(t, u) :(t, u) K} E fin. F {u h(t, u) :(t, u) K} E fin and {u Fh(t, u) :(t, u) K} E fin. Then we can choose a finite-dimensional subspace E fin such that p E fin and that Fh :[, 1] ( Q E fin ) E fin. We claim that Fh(t, u) p for all u ( Q E fin )= Q E fin and t [, 1]. By way of negation, if there exist t [, 1] and u Q E fin such that Fh(t,u )=p, i.e., h(t,u ) B. It follows that H 1 (u ) H 1 (h(t,u )) inf H 1 > sup H 1, B Q

6 66 M. SCHECHTER AND W. ZOU which contradicts the assumption (d). Thus, our claim is true. By the homotopy invariance of Brouwer degree, we get that deg(fh(1, ),Q E fin,p )=deg(fh(, ),Q E fin,p ) =deg(id, Q E fin,p ) =1. Therefore, there exists u Q E fin such that Fh(1,u )=p. Step 2. Evidently, λ C λ is nonincreasing, hence C λ = dc λ exists for almost every λ [1, 2]. We consider dλ those λ [1, 2] where C λ exists and use the monotonicity trick (see e.g. [21]). Let λ n [1, 2] be a strictly increasing sequence such that λ n λ. Then there exists n(λ) large enough such that C λ 1 C λ n C λ C λ +1 for n n(λ). (2.1) λ λ n Step 3. There exists a sequence h n Γ, k := k(λ) > such that h n (1,u) k if H λ (h n (1,u)) C λ (λ λ n ). In fact, by the definition of C λn,leth n Γ be such that sup H λn (h n (1,u)) C λn +(λ λ n ). (2.2) u Q Therefore, if H λ (h n (1,u)) C λ (λ λ n )forsomeu Q, thenforn n(λ)(large enough), by (2.1) and (2.2), and By assumption (b), h n (1,u) k := k(λ). Step 4. By step 2 and (2.2) sup u Q Step 5. For ε>, define J(h n (1,u)) = H λ n (h n (1,u)) H λ (h n (1,u)) λ λ n C λ n C λ λ λ n +2 C λ +3 I(h n (1,u)) = H λn (h n (1,u)) + λ n J(h n (1,u)) C λn +(λ λ n )+λ n ( C λ +3) C λ λc λ +3λ. H λ (h n (1,u)) suph λn (h n (1,u)) C λ +(2 C λ )(λ λ n). u Q F ε (λ) :={u E : u k +4, H λ (u) C λ ε} (2.3) Then we claim, for ε small enough, that inf{ H λ (u) : u F ε(λ)} =. Otherwise, there exists ε > such that H λ (u) ε for all u F ε (λ). Let h n Γ be as in Steps 3, 4 and n be large enough such that λ λ n ε and (2 C λ )(λ λ n) ε. Define F ε (λ) :={u E : u k +4,C λ (λ λ n ) H λ (u) C λ + ε } (2.4)

7 WEAK LINKING THEOREMS AND SCHRÖDINGER EQUATIONS WITH CRITICAL SOBOLEV EXPONENT 67 Clearly, F ε (λ) F ε (λ). Now, we consider and F ε (λ) F (λ). Since H λ (u) ε for u F ε (λ), we let F (λ) :={u E : H λ (u) <C λ (λ λ n )} (2.5) h λ (u) := 2H λ (u) H λ (u) 2 for u F ε (λ). Then H λ (u),h λ(u) =2foru Fε (λ). Since H λ is weakly sequentially continuous, if {u n} is -bounded w and u n ū, then un ū in E, hence H λ (u n),h λ (u) H λ (ū),h λ(u) as n. It follows that H λ ( ),h λ(u) is w -continuous on sets bounded in E. Therefore, there is an open w -neighborhood N u of u such that H λ (v),h λ(u) > 1 for v N u,u F ε (λ). On the other hand, since H λ is w -upper semi-continuous, F (λ) is w -open. Consequently, N λ := {N u : u F ε (λ)} F (λ) is an open cover of F ε (λ) F (λ). Nowwemayfinda w -locally finite and w open refinement (U j ) j J with a corresponding w -Lipschitz continuous partition of unity (β j ) j J. For each j, we can either find u j F ε (λ) such that U j N uj, or if such u does not exist, then we have U j F (λ). In the first case we set w j (u) =h λ (u j ); in the second case, w j (u) =. Let U = j J U j, then U is w -open and F ε (λ) F (λ) U. Define Y λ (u) := j J β j (u)w j (u), (2.6) then Y λ : U E is a vector field which has the following properties: (1) Y λ is locally Lipschitz continuous in both and w topology; (2) H λ (u),y λ(u), u U ; (3) H λ (u),y λ(u) 1, u Fε (λ); (4) Y λ (u) w Y λ (u) 2/ε for u U and all λ [1, 2]. Consider the following initial value problem dη(t, u) dt = Y λ (η), η(,u)=u, for all u F (λ) F (λ, ε ), where F (λ) is given by (2.5) and F (λ, ε ):={u E : u k, C λ (λ λ n ) H λ (u) C λ + ε } F ε (λ). (2.7) Then by classical theory of ordinary differential equations and the properties of Y λ,foreachu as above, there exists a unique solution η(t, u) as long as it does not approach the boundary of U. Furthermore, t H λ (η(t, u)) is nonincreasing.

8 68 M. SCHECHTER AND W. ZOU Step 6. We prove that η(t, u) is w -continuous for t [, 2ε ],u F (λ, ε ) F (λ). For fixed t [, 2ε ],u F (λ, ε ) F (λ), we see that t ( ) η(t, u) η(t, u )=u u + Y λ (η(s, u )) Y λ (η(s, u)) ds. (2.8) Since the set Λ := η([, 2ε ] {u }) is compact and w -compact and Y λ is w -locally w -Lipschitz, there exist r 1 >,r 2 > such that {u E : inf e Λ u e w <r 1 } U and Y λ (u) Y λ (v) w r 2 u v w for any u, v Λ. Suppose that η(s, u) U for s t. Then by (2.8), η(t, u) η(t, u ) w u u w + By the Gronwall inequality (see e.g., Lem. 6.9 of [26]), t Y λ (η(s, u )) Y λ (η(s, u)) w ds t u u w + r 2 η(s, u ) η(s, u) w ds. η(t, u) η(t, u ) w u u w e r2t u u w e r2. If u u w <δ,where<δ<r 1 e r2, then η(t, u) η(t, u ) w <r 1. Therefore, if t t <δ, η(t, u) η(t,u ) w η(t, u) η(t, u ) w + η(t, u ) η(t,u ) w t η(t, u) η(t, u ) w + Y λ (η(s, u ))ds t w δe r2 + δc as δ. Step 7. Consider We prove that η Γ. η (t, u) = { hn (2t, u) t 1/2 η(4ε t 2ε,h n (1,u)) 1/2 t 1. Evidently, for u Q, we have either h n (1,u) F (λ) orc λ (λ λ n ) H λ (h n (1,u)). For the later case, we observe that h n (1,u) k by Step 3 and H λ (h n (1,u)) C λ + ε by Step 4, hence, h n (1,u) F (λ, ε ). In view of Step 6, η is w -continuous satisfying η (,u)=u and H(η (t, u)) H(u). Now for any (s,u ) [, 1] Q, since h n Γ, we first find a w -neighborhood U(s 1,u ) such that {u h n (s, u) :(s, u) U 1 (s,u ) ([, 1] Q)} E fin. (2.9) Furthermore, it is easy to see that there exists a w -neighborhood U 2 (s,u ) of (s,u ) such that {h n (s, u) h n (2s, u) :(s, u) U 2 (s,u ) ([, 1] Q)} E fin. (2.1) Next, we have to estimate h n (t, u) η(4ε t 2ε,h n (1,u)) for t [1/2, 1]. If H λ (h n (1,u )) <C λ (λ λ n ), then H λ (η(t, h n (1,u ))) H λ (h n (1,u )) <C λ (λ λ n ), for t [, 2ε ]. (2.11) Particularly, η(t, h n (1,u )) F (λ) (see (2.5)).

9 WEAK LINKING THEOREMS AND SCHRÖDINGER EQUATIONS WITH CRITICAL SOBOLEV EXPONENT 69 Since If H λ (h n (1,u )) C λ (λ λ n ), then by Step 3, h n (1,u ) k andbystep4, h n (1,u ) F (λ, ε ) F ε (λ). (2.12) η(t, h n (1,u )) h n (1,u ) = t t dη(s, h n (1,u )) Y λ (η(s, h n (1,u ))) ds 2t, ε hence η(t, h n (1,u )) h n (1,u ) + 2t k +4, for t [, 2ε ]. (2.13) ε Further, by Step 4, H λ (η(t, h n (1,u )) H λ (h n (1,u )) C λ + ε. Therefore, for this case, η(t, h n (1,u )) F ε (λ) F (λ), t [, 2ε ]. (2.14) Consider Λ 1 := {η([, 2ε ],h n (1,u ))}, which is w -compact and contained in U of Step 5 because of (2.11) and (2.14). Moreover, there are r 3 >,r 4 > such that Λ 2 := {u E : u Λ 1 w <r 3 } U ; Y λ (u) Y λ (v) w r 4 u v w, u, v Λ 2 ; Y λ (Λ 2 ) E fin. Evidently, by the w continuity of Y λ,η, and h n, there exists a w -neighborhood U(s 3,u ) such that for t [, 2ε ]andu U(s 3,u ). For t [1/2, 1], note that we conclude by (2.15) that According to the definition of η, h n (t, u) η(4ε t 2ε,h n (1,u)) = h n (t, u) h n (1,u)+ η(t, h n (1,u)) Λ 2 (2.15) 4εt 2ε Y λ (η(s, h n (1,u)))ds, {h n (t, u) η(4ε t 2ε,h n (1,u)) : (t, u) U 3 (s,u ) ([1/2, 1] Q)} E fin. (2.16) u η (t, u) =u h n (t, u)+h n (t, u) h n (2t, u), t [, 1/2]; u η (t, u) =u h n (t, u)+h n (t, u) η(4ε t 2ε,h n (1,u)), t [1/2, 1]. Therefore, by combining (2.9, 2.1) and (2.16), we obtain that {u η (t, u) :(t, u) Ũ (s,u ) ([, 1] Q)} E fin, which implies that η Γ, where Ũ (s,u ) = U 1 (s,u ) U 2 (s,u ) or Ũ (s,u ) = U 1 (s,u ) U 3 (s,u )

10 61 M. SCHECHTER AND W. ZOU Step 8. We will get a contradiction in this step. Case 1: if H λ (h n (1,u)) <C λ (λ λ n )forsomeu Q, then h n (1,u) F (λ) (see (2.5)) and H λ (η (1,u)) = H λ (η(2ε,h n (1,u)) H λ (h n (1,u))) <C λ (λ λ n ). (2.17) Case 2: if H λ (h n (1,u)) C λ (λ λ n )forsomeu Q, thenbystep3andstep4, h n (1,u) k and sup u Q H λ (h n (1,u)) C λ + ε. Then, h n (1,u) F ε (λ). Assume that H λ (η (1,u)) C λ (λ λ n ), then for t 2ε,wehave, Furthermore, for any t [, 2ε ], by Property (4) of Y λ (see (2.6)), C λ (λ λ n ) H λ (η (1,u)) = H λ (η(2ε,h n (1,u))) H λ (η(t, h n (1,u))) H λ (η(,h n (1,u))) = H λ (h n (1,u)) C λ + ε. (2.18) η(t, h n (1,u)) h n (1,u) = t t 2t/ε, dη(s, h n (1,u)) ds ds Y λ (η(s, h n (1,u)) ds it follows that η(t, h n (1,u)) 2t/ε + h n (1,u) k +4 fort [, 2ε ]. (2.19) Hence, equations (2.18) and (2.19) imply that η(t, h n (1,u)) Fε (λ) for t [, 2ε ]. Since on Fε (λ), H λ (u),y λ(u) > 1, then Therefore, by Step 4, H λ (η(2ε,h n (1,u))) H λ (h n (1,u))) = = 2ε 2ε 2ε. d dt H λ(η(t, h n (1,u)))dt H λ (η(t, h n(1,u))),y λ (η(t, h n (1,u))) dt Combining (2.17) and (2.2), we find H λ (η(2ε,h n (1,u))) H λ (h n (1,u)) 2ε C λ ε C λ (λ λ n ). (2.2) H λ (η (1,u)) = H λ (η(2ε,h n (1,u))) C λ (λ λ n ) for any (t, u) [, 1] Q, which contradicts the definition of C λ.

11 WEAK LINKING THEOREMS AND SCHRÖDINGER EQUATIONS WITH CRITICAL SOBOLEV EXPONENT Schrödinger equation Let E := W 1,2 ( ). It is well known that there is a one-to-one correspondence between solutions of (S) and critical points of the C 1 (E,R)-functional H(u) := 1 2 ( u 2 + V (x)u 2 )dx 1 2 K(x) u 2 dx G(x, u)dx. (3.1) Let (E(λ)) λ R be the spectral family of +V in L 2 ( ). Let E := E()L 2 E and E + := (id E())L 2 E, then the quadratic form R ( u 2 + Vu 2 )dx is positive definite on E + and negative definite on E (cf. [22]). N By introducing a new inner product, in E, the corresponding norm is equivalent to 1,2, the usual norm of W 1,2 ( ). Moreover, R ( u 2 + Vu 2 )dx = u + 2 u 2, where u ± E ±. Then functional (3.1) N can be rewritten as H(u) = 1 2 u u K(x) u 2 dx G(x, u)dx. (3.2) In order to use Theorem 2.1, we consider the family of functional defined by H λ (u) = 1 ( 1 2 u+ 2 λ 2 u ) 2 K(x) u 2 dx + G(x, u)dx (3.3) for λ [1, 2]. Lemma 3.1. H λ is w -upper semicontinuous. H λ is weakly sequentially continuous. w Proof. Noting that u n := u n + u+ n u implies that un uweakly in E and u + n u+ strongly in E, then the proof is the same as that in [23] (see also [6, 12]). The second conclusion is due to [6]. Let ϕ ε (x) := c N ψ(x)ε (N 2)/2 (ε 2 + x 2 ), (N 2)/2 where c N =(N(N 2)) (N 2)/4,ε >andψ C (RN, [, 1]) with ψ(x) =1if x r/2; ψ(x) =if x r, r small enough (cf. e.g. pp. 35 and 52 of [26]). Write ϕ ε = ϕ + ε + ϕ ε with ϕ + ε E +,ϕ ε E. Then ϕ ε, ϕ + ε 2 2 SN/2 as ε (cf. Prop. 4.2 of [6]), where u 2 2 S := inf u E\{} u 2 2 The following lemma can be found in Proposition 4.2 of [6]. Lemma 3.2. Set then for ε small enough, where Z ε := E Rϕ + ε. I 1 (u) := 1 2 u u K(x) u 2 dx, u E, (3.4) sup Z ε I 1 <c := To carry forward, we prepare an auxiliary results. S N/2 N K (N 2)/2,

12 612 M. SCHECHTER AND W. ZOU Lemma 3.3. Assume that g(x, u)/u as u uniformly for x and that g is of subcritical Sobolev exponent growth. If a bounded sequence (w n ) E and λ n [1, 2] satisfy where <c(λ) <c λ := λ n λ, H λ n (w n ), H λn (w n ) c(λ), S N/2 N λk (N 2)/2, then (w n ) is nonvanishing, i.e., there exist r, η > and a sequence (y n ),asequenceofopenball(b(y n,r)) centered at y n with radius r, such that lim sup wndx 2 η. n B(y n,r) Proof. The idea is essentially due to Proposition 4.1 of [6]. We give the sketch for the reader s convenience. If (w n ) is not nonvanishing, then w n inl r ( )for2<r<2 by Lions lemma ([16], Lem 1.21). By standard arguments, g(x, w n )v n dx and G(x, w n )dx (3.5) whenever (v n ) E is bounded. Hence H λn (w n ) 1 2 H λ n (w n ),w n = λ n N K(x) w n 2 dx + o(1) c(λ). (3.6) For any δ>, we choose µ> V (1 + δ)/δ. Write w n = w n + + wn E + E, and let w n + = w n + z n,with w n E(µ)L 2, z n (id E(µ))L 2, where (E(λ)) λ R is the spectral family of +V in L 2. By Proposition 2.4 of [6], w n E and wn q c wn 2 c w n and w n q c w n 2 c w n, (3.7) where q =2N/(N 4) if N>4andq may be chosen arbitrarily large if N = 4. Therefore, λ n wn 2 = H λ n (w n ),wn λ n K(x) w n 2 w n wn dx λ n g(x, w n )wn dx 2 K w n 2 1 r wn q + o(1), where r satisfies (2 1)/r +1/q =1, hence 2 <r<2. By the same reasoning, w n, hence, w n z n. (3.8) It follows that z n 2 = ( z n 2 + V z n)dx 2 = λ n K(x) w n 2 2 w n z n dx + o(1) R N = λ n K(x) w n 2 dx (3.9) On the other hand, by (4.6) of [6], for any δ>andµ> V (1 + δ)/δ, wehavethat (1 δ) z n 2 dx ( z n 2 + V z n 2 )dx. (3.1)

13 WEAK LINKING THEOREMS AND SCHRÖDINGER EQUATIONS WITH CRITICAL SOBOLEV EXPONENT 613 By (3.9, 3.8) and (3.1), we have that ( ) 2/2 λ K(x) w n 2 dx (λ K ) 2/2 w n 2 2 If we let n and use (3.6), it follows that =(λ K ) 2/2 z n o(1) (λ K ) 2/2 z n 2 2/S + o(1) (λ K ) 2/2 λ K(x) w n 2 dx + o(1). S(1 δ) (Nc(λ)) 2/2 (λ K ) 2/2 Nc(λ), S(1 δ) which implies that either c(λ) =orc(λ) (1 δ) N/2 c λ. Either way, we get a contradiction since δ is chosen arbitrarily. Choose z := ϕ + ε / ϕ+ ε E+. For R>, set Q := {u = u + sz : u <R,u E,s R + }. Let p = s z Q, s >. For any u E, we write u = u + sz + w with u E,w (E Rz ),s R. Consider a map F : E E Rz defined by F (u + sz + w) =u + sz + w z. Let B := F 1 (p ), then B = {u = sz + w : w (E Rz ), u = s } It is easy to check that F, p,b satisfy the basic assumptions in Section 2. By hypotheses (S 4 )and(s 5 ), the proof of the next lemma is trivial. Lemma 3.4. There exist R>,s >, such that inf B H λ >, sup H λ, for all λ [1, 2]. Q Lemma 3.5. For almost all λ [1, 2], thereexists{u n } E such that sup u n <, H λ (u n) and H λ (u n ) C λ, n where C λ [inf H λ, sup H]. Furthermore, there exists δ > small enough such that, for almost all λ [1, 1+δ ], B Q there exists u λ such that H λ(u λ )=, H λ (u λ ) sup H. Q Proof. The first conclusion follows immediately from Lemmas 3.1, 3.4, 3.5 and Theorem 2.1. Now we prove the second conclusion. Since g(x, u)u and Q Z ε,wegetthat <C λ sup H sup I 1 <c, (3.11) Q Z ε where I 1, c and Z ε come from Lemma 3.2. Therefore, there exists δ > such that <C λ <c λ for almost all λ [1, 1+δ ], where c λ comes from Lemma 3.3. For those λ, by Lemma 3.3, {u n} is nonvanishing, that is,

14 614 M. SCHECHTER AND W. ZOU there exist y n,α>,r 1 > such that lim sup n B(y n,r 1) u n 2 dx α>. We find ȳ n Z N such that lim sup v n 2 dx α>, n B(,2R 1) where v n (x) :=u n (x +ȳ n ). By the periodicity of V,K and g, {v n } is still bounded and lim H λ(v n ) n [ ] inf H λ, sup H B Q, lim n H λ (v n)=. We may suppose that v n u λ. Since E is embedded compactly in L t loc (RN )for2 t<2, then <α lim v n 2 dx = u λ 2 dx u λ 2 n 2, B(,2R 1) B(,2R 1) therefore, u λ. Since H λ is weakly sequentially continuous, H λ (u λ)=. Finally, by Fatou s lemma, H λ (u λ )=H λ (u λ ) 1 2 H λ(u λ ),u λ ( 1 = λ R 2 (K(x) u λ 2 + g(x, u λ )u λ ) 1 ) 2 K(x) u λ 2 G(x, u λ ) dx N ( 1 = λ lim n 2 (K(x) v n 2 + g(x, v n )v n ) 1 ) 2 K(x) v n 2 G(x, v n ) dx ( lim H λ (v n ) 1 ) n 2 H λ (v n),v n lim H λ(v n ) n sup H. Q Lemma 3.6. There exist λ n [1, 1+δ ] with λ n 1, and z n E\{} such that H λ n (z n )=, H λn (z n ) sup H. Q Proof. It is an immediately consequence of Lemma 3.5. Lemma 3.7. {z n } is bounded. Proof. Let g 1 (x, u) :=K(x) u 2 2 u + g(x, u) andg 1 (x, u) := u g 1(x, s)ds. Then by the assumption (S 4 ), we see that g 1 (x, u)u lim u G 1 (x, u) =2 uniformly for x. Let ε 1 > be such that 2 ε 1 > 2. Hence, there exists R 1 > such that g 1 (x, u)u (2 ε 1 )G 1 (x, u), for x, u R 1. (3.12)

15 WEAK LINKING THEOREMS AND SCHRÖDINGER EQUATIONS WITH CRITICAL SOBOLEV EXPONENT 615 On the other hand, since g(x, u) is of subcritical growth, g 1 (x, u)u 2G 1 (x, u) lim =(1 2 )K(x) c> (3.13) u u 2 2 uniformly for x. Furthermore, condition (1.2) implies that <g(x, u)u 2 N 2 k u 2 for all x,u, hence g 1 (x, u)u 2G 1 (x, u) > for all x,u. (3.14) Therefore (3.13) and (3.14) imply that there exists c small enough, such that Recall that H λn (z n ) sup H and H λ n (z n )=, then Q g 1 (x, u)u 2G 1 (x, u) c u 2 for all x, u R 1. (3.15) ( ) ( z + 2 n 2 λ n zn 2) ( 1 + λ n ε ) ε 1 2 K(x) z n 2 dx R ( N 1 + λ n 2 g(x, z n )z n G(x, z n ) ε 1 ) dx sup H. (3.16) Q By (3.12, 3.14) and (3.16), ( ) ( ) ( ) ( z + 2 n ε 2 λ n zn 2) 1 c + c + G 1 (x, z n ) 1 z n R 1 z n R 1 2 g 1 (x, z n )z n dx ε 1 ( ) 1 c + c G 1 (x, z n ) z n R 1 2 g 1 (x, z n )z n dx ε 1 ( ) 1 c + c z n R 1 2 g 1 1(x, z n )z n 2 g 1 (x, z n )z n dx ε 1 = c + c g 1 (x, z n )z n dx. (3.17) z n R 1 Since, by (S 4 ), g(x, z)z c z 2 for all (x, z) R, (3.17) implies that z n + 2 λ n zn 2 c + c g 1 (x, z n )z n dx z n R 1 ) c + c (K(x) z n 2 + g(x, z n )z n dx z n R 1 c + c z n 2 dx. (3.18) z n R 1

16 616 M. SCHECHTER AND W. ZOU However (3.14) and (3.15) imply that Then, combining (3.18) and (3.19), we obtain that Noting that H λ n (z n ),z n =, we see that z n + 2 λ n zn 2 = λ n sup H H λn (z n ) 1 Q 2 H λ n (z n ),z n ( ) 1 = R 2 g 1(x, z n )z n G 1 (x, z n ) dx N ( ) 1 z n R 1 2 g 1(x, z n )z n G 1 (x, z n ) dx c z n 2 dx. (3.19) z n R 1 z + n 2 λ n z n 2 c. (3.2) (K(x) z n 2 + g(x, z n )z n ) dx c z n 2 dx. (3.21) So, by (3.2) and (3.21), z n 2 dx c. Noting that H λ n (z n ),z n + =and(s 4), we obtain, by Hölder s inequality and (3.21), that z n + 2 = λ n K(x) z n 2 2 z n z n + dx + λ n g(x, z n )z n + dx R N c z n 2 1 z n + c z n z+ n 2 c z n +. Therefore z n + c, and hence, z n c by (3.21). Lemma 3.8. {z n } is nonvanishing. Proof. Since (z n ) is bounded, we may assume that H λn (z n ) c 1 sup H<c (cf. (3.11)). (3.22) Q If {z n } is not nonvanishing (i.e., is vanishing), then it follows from Lions lemma (cf. [16], Lem. 1.21) that z n inl r whenever 2 <r<2. The assumption (S 4 ) implies that g(x, z n )z n dx, G(x, z n )dx, (3.23) and consequently H λn (z n ) 1 2 H λ n (z n ),z n = λ n K(x) z n 2 dx + o(1) c 1. N (3.24)

17 WEAK LINKING THEOREMS AND SCHRÖDINGER EQUATIONS WITH CRITICAL SOBOLEV EXPONENT 617 Since K(x) >, c 1. Case 1: If c 1 >, then by (3.22) and Lemma 3.3, z n is nonvanishing. Case 2: If c 1 =, then (3.24) implies that z n 2 dx. (3.25) Since H λ n (z n )=, for any ε>, by (S 4 ), we have that ( ) z n + 2 = λ n K(x) z n 2 2 z n z n + + g(x, z n )z n + dx R N c z n 2 1 z n + dx + ε z n z n + + c z n p p 1 z n + c z n 2 1 z n + + ε z n 2 + ε z n zn + z n + p 1 z n +. Since z n z + n (see (3.21)) and ε is arbitrary, c z + n 2 c z + n p + c z + n 2, which implies that z n + c>. But, H λ n (z n ) =, and (S 4 ) implies that ( ) z n + 2 = λ n K(x) z n 2 2 z n z n + + g(x, z n)z n + dx c z n z+ n 2 + εc z n z n + + c z n p 1 p z n +. By the vanishing of {z n } and (3.25), z n +, a contradiction. Therefore, {z n} is nonvanishing. Proof of Theorem 1.1. Since {z n } is nonvanishing, there exist r>,α>andy n such that lim sup zndx 2 α. (3.26) n B(y n,r) We may assume that y n Z N by taking a large r if necessary. Now set z n (x) :=z n (x + y n ), since H λ is invariant with respect to the translation of x by elements of Z N (i.e., H λ (u( )) = H λ (u( + y)) whenever y Z N ), z n = z n,h λn (z n )=H λn ( z n ). Without loss of generality, we may suppose, up to a subsequence, that z n z, then (3.26) implies that z andh 1 (z )=,i.e., H (z )=. 4. Appendix In this Appendix, we give the proof of the existence of the new norm w satisfying v w v, v N and such that the topology induced by this norm is equivalent to the weak topology of N on bounded subset of N, more details can be found in [9]. Let {e k } be an orthonormal basis for N. Define v w = k=1 (v, e k ) 2 k, v N. Then v w is a norm on N and satisfies v w v, v N. Ifv j v weakly in N, then there is a C>such that v j, v C, j >.

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