Existence and number of solutions for a class of semilinear Schrödinger equations

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1 Existence numer of solutions for a class of semilinear Schrödinger equations Yanheng Ding Institute of Mathematics, AMSS, Chinese Academy of Sciences Beijing, China Andrzej Szulkin Deartment of Mathematics, Stockholm University Stockholm, Sweden Dedicated to Djairo G. de Figueiredo on the occasion of his 70th irthday Astract Using an argument of concentration-comactness tye we study the rolem u + λv (x)u = u 2 u, x R, where 2 < < 2 the set {x R : V (x) < } is nonemty has finite measure for some > 0. In articular, we show that if V 1 (0) has nonemty interior, then the numer of solutions increases with λ. We also study concentration of solutions on the set V 1 (0) as λ. 1 Introduction The urose of this aer is to resent simle roofs of some results concerning the existence the numer of decaying solutions for the Schrödinger equation (1.1) u + V (x)u = u 2 u, x R, for the related equations (1.2) u + λv (x)u = u 2 u, x R (1.3) ε 2 u + V (x)u = u 2 u, x R, resectively as λ ε 0. In a concluding section we shall also consider concentration of solutions as λ or ε 0. We shall assume throughout that V satisfy the following assumtions: (V 1 ) V C(R ) V is ounded elow. (V 2 ) There exists > 0 such that the set {x R : V (x) < } is nonemty has finite measure. (P ) (2, 2 ), where 2 := 2/( 2) if 3 2 := + if = 1 or 2. 1

2 Assumtion (V 1 ) is only for simlicity. In Sections 2 3 it can e relaced y (V 1 ) V L1 loc (R ) V := max{ V, 0} L q (R ), where q = /2 if 3, q > 1 if = 2 q = 1 if = 1 while in Section 4 we also need V L q loc (R ). Such an extension requires nothing more than a simle modification of our arguments. ote that if ε 2 = λ 1, then u is a solution of (1.2) if only if v = λ 1/( 2) u is a solution of (1.3), hence as far as the existence the numer of solutions are concerned, these two rolems are equivalent. Prolem (1.3) with V 0 a more general right-h side has een studied extensively y several authors, see e.g. [5, 10, 11] the references therein. For a rolem similar to (1.2), again with V 0 a more general right-h side, see [2]. In a recent work [6] it has een shown that for a certain class of functions V which may change sign, (1.1) has infinitely many solutions, see Remark 3.6 elow. The results of the resent aer extend comlement those mentioned aove. In articular, our assumtions on V are rather weak, ut erhas more imortant, our roofs seem to e new simler. On the other h, contrary to [5, 10, 11], we do not study single- or multisike solutions of (1.3) as ε 0. In a forthcoming aer we shall consider (1.2) for a much more general class of nonlinearities. However, this will e done at the exense of the simlicity of arguments. Below u will denote the usual L (R )-norm V ± (x) := max{±v (x), 0}. B ρ S ρ will resectively denote the oen all the shere of radius ρ center at the origin. It is well known that the functional Φ(u) := 1 ( u 2 + V (x)u 2 ) dx 1 u dx 2 is of class C 1 in the Soolev sace (1.4) E = {u H 1 (R ) : u 2 := ( u 2 + V + (x)u 2 ) dx < } critical oints of Φ corresond to solutions u of (1.1). Moreover, u(x) 0 as x. It is easy to see that if ( u 2 + V (x)u 2 ) dx (1.5) M := inf u E\{0} u 2 is attained at some ū M is ositive, then u = M 1/( 2) ū/ ū is a solution of (1.1) u(x) 0 as x. Such u is called a ground state. We note for further reference that (V 1 ), (V 2 ) the Poincaré inequality imly E is continuously emedded in H 1 (R ). For asic critical oint theory in a setting suitale for our uroses the reader is referred e.g. to [7, 14]. That u(x) 0 as x can e seen as follows. If = 1 u H 1 (R), then u(x) 0 as x. Suose 2, let u e a solution of (1.1) set W (x) = V (x) u(x) 2. Since V is continuous, ounded elow u 2 L r (R ) for some r > /2, it is easy to verify that W + K loc W K, where K K loc are the Kato classes as defined in Section A2 of [13]. Since u + W (x)u = 0, u(x) 0 according to Theorem C.3.1 in [13]. An alternative roof, for a much more general class of Schrödinger equations including those with V satisfying (V 1 ) instead of (V 1), may e found in [8]. 2

3 2 Comactness In this section we study the comactness of minimizing sequences of Palais-Smale sequences. We adat well known arguments (see e.g. [7, 14]) to our situation. Let V (x) := max{v (x), }, (2.1) M := inf u E\{0} ( u 2 + V (x)u 2 ) dx u 2. Denote the sectrum of + V in L 2 (R ) y σ( + V ) recall the definition (1.5) of M. Theorem 2.1 Suose (V 1 ), (V 2 ), (P ) are satisfied σ( + V ) (0, ). If M < M, then each minimizing sequence for M has a convergent susequence. So in articular, M is attained at some u E \ {0}. Proof Let (u m ) e a minimizing sequence. We may assume u m = 1. Since V < 0 on a set of finite measure, (u m ) is ounded in the norm of E given y (1.4). Passing to a susequence we may assume u m u in E y the continuity of the emedding E H 1 (R ), u m u in L 2 loc (R ), L loc (R ) a.e. in R. Let u m = v m + u. Then (2.2) ( u m 2 + V (x)u 2 m) dx = ( v m 2 + V (x)vm) 2 dx + ( u 2 + V (x)u 2 ) dx + o(1) y the Brézis-Lie lemma [4], [14, Lemma 1.32], (2.3) u m dx = v m dx + u dx + o(1). Moreover, y (V 2 ) since v m 0, (2.4) (V (x) V (x))vm 2 dx 0. Using (2.2)-(2.4) the definitions of M, M we otain ( u 2 + V (x)u 2 ) dx + ( v m 2 + V (x)vm) 2 dx + o(1) = M = M u m 2 = M( u + v m ) 2/ + o(1) M( u 2 + v m 2 ) + o(1) ( u 2 + V (x)u 2 ) dx + MM 1 ( v m 2 + V (x)vm) 2 dx + o(1) ( u 2 + V (x)u 2 ) dx + MM 1 ( v m 2 + V (x)vm 2 ) dx + o(1). Since MM 1 < 1 V (x)vm 2 dx 0, it follows that v m 0 therefore u m u. It is clear that u 0. Remark 2.2 If M = M, then all inequalities in the last formula aove ecome equalities after assing to the limit. Therefore either u = 0 or u m u in L (R ). In the latter case M is attained. 3

4 From the aove theorem it follows that if σ( + V ) (0, ) M < M, then there exists a ground state solution of (1.1). We shall also need to work with the functional Φ. Recall that (u m ) is called a Palais-Smale sequence at the level c (a (P S) c -sequence) if Φ (u m ) 0 Φ(u m ) c. If each (P S) c -sequence has a convergent susequence, then Φ is said to satisfy the (P S) c -condition. Theorem 2.3 If (V 1 ), (V 2 ) (P ) hold, then Φ satisfies (P S) c for all ( 1 c < 2 1 ) M /( 2). Proof Let (u m ) e a (P S) c -sequence with c satisfying the inequality aove. First we show that (u m ) is ounded. We have (2.5) d 1 + d 2 u m Φ(u m ) 1 ( 1 2 Φ (u m ), u m = 2 1 ) u m (2.6) d 1 + d 2 u m Φ(u m ) 1 Φ (u m ), u m = ( ) ( 1 u m ) V (x)u 2 m dx for some constants d 1, d 2 > 0. Suose u m let w m := u m / u m. Dividing (2.5) y u m we see that w m 0 in L (R ) therefore w m 0 in E after assing to a susequence. Hence V (x)wm 2 dx 0 (recall V is ounded; in fact it suffices that V L q (R ), where q is as in (V 1 )). So dividing (2.6) y u m 2, it follows that w m 0 in E, a contradiction. Thus (u m ) is ounded. As in the receding roof, we may assume u m u in E u m u in L 2 loc (R ), L loc (R ) a.e. in R. Set u m = v m + u. Since Φ (u) = 0 Φ(u) = Φ(u) 1 2 Φ (u), u = ( ) u 0, it follows from (2.2), (2.3) that (2.7) 0 = Φ (u m ), u m + o(1) = Φ (v m ), v m + Φ (u), u + o(1) = Φ (v m ), v m + o(1) (2.8) c = Φ(u m ) + o(1) = Φ(v m ) + Φ(u) + o(1) Φ(v m ) + o(1). By (2.7), (2.9) lim ( v m 2 + V (x)vm) 2 dx = lim v m dx =: γ, ossily after assing to a susequence, therefore it follows from (2.8) that ( 1 (2.10) c 2 1 ) γ. By (2.4), lim ( v m 2 + V (x)vm) 2 dx = lim ( v m 2 + V (x)vm) 2 dx = γ. 4

5 On the other h, v m 2 M 1 ( v m 2 + V (x)vm 2 ) dx y the definition (2.1) of M ; therefore γ 2/ M 1 γ. Comining this with (2.10) we see that either γ = 0 or ( 1 c 2 1 ) M /( 2), hence γ must e 0 y the assumtion on c. So according to (2.9), lim ( v m 2 + V + (x)v 2 m ) dx = lim ( v m 2 + V (x)v m 2 ) dx = 0. Therefore v m 0 u m u in E. 3 Existence of solutions Theorem 3.1 Suose (V 1 ) (P ) are satisfied, σ( + V ) (0, ), su x V (x) = > 0 the measure of the set {x R : V (x) < ε} is finite for all ε > 0. Then the infimum in (1.5) is attained at some u 0. If V 0, then u > 0 in R. Proof Since V + is ounded, E = H 1 (R ) here. Let u e the radially symmetric ositive solution of the equation u + u = u 2 u, x R. It is well known that such u exists, is unique minimizes ( u 2 + u 2 ) dx (3.1) := inf u E\{0} u 2 (see e.g. [7, Section 8.4] or [14, Section 1.7]). So if V, we are done. Otherwise we may assume without loss of generality that V (0) <. Then ( u 2 + V (x)u 2 ) dx ( u 2 + V (x)u 2 M = inf u E\{0} u 2 ) dx u 2 < ( u 2 + u 2 ) dx u 2 = = M, where the last equality follows from the fact that V =. In order to aly Theorem 2.1 we need to show that M < M ε for some ε > 0 (M < M does not suffice ecause the set {x R : V (x) < } may have infinite measure). A simle comutation shows that if λ > 0, then λ is attained at u λ (x) = λ 1/( 2) u ( λx) (3.2) λ = λ r, where r = > 0. Choosing λ = ( ε)/ we see that ε < ε as ε 0. So for ε small enough we have ( u 2 + ( ε)u 2 ) dx ( u 2 + V ε (x)u 2 ) dx M < ε = inf u E\{0} u 2 inf u E\{0} u 2 = M ε. 5

6 Hence M is attained at some u. Since the exression on the right-h side of (1.5) does not change if u is relaced y u, we may assume u 0. By the maximum rincile, if V 0, then u > 0 in R. Theorem 3.2 Suose V 0 (V 1 ), (V 2 ), (P ) are satisfied. Then there exists Λ > 0 such that for each λ Λ the infimum in (1.5) (with V (x) relaced y λv (x)) is attained at some u λ > 0. Proof Here V = V +. Let e as in (V 2 ) (3.3) M λ ( u 2 + λv (x)u 2 ) dx := inf u E\{0} u 2, M λ := inf u E\{0} ( u 2 + λv (x)u 2 ) dx u 2. It suffices to show that M λ < M λ for all λ large enough. We may assume V (0) < choose ε, δ > 0 so that V (x) < ε whenever x < 2δ. Let ϕ C0 (R, [0, 1]) e a function such that ϕ(x) = 1 for x δ ϕ(x) = 0 for x 2δ. Set w λ (x) := ϕ(x)u λ (x) λ 1/( 2) u ( λx)ϕ(x), where u is as in the roof of Theorem 3.1. Then for all sufficiently large λ some c 0 > 0, M λ ( w λ 2 + λv (x)wλ 2 ) dx ( w λ 2 + λ( ε)wλ 2 w λ 2 ) dx w λ 2 ( = λ r ( u 2 + u 2 ) dx ε u 2 dx ) u 2 + o(1) λ r ( c 0 ε) ( is defined in (3.1) r in (3.2)). Using (3.2) (3.3) we also see that (3.4) M λ inf ( u 2 + λu 2 ) dx u E\{0} u 2 = λ = λ r, hence M λ < M λ (the infimum aove is equal to λ also when E is a roer susace of H 1 (R ) ecause C0 (R ), hence also E, is dense in H 1 (R )). By the argument at the end of the roof of Theorem 3.1, the infimum is attained at some u λ > 0. Remark 3.3 If (V 1 ) is relaced y (V 1 ), then we need to assume that the set {x R : V (x) < ε} aearing in Theorem 3.1 has nonemty interior for each ε > 0. Likewise, in Theorem 3.2 the set {x R : V (x) < } should have nonemty interior. ext we shall consider the existence of multile solutions under the hyothesis that V 1 (0) has nonemty interior. Theorem 3.4 Suose V 0, V 1 (0) has nonemty interior (V 1 ), (V 2 ), (P ) are satisfied. For each k 1 there exists Λ k > 0 such that if λ Λ k, then (1.2) has at least k airs of nontrivial solutions in E. Proof For a fixed k we can find ϕ 1,..., ϕ k C0 (R ) such that su ϕ j, 1 j k, is contained in the interior of V 1 (0) su ϕ i su ϕ j = whenever i j. Let F k := san{ϕ 1,..., ϕ k }. 6

7 Since V 0, Φ(u) = 1 2 u 2 1 u therefore there exist α, ρ > 0 such that Φ Sρ α. Denote the set of all symmetric (in the sense that A = A) closed susets of E y Σ, for each A Σ let γ(a) e the Krasnoselski genus i(a) := min h Γ γ(h(a) S ρ), where Γ is the set of all odd homeomorhisms h C(E, E). Then i is a version of Benci s seudoindex [1, 3]. Let Φ λ (u) := 1 ( u 2 + λv (x)u 2 ) dx 1 u dx, λ 1 2 c j := inf i(a) j su Φ λ (u), 1 j k. u A Since Φ λ (u) Φ(u) α for all u S ρ since i(f k ) = dim F k = k (see [1, 3]), α c 1... c k su u F k Φ λ (u) =: C. It is clear that C deends on k ut not on λ. As in (3.4), we have M λ λ = λ r, where r > 0, therefore M λ. Hence C < ( )(M λ)/( 2) whenever λ is large enough it follows from Theorem 2.3 that for such λ the Palais-Smale condition is satisfied at all levels c C. By the usual critical oint theory, all c j are critical levels Φ λ has at least k airs of nontrivial critical oints. ext we extend the aove result to the case of V 0. As in [9], we consider the eigenvalue rolem (3.5) u + λv + (x)u = µλv (x)u, u E (here λ 1 is fixed). An equivalent norm u λ in E is given y the inner roduct u, v λ := ( u v + λv + (x)uv) dx. Since V > 0 on a set of finite measure, the linear oerator u λv (x)u dx is comact. It follows that there are finitely many eigenvalues µ 1 the quadratic form u ( u 2 + λv (x)u 2 ) dx is negative semidefinite on the sace E sanned y the corresonding eigenfunctions. It is easy to see that dim E as λ. Theorem 3.5 Suose V 0, V 1 (0) has nonemty interior (V 1 ), (V 2 ), (P ) are satisfied. For each k 1 there exists Λ k > 0 such that if λ Λ k, then (1.2) has at least k airs of nontrivial solutions in E. 7

8 Proof We need to modify the argument of Theorem 3.4. Let ϕ j F k e as efore. If e is an eigenfunction of (3.5) µ a corresonding eigenvalue, then (3.6) e, ϕ j λ = µλ V (x)eϕ j dx = 0 ecause su ϕ j V 1 (0). Hence E k := E + F k = E F k. Let l = dim E c j := Write u = e + f, e E, f F k. L (R ) F k, for some C 1. Thus Φ λ (u) 1 2 inf i(a) l+j su Φ λ (u), 1 j k. u A By (3.6) since there exists a continuous rojection f 2 dx C c k su u E k Φ λ (u) = C, f dx where C is indeendent of λ. If i(a) l + 1, then γ(h(a) S ρ ) l + 1 for each h Γ therefore h(a) S ρ intersects any susace of codimension l. The sace E has an orthogonal decomosition E = E + E F (with resect to the inner roduct.,. λ ), where E + corresonds to the eigenvalues µ > 1 of (3.5) F is the susace of functions u E whose suort is contained in V 1 ([0, )). It is clear that the quadratic art of Φ λ is ositive definite on E +, it is also ositive definite on F ecause V 1 (0) has finite measure. Hence there exist α, ρ > 0 (ossily deending on λ) such that Φ λ Sρ (E + F ) α. Since codim(e + F ) = l, it follows that h(a) S ρ (E + F ) c 1 α. ow it remains to reeat the argument at the end of the receding roof. Remark 3.6 (i) If V (x) as x, then it is well known that the emedding H 1 (R ) L q (R ), 2 q < 2 is comact, see e.g. [2]. Therefore the Palais-Smale condition holds at all levels (1.1) has infinitely many solutions. (ii) It has een shown in [6] that if V C 1 (R ) satisfies certain growth conditions at infinity (which are much weaker than the requirement that V (x) as x ), then (1.1) has infinitely many solutions. 4 Concentration of solutions Theorem 4.1 Suose (V 1 ), (V 2 ), (P ) are satisfied V 1 (0) has nonemty interior Ω. u m E e a solution of the equation Let (4.1) u + λ m V (x)u = u 2 u, x R. If λ m u m λm C for some C > 0, then, u to a susequence, u m ū in L (R ), where ū is a weak solution of the equation (4.2) u = u 2 u, x Ω, ū = 0 a.e. in R \ V 1 (0). If moreover V 0, then u m ū in E. 8

9 We note that ū H 1 0 (Ω) if V 1 (0) = Ω Ω is locally Lischitz continuous (cf. [2]). Before roving the aove theorem we oint out some of its consequences. Corollary 4.2 Suose (V 1 ), (V 2 ), (P ) are satisfied, V 1 (0) has nonemty interior, V 0, u m E is a solution of (4.1), λ m Φ λm (u m ) is ounded ounded away from 0. Then the conclusion of Theorem 4.1 is satisfied ū 0. Proof We have Φ λm (u m ) = 1 2 u m 2 λ m 1 u m Φ λm (u m ) = Φ λm (u m ) 1 2 Φ λ m (u m ), u m = ( ) u m. Hence u m, therefore also u m λm is ounded. So the conclusion of Theorem 4.1 holds. Moreover, as u m is ounded away from 0, ū 0. ote that as a consequence of this corollary, if k is fixed, then any sequence of solutions u m of (1.2) with λ = λ m otained in Theorem 3.4 contains a susequence concentrating at some ū 0. Moreover, it is ossile to otain a ositive solution for each λ, either via Theorem 3.1 or y the mountain ass theorem. It follows that each sequence (u m ) of such solutions with λ m has a susequence concentrating at some ū which is ositive in Ω. Corresonding to u m are solutions v m = ε 2/( 2) m u m of (1.3), where ε 2 m = λ 1 m. Then v m 0 ε 2/( 2) m v m ū. This should e comared with (iii) of Theorem 1 in [5] where it was shown that lim ε 2/( 2) m v m > 0. It will ecome clear from the roof of Theorem 4.1 that if V 1 (0) has emty interior, then ū 0 which is imossile under the assumtions of Corollary 4.2. Since σ( + λv ) (a, ) for some a > 0 (indeendent of λ if λ is ounded away from 0), u = 0 is the only critical oint of Φ λ in B r for some r > 0. Hence in this case Φ λm (u m ) u m if u m is a nontrivial solution of (1.2) with λ = λ m. If V 0, we do not know whether u m ū in E or whether a result corresonding to Corollary 4.2 is true. However, if V 1 (0) has emty interior, then it follows from Theorem 4.1 that either u m 0 in L (R ) or u m λm. Proof of Theorem 4.1 We modify the argument in [2]. Since λ m 1, u m u m λm C. Passing to a susequence, u m ū in E u m ū in L loc (R ). Since Φ λ m (u m ), ϕ = 0, we see that V (x)u m ϕ dx 0 V (x)ūϕ dx = 0 for all ϕ C0 (R ). Therefore ū = 0 a.e. in R \ V 1 (0). We claim that u m ū in L (R ). Assuming the contrary, it follows from P.L. Lions vanishing lemma (see [12, Lemma I.1] or [14, Lemma 1.21]) that (u m ū) 2 dx γ B ρ(x m) for some (x m ) R, ρ, γ > 0 almost all m (B ρ (x) denotes the oen all of radius ρ center x). Since u m ū in L 2 loc (R ), x m. Therefore the measure of the set B ρ (x m ) {x R : V (x) < } tends to 0 u m 2 λ m λ m u 2 m dx = λ m B ρ(x m) {V } 9 ( (u m ū) 2 dx + o(1) B ρ(x m) ),

10 a contradiction. Let now V 0. Since u m satisfies (4.1), Φ λ m (u m ), ū = 0 ū(x) = 0 whenever V (x) > 0, it follows that u m 2 u m 2 λ m = u m ū 2 = ū 2 λ m = ū. Hence lim su u m 2 ū = ū 2 therefore u m ū in E. References [1] P. Bartolo, V. Benci D. Fortunato, Astract critical oint theorems alications to some nonlinear rolems with strong resonance at infinity, onlin. Anal. 7 (1983), [2] T. Bartsch, A. Pankov Z.Q. Wang, onlinear Schrödinger equations with stee otential well, Comm. Contem. Math. 3 (2001), [3] V. Benci, On critical oint theory of indefinite functionals in the resence of symmetries, Trans. Amer. Math. Soc. 274 (1982), [4] H. Brézis, E. Lie, A relation etween ointwise convergence of functions convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), [5] J. Byeon, Z.Q. Wang, Sting waves with a critical frequency for nonlinear Schrödinger equations, II, Calc. Var. PDE 18 (2003), [6] G. Cerami, G. Devillanova S. Solimini, Infinitely many ound states for some nonlinear scalar field equations, Prerint. [7] J. Charowski, Variational Methods for Potential Oerator Equations, de Gruyter, Berlin [8] J. Charowski A. Szulkin, On the Schrödinger equation involving a critical Soolev exonent magnetic field, To. Meth. onl. Anal., to aear. [9] S.W. Chen Y.Q. Li, ontrivial solution for a semilinear ellitic equation in unounded domain with critical Soolev exonent, J. Math. Anal. Al. 272 (2002), [10] M. del Pino P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann. 324 (2002), [11] L. Jeanjean K. Tanaka, Singularly ertured ellitic rolems with suerlinear or asymtotically linear nonlinearities, Calc. Var. PDE 21 (2004), [12] P.L. Lions, The concentration-comactness rincile in the calculus of variations. The locally comact case. Part I, Ann. IHP, Analyse on Linéaire 1 (1984), [13] B. Simon, Schrödinger semigrous, Bull. Amer. Math. Soc. (ew Series) 7, (1982), [14] M. Willem, Minimax Theorems, Birkhäuser, Boston