Nonlinear scalar field equations with L 2 constraint: Mountain pass and symmetric mountain pass approaches
aa r X i v : . [ m a t h . A P ] M a r Nonlinear scalar field equations with L constraint:Mountain pass and symmetric mountain pass approaches Jun Hirata and Kazunaga Tanaka ∗ Department of Mathematics, School of Science and EngineeringWaseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan
Abstract:
We study the existence of radially symmetric solutions of the followingnonlinear scalar field equations in R N ( N ≥ ∗ ) m − ∆ u = g ( u ) − µu in R N , k u k L ( R N ) = m,u ∈ H ( R N ) , where g ( ξ ) ∈ C ( R , R ), m > µ ∈ R is a Lagrangemultiplier.We introduce a new approach using a Lagrange formulation of the prob-lem ( ∗ ) m . We develop a new deformation argument under a new version of thePalais-Smale condition. For a general class of nonlinearities related to [ BL1 , BL2 , HIT ], it enables us to apply minimax argument for L constraint problemsand we show the existence of infinitely many solutions as well as mountain passcharacterization of a minimizing solution of the problem:inf { Z R N |∇ u | − G ( u ) dx ; k u k L ( R N ) = m } , G ( ξ ) = Z ξ g ( τ ) dτ.
0. Introduction
In this paper, we study the existence of radially symmetric solutions of the followingnonlinear scalar field equations in R N ( N ≥ ∗ ) m − ∆ u = g ( u ) − µu in R N , k u k L ( R N ) = m,u ∈ H ( R N ) , ∗ The second author is partially supported by JSPS Grants-in-Aid for Scientific Research(B) (25287025) and (B) (17H02855). 1here g ( ξ ) ∈ C ( R , R ), m > µ ∈ R is a Lagrange multiplier.Solutions of ( ∗ ) m can be characterized as critical points of the constraint problem: F ( u ) = 12 Z R N |∇ u | − Z R N G ( u ) : S m → R , where S m = { u ∈ H r ( R N ); k u k L ( R N ) = m } and G ( ξ ) = R ξ g ( τ ) dτ .When g ( ξ ) has L -subcritical growth, Cazenave-Lions [ CL ] and Shibata [ S1 ] success-fully found a solution of ( ∗ ) m via minimizing method: I m = inf u ∈ S m F ( u ) . (0 . CL ] dealt with g ( ξ ) = | ξ | q − ξ (1 < q < N ) and [ S1 ] dealt with a class of more generalnonlinearities, which satisfy conditions:( g g ( ξ ) ∈ C ( R , R ),( g
2) lim ξ → g ( ξ ) ξ = 0,( g
3) lim | ξ |→∞ | g ( ξ ) || ξ | p = 0, where p = 1 + N ,( g
4) There exists ξ > G ( ξ ) > S1 ] showed(i) There exists m S ≥ m > m S . I m defined in (0.1) is achieved and ( ∗ ) m has at least one solution for m > m S .(ii) m S = 0 if and only if lim ξ → g ( ξ ) | ξ | N ξ = ∞ . (0 . S1 , CL ] they also studied orbital stability of the minimizer. We alsorefer to Jeanjean [ J ] and Bartsch-de Valeriola [ BV ] for the study of L -supercritical case(e.g. g ( ξ ) ∼ | ξ | p − ξ with p ∈ (1 + N , N +2 N − )).We note that the conditions ( g g
4) are related to those in Berestycki-Lions [
BL1 , BL2 ] (see also [
BGK , HIT ]) as almost necessary and sufficient conditions for the existenceof solutions of nonlinear scalar field equations: ( − ∆ u = g ( u ) in R N ,u ∈ H ( R N ) . (0 . g
2) by lim sup ξ → g ( ξ ) ξ < p with N +2 N − in ( g g ( ξ ) inaddition: 2 g g ( − ξ ) = − g ( ξ ) for all ξ ∈ R .We remark that if g ( ξ ) satisfies ( g g g ( ξ ) = g ( ξ ) − µξ satisfies the conditions of[ BL1 , BL2 ] for µ ∈ (0 , ∞ ) small.In [ CL , S1 ], to show the achievement of I m on S m , the following sub-additivity in-equality plays an important role. I m < I s + I m − s for all s ∈ (0 , m ) , (0 . I m .In this paper, we take another approach to ( ∗ ) m and we try to apply minimax methodsto a Lagrange formulation of the problem ( ∗ ) m : L ( µ, u ) = 12 Z R N |∇ u | − Z R N G ( u ) + µ (cid:16) k u k L ( R N ) − m (cid:17) : R × H r ( R N ) → R . (0 . S1 ]; we take an approach related toHirata-Ikoma-Tanaka [ HIT ] and Jeanjean [ J ], which made use of the scaling properties ofthe problems to generate Palais-Smale sequences in augmented spaces with extra propertiesrelated to the Pohozaev identities. We remark that such approaches were successfully ap-plied to other problems with suitable scaling properties. See Azzollini-d’Avenia-Pomponio[ AdAP ], Byeon-Tanaka [ BT ], Chen-Tanaka [ CT ] and Moroz-Van Schaftingen [ MVS ].We also give a mountain pass characterization of the minimizing value I m through thefunctional (0.5), which we expect to be useful in the study of singular perturbation prob-lems. We remark that a mountain pass characterization of the least energy solutions fornonlinear scalar field equations (0.3) was given in Jeanjean-Tanaka [ JT ]. Theorem 0.1.
Assume ( g – ( g . Then we have(i) There exists m ∈ [0 , ∞ ) such that for m > m , ( ∗ ) m has at least one solution.(ii) Assume (0.2) in addition to ( g – ( g . Then ( ∗ ) m has at least one solution for all m > .(iii) In the setting of (i)–(ii), a solution is obtained through a mountain pass minimaxmethod: b mp = inf γ ∈ Γ mp max t ∈ [0 , I ( γ ( t )) . See Section 5 for a precise definition of the minimax class Γ mp . We also have b mp = I m , here I m defined in (0.1) . We will give a presentation of m using least energy levels of − ∆ u + µu = g ( u ) in Section5. We also show m > m if and only if I m < g ( ξ ).It seems that the existence of infinitely many solutions for the L -constraint problem isnot well-studied. Our main result is the following Theorem 0.2.
Assume ( g – ( g and ( g . Then(i) For any k ∈ N there exists m k ≥ such that for m > m k , ( ∗ ) m has at least k solutions.(ii) Assume (0.2) in addition to ( g – ( g . Then for any m > , ( ∗ ) m has countably manysolutions ( u n ) ∞ n =1 , which satisfies F ( u n ) < for all n ∈ N , F ( u n ) → as n → ∞ . To show Theorem 0.2, we develop a version of symmetric mountain pass methods, in whichgenus plays an important role.In the following sections, we give proofs to our Theorems 0.1 and 0.2. Since theexistence part of Theorem 0.1 is already known by [ CL , S1 ]. We mainly deal with Theorem0.2 in Sections 1–4.In Section 1, first we give a variational formulation of the problem ( ∗ ) m . For a technicalreason, we write µ = e λ ( λ ∈ R ) and we try to find critical points of I ( λ, u ) = Z R N |∇ u | − G ( u ) + e λ (cid:18)Z R N | u | − m (cid:19) ∈ C ( R × H r ( R N ) , R ) . We also setup function spaces. Second for a fixed λ ∈ R , we study the symmetric mountainpass value a k ( λ ) of u b I ( u ) = Z R N |∇ u | + e λ u − G ( u ) . Behavior of a k ( λ ) is important in our study. In particular, m k in Theorem 0.2 is given by m k = 2 inf λ ∈ ( −∞ ,λ ) a k ( λ ) e λ . In Sections 2–3, we find that I ( λ, u ) : R × H r ( R N ) → R has a kind of mountain passgeometry and we give a family of minimax sets for I ( λ, u ), which involves the notion ofgenus under Z -invariance: I ( λ, − u ) = I ( λ, u ).4n Section 4, we develop a new deformation argument to justify the minimax methodsin Section 2. Usually deformation theories are developed under so-called Palais-Smalecondition. However, under the conditions ( g g I ( λ, u ). We introduce a new version of Palais-Smale condition,which is inspired by our earlier work [ HIT ] and Jeanjean [ J ]. Here we extend this ideafurther to generate a deformation flow, which is different from the standard one; our flowdoes not come from ODE in R × H r ( R N ) and in general it is not of class C .In Section 5, we deal with Theorem 0.1 and we study the minimizing problem (0.1).Applying mountain pass approach to I ( λ, u ), we give another proof of the existence resultas well as a mountain pass characterization of I m in R × H r ( R N ).Our new deformation argument also works for the nonlinear scalar field equations(0.3); we can give a simpler proof to the results in [ HIT ]. We believe that it is of interestand the idea is applicable to other problems with scaling properties.
1. Preliminaries1.a. Functional settings
In Sections 1–4, we deal with Theorem 0.2 and we assume ( g g H r ( R N ) the space of radially symmetric functions u ( x ) = u ( | x | ) which satisfy u ( x ), ∇ u ( x ) ∈ L ( R N ). We also use notation k u k r = (cid:18)Z R N | u ( x ) | r (cid:19) /r for r ∈ [1 , ∞ ) and u ∈ L r ( R N ) , k u k H = ( k∇ u k + k u k ) / for u ∈ H r ( R N ) . We also write ( u, v ) = Z R N uv for u, v ∈ L ( R N ) . In what follows, we denote by p the L critical exponent, i.e., p = 1 + 4 N . (1 . p + 1 p − − N , (1 . µ = e λ in (0.5) and we set for a given m > I ( λ, u ) = 12 k∇ u k − Z R N G ( u ) + e λ (cid:0) k u k − m (cid:1) : R × H r ( R N ) → R .
5t is easy to see that I ( λ, u ) ∈ C ( R × H r ( R N ) , R ) and solutions of ( ∗ ) m can be charac-terized as critical points of I ( λ, u ), that is, ( µ, u ) with µ = e λ > ∗ ) m if and onlyif ∂ λ I ( λ, u ) = 0 and ∂ u I ( λ, u ) = 0. We also have I ( λ, − u ) = I ( λ, u ) for all ( λ, u ) ∈ R × H r ( R N ) . (1 . b I ( λ, u ) = 12 k∇ u k + e λ k u k − Z R N G ( u ) : R × H r ( R N ) → R , (1 . P ( λ, u ) = N − k∇ u k + N (cid:18) e λ k u k − Z R N G ( u ) (cid:19) : R × H r ( R N ) → R . (1 . λ ∈ R , u b I ( λ, u ) is corresponding to ( − ∆ u + e λ u = g ( u ) in R N ,u ∈ H r ( R N ) . (1 . I ( λ, u ) = b I ( λ, u ) − e λ m for all ( λ, u ) . (1 . P ( λ, u ) is related to the Pohozaev identity. It is well-known that for λ ∈ R if u ( x ) ∈ H r ( R N ) solves (1.6), then P ( λ, u ) = 0. b I ( λ, u )First we observe that for λ ≪ u b I ( λ, u ) satisfies the assumptions of [ BL1 , BL2 , BGK , HIT ] and possesses the symmetric mountain pass geometry. In what follows, wewrite S k − = { ξ ∈ R k ; | ξ | = 1 } , D k = { ξ ∈ R k ; | ξ | ≤ } . and set λ = log ξ =0 G ( ξ ) ξ ! if sup ξ =0 G ( ξ ) ξ < ∞ , ∞ if sup ξ =0 G ( ξ ) ξ = ∞ . (1 . Lemma 1.1. i) For λ ∈ ( −∞ , λ ) , G ( ξ ) − e λ | ξ | > for some ξ > . In particular, e g ( ξ ) = g ( ξ ) − e λ ξ satisfies the assumptions of [ BL1 , BL2 , HIT ], that is, e g ( ξ ) satisfies ( g , ( g – ( g and lim ξ → ˜ g ( ξ ) ξ < . (ii) For any λ ∈ ( −∞ , λ ) and for any k ∈ N , there exists a continuous odd map ζ : S k − → H r ( R N ) such that b I ( λ, ζ ( ξ )) < for all ξ ∈ S k − . (iii) When λ < ∞ , for λ ≥ λ we have G ( ξ ) − e λ | ξ | ≤ for all ξ ∈ R . In particular, b I ( λ, u ) ≥ for all u ∈ H r ( R N ) . Proof.
By ( g g
5) and the definition (1.8) of λ , we can easily see (i) and (iii). By thearguments in [ BL2 , HIT ], we can observe that u b I ( λ, u ) has the property (ii).For k ∈ N and λ ∈ ( −∞ , λ ), we set b Γ k ( λ ) = { ζ ∈ C ( D k , H r ( R N )); ζ ( − ξ ) = − ζ ( ξ ) for ξ ∈ D k , b I ( λ, ζ ( ξ )) < ξ ∈ ∂D k = S k − } , (1 . a k ( λ ) = inf ζ ∈ b Γ k ( λ ) max ξ ∈ D k b I ( λ, ζ ( ξ )) . (1 . b Γ k ( λ ) = ∅ by Lemma 1.1 (ii). Since b I ( λ, u ) = ( k∇ u k + e λ k u k ) + o ( k u k H )as k u k H ∼ a k ( λ ) > λ ∈ ( −∞ , λ ) and k ∈ N . By the results of [
HIT ], we observe that a k ( λ ) is a critical value of u b I ( λ, u ). See alsoSection 6.We also have0 < a ( λ ) ≤ a ( λ ) ≤ · · · ≤ a k ( λ ) ≤ a k +1 ( λ ) ≤ · · · for all λ ∈ ( −∞ , λ ) , (1 . a k ( λ ) ≤ a k ( λ ′ ) for all λ < λ ′ < λ and k ∈ N . a k ( λ ) as λ → −∞ , the condition (0.2) is important. We have Lemma 1.2.
Assume ( g – ( g .(i) Assume (0.2) in addition. Then for any k ∈ N lim λ →−∞ a k ( λ ) e λ = 0 . (ii) If lim sup ξ → | g ( ξ ) || ξ | p < ∞ , (1 . then for any k ∈ N lim inf λ →−∞ a k ( λ ) e λ > . Proof. (i) Choose r ∈ (1 + N , N +2 N − ). By (0.2) and ( g L > C L > ξg ( ξ ) ≥ L | ξ | p +1 − C L | ξ | r +1 for all ξ ∈ R , from which we have G ( ξ ) ≥ Lp + 1 | ξ | p +1 − C L r + 1 | ξ | r +1 for all ξ ∈ R , b I ( λ, u ) ≤ k∇ u k + e λ k u k − Lp + 1 k u k p +1 p +1 + C L r + 1 k u k r +1 r +1 for all u ∈ H r ( R N ) . Setting u ( x ) = e λp − v ( e λ/ x ), v ( x ) ∈ H r ( R N ), we have from (1.2) b I ( λ, u ) ≤ e λ (cid:18) k v k H − Lp + 1 k v k p +1 p +1 + C L r + 1 e r − pp − λ k v k r +1 r +1 (cid:19) . We note that I ( v ) = 12 k v k H − p + 1 k v k p +1 p +1 : H r ( R N ) → R has the symmetric mountain pass geometry and thus there exists an odd continuousmap ζ ( ξ ) : D k → H r ( R N ) such that I ( ζ ( ξ )) < ξ ∈ ∂D k . By (1.7), ζ λ ( ξ ) = e λp − ζ ( ξ )( e λ/ x ) satisfies for L ≥ b I ( λ, ζ λ ( ξ )) ≤ e λ (cid:18) I ( ζ ( ξ )) − L − p + 1 k ζ ( ξ ) k p +1 p +1 + C L r + 1 e r − pp − λ k ζ ( ξ ) k r +1 r +1 (cid:19) . Thus for λ ≪
0, we have ζ λ ( ξ ) ∈ b Γ k ( λ ) and we havelim sup λ →−∞ a k ( λ ) e λ ≤ max ξ ∈ D k (cid:18) k ζ ( ξ ) k H − Lp + 1 k ζ ( ξ ) k p +1 p +1 (cid:19) . L ≥ g C > G ( ξ ) ≤ C | ξ | p +1 for all ξ ∈ R . Thus we have b I ( λ, u ) ≥ k∇ u k + e λ k u k − Cp + 1 k u k p +1 p +1 . As in (i), b I ( λ, e λp − u ( e λ/ x )) ≥ e λ (cid:18) k u ( x ) k H − Cp + 1 k u ( x ) k p +1 p +1 (cid:19) , from which we deduce that a k ( λ ) /e λ is estimated from below by the mountain pass mini-max value for u k u ( x ) k H − Cp +1 k u ( x ) k p +1 p +1 . Thus (ii) holds.We define for k ∈ N m k = 2 inf λ ∈ ( −∞ ,λ ) a k ( λ ) e λ ≥ . (1 . ≤ m ≤ m ≤ · · · ≤ m k ≤ m k +1 ≤ · · · . (1 . m > m k arbitrary and try to show that I ( λ, u ) has at least k pairsof critical points.As a corollary to Lemma 1.2, we have Corollary 1.3. (i) Under the condition (0.2) , m k = 0 for all k ∈ N . (ii) Under the condition (1.12) , m k > for all k ∈ N . By ( g
2) and ( g δ > C δ > ξg ( ξ ) ≤ C δ | ξ | + δ | ξ | p +1 for all ξ ∈ R . (1 . G ( ξ ) ≤ C δ | ξ | + δp + 1 | ξ | p +1 for all ξ ∈ R . I ( λ, u ) = 12 k∇ u k + 12 ( e λ − C δ ) k u k − δp + 1 k u k p +1 p +1 , we have b I ( λ, u ) ≥ I ( λ, u ) for all ( λ, u ) ∈ R × H r ( R N ) . (1 . u I ( λ, u ) has a typical mountain pass geometry if e λ > C δ + 1, whichenables us to give an estimate of b I ( λ, u ) from below.In what follows, we denote by E > − ∆ u + u = | u | p − u in R N , that is, E = inf { k∇ u k + 12 k u k − p + 1 k u k p +1 p +1 ; u = 0 , k∇ u k + k u k = k u k p +1 p +1 } . Lemma 1.4.
For e λ ≥ C δ + 1 , b I ( λ, u ) ≥ δ − p − ( e λ − C δ ) E if u = 0 , k∇ u k + ( e λ − C δ ) k u k = δ k u k p +1 p +1 , (1 . b I ( λ, u ) ≥ if k∇ u k + ( e λ − C δ ) k u k ≥ δ k u k p +1 p +1 . (1 . Proof.
Let ω ( x ) be the least energy solution of − ∆ u + u = | u | p − u . Then it is easy tosee that u λ,δ ( x ) = (cid:18) e λ − C δ δ (cid:19) p − ω (( e λ − C δ ) / x )is a least energy solution of − ∆ u + ( e λ − C δ ) u = δ | u | p − u in R N . Set S λ,δ = { u ∈ H r ( R N ) \ { } ; k∇ u k + ( e λ − C δ ) k u k = δ k u k p +1 p +1 } . By (1.2), it is easy to see that for e λ ≥ C δ + 1 I ( λ, u ) ≥ δ − p − ( e λ − C δ ) E for u ∈ S λ,δ . Thus we get (1.17) from (1.16). Noting { u ∈ H r ( R N ); k∇ u k + ( e λ − C δ ) k u k ≥ δ k u k p +1 p +1 } = { tu ; t ∈ [0 , , u ∈ S λ,δ } and that for u ∈ S λ,δ , I ( λ, tu ) is increasing for t ∈ (0 ,
2. Minimax methods for I ( λ, u ) We fix k ∈ N and m > m > m k , (2 . m k ≥ I ( λ, u ) has at least k pairs of criticalpoints.We choose δ m > δ − p − m E > m . C δ m > λ m = log( C δ m + 1) , we set Ω m = int { ( λ, u ); λ ≥ λ m , k∇ u k + ( e λ − C δ m ) k u k ≥ δ m k u k p +1 p +1 } . (2 . m = S λ ∈ [ λ m , ∞ ) ( { λ } × D λ ) is a domain whose section D λ ⊂ H r ( R N ) is aset surrounded by the Nehari manifold { u ∈ H r ( R N ) \ { } ; k∇ u k + ( e λ − C δ m ) k u k = δ m k u k p +1 p +1 } . In particular [ λ m , ∞ ) × { } ⊂ Ω m .Using Lemma 1.4, we have Lemma 2.1. (i) B m ≡ inf ( λ,u ) ∈ ∂ Ω m I ( λ, u ) > −∞ .(ii) b I ( λ, u ) ≥ for ( λ, u ) ∈ Ω m . Proof.
Note that ∂ Ω m = C ∪ C , where C = { ( λ, u ) ∈ R × ( H r ( R N ) \ { } ); λ ≥ λ m , k∇ u k + ( e λ − C δ m ) k u k = δ m k u k p +1 p +1 } , C = { ( λ m , u ) ∈ R × H r ( R N ); k∇ u k + ( e λ m − C δ m ) k u k ≥ δ m k u k p +1 p +1 } , By Lemma 1.4, I ( λ, u ) = b I ( λ, u ) − e λ m ≥ δ − p − m ( e λ − C δ m ) E − e λ m for ( λ, u ) ∈ C ,I ( λ, u ) = b I ( λ, u ) − e λ m ≥ − e λ m m for ( λ, u ) ∈ C . By our choice (2.2) of δ m , we have inf ( λ,u ) ∈ ∂ Ω m I ( λ, u ) > −∞ and (i) holds. (ii) is alsoclear.We introduce a family of minimax methods. For j ∈ N we setΓ j = { γ ( ξ ) = ( ϕ ( ξ ) , ζ ( ξ )) ∈ C ( D j , R × H r ( R N )); γ ( ξ ) satisfies conditions ( γ γ
3) below } , γ ϕ ( − ξ ) = ϕ ( ξ ), ζ ( − ξ ) = − ζ ( ξ ) for all ξ ∈ D j .( γ
2) There exists λ ∈ ( −∞ , λ ) such that ϕ ( ξ ) = λ, I ( λ, ζ ( ξ )) ∈ ( R × H r ( R N )) \ Ω m , I ( λ, ζ ( ξ )) ≤ B m − ξ ∈ ∂D j . ( γ ϕ (0) ∈ [ λ m , ∞ ) and ζ (0) = 0. Moreover I ( ϕ (0) , ζ (0)) = − e ϕ (0) m ≤ B m − . We note that(i) for λ ∈ ( −∞ , λ ), u b I ( λ, u ) has the symmetric mountain pass geometry.(ii) I ( λ,
0) = − e λ m → −∞ as λ → ∞ .From these facts, we have Γ j = ∅ for all j ∈ N .We remark that Γ j is a family of j -dimensional symmetric mountain paths joiningpoints in [ λ m , ∞ ) × { } ⊂ Ω m and ( R × H r ( R N )) \ Ω m .We set b j = inf γ ∈ Γ j max ξ ∈ D j I ( γ ( ξ )) for j ∈ N . Proposition 2.2. (i) b j ≥ B m for all j ∈ N .(ii) b j < for j = 1 , , · · · , k . To show Proposition 2.2, we need
Lemma 2.3. b j ≤ a j ( λ ) − e λ m for λ ∈ ( −∞ , λ ) . (2 . Proof.
First we note that by (ii) of Lemma 2.1 that( λ, ζ ( ξ )) ∈ ( R × H r ( R N )) \ Ω m for ζ ∈ b Γ( λ ) and ξ ∈ ∂D j . Second we remark that we may assume for ζ ( ξ ) ∈ b Γ j ( λ ) I ( λ, ζ ( ξ )) ≤ B m − ξ ∈ ∂D j . (2 . u ∈ H r ( R N ) and ν > b I ( λ, u ( x/ν )) = 12 ν N − k∇ u k + ν N (cid:18) e λ k u k − Z R N G ( u ) (cid:19) , b I ( λ, u ( x )) <
0, then ν b I ( λ, u ( x/ν )); [1 , ∞ ) → R isdecreasing and lim ν →∞ b I ( λ, u ( x/ν )) = −∞ . Thus, for a given ζ ( ξ ) ∈ b Γ j ( λ ), setting e ζ ( ξ )( x ) = (cid:26) ζ (2 ξ ) for | ξ | ∈ [0 , ], ζ ( ξ/ | ξ | )( xL (2 | ξ |− ) for | ξ | ∈ ( , L ≫ e ζ ( ξ ) ∈ b Γ j ( λ ) andmax ξ ∈ D j b I ( λ, e ζ ( ξ )) = max ξ ∈ D j b I ( λ, ζ ( ξ )) ,I ( λ, e ζ ( ξ )) ≤ B m − ξ ∈ ∂D j . Thus we may assume (2.5) for ζ ( ξ ) ∈ b Γ j ( λ ).Next we show (2.4). For ζ ( ξ ) ∈ b Γ j ( λ ) with (2.5), we set ˇ γ ( ξ ) = ( ˇ ϕ ( ξ ) , ˇ ζ ( ξ )) byˇ ϕ ( ξ ) = (cid:26) λ + R (1 − | ξ | ) for | ξ | ∈ [0 , ], λ for | ξ | ∈ ( , ζ ( ξ ) = ( | ξ | ∈ [0 , ], ζ ( ξ | ξ | (2 | ξ | − | ξ | ∈ ( , R large, we have ˇ γ ( ξ ) ∈ Γ j and I (ˇ γ ( ξ )) = I ( λ + R (1 − | ξ | ) ,
0) = − e λ + R (1 − | ξ | ) m ≤ − e λ m for | ξ | ∈ [0 ,
12 ] ,I (ˇ γ ( ξ )) = I ( λ, ˇ ζ ( ξ )) = b I ( λ, ˇ ζ ( ξ )) − e λ m ≤ max ξ ∈ D j b I ( λ, ζ ( ξ )) − e λ m for | ξ | ∈ ( 12 , . Since ζ ( ξ ) ∈ b Γ j ( λ ) is arbitrary, we have (2.4).Now we give a proof to Proposition 2.2. Proof of Proposition 2.2. (i) By ( γ γ γ ( ∂D j ) ∩ Ω m = ∅ and γ (0) ∈ Ω m for all γ ∈ Γ j . Thus γ ( D j ) ∩ ∂ Ω m = ∅ for all γ ∈ Γ j and it follows from Lemma 2.1 (i) thatmax ξ ∈ D j I ( γ ( ξ )) ≥ inf ( λ,u ) ∈ ∂ Ω m I ( λ, u ) ≡ B m . Since γ ∈ Γ j is arbitrary, we have (i).(ii) By Lemma 2.3, for any λ ∈ ( −∞ , λ ) b j e λ ≤ a j ( λ ) e λ − m . λ ∈ ( −∞ ,λ ) (cid:18) a j ( λ ) e λ − m (cid:19) = m j − m, the conclusion (ii) follows from (1.14) and (2.1).In Section 3, we will see that I ( λ, u ) satisfies a version of Palais-Smale type condition( P SP ) b for b <
0, which enables us to develop a deformation argument and to show b j ( j = 1 , , · · · , k ) are critical values of I ( λ, u ). However to show multiplicity, i.e., to dealwith the case b i = · · · = b i + ℓ , we need another family of minimax methods, which involvethe notion of genus. In this section, we use an idea from Rabinowitz [ R ] to define another family of minimaxmethods. Here the notion of genus plays a role. Definition.
Let E be a Banach space. For a closed set A ⊂ E \ { } , which is symmetricwith respect to 0, i.e., − A = A , we define genus( A ) = n if and only if there exists an oddmap ϕ ∈ C ( A, R n \{ } ) and n is the smallest integer with this property. When there areno odd map ϕ ∈ C ( A, R n \{ } ) with this property for any n ∈ N , we define genus( A ) = ∞ .Finally we set genus( ∅ ) = 0.We refer to [ R ] for fundamental properties of the genus.Our setting is different from [ R ]; our functional is invariant under the following Z -action: Z × R × H r ( R N ) → R × H r ( R N ); ( ± , λ, u ) ( λ, ± u ) , (2 . I ( λ, − u ) = I ( λ, u ). Remarking that there is no critical points in the Z -invariants { ( λ, λ ∈ R } , we modify the arguments in [ R ].We define our second family of minimax sets as follows:Λ j = { γ ( D j + ℓ \ Y ); ℓ ≥ , γ ∈ Γ j + ℓ , Y ⊂ D j + ℓ \ { } is closed,symmetric with respect to 0 and genus( Y ) ≤ ℓ } ,c j = inf A ∈ Λ j max ( λ,u ) ∈ A I ( λ, u ) . Here we summarize fundamental properties of Λ j . Here we use a projection P : R × H r ( R N ) → H r ( R N ) defined by P ( λ, u ) = u for ( λ, u ) ∈ R × H r ( R N ) . Lemma 2.4. i) Λ j = ∅ for all j ∈ N .(ii) Λ j +1 ⊂ Λ j for all j ∈ N .(iii) Let ψ ( λ, u ) = ( ψ ( λ, u ) , ψ ( λ, u )) : R × H r ( R N ) → R × H r ( R N ) be a continuous mapwith properties ψ ( λ, − u ) = ψ ( λ, u ) , ψ ( λ, − u ) = − ψ ( λ, u ) for all ( λ, u ) ∈ R × H r ( R N ) , (2 . ψ ( λ, u ) = ( λ, u ) if I ( λ, u ) ≤ B m − . (2 . Then for A ∈ Λ j , we have ψ ( A ) ∈ Λ j .(iv) For A ∈ Λ j and a closed set Z , which is invariant under Z -action (2.6) , i.e., ( λ, − u ) ∈ Z for all ( λ, u ) ∈ Z , with P ( Z ) , A \ Z ∈ Λ j − i , where i = genus( P ( Z )) . (v) A ∩ ∂ Ω m = ∅ for any A ∈ Λ j . Here Ω m is defined in (2.3) . Proof. (i), (ii) follow from the definition of Λ j .(iii) Suppose ψ ( λ, u ) : R × H r ( R N ) → R × H r ( R N ) satisfies (2.7)–(2.8). Then it is easyto see ψ ◦ γ ∈ Γ j for all γ ∈ Γ j . Thus (iii) holds.(iv) Following and modifying the argument in Sections 7–8 of [ R ], we can show (iv). Forthe sake of completeness, we give a proof in the Appendix.(v) Suppose A = γ ( D j + ℓ \ Y ) ∈ Λ j , where γ ∈ Γ j + ℓ , genus( Y ) ≤ ℓ , and let U be theconnected component of O = γ − (int Ω m ) containing 0. It is easy to see0 ∈ U, U ⊂ int D j + ℓ , from which we have genus( ∂U ) = j + ℓ . Thusgenus( ∂U \ Y ) ≥ genus( ∂U ) − genus( Y ) ≥ j. In particular, ∂U \ Y = ∅ . Since γ ( ∂U \ Y ) ⊂ A ∩ ∂ Ω m , we have A ∩ ∂ Ω m = ∅ .As fundamental properties of c n , we have Lemma 2.5. (i) B m ≤ c ≤ c ≤ · · · ≤ c j ≤ c j +1 ≤ · · · .(ii) c j ≤ b j for all j ∈ N . Proof. (i) By (v) of Lemma 2.4, we have for any A ∈ Λ j ,max ( λ,u ) ∈ A I ( λ, u ) ≥ inf ( λ,u ) ∈ ∂ Ω m I ( λ, u ) = B m , c j ≥ B m for all j ∈ N . (ii) of Lemma 2.4 implies c j ≤ c j +1 .(ii) It is easy to see γ ( D j ) ∈ Λ j for any γ ∈ Γ j . Thus we have c j ≤ b j .In the following section, we use a special deformation lemma to show c j ( j =1 , , · · · , k ) are attained by critical points.
3. Deformation argument and existence of critical points
In this section we introduce a deformation result for I ( λ, u ) and we show that c j ( j =1 , , · · · , k ) introduced in the previous section are achieved. I ( λ, u )For b ∈ R we set K b = { ( λ, u ) ∈ R × H r ( R N ); I ( λ, u ) = b, ∂ λ I ( λ, u ) = 0 , ∂ u I ( λ, u ) = 0 , P ( λ, u ) = 0 } . (3 . P ( λ, u ) is introduced in (1.5). We note that ∂ u I ( λ, u ) = 0 implies P ( λ, u ) = 0. Wealso use the following notation:[ I ≤ c ] = { ( λ, u ) ∈ H r ( R N ); I ( λ, u ) ≤ c } for c ∈ R . We have the following deformation result.
Proposition 3.1.
Assume ( g – ( g and b < . Then(i) K b is compact in R × H r ( R N ) and K b ∩ ( R ×{ } ) = ∅ .(ii) For any open neighborhood O of K b and ε > there exist ε ∈ (0 , ε ) and a continuousmap η ( t, λ, u ) : [0 , × R × H r ( R N ) → R × H r ( R N ) such that ◦ η (0 , λ, u ) = ( λ, u ) for all ( λ, u ) ∈ R × H r ( R N ) . ◦ η ( t, λ, u ) = ( λ, u ) if ( λ, u ) ∈ [ I ≤ b − ε ] . ◦ I ( η ( t, λ, u )) ≤ I ( λ, u ) for all ( t, λ, u ) ∈ [0 , × R × H r ( R N ) . ◦ η (1 , [ I ≤ b + ε ] \ O ) ⊂ [ I ≤ b − ε ] , η (1 , [ I ≤ b + ε ]) ⊂ [ I ≤ b − ε ] ∪ O . ◦ If K b = ∅ , η (1 , [ I ≤ b + ε ]) ⊂ [ I ≤ b − ε ] . ◦ Writing η ( t, λ, u ) = ( η ( t, λ, u ) , η ( t, λ, u )) , we have η ( t, λ, − u ) = η ( t, λ, u ) , η ( t, λ, − u ) = − η ( t, λ, u ) for all ( t, λ, u ) ∈ [0 , × R × H r ( R N ) . Such a deformation result is usually obtained under the Palais-Smale compactness con-dition. However it seems difficult to verify the standard Palais-Smale condition under16 g g P SP ) of Palais-Smale condition andwe develop a new deformation argument to prove Proposition 3.1. We postpone a proof ofProposition 3.1 until Section 4 and in this section we show c j ( j = 1 , , · · · , k ) are attained. As an application of our Proposition 3.1 we show the following
Proposition 3.2. (i) For j = 1 , , · · · , k , c j < c j is a critical value of I ( λ, u ).(ii) If c j = c j +1 = · · · = c j + q = b < j + q ≤ k ), thengenus( P ( K b )) ≥ q + 1 . In particular, K b ) = ∞ if q ≥ Proof. c j < j = 1 , , · · · , k ) follow from Proposition 2.2 and Lemma 2.5 (ii). Theargument for the fact that K c j = ∅ is similar to the proof of (ii). So we omit it.(ii) Suppose that c j = c j +1 = · · · = c j + q = b <
0. Since K b is compact and K b ∩ ( R ×{ } ) = ∅ , the projection P ( K b ) of K b onto H r ( R N ) is compact, symmetric with respect to 0 and0 P ( K b ). Thus by the fundamental property of the genus,genus( P ( K b )) < ∞ , there exists δ > P ( N δ ( K b ))) = genus( P ( K b )) . Here we denote δ -neighborhood of a set A ⊂ R × H r ( R N ) by N δ ( A ), i.e., N δ ( A ) = { ( λ, u ); dist (( λ, u ) , A ) ≤ δ } , where dist (( λ, u ) , A ) = inf ( λ ′ ,u ′ ) ∈ A q | λ − λ ′ | + k u − u ′ k H . By Proposition 3.1, there exist ε > η : [0 , × R × H r ( R N ) → R × H r ( R N )such that η (1 , [ I ≤ b + ε ] \ N δ ( K b )) ⊂ [ I ≤ b − ε ] ,η ( t, λ, u ) = ( λ, u ) if I ( λ, u ) ≤ b − . We note that B m − ≤ b − .We take A ∈ Λ j + q such that A ⊂ [ I ≤ b + ε ]. Then η (1 , A \ N δ ( K b )) ⊂ [ I ≤ b − ε ] . (3 . P ( K b )) ≤ q , we have genus( P ( N δ ( K b ))) ≤ q . By (iv) of Lemma 2.4, A \ N δ ( K b ) ∈ Λ j . (3 . c j ≤ b − ε , which is a contradiction. Thus genus( P ( K b )) ≥ q + 1.Now we can show Proof of (i) of Theorem 0.2.
Clearly (i) of Theorem 0.2 follows from Proposition 3.2.
Proof of (ii) of Theorem 0.2.
Under the condition (0.2), we have m k = 0 for all k ∈ N .Thus we have c j ≤ b j < j ∈ N and c j ( j ∈ N ) are critical values of I ( λ, u ). Weneed to show c j → j → ∞ .Arguing indirectly, we assume c j → c < j → ∞ . Then K c is compact and K c ∩ ( R ×{ } ) = ∅ . Set q = genus( P ( K c )) < ∞ and choose δ > P ( N δ ( K c ))) = genus( P ( K c )) = q. As in the proof of Proposition 3.2, there exist ε > η : [0 , × R × H r ( R N ) → R × H r ( R N ) such that η (1 , [ I ≤ c + ε ] \ N δ ( K c )) ⊂ [ I ≤ c − ε ] , (3 . η ( t, λ, u ) = ( λ, u ) if I ( λ, u ) ≤ B m − . We choose j ≫ c j ≥ c − ε and take A ∈ Λ j such that A ⊂ [ I ≤ c + ε ]. Then wehave A \ N δ ( K c ) ∈ Λ j − q . (3 . c j − q ≤ c − ε . Since we can take j arbitrary large, we have lim j →∞ c j ≤ c − ε . This is a contradiction. ( P SP ) condition and construction of a flow In this section we give a new type of deformation argument for our functional I ( λ, u ). Ourdeformation argument is inspired by our previous work [ HIT ].18 .a. ( P SP ) condition Since it is difficult to verify the standard Palais-Smale condition for I ( λ, u ) under theconditions ( g g P SP ) b , which isweaker than the standard Palais-Smale condition and which takes the scaling property of I ( λ, u ) into consideration through the Pohozaev functional P ( λ, u ) . Definition.
For b ∈ R , we say that I ( λ, u ) satisfies ( P SP ) b condition if and only if thefollowing holds.( P SP ) b If a sequence ( λ n , u n ) ∞ n =1 ⊂ R × H r ( R N ) satisfies as n → ∞ I ( λ n , u n ) → b, (4 . ∂ λ I ( λ n , u n ) → , (4 . ∂ u I ( λ n , u n ) → H r ( R N )) ∗ , (4 . P ( λ n , u n ) → , (4 . λ n , u n ) ∞ n =1 has a strongly convergent subsequence in R × H r ( R N ).First we observe that ( P SP ) b holds for I ( λ, u ) for b < Proposition 4.1.
Assume ( g – ( g . Then I ( λ, u ) satisfies ( P SP ) b for b < . Proof.
Let b < λ n , u n ) ∞ n =1 satisfies (4.1)–(4.4). We will show that( λ n , u n ) ∞ n =1 has a strongly convergent subsequence. Proof consists of several steps. Step 1: λ n is bounded from below as n → ∞ . Since P ( λ n , u n ) = N ( I ( λ n , u n ) + m e λ n ) − k∇ u n k , we have from (4.1), (4.4) that m n →∞ e λ n ≥ − b > . Thus λ n is bounded from below as n → ∞ . Step 2: k u n k → m as n → ∞ . Since ∂ λ I ( λ n , u n ) = e λn ( k u n k − m ), it follows from (4.2) and Step 1 that k u n k → m . Step 3: k∇ u n k and λ n are bounded as n → ∞ . We have ∂ u I ( λ n , u n ) u n = k∇ u n k − Z R N g ( u n ) u n + e λ n k u n k . (4 . g
2) and ( g δ > C δ > | g ( ξ ) ξ | ≤ C δ | ξ | + δ | ξ | p +1 for all ξ ∈ R . Thus (cid:12)(cid:12)(cid:12)(cid:12)Z R N g ( u ) u (cid:12)(cid:12)(cid:12)(cid:12) ≤ C δ k u k + δ k u k p +1 p +1 for all u ∈ H r ( R N ) . Since p = 1 + N , by Gagliardo-Nirenberg inequality there exists C N > k u k p +1 p +1 ≤ C N k∇ u k k u k p − for all u ∈ H r ( R N ) . Thus it follows from (4.5) that k∇ u n k − C δ k u n k − δC N k∇ u n k k u n k p − + e λ n k u n k ≤ ε n q k∇ u n k + k u n k , where ε n = k ∂ u I ( λ n , u n ) k ( H r ( R N )) ∗ → (cid:16) − δC N ( m + o (1)) p − (cid:17) k∇ u n k + ( e λ n − C δ )( m + o (1)) ≤ ε n q k∇ u n k + m + o (1) . Choosing δ > δC N m p − < , we observe that k∇ u n k and e λ n are boundedas n → ∞ . Step 4: Conclusion
By Steps 1–3, ( λ n , u n ) ∞ n =1 is a bounded sequence in R × H r ( R N ). After extracting asubsequence — still denoted by ( λ n , u n ) ∞ n =1 —, we may assume that λ n → λ and u n ⇀ u weakly in H r ( R N ) for some ( λ , u ) ∈ R × H r ( R N ). By ( g g Z R N g ( u n ) u → Z R N g ( u ) u , Z R N g ( u n ) u n → Z R N g ( u ) u . Thus, we deduce from ∂ u I ( λ n , u n ) u n → ∂ u I ( λ n , u n ) u → k∇ u n k + e λ k u n k → k∇ u k + e λ k u k , which implies u n → u strongly in H r ( R N ). Remark 4.2.
For b = 0, ( P SP ) does not hold for I ( λ, u ). In fact, for a sequence( λ n , ∞ n =1 with λ n → −∞ , we have I ( λ n ,
0) = − e λ n m → , ∂ λ I ( λ n ,
0) = − e λ n m → ,∂ u I ( λ n ,
0) = 0 , P ( λ n ,
0) = 0 . λ n , ∞ n =1 has no convergent subsequences.As a corollary to Proposition 4.1, we have Corollary 4.3.
For b < , K b defined in (3.1) is compact in R × H r ( R N ) and satisfies K b ∩ ( R ×{ } ) = ∅ . Proof. K b is compact since I ( λ, u ) satisfies ( P SP ) b . K b ∩ ( R ×{ } ) = ∅ follows from thefact that ∂ λ I ( λ,
0) = − e λ m = 0. J ( θ, λ, u )To construct a deformation flow we need an augmented functional J ( θ, λ, u ) : R × R × H r ( R N ) → R defined by J ( θ, λ, u ) = 12 e ( N − θ k∇ u k − e Nθ Z R N G ( u ) + e λ (cid:0) e Nθ k u k − m (cid:1) . We introduce J ( θ, λ, u ) to make use of the scaling property of I ( λ, u ). As a basic propertyof J ( θ, λ, u ) we have I ( λ, u ( x/e θ )) = J ( θ, λ, u ) for all ( θ, λ, u ) ∈ R × R × H r ( R N ) . (4 . I ( λ, u ) through a deformation flow for J ( θ, λ, u ). J ( θ, λ, u ) satisfies the following properties. Lemma 4.4.
For all ( θ, λ, u ) ∈ R × R × H r ( R N ) , h ∈ H r ( R N ) and β ∈ R , ∂ θ J ( θ, λ, u ( x )) = P ( λ, u ( x/e θ )) , (4 . ∂ λ J ( θ, λ, u ( x )) = ∂ λ I ( λ, u ( x/e θ )) , (4 . ∂ u J ( θ, λ, u ( x )) h ( x ) = ∂ u I ( λ, u ( x/e θ )) h ( x/e θ ) , (4 . J ( θ + β, λ, u ( e β x )) = J ( θ, λ, u ( x )) . (4 . roof. We compute that ∂ θ J ( θ, λ, u ( x )) = N − e ( N − θ k∇ u k + N e Nθ (cid:18) e λ k u k − Z R N G ( u ) (cid:19) = N − k∇ ( u ( x/e θ x )) k + N (cid:18) e λ k u ( x/e θ ) k − Z R N G ( u ( x/e θ )) (cid:19) = P ( λ, u ( x/e θ )) ,∂ λ J ( θ, λ, u ( x )) = e λ (cid:0) e Nθ k u k − m (cid:1) = e λ (cid:0) k u ( x/e θ ) k − m (cid:1) = ∂ λ I ( λ, u ( x/e θ )) ,∂ u J ( θ, λ, u ( x )) h ( x ) = e ( N − θ ( ∇ u, ∇ h ) + e λ e Nθ ( u, h ) − e Nθ Z R N g ( u ( x )) h ( x )= ( ∇ u ( x/e θ ) , ∇ h ( x/e θ )) + e λ ( u ( x/e θ ) , h ( x/e θ )) − Z R N g ( u ( x/e θ )) h ( x/e θ )= ∂ u I ( λ, u ( x/e θ )) h ( x/e θ ) . Thus we have (4.7)–(4.9). (4.10) follows from (4.6).To analyze J ( θ, λ, u ), it is natural to regard R × R × H r ( R N ) as a Hilbert manifoldwith a metric related to (4.6). More precisely, we write M = R × R × H r ( R N ). We notethat T ( θ,λ,u ) M = R × R × H r ( R N ) for ( θ, λ, u ) ∈ M and we introduce a metric h· , ·i ( θ,λ,u ) on T ( θ,λ,u ) M by h ( α, ν, h ) , ( α ′ , ν ′ , h ′ ) i ( θ,λ,u ) = αα ′ + νν ′ + e ( N − θ ( ∇ h, ∇ h ′ ) + e Nθ ( h, h ′ ) , k ( α, ν, h ( x )) k ( θ,λ,u ) = q h ( α, ν, h ) , ( α, ν, h ) i ( θ,λ,u ) for ( α, ν, h ), ( α ′ , ν ′ , h ′ ) ∈ T ( θ,λ,u ) M . We also denote the dual norm of k·k ( θ,λ,u ) by k·k ( θ,λ,u ) , ∗ , that is, k f k ( θ,λ,u ) , ∗ = sup k ( α,ν,h ) k ( θ,λ,u ) ≤ | f ( α, ν, h ) | for f ∈ T ∗ ( θ,λ,u ) ( M ) . (4 . M, h· , ·i ) is a complete Hilbert manifold. We note that h· , ·i ( θ,λ,u ) and k·k ( θ,λ,u ) depend only on θ . So sometimes we denote them by h· , ·i ( θ, · , · ) , k·k ( θ, · , · ) . Wehave k ( α, ν, h ) k ( θ, · , · ) = α + ν + e ( N − θ k∇ h k + e Nθ k h k = α + ν + k h ( x/e θ ) k H = k ( α, ν, h ( x/e θ )) k (0 , · , · ) for ( α, ν, h ) ∈ T ( θ,λ,u ) M. (4 . α, ν, h ) ∈ T ( θ, · , · ) M and β ∈ R k ( α, ν, h ( e β x )) k ( θ + β, · , · ) = k ( α, ν, h ( x )) k ( θ, · , · ) . (4 . M ( · , · ) on M bydist M (( θ , λ , h ) , ( θ , λ , h ))= inf nZ k ˙ σ ( t ) k σ ( t ) dt ; σ ( t ) ∈ C ([0 , , M ) ,σ (0) = ( θ , λ , h ) , σ (1) = ( θ , λ , h ) o . (4 . β ∈ R dist M (( θ + β, λ , u ( e β x )) , ( θ + β, λ , u ( e β x ))) = dist M (( θ , λ , u ( x )) , ( θ , λ , u ( x ))) . (4 . D = ( ∂ θ , ∂ λ , ∂ u ) , we have Lemma 4.5.
For ( θ, λ, u ) ∈ M , we have kD J ( θ, λ, u ) k ( θ,λ,u ) , ∗ = (cid:16) | P ( λ, u ( x/e θ )) | + | ∂ λ I ( λ, u ( x/e θ )) | + k ∂ u I ( λ, u ( x/e θ )) k H r ( R N )) ∗ (cid:17) / . Proof.
By Lemma 4.4, we have D J ( θ, λ, u )( α, ν, h )= P ( λ, u ( x/e θ )) α + ∂ λ I ( λ, u ( x/e θ )) ν + ∂ u I ( λ, u ( x/e θ )) h ( x/e θ ) . Noting (4.12), the conclusion of Lemma 4.5 follows from the definition (4.11).For b ∈ R , we use notation e K b = { ( θ, λ, u ) ∈ M ; J ( θ, λ, u ) = b, D J ( θ, λ, u ) = (0 , , } . By (4.6)–(4.9), we observe that e K b = { ( θ, λ, u ( e θ x )); θ ∈ R , ( λ, u ) ∈ K b } . We also use notation for ( θ, λ, u ) ∈ M and e A ⊂ M dist M (( θ, λ, u ) , e A ) = inf ( θ ′ ,λ ′ ,u ′ ) ∈ e A dist M (( θ, λ, u ) , ( θ ′ , λ ′ , u ′ )) . P SP ) b condition for I ( λ, u ), we deduce the following Proposition 4.6.
For b < , J ( θ, λ, u ) satisfying the following property: ( g P SP ) b For any sequence ( θ n , λ n , u n ) ∞ n =1 ⊂ M with J ( θ n , λ n , u n ) → b, (4 . kD J ( θ n , λ n , u n ) k ( θ n ,λ n ,u n ) , ∗ → as n → ∞ , (4 . we have dist M (( θ n , λ n , u n ) , e K b ) → . (4 . Proof.
Suppose that ( θ n , λ n , u n ) ∞ n =1 satisfies (4.16)–(4.17). It suffices to show that( θ n , λ n , u n ) ∞ n =1 has a subsequence with the property (4.18).Setting ˆ u n ( x ) = u n ( x/e θ n ), we have by Lemma 4.5 that I ( λ n , ˆ u n ) → b < ,P ( λ n , ˆ u n ) → , ∂ λ I ( λ n , ˆ u n ) → , ∂ u I ( λ n , ˆ u n ) → H r ( R N )) ∗ . Thus by Proposition 4.1, there exists a subsequence — still denoted by ( λ n , ˆ u n ) ∞ n =1 — and( λ , ˆ u ) ∈ R × H r ( R N ) such that λ n → λ and ˆ u n → ˆ u strongly in H r ( R N ) . Note that ( λ , ˆ u ) ∈ K b and thus ( θ n , λ , ˆ u ( e θ n x )) ∈ e K b . By (4.15), we havedist M (( θ n , λ n , u n ) , e K b ) ≤ dist M (( θ n , λ n , u n ) , ( θ n , λ , ˆ u ( e θ n x )))=dist M ((0 , λ n , ˆ u n ) , (0 , λ , ˆ u ( x )) ≤ (cid:0) | λ n − λ | + k ˆ u n − ˆ u k H (cid:1) / → n → ∞ . As a corollary to Proposition 4.6, we have the following uniform estimate of D J ( θ, λ, u )outside of ρ -neighborhood of e K b . Corollary 4.7.
Assume b < . Then for any ρ > there exists δ ρ > such that for ( θ, λ, u ) ∈ M | J ( θ, λ, u ) − b | < δ ρ and dist M (( θ, λ, u ) , e K b ) ≥ ρ imply kD J ( θ, λ, u ) k ( θ,λ,u ) , ∗ ≥ δ ρ . We remark that e K b is not compact in M but Corollary 4.7 gives us a uniform lowerbound of kD J ( θ, λ, u ) k ( θ,λ,u ) , ∗ outside of ρ -neighborhood of e K b , which enables us to con-struct a deformation flow for J ( θ, λ, u ). 24 .c. Deformation flow for J ( θ, λ, u )In this section we give a deformation result for J ( θ, λ, u ). We need the following notation:[ J ≤ c ] M = { ( θ, λ, u ) ∈ M ; J ( θ, λ, u ) ≤ c } for c ∈ R , e N ρ ( e A ) = { ( θ, λ, u ) ∈ M ; dist M (( θ, λ, u ) , e A ) ≤ ρ } for e A ⊂ M and ρ > . We have the following deformation result.
Proposition 4.8.
Assume b < . Then for any ε > and ρ > there exist ε ∈ (0 , ε ) anda continuous map e η ( t, θ, λ, u ) : [0 , × M → M such that ◦ e η (0 , θ, λ, u ) = ( θ, λ, u ) for all ( θ, λ, u ) ∈ M . ◦ e η ( t, θ, λ, u ) = ( θ, λ, u ) if ( θ, λ, u ) ∈ [ J ≤ b − ε ] M . ◦ J ( e η ( t, θ, λ, u )) ≤ J ( θ, λ, u ) for all ( t, θ, λ, u ) ∈ [0 , × M . ◦ e η (1 , [ J ≤ b + ε ] M \ e N ρ ( e K b )) ⊂ [ J ≤ b − ε ] M , e η (1 , [ J ≤ b + ε ] M ) ⊂ [ J ≤ b − ε ] M ∪ e N ρ ( e K b ) . ◦ If K b = ∅ , e η (1 , [ J ≤ b + ε ] M ) ⊂ [ J ≤ b − ε ] M . ◦ We write e η ( t, θ, λ, u ) = ( e η ( t, θ, λ, u ) , e η ( t, θ, λ, u ) , e η ( t, θ, λ, u )) . Then e η ( t, θ, λ, u ) , e η ( t, θ, λ, u ) are even in u and e η ( t, θ, λ, u ) is odd in u . That is, for all ( t, θ, λ, u ) ∈ [0 , × M e η ( t, θ, λ, − u ) = e η ( t, θ, λ, u ) , e η ( t, θ, λ, − u ) = e η ( t, θ, λ, u ) , e η ( t, θ, λ, − u ) = − e η ( t, θ, λ, u ) . Proof.
Let M ′ = { ( θ, λ, u ) ∈ M ; D J ( θ, λ, u ) = (0 , , } . It is well-known that thereexists a pseudo-gradient vector field V : M ′ → T M such that for ( θ, λ, u ) ∈ M ′ (1) kV ( θ, λ, u ) k ( θ,λ,u ) ≤ kD J ( θ, λ, u ) k ( θ,λ,u ) , ∗ ,(2) D J ( θ, λ, u ) V ( θ, λ, u ) ≥ kD J ( θ, λ, u ) k θ,λ,u ) , ∗ ,(3) V : M ′ → R × R × H r ( R N ) is locally Lipschitz continuous.We can also have(4) V ( θ, λ, u ) = ( V ( θ, λ, u ) , V ( θ, λ, u ) , V ( θ, λ, u )) satisfies V ( θ, λ, − u ) = V ( θ, λ, u ) , V ( θ, λ, − u ) = V ( θ, λ, u ) , V ( θ, λ, − u ) = −V ( θ, λ, u ) . For a given ρ > δ ρ > | J ( θ, λ, u ) − b | < δ ρ and ( θ, λ, u ) e N ρ/ ( e K b ) imply kD J ( θ, λ, u ) k ( θ,λ,u ) , ∗ ≥ δ ρ . (4 . ϕ : M → [0 ,
1] such that ϕ ( θ, λ, u ) = 1 for ( θ, λ, u ) ∈ M \ e N ρ ( e K b ) ,ϕ ( θ, λ, u ) = 0 for ( θ, λ, u ) ∈ e N ρ ( e K b ) ,ϕ ( θ, λ, − u ) = ϕ ( θ, λ, u ) for all ( θ, λ, u ) ∈ M. For ε > ε ∈ (0 , δ ρ ) and we choose a locally Lipschitz continuous function ψ : R → [0 ,
1] such that ψ ( s ) = (cid:26) s ∈ [ b − ε , b + ε ],0 for s ∈ R \ [ b − ε, b + ε ].We consider the following ODE in M : d e ηdt = − ϕ ( e η ) ψ ( J ( e η )) V ( e η ) kV ( e η ) k e η , e η (0 , θ, λ, u ) = ( θ, λ, u ) . For ε ∈ (0 , ε ) small, e η ( t, θ, λ, u ) has the desired properties 1 ◦ –6 ◦ . We show just the firstpart of 4 ◦ : e η (1 , [ J ≤ b + ε ] M \ e N ρ ( e K b )) ⊂ [ J ≤ b − ε ] M . (4 . ◦ –3 ◦ easily and we use them in what follows. We also note that k d e ηdt ( t ) k e η ( t ) ≤ t. (4 . ε ∈ (0 , ε ), which we choose later, we assume e η ( t ) = e η ( t, θ, λ, u ) satisfies e η (0) ∈ [ J ≤ b + ε ] M \ e N ρ ( e K b ) . If e η (1) [ J ≤ b − ε ] M , we have J ( e η ( t )) ∈ [ b − ε, b + ε ] for all t ∈ [0 , e η ( t ) e N ρ ( e K b ) for all t ∈ [0 , e η ( t ) ∈ e N ρ ( e K b ) for some t ∈ [0 , kD J ( e η ( t )) k e η ( t ) , ∗ ≥ δ ρ for all t ∈ [0 , . By our choice of ϕ , ψ , we have ddt J ( e η ( t )) = D J ( e η ( t )) d e ηdt ( t ) = −D J ( e η ( t )) V ( e η ( t )) kV ( e η ( t )) k e η ( t ) ≤ − kD J ( e η ( t )) k e η ( t ) , ∗ ≤ − δ ρ . J ( e η (1)) = J ( e η (0)) + Z ddt J ( e η ( t )) dt ≤ J ( e η (0)) − δ ρ ≤ b + ε − δ ρ . If Case 2 takes a place, we can find an interval [ α, β ] ⊂ [0 ,
1] such that e η ( α ) ∈ ∂ e N ρ ( e K b ) , e η ( β ) ∈ ∂ e N ρ ( e K b ) , e η ( t ) ∈ e N ρ ( e K b ) \ e N ρ ( e K b ) for all t ∈ [ α, β ) . By (4.21), β − α ≥ Z βα k d e ηdt ( t ) k e η ( t ) dt ≥ dist M ( e η ( α ) , e η ( β )) ≥ ρ. Thus, J ( e η (1)) ≤ J ( e η ( β )) = J ( e η ( α )) + Z βα ddt J ( e η ( t )) dt ≤ J ( e η (0)) + Z βα ddt J ( e η ( t )) dt ≤ J ( e η (0)) + Z βα − δ ρ dt ≤ J ( e η (0)) − δ ρ β − α ) ≤ b + ε − δ ρ ρ . Choosing ε < min { ε , δ ρ , δ ρ ρ } , we have J ( e η (1)) ≤ b − ε in both cases. This is a contra-diction and we have (4.20).In the following section, we can construct a deformation flow for I ( λ, u ) using e η ( t, θ, λ, u ). I ( λ, u )In this section, we construct a deformation flow for I ( λ, u ) and give a proof to our Propo-sition 3.1.We use the following maps: π : M → R × H r ( R N ); ( θ, λ, u ( x )) ( λ, u ( x/e θ )) ,ι : R × H r ( R N ) → M ; ( λ, u ( x )) (0 , λ, u ( x ))and we construct a deformation flow η ( t, λ, u ) : [0 , × R × H r ( R N ) → R × H r ( R N ) as acomposition π ◦ e η ( t, · ) ◦ ι ; η ( t, λ, u ) = π ( e η ( t, ι ( λ, u ))) = π ( e η ( t, , λ, u )) . (4 . π and ι , we have π ( ι ( λ, u )) = ( λ, u ) for all ( λ, u ) ∈ R × H r ( R N ) ,ι ( π ( θ, λ, u )) = (0 , λ, u ( x/e θ )) for all ( θ, λ, u ) ∈ M,J ( θ, λ, u ) = I ( π ( θ, λ, u )) for all ( θ, λ, u ) ∈ M. π ( e K b ) = K b . The following lemma gives us a relation between π ( e N ρ ( e K b )) and N ρ ( K b ). Lemma 4.9.
For any ρ > there exists R ( ρ ) > such that π ( e N ρ ( e K b )) ⊂ N R ( ρ ) ( K b ) , (4 . ι (cid:16) (( R × H r ( R N )) \ N R ( ρ ) ( K b )) (cid:17) ⊂ M \ e N ρ ( e K b ) . (4 . Moreover R ( ρ ) → as ρ → . (4 . Proof.
For ρ >
0, suppose that ( λ , u ) ∈ R × H r ( R N ) satisfies dist M ((0 , λ , u ) , e K b ) ≤ ρ .First we showdist M (( λ , u ) , K b ) ≤ e Nρ ρ + sup {k ω ( e α x ) − ω ( x ) k H ; | α | ≤ ρ, ω ∈ P ( K b ) } . (4 . ε > σ ( t ) = ( θ ( t ) , λ ( t ) , u ( t )) ∈ C ([0 , , M ) such that σ (0) =(0 , λ , u ), σ (1) ∈ e K b and Z k ˙ σ ( t ) k σ ( t ) dt ≤ ρ + ε. In particular, since θ (0) = 0, for any t ∈ [0 , | θ ( t ) | ≤ Z | ˙ θ ( t ) | dt ≤ Z k ˙ σ ( t ) k σ ( t ) dt ≤ ρ + ε. Thus k ( λ (0) , u (0)) − ( λ (1) , u (1)) k R × H r ( R N ) ≤ Z (cid:16) | ˙ λ ( t ) | + k ˙ u ( t ) k H (cid:17) / dt ≤ e N ( ρ + ε )2 Z (cid:16) | ˙ θ ( t ) | + | ˙ λ ( t ) | + e ( N − θ ( t ) k∇ ˙ u ( t ) k + e Nθ ( t ) k ˙ u ( t ) k (cid:17) / dt = e N ( ρ + ε )2 Z k ( ˙ θ ( t ) , ˙ λ ( t ) , ˙ u ( t )) k σ ( t ) dt ≤ e N ( ρ + ε )2 ( ρ + ε ) . On the other hand, since ( θ (1) , λ (1) , u (1)) ∈ e K b , we have ( λ (1) , u (1)( x/e θ (1) )) ∈ K b , i.e., u (1)( x/e θ (1) ) ∈ P ( K b ). Thusdist (( λ , u ) , K b ) ≤ k ( λ (0) , u (0)) − ( λ (1) , u (1)( x/e θ (1) )) k R × H r ( R N ) ≤k ( λ (0) , u (0)) − ( λ (1) , u (1)) k R × H r ( R N ) + k ( λ (1) , u (1)( x )) − ( λ (1) , u (1)( x/e θ (1) )) k R × H r ( R N ) ≤k ( λ (0) , u (0)) − ( λ (1) , u (1)) k R × H r ( R N ) + k u (1)( x ) − u (1)( x/e θ (1) ) k H ≤ e N ( ρ + ε )2 ( ρ + ε ) + sup {k ω ( e α x ) − ω ( x ) k H ; | α | ≤ ρ + ε, ω ∈ P ( K b ) } . ε > R ( ρ ) = e Nρ ρ + sup {k ω ( e α x ) − ω ( x ) k H ; | α | ≤ ρ, ω ∈ P ( K b ) } . Then dist M ((0 , λ , u ) , e K b ) ≤ ρ implies dist (( λ , u ) , K b ) ≤ R ( ρ ) . (4 . P ( K b ) is compact in H r ( R N ), we havesup {k ω ( e α x ) − ω ( x ) k H ; | α | ≤ ρ, ω ∈ P ( K b ) } → ρ → , which implies (4.25). Noting dist M (( θ, λ, u ) , e K b ) = dist M ((0 , λ, u ( x/e θ )) , e K b ), (4.27) im-plies (4.23) and (4.24).Now we can give a proof of Proposition 3.1. Proof of Proposition 3.1.
Let O be a given neighborhood of K b and let ε > ρ > N R ( ρ ) ( K b ) ⊂ O . By Proposition 4.8, there exist ε ∈ (0 , ε ) and e η : [0 , × M → M such that 1 ◦ –6 ◦ in Proposition 4.8 hold. We define η ( t, λ, u ) : [0 , × R × H r ( R N ) → R × H r ( R N ) by (4.22).We can check that η ( t, λ, u ) satisfies the properties 1 ◦ –6 ◦ of Proposition 3.1. Here wejust prove η (1 , [ I ≤ b + ε ] \ O ) ⊂ [ I ≤ b − ε ] . (4 . I ≤ b + ε ] \ O ⊂ [ I ≤ b + ε ] \ N R ( ρ ) ( K b ), we have from (4.24) ι ([ I ≤ b + ε ] \ O ) ⊂ [ J ≤ b + ε ] M \ e N ρ ( e K b ) . (4 . ◦ of Proposition 4.8, e η (1 , [ J ≤ b + ε ] M \ e N ρ ( e K b )) ⊂ [ J ≤ b − ε ] M . (4 . π and (4.6), π ([ J ≤ b − ε ] M ) ⊂ [ I ≤ b − ε ] . (4 . Remark 4.10.
By our construction, t e η ( t, θ, λ, u ); [0 , → R × R × H r ( R N ) is ofclass C . However, t u ( x/e t ); R → H r ( R N ) is continuous but not of class C for u ∈ H r ( R N ) \ H ( R N ) and thus t η ( t, λ, u ) = π ( e η ( t, , λ, u )); [0 , → R × H r ( R N ) iscontinuous but not of class C . 29 . Minimizing problem In this section we assume ( g g
4) (without ( g I m <
0, the existence of a solution is shown by Shibata [ S1 ], that is, heshowed that I m is achieved by a solution of ( ∗ ) m . First we give an approach using ourfunctional I ( λ, u ). Under the condition ( g g λ ∈ ( −∞ , ∞ ] by (1.8).(i) For λ < λ , u b I ( λ, u ) has the mountain pass geometry.(ii) When λ < ∞ , b I ( λ, u ) ≥ λ ≥ λ and u ∈ H r ( R N ).We set for λ < λ b Γ mp ( λ ) = { ζ ( τ ) ∈ C ([0 , , H r ( R N )); ζ (0) = 0 , b I ( λ, ζ (1)) < } ,a mp ( λ ) = inf ζ ∈ b Γ mp ( λ ) max τ ∈ [0 , b I ( λ, ζ ( τ )) . (5 . g
5) holds, a mp ( λ ) coincides with a ( λ ) defined in (1.9)–(1.10). By theresult of [ HIT ], we see that a mp ( λ ) is attained by a critical point of u b I ( λ, u ). This factcan also be shown via our new deformation argument. See Section 6.We set m = 2 inf λ ∈ ( −∞ ,λ ) a mp ( λ ) e λ . (5 . Theorem 5.1.
Assume ( g – ( g . Suppose m > m . Then ( ∗ ) m has at least one solution ( λ ♯ , u ♯ ) , which is characterized by the following minimax method; I ( λ ♯ , u ♯ ) = b mp < , where b mp = inf γ ∈ Γ mp max τ ∈ [0 , I ( γ ( τ )) , Γ mp = { γ ( τ ) ∈ C ([0 , , R × H r ( R N )); γ (0) ∈ [ λ m , ∞ ) × { } , I ( γ (0)) ≤ B m − ,γ (1) ∈ ( R × H r ( R N )) \ Ω m , I ( γ (1)) ≤ B m − } . Here λ m ∈ R , Ω m ⊂ [ λ m , ∞ ) × H r ( R N ) and B m = inf ( λ,u ) ∈ ∂ Ω m I ( λ, u ) > −∞ are chosenas in Section 2. As a corollary, we have 30 orollary 5.2.
Assume ( g – ( g and suppose m > m . Then I m < . Proof.
The critical point ( λ ♯ , u ♯ ) obtained in Theorem 5.1 satisfies k u ♯ k = m and F ( u ♯ ) = I ( λ ♯ , u ♯ ) = b mp < . Thus I m = inf k u k = m F ( u ) ≤ F ( u ♯ ) < Theorem 5.3.
Under the assumption of Theorem 5.1, there exists γ ∈ Γ mp such that b mp = max τ ∈ [0 , I ( γ ( τ )) . Proof.
Let ( λ ♯ , u ♯ ) be the critical point corresponding to b mp . In Jeanjean-Tanaka [ JT ],we find a path ζ ( τ ) ∈ b Γ mp ( λ ♯ ) such that u ♯ ∈ ζ ([0 , b mp = b I ( λ ♯ , u ♯ ) = max τ ∈ [0 , b I ( λ ♯ , γ ( τ )) . As in the proof of Lemma 2.3, we may assume b I ( λ ♯ , ζ (1)) ≤ B m −
1. Joining paths[0 , → R × H r ( R N ); τ ( λ ♯ τ + L (1 − τ ) , , → R × H r ( R N ); τ ( λ ♯ , ζ ( τ )) , we find the desired path γ ∈ Γ mp . I m Next we consider the problem ( ∗ ) m under the conditions ( g g
4) and I m < S1 ] showed the following Theorem 5.4 ([ S1 ]) . There exists m S ∈ [0 , ∞ ) such that(i) I m = 0 for m ∈ (0 , m S ] , I m < for m ∈ ( m S , ∞ ) .(ii) If I m < , I m is attained and the minimizer is a solution of ( ∗ ) m . In what follows, we will show that m given in (5.2) coincides with m S and I m = b mp .Precisely, 31i) m > m if and only if I m < m > m , I m = b mp .First we show the minimizer of I m satisfies the following properties. Lemma 5.5.
Suppose I m < and let ( µ ∗ , u ∗ ) be the corresponding minimizer of I m , i.e., F ( u ∗ ) = I m , k u ∗ k = m . Then(i) µ ∗ > .(ii) N − k∇ u ∗ k + N (cid:16) µ ∗ k u ∗ k − R R N G ( u ∗ ) (cid:17) = 0 . Proof.
First we show (ii). We set u ∗ θ ( x ) = θ N/ u ∗ ( θx ) for θ >
0. Since u ∗ is a minimizerof F ( u ) under the constraint k u k = m and k u ∗ θ k = m for all θ >
0, we have ddθ (cid:12)(cid:12)(cid:12) θ =1 F ( u ∗ θ ) = 0 , that is, k∇ u ∗ k + N Z R N G ( u ∗ ) − N Z R N g ( u ∗ ) u ∗ = 0 . (5 . µ ∗ , u ∗ ) solves ( ∗ ) m , we also have k∇ u ∗ k + µ ∗ k u ∗ k = Z R N g ( u ∗ ) u ∗ . (5 . µ ∗ N m = µ ∗ N k u ∗ k = − N F ( u ∗ ) + k∇ u ∗ k ≥ − N I m > . Thus we have µ ∗ > λ ∗ = log µ ∗ , ( λ ∗ , u ∗ ) is a critical point of I ( λ, u ) with I ( λ ∗ , u ∗ ) = I m and P ( λ ∗ , u ∗ ) = 0 . Next we show
Proposition 5.6.
Suppose I m < and let ( λ ∗ , u ∗ ) be a critical point corresponding to I m . Then we have(i) u b I ( λ ∗ , u ) has the mountain pass geometry, that is, λ ∗ < λ .(ii) b I ( λ ∗ , u ∗ ) ≥ a mp ( λ ∗ ) . iii) m > m , where m is given in (5.2) . Proof. (i) It suffices to show b I ( λ ∗ , u ) < u ∈ H r ( R N ). We set b G ( ξ ) = G ( ξ ) − e λ ∗ ξ for ξ ∈ R . Then we have for some v ∈ H r ( R N ) Z R N b G ( v ) > . (5 . N ≥
3, it follows from P ( λ ∗ , u ∗ ) = 0 that (5.5) holds with v = u ∗ .When N = 2, we have by P ( λ ∗ , u ∗ ) = 0 Z R N b G ( u ∗ ) = 0 . We also have from ( ∗ ) m dds (cid:12)(cid:12)(cid:12) s =1 Z R N b G ( su ∗ ) = Z R N g ( u ∗ ) u ∗ − e λ ∗ k u ∗ k = k∇ u ∗ k > . Thus (5.5) holds with v = su ∗ for s > b I ( λ ∗ , v ( x/θ )) = 12 θ N − k∇ v k − θ N Z R N b G ( v ) < θ ≫ , (i) holds.(ii) By the result of [ JT ], the mountain pass minimax value a mp ( λ ∗ ) gives the least energylevel for b I ( λ ∗ , u ). Thus b I ( λ ∗ , u ∗ ) ≥ a mp ( λ ∗ ).(iii) (ii) implies e λ ∗ m = e λ ∗ k u ∗ k = b I ( λ ∗ , u ∗ ) − F ( u ∗ ) ≥ a mp ( λ ∗ ) − I m > a mp ( λ ∗ ) . Thus m > a mp ( λ ∗ ) e λ ∗ ≥ m . Proposition 5.7.
Suppose I m < . Then I m = b mp . Proof.
As in Lemma 2.3, we can show b mp ≤ a mp ( λ ) − e λ m for all λ ∈ ( −∞ , λ ) . λ ∗ , u ∗ ) corresponding to I m , I m = I ( λ ∗ , u ∗ ) = b I ( λ ∗ , u ∗ ) − e λ ∗ m ≥ a mp ( λ ∗ ) − e λ ∗ m ≥ b mp . On the other hand, it follows from (iii) of Proposition 5.6 that m > m and b mp is attainedby a critical point ( λ ♯ , u ♯ ) ∈ R × H r ( R N ). Thus k u ♯ k = m, F ( u ♯ ) = I ( λ ♯ , u ♯ ) = b mp . Thus I m = inf k u k = m F ( u ) ≤ F ( u ♯ ) = b mp . Therefore we have I m = b mp .We also have Corollary 5.8. I m < if and only if m > m . Proof. “if” part follows from Theorem 5.1 and “only if” part follows from Proposition5.6.
End of the proof of Theorem 0.1.
Theorem 0.1 follows from Theorem 5.1, Propositions5.6, 5.7 and Corollary 5.8.
6. Deformation lemma for scalar field equations
In this section we study the following nonlinear scalar field equations: ( − ∆ u + µu = g ( u ) in R N ,u ∈ H ( R N ) , (6 . N ≥ µ > g ( ξ ) ∈ C ( R , R ) satisfies ( g g g
3) with p = N +2 N − ( N ≥ p ∈ (1 , ∞ ) ( N = 2). Solutions of (6.1) are characterized as critical points of the followingfunctional: I ( u ) = 12 k∇ u k + µ k u k − Z R N G ( u ) ∈ C ( H r ( R N ) , R ) . Here we use notation different from previous sections. We also write P ( u ) = N − k∇ u k + N (cid:18) µ k u k − Z R N G ( u ) (cid:19) . In this section we give a new deformation result for (6.1) using ideas in Sections 3–4.34 key of our argument is the following
Proposition 6.1.
For any b ∈ R , I ( u ) satisfies the following ( P SP ′ ) b : ( P SP ′ ) b If a sequence ( u n ) ∞ n =1 ⊂ H r ( R N ) satisfies as n → ∞ I ( u n ) → b, (6 . ∂ u I ( u n ) → strongly in ( H r ( R N )) ∗ , (6 . P ( u n ) → , (6 . then ( u n ) ∞ n =1 has a strongly convergent subsequence in H r ( R N ) . Proof.
First we note by ( g g
3) with p = N +2 N − ( N ≥ p ∈ (0 , ∞ ) ( N = 2) that u m ⇀ u weakly in H r ( R N ) implies for any ϕ ∈ H r ( R N ) Z R N g ( u n ) ϕ → Z R N g ( u ) ϕ, Z R N g ( u n ) u n → Z R N g ( u ) u (6 . HIT ] (Propo-sitions 5.1 and 5.3).
Step 1: k∇ u n k is bounded as n → ∞ . Since k∇ u n k = N I ( u n ) − P ( u n ), Step 1 follows from (6.2) and (6.4).From now on we prove that k u n k is bounded as n → ∞ . We argue indirectly and weassume t n = k u n k − /N → n → ∞ . We set v n ( x ) = u n ( x/t n ). Since k v n k = 1 and k∇ v n k = t N − n k∇ u n k , (6 . v n ) ∞ n =1 is bounded in H r ( R N ). Thus we may assume after extracting a subsequence that v n ⇀ v weakly in H r ( R N ) . Step 2: v = 0 . Denoting ε n ≡ k ∂ u I ( u n ) k ( H r ( R N )) ∗ →
0, we have (cid:12)(cid:12)(cid:12)(cid:12) ( ∇ u n , ∇ ζ ) + µ ( u n , ζ ) − Z R N g ( u n ) ζ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε n k ζ k H for any ζ ∈ H r ( R N ) . u n ( x ) = v n ( t n x ), ζ ( x ) = ϕ ( t n x ), where ϕ ∈ H r ( R N ), (cid:12)(cid:12)(cid:12)(cid:12) t − ( N − n ( ∇ v n , ∇ ϕ ) + µt − Nn ( v n , ϕ ) − t − Nn Z R N g ( v n ) ϕ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε n (cid:16) t − ( N − n k∇ ϕ k + t − Nn k ϕ k (cid:17) / . Thus (cid:12)(cid:12)(cid:12)(cid:12) t n ( ∇ v n , ∇ ϕ ) + µ ( v n , ϕ ) − Z R N g ( v n ) ϕ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε n t N/ n (cid:0) t n k∇ ϕ k + k ϕ k (cid:1) / , (6 . Z R N ( µv − g ( v )) ϕ = 0 for any ϕ ∈ H r ( R N ) . Thus µv − g ( v ) = 0. Since ξ = 0 is an isolated solution of µξ − g ( ξ ) = 0 by ( g v ( x ) ≡ Step 3: k u n k is bounded as n → ∞ . Setting ϕ = v n in (6.7), (cid:12)(cid:12)(cid:12)(cid:12) t n k∇ v n k + µ k v n k − Z R N g ( v n ) v n (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε n t N/ n (cid:0) t n k∇ v n k + k v n k (cid:1) / . Thus, by (6.5), k v n k → n → ∞ , which contradicts with (6.6). Thus ( u n ) ∞ n =1 isbounded in H r ( R N ). Step 4: Conclusion.
By Step 1 and Step 3, ( u n ) ∞ n =1 is bounded in H r ( R N ). After extracting a subsequence,we may assume that u n ⇀ u weakly in H r ( R N ) for some u . Since ∂ u I ( u n ) u n → ∂ u I ( u n ) u →
0, we deduce from (6.5) thatlim n →∞ ( k∇ u n k + µ k u n k ) = k∇ u k + µ k u k . Thus u n → u strongly in H r ( R N ).Arguing as in Sections 3–4, we have Proposition 6.2.
Under the assumption of Proposition 6.1, for any b ∈ R we have(i) K b = { u ∈ H r ( R N ); I ( u ) = b, ∂ u I ( u ) = 0 , P ( u ) = 0 } is compact in H r ( R N ) .(ii) For any open neighborhood O of K b and ε > there exist ε ∈ (0 , ε ) and a continuousmap η ( t, u ) : [0 , × H r ( R N ) → H r ( R N ) such that ◦ η (0 , u ) = u for all u ∈ H r ( R N ) . ◦ η ( t, u ) = u if u ∈ [ I ≤ b − ε ] . ◦ I ( η ( t, u )) ≤ I ( u ) for all ( t, u ) ∈ [0 , × H r ( R N ) . ◦ η (1 , [ I ≤ b + ε ] \ O ) ⊂ [ I ≤ b − ε ] , η (1 , [ I ≤ b + ε ]) ⊂ [ I ≤ b − ε ] ∪ O . ◦ If K b = ∅ , η (1 , [ I ≤ b + ε ]) ⊂ [ I ≤ b − ε ] .Here we use notation: [ I ≤ c ] = { u ∈ H r ( R N ); I ( u ) ≤ c } for c ∈ R . Using Proposition 6.2, we can show that a mp ( λ ) given in (5.1) is a critical value of u b I ( λ, u ). A. Appendix: Proof of (iv) of Lemma 2.4
In this appendix we give a proof to (iv) of Lemma 2.4.
Proof of (iv) of Lemma 2.4.
Suppose that a closed set Z is invariant under Z -action(2.6) and satisfies 0 P ( Z ). Then P ( Z ) ⊂ H r ( R N ) is symmetric with respect to 0 andgenus( P ( Z )) is well-defined.For A = γ ( D j + ℓ \ Y ), γ ∈ Γ j + ℓ , genus( Y ) ≤ ℓ , we have A \ Z = γ ( D j + ℓ \ ( Y ∪ γ − ( Z ))) . (A . γ ( D j + ℓ \ ( Y ∪ γ − ( Z ))) = γ ( D j + ℓ \ Y ) \ Z ⊂ γ ( D j + ℓ \ Y ) \ Z = A \ Z. (A . B \ C ⊂ B \ C for a set B and a closed set C , we have A \ Z = γ ( D j + ℓ \ Y ) \ Z = γ ( D j + ℓ \ Y \ γ − ( Z )) ⊂ γ ( D j + ℓ \ ( Y ∪ γ − ( Z ))) . (A . P ◦ γ : γ − ( Z ) → P ( Z ) is an odd map,genus( γ − ( Z )) ≤ genus( P ( Z )) = i. Thus, genus( Y ∪ γ − ( Z )) ≤ genus( Y ) + genus( γ − ( Z )) ≤ genus( Y ) + genus( P ( Y )) ≤ ℓ + i. Therefore, by (A.1) we have A \ Z ∈ Λ j − i . 37 cknowledgments The authors would like to thank Professor Tohru Ozawa for helpful discussions and com-ments.
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