Instability of ground states for the NLS equation with potential on the star graph
aa r X i v : . [ m a t h . A P ] F e b INSTABILITY OF GROUND STATES FOR THE NLS EQUATIONWITH POTENTIAL ON THE STAR GRAPH
Alex H. Ardila
Universidade Federal de Minas GeraisCEP 30123-970, Belo Horizonte-MG, Brazil
Liliana Cely
Universidade de S˜ao PauloCEP 05508-090, Cidade Universit´aria, S˜ao Paulo-SP, Brazil
Nataliia Goloshchapova
Universidade de S˜ao PauloCEP 05508-090, Cidade Universit´aria, S˜ao Paulo-SP, Brazil
Abstract.
We study the nonlinear Schr¨odinger equation with an arbitrary real poten-tial V ( x ) ∈ ( L + L ∞ )(Γ) on a star graph Γ. At the vertex an interaction occurs describedby the generalized Kirchhoff condition with strength − γ <
0. We show the existence ofground states ϕ ω ( x ) as minimizers of the action functional on the Nehari manifold underadditional negativity and decay conditions on V ( x ). Moreover, for V ( x ) = − βx α , in thesupercritical case, we prove that the standing waves e iωt ϕ ω ( x ) are orbitally unstable in H (Γ) when ω is large enough. Analogous result holds for an arbitrary γ ∈ R when thestanding waves have symmetric profile. Introduction
We consider the following focusing nonlinear Schr¨odinger equation on an infinite stargraph Γ:(1.1) ( i∂ t u ( t, x ) = − ∆ γ u ( t, x ) + V ( x ) u ( t, x ) − | u ( t, x ) | p − u ( t, x ) , ( t, x ) ∈ R × Γ ,u (0 , x ) = u ( x ) , where γ > p > u : R × Γ → C N , and ∆ γ is the Laplace operator with the generalizedKirchhoff condition at the vertex of Γ ( · ′ stands for spatial derivative): v (0) = . . . = v N (0) , N X e =1 v ′ e (0) = − γv (0) . We assume that the potential V ( x ) = ( V e ( x )) Ne =1 is real-valued and satisfies the Assump-tions (see notation section): Self-adjointness assumption : V ( x ) ∈ L (Γ) + L ∞ (Γ) . Weak continuity assumption : lim x →∞ V e ( x ) = 0 . Mathematics Subject Classification.
Primary: 35Q55; Secondary: 35Q40.
Key words and phrases.
Nonlinear Schr¨odinger equation, linear potential, generalized Kirchhoff’s con-dition, ground state, orbital stability. Minimizing assumption : R R + V e ( x ) | φ ( x ) | dx < φ ( x ) ∈ H ( R + ) \ { } . Virial identity assumption : xV ′ ( x ) ∈ L (Γ) + L ∞ (Γ).Notice that Assumption 3 essentially guarantees (
V u, u ) < , u ∈ H (Γ) \ { } , and V ( x ) ≤ i∂ t u ( t, x ) = − ∆ u ( t, x ) + V ( x ) u ( t, x ) − | u ( t, x ) | p − u ( t, x ) , ( t, x ) ∈ R × R n , /n ≤ p < / ( n − , was initiated in [27]. More precisely, the authors proved orbital stability of e iωt ϕ ω ( x ) for ω sufficiently close to minus the smallest eigenvalue of the operator − ∆ + V (under theassumptions V ( x ) ∈ L ∞ ( R n ), lim | x |→∞ V ( x ) = 0.) In [15], the stability results obtainedby [27] were improved for V ( x ) satisfying more general assumptions.Recently in [25], the author studied strong instability (by blow-up) of the standing wavesin the case of harmonic potential V ( x ) = | x | . In particular, he proved strong instabilityunder certain concavity condition for the associated action functional (cf. Theorem 1.4below). The same idea was applied in [13] to investigate strong instability for V ( x ) = − β | x | α , < α < min { , n } , β > . The reader is also referred to [24] for more informationabout NLS near soliton dynamics.In the case V ( x ) ≡
0, the well-posedness in H (Γ), variational and stability/instabilityproperties of (1.1) have been extensively studied during the last decade. The well-posedness results were obtained in [2, 18], whereas the existence, stability and variationalproperties of ground states were studied in [1–4, 20]. Moreover, the regularity and stronginstability results were elaborated in [18].On the other hand, the NLS with potential on graphs is little studied. To our knowledge,the only results concerning the existence and stability of standing waves were obtainedin [5, 9, 10]. In the subcritical (1 < p <
5) and critical ( p = 5) case orbitally stablestanding waves e iωt ϕ ω ( x ) were constructed in [9, 10] under specific conditions on V ( x ) . Subsequently in [5] the orbital stability of e iωt ϕ ω ( x ) was studied in the supercritical case( p > e iωt ϕ ω ( x ) is stable when the mass of ϕ ω ( x ) is sufficiently small.In this paper, we show the existence and orbital instability of the standing wave solu-tions to (1.1) relying on methods developed in [13, 16]. Moreover, we state regularity ofthe solutions to the Cauchy problem for the initial data from the domain of the operator − ∆ γ + V ( x ) . This result is used to show virial identity which is the key ingredient in theproof of the instability result.1.1.
Notation.
We consider a graph Γ consisting of a central vertex ν and N infinitehalf-lines attached to it. One may identify Γ with the disjoint union of the intervals I e = (0 , ∞ ), e = 1 , . . . , N , augmented by the central vertex ν = 0. Given a function NSTABILITY OF GROUND STATES 3 v : Γ → C N , v = ( v e ) Ne =1 , where v e : (0 , ∞ ) → C denotes the restriction of v to I e . Wedenote by v e (0) and v ′ e (0) the limits of v e ( x ) and v ′ e ( x ) as x → + .We say that a function v is continuous on Γ if every restriction v e is continuous on I e and v (0) = . . . = v N (0). The space of continuous functions is denoted by C (Γ) . The natural Hilbert space associated to the Laplace operator ∆ γ is L (Γ), which isdefined as L (Γ) = L Ne =1 L ( R + ), and is equipped with the norm k v k = Z Γ | v | dx = N X e =1 ∞ Z | v e ( x ) | dx. The inner product in L (Γ) is denoted by ( · , · ) . The space L q (Γ) for 1 ≤ q ≤ ∞ is definedanalogously, and k · k q stands for its norm. The Sobolev spaces H (Γ) and H (Γ) aredefined as H (Γ) = (cid:8) v ∈ C (Γ) : v e ∈ H ( R + ) , e = 1 , . . . , N (cid:9) ,H (Γ) = (cid:8) v ∈ C (Γ) : v e ∈ H ( R + ) , e = 1 , . . . , N (cid:9) . We consider the self-adjoint operator H γ,V on L (Γ):( H γ,V v ) e = − (∆ γ v ) e + V e v e = − v ′′ e + V e v e , dom( H γ,V ) = ( v ∈ H (Γ) : − v ′′ e + V e v e ∈ L ( R + ) , N X e =1 v ′ e (0) = − γv (0) ) . (1.2)When γ = 0, the condition at the vertex in (1.2) is usually referred as free or Kirchhoffboundary condition. For γ ∈ R the operator H γ,V has a precise interpretation as the self-adjoint operator on L (Γ) uniquely associated with the closed semibounded quadraticform F γ,V defined on H (Γ) by (see Lemma 4.10 in Appendix) F γ,V ( v ) = k v ′ k − γ | v (0) | + ( V v, v ) = N X e =1 ∞ Z | v ′ e ( x ) | dx − γ | v (0) | + N X e =1 ∞ Z V e ( x ) | v e ( x ) | dx. (1.3)Note that we can formally rewrite (1.1) as i∂ t u ( t ) = E ′ ( u ( t )) , where E is the energy functional defined by E ( u ) = 12 F γ,V ( u ) − p + 1 k u k p +1 p +1 . The energy functional is well-defined on H (Γ) since the potential V ( x ) belongs to ( L + L ∞ )(Γ) (see Lemma 4.10 in Appendix). ALEX H. ARDILA, LILIANA CELY, AND NATALIIA GOLOSHCHAPOVA
Standing waves and instability results.
By a standing wave of (1.1), we mean asolution of the form e iωt ϕ ( x ), where ω ∈ R and ϕ is a solution of the stationary equation(1.4) H γ,V φ + ωφ − | φ | p − φ = 0 . We define two functionals on H (Γ): S ω ( v ) : = 12 F γ,V ( v ) + ω k v k − p + 1 k v k p +1 p +1 ( action functional ) ,I ω ( v ) : = F γ,V ( v ) + ω k v k − k v k p +1 p +1 . Observe that (1.4) is equivalent to S ′ ω ( φ ) = 0 (see [2, Theorem 4]) and I ω ( v ) = ∂ λ S ω ( λv ) | λ =1 = h S ′ ω ( v ) , v i . Denote the set of non-trivial solutions to (1.4) by B ω = n v ∈ H (Γ) \{ } : S ′ ω ( v ) = 0 o . A ground state for (1.4) is a function ϕ ∈ B ω that minimizes S ω on B ω , and the set ofground states is given by G ω = n φ ∈ B ω : S ω ( φ ) ≤ S ω ( v ) for all v ∈ B ω o . We consider the minimization problem on the Nehari manifold d ω = inf (cid:8) S ω ( v ) : v ∈ H (Γ) \{ } , I ω ( v ) = 0 (cid:9) , and the set of minimizers M ω = (cid:8) φ ∈ H (Γ) \{ } : S ω ( φ ) = d ω , I ω ( φ ) = 0 (cid:9) . We now state the first result, which provides the existence of the minimizer for d ω whenthe strength − γ is sufficiently strong. Denote (see Lemma 4.13)(1.5) − ω := inf σ ( H γ,V ) = min σ p ( H γ,V ) < . Proposition 1.1.
Let p > , ω > ω , and V ( x ) = V ( x ) satisfy Assumptions 1-3. Thenthere exists γ ∗ > such that the set G ω is not empty for any γ > γ ∗ , in particular, G ω = M ω . If ϕ ω ∈ G ω , then there exist θ ∈ R and a positive function φ ∈ dom( H γ,V ) such that ϕ ω ( x ) = e iθ φ ( x ) . To be precise, γ ∗ is given in [2] by(1.6) Z (1 − t ) p − dt = N Z γ ∗ N √ ω (1 − t ) p − dt. The condition γ > γ ∗ guarantees that the action functional S ω constrained to the Neharimanifold admits an absolute minimum when V ( x ) ≡ Remark 1.2.
The proof of the last assertion of Proposition 1.1 essentially uses that V ( x ) ≤ a.e. on Γ , which is a consequence of Assumption 3.To show this one observes that R R + − V e ( x ) φ ( x ) dx ≥ for all nonnegative functions φ ( x ) from C c ( R + ) (the set of continuous functions with compact support). Indeed, let ˜ φ ( x ) be an extension onto R by zero of a nonnegative function φ ( x ) ∈ C c ( R + ) . Take NSTABILITY OF GROUND STATES 5 { φ n ( x ) } ⊂ C ∞ c ( R ) such that φ n −→ n →∞ q ˜ φ uniformly, and supp ˜ φ, supp φ n ⊂ K ⊂ R + ,where K is a compact set. Then φ n −→ n →∞ ˜ φ uniformly, and, by the Dominated ConvergenceTheorem, we get − Z R + V e ( x ) φ n ( x ) dx −→ n →∞ − Z R + V e ( x ) φ ( x ) dx ≥ . Now, since f ( φ ) = − R R + V e ( x ) φ ( x ) dx is a positive linear functional on C c ( R + ) , then,by the Riesz–Markov–Kakutani representation theorem for positive linear functionals, weconclude the existence of a unique Radon measure µ on R + such that f ( φ ) = R R + φ ( x ) dµ ( x ) .On the other hand, f ( φ ) = R R + v ( x ) φ ( x ) dν ( x ) , where ν ( A ) = R A | V e | dx for A from the Borel σ -algebra on R + , and v ( x ) = (cid:26) V e ( x ) | V e ( x ) | , x ∈ { x : V e ( x ) = 0 } , otherwise . Finally, from the uniquenessstated in [12, Theorem 2.5.12] it follows that µ = ν and v = 1 ν -a.e. on R + , hence − V e ≥ ν -a.e. on R + . This implies − V e ≥ Lebesgue-a.e. on R + since the Lebesguemeasure and the measure ν are mutually absolutely continuous on the set { x : V e ( x ) = 0 } . The next step in the study of ground states for (1.4) is to investigate their stabilityproperties. We define orbital stability as follows.
Definition 1.3.
For ϕ ω ∈ G ω , we set (1.7) N δ ( ϕ ω ) := (cid:8) v ∈ H (Γ) : inf θ ∈ R (cid:13)(cid:13) v − e iθ ϕ ω (cid:13)(cid:13) H (Γ) < δ (cid:9) . We say that a standing wave solution e iωt ϕ ω ( x ) of (1.1) is orbitally stable in H (Γ) if forany ε > there exists δ > such that for any u ∈ N δ ( ϕ ω ) , the solution u ( t ) of (1.1) satisfies u ( t ) ∈ N ε ( ϕ ω ) for all t ≥ . Otherwise, e iωt ϕ ω ( x ) is said to be orbitally unstablein H (Γ) . Using the ideas developed in [13, 16], we obtain a sufficient condition for the instabilityof standing waves when p >
Theorem 1.4.
Assume that p > , γ > γ ∗ , ω > ω , and V ( x ) = V ( x ) satisfies Assump-tions 1-4. If ϕ ω ( x ) ∈ G ω and ∂ λ E ( ϕ λω ) | λ =1 < , where ϕ λω ( x ) := λ / ϕ ω ( λx ) for λ > ,then the standing wave solution e iωt ϕ ω ( x ) of (1.1) is orbitally unstable in H (Γ) . To prove Theorem 1.4 we use the variational characterization given in Proposition 1.1and virial identity (2.4). Notice that the standing wave solution e iωt ϕ ω ( x ) of (1.1) with γ > V ( x ) ≡ H (Γ) when p > ω is large enough (see [2, Remark6.1] and also [18, Theorem 1.4]). Below we state that this also holds true for γ > V ( x ) = − βx α , < α < , β > ∂ λ E ω ( ϕ λω ) | λ =1 < ω ). The choice of the potential is due to its “homogeneity” property,which is principal for the proof (see formula (4.7)). ALEX H. ARDILA, LILIANA CELY, AND NATALIIA GOLOSHCHAPOVA
Corollary 1.5.
Assume that V ( x ) = − βx α , β > , < α < , γ > γ ∗ , p > . If ϕ ω ( x ) ∈ G ω , then there exists ω ∗ = ω ∗ ( β, α, γ, p ) ∈ ( ω , ∞ ) such that for any ω ∈ ( ω ∗ , ∞ ) the standing wave solution e iωt ϕ ω ( x ) of (1.1) is orbitally unstable in H (Γ) . As far as we know, these are the first results on instability of ground states for the NLSwith potential on graphs. In Subsection 4.3, we state the counterparts to Proposition1.1, Theorem 1.4, Corollary 1.5 in the space H (Γ) of symmetric functions and arbitrary γ ∈ R . The paper is organized as follows. In Section 2, we prove Proposition 2.2 that concernslocal well-posedness in the energy domain. In Section 3, we provide the proof of Propo-sition 1.1 . Section 4 is devoted to the proof of Theorem 1.4 and Corollary 1.5. In theAppendix we discuss some properties of the operator H γ,V .2. local existence results and virial identity We start with the proof of the following key lemma involving the estimate of H -normof the unitary group generated by the self-adjoint operator H γ,V . Lemma 2.1.
Let e − iH γ,V t be a unitary group generated by H γ,V . Then e − iH γ,V t H (Γ) ⊆ H (Γ) and (2.1) k e − iH γ,V t v k H (Γ) ≤ C k v k H (Γ) . Proof.
The idea of the proof was given in [10] (see formula (2.5)). However, some addi-tional technical details seem useful.Let m > ω , where ω is given by (1.5). Remark that H (Γ) = dom ( F γ,V ) =dom(( H γ,V + m ) / ) (see, for instance, [21, Chapter VI, Problem 2.25]). Since e − iH γ,V t isbounded, we get for v ∈ H (Γ) e − iH γ,V t ( H γ,V + m ) / v = ( H γ,V + m ) / e − iH γ,V t v. Hence e − iH γ,V t v ∈ H (Γ) and e − iH γ,V t H (Γ) ⊆ H (Γ) . Further, using L -unitarity of e − iH γ,V t , we obtain for v ∈ H (Γ) F γ,V ( v ) + m k v k = (cid:0) ( H γ,V + m ) / v, ( H γ,V + m ) / v (cid:1) = (cid:0) e − iH γ,V t ( H γ,V + m ) / v, e − iH γ,V t ( H γ,V + m ) / v (cid:1) = (cid:0) ( H γ,V + m ) / e − iH γ,V t v, ( H γ,V + m ) / e − iH γ,V t v (cid:1) = F γ,V ( e − iH γ,V t v ) + m k e − iH γ,V t v k . From the proof of Lemma 4.13-( ii ) we get C k e − iH γ,V t v k H (Γ) ≤ F γ,V ( e − iH γ,V t v ) + m k e − iH γ,V t v k = F γ,V ( v ) + m k v k ≤ C k v k H (Γ) , and (2.1) follows easily. (cid:3) The proposition below states the local well-posedness of (1.1).
Proposition 2.2.
For any u ∈ H (Γ) , there exist T = T ( u ) > and a unique solution u ( t ) ∈ C ([0 , T ] , H (Γ)) ∩ C ([0 , T ] , ( H (Γ)) ′ ) of problem (1.1) . For each T ∈ (0 , T ) themapping u ∈ H (Γ) u ( t ) ∈ C ([0 , T ] , H (Γ)) is continuous. Moreover, problem (1.1) NSTABILITY OF GROUND STATES 7 has a maximal solution defined on an interval of the form [0 , T H ) , and the following“blow-up alternative” holds: either T H = ∞ or T H < ∞ and lim t → T H k u ( t ) k H (Γ) = ∞ . Finally, the conservation of energy and charge holds: for t ∈ [0 , T H ) E ( u ( t )) = 12 F γ,V ( u ( t )) − p + 1 k u ( t ) k p +1 p +1 = E ( u ) , k u ( t ) k = k u k . (2.2) Proof.
A sketch of the proof was given in [10]. However, the rigorous proof (which servesfor p >
1) might be obtained repeating the one of [11, Theorem 4.10.1]. In particular,one needs to use the fact that g ( u ) = | u | p − u ∈ C ( C , C ) (i.e. Im( g ) and Re( g ) are C -functions of Re u, Im u ) for p > (cid:3) Remark 2.3. ( i ) For p ≥ , the conservation laws follow easily from Proposition 2.4below and continuous dependence on initial data. ( ii ) For < p < , problem (1.1) is globally well-posed in H (Γ) . To see that one mightrepeat the proof of [11, Theorem 3.4.1], where condition (3.4.1) follows from k u k p +1 p +1 − ( V u, u ) + γ | u (0) | ≤ C k u ′ k p − k u k p +32 + 2 ε k u ′ k + C k u k ≤ ε k u ′ k + C k u k p +3)5 − p + C k u k ≤ ε k u k H (Γ) + C ( k u k ) . The above estimate is induced by the conservation of charge, estimate (4.19) , the Gagliardo-Nirenberg inequality (see (2.1) in [10]), and the Young inequality ab ≤ δa q + C δ b q ′ , q + q ′ =1 , q, q ′ > , a, b ≥ . Observe that the key point is that q = p − > for < p < . Now, let m ≥ ω . Introduce the norm k v k H γ,V := k ( H γ,V + m ) v k that endowsdom( H γ,V ) with the structure of a Hilbert space. We denote D H γ,V = (dom( H γ,V ) , k ·k H γ,V ). Proposition 2.4.
Let p ≥ and u ∈ dom( H γ,V ) . Then there exists T > such thatproblem (1.1) has a unique solution u ( t ) ∈ C ([0 , T ] , D H γ,V ) ∩ C ([0 , T ] , L (Γ)) . Moreover,problem (1.1) has a maximal solution defined on an interval of the form [0 , T H γ,V ) , andthe following “blow-up alternative” holds: either T H γ,V = ∞ or T H γ,V < ∞ and lim t → T Hγ,V k u ( t ) k H γ,V = ∞ . Proof.
The proof repeats the one of [18, Theorem 2.3] observing that dom( H γ,V ) ⊂ H (Γ) = dom(( H γ,V + m ) / ) and, by m ≥ ω , k u k ∞ ≤ C k u k H (Γ) ≤ C k ( H γ,V + m ) / u k ≤ C k ( H γ,V + m ) u k . (cid:3) Remark 2.5.
Notice that due to estimate (4.19) , Propositions 2.2 and 2.4 hold for any γ ∈ R and V ( x ) ∈ ( L + L ∞ )(Γ) . ALEX H. ARDILA, LILIANA CELY, AND NATALIIA GOLOSHCHAPOVA
Set P ( v ) = k v ′ k − Z Γ xV ′ ( x ) | v ( x ) | dx − γ | v (0) | − p − p + 1) k v k p +1 p +1 , v ∈ H (Γ) . Proposition 2.6.
Let
Σ(Γ) = { v ∈ H (Γ) : xv ∈ L (Γ) } . Assume that u ∈ Σ(Γ) , and u ( t ) is the corresponding maximal solution to (1.1) . Then u ( t ) ∈ C ([0 , T H ) , Σ(Γ)) , andthe function f ( t ) := Z Γ x | u ( t, x ) | dx = k xu ( t ) k belongs to C [0 , T H ) . Moreover, (2.3) f ′ ( t ) = 4 Im Z Γ xu∂ x u dx, and (2.4) f ′′ ( t ) = 8 P ( u ( t )) , t ∈ [0 , T H ) . (virial identity)Proof. The proof is similar to the one of [11, Proposition 6.5.1]. We provide the detailssince the virial identity is the key ingredient in the instability analysis. Firstly we show(2.3), secondly we prove (2.4) for u ∈ dom( H γ,V ), then we conclude that (2.4) holds for u ∈ H (Γ) using continuous dependence on the initial data. Step 1.
Let ε >
0, define f ε ( t ) = k e − εx xu ( t ) k , for t ∈ [0 , T ] , T ∈ (0 , T H ). Then,observing that e − εx x u ( t ) ∈ H (Γ) and taking ( H ) ′ − H duality product of equation(1.1) with ie − εx x u ( t ), we get f ′ ε ( t ) = 2 Im Z Γ (cid:16) ∂ x u ∂ x ( e − εx x u ) − e − εx x | u | p +1 (cid:17) dx = 4 Im Z Γ n e − εx (1 − εx ) o uxe − εx ∂ x u dx. (2.5)Remark that | e − εx (1 − εx ) | ≤ x . From (2.5), by the Cauchy-Schwarz inequality,we obtain | f ′ ε ( t ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Γ n e − εx (1 − εx ) o uxe − εx ∂ x u dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Γ | e − εx xu∂ x u | dx ≤ N X j =1 k ∂ x u j k k e − εx xu j k ≤ C k u k H (Γ) p f ε ( t ) . (2.6)From (2.6) one implies t Z f ′ ε ( s ) p f ε ( s ) ds ≤ C t Z k u ( s ) k H (Γ) ds, NSTABILITY OF GROUND STATES 9 and therefore p f ε ( t ) ≤ k xu k + C t Z k u ( s ) k H (Γ) ds, t ∈ [0 , T ] . Letting ε ↓ xu ( t ) ∈ L (Γ) and f ( t ) is boundedin [0 , T ] . Observe that from (2.5) one induces(2.7) f ε ( t ) = f ε (0) + 4 Im t Z Z Γ n e − εx (1 − εx ) o uxe − εx ∂ x u dx ds. We have the following estimates for any positive x and ε : e − εx x | u ( t ) | ≤ x | u ( t ) | ,e − εx x | u | ≤ x | u | , | e − εx (1 − εx ) uxe − εx ∂ x u | ≤ | ∂ x u k xu | . (2.8)Having pointwise convergence, and using (2.8), by the Dominated Convergence Theoremwe get from (2.7) f ( t ) = k xu ( t ) k = k xu k + 4 Im t Z Z Γ xu∂ x u dx ds. Since u ( t ) is strong H -solution, f ( t ) is C -function, and (2.3) holds for any t ∈ [0 , T H ) . Using continuity of k xu ( t ) k and the inclusion u ( t ) ∈ C ([0 , T H ) , H (Γ)), by the Brezis-Lieb lemma [8], we get for t , t n ∈ [0 , T H )lim t n → t k xu ( t n ) − xu ( t ) k = lim t n → t k xu ( t n ) k − k xu ( t ) k = 0 , hence u ( t ) ∈ C ([0 , T H ) , Σ(Γ)) . Step 2.
Let u ∈ dom( H γ,V ). By Proposition 2.4, the solution u ( t ) to the correspondingCauchy problem belongs to C ([0 , T H γ,V ) , D H γ,V ) ∩ C ([0 , T H γ,V ) , L (Γ)).Let ε > θ ε ( x ) = e − εx . Define(2.9) h ε ( t ) = Im Z Γ θ ε xu∂ x u dx for t ∈ [0 , T ] , T ∈ (0 , T H γ,V ) . First, let us show that(2.10) h ′ ε ( t ) = − Im Z Γ ∂ t u (cid:8) θ ε x∂ x u + ( θ ε + xθ ′ ε ) u (cid:9) dx or equivalently(2.11) h ε ( t ) = h ε (0) − Im t Z Z Γ ∂ s u (cid:8) θ ε x∂ x u + ( θ ε + xθ ′ ε ) u (cid:9) dx ds. Let us prove that identity (2.11) holds for u ( t ) ∈ C ([0 , T ] , H (Γ)) ∩ C ([0 , T ] , L (Γ)) . Note that by density argument it is sufficient to show (2.11) for u ( t ) ∈ C ([0 , T ] , H (Γ)) ∩ C ([0 , T ] , L (Γ)) . From (2.9), it follows(2.12) h ′ ε ( t ) = − Im Z Γ n θ ε x∂ t u∂ x u + θ ε xu∂ xt u o dx. Note that θ ε xu∂ xt u = θ ε xu∂ tx u = ∂ x (cid:0) θ ε xu∂ t u (cid:1) − θ ε u∂ t u − θ ε x∂ x u∂ t u − xθ ′ ε u∂ t u, which induces Z Γ θ ε xu∂ xt u dx = − Z Γ ∂ t u { θ ε ( u + x∂ x u ) + xθ ′ ε u } dx. Therefore, from (2.12) we get h ′ ε ( t ) = − Im Z Γ (cid:8) θ ε x∂ t u∂ x u + ∂ t u (cid:0) θ ε ( u + x∂ x u ) + xθ ′ ε u (cid:1)(cid:9) dx. Consequently we obtain (2.11) for u ( t ) ∈ C ([0 , T ] , H (Γ)) ∩ C ([0 , T ] , L (Γ)) and hencefor u ( t ) ∈ C ([0 , T ] , H (Γ)) ∩ C ([0 , T ] , L (Γ)) which implies (2.10).Since u ( t ) ∈ C ([0 , T H γ,V ) , D H γ,V ), from (2.10) we get(2.13) h ′ ε ( t ) = Re Z Γ ( H γ,V u − | u | p − u ) (cid:8) θ ε x∂ x u + ( xθ ε ) ′ u (cid:9) dx. Below we will consider separately linear and nonlinear part of identity (2.13). Integratingby parts, we obtain − Re Z Γ ∆ γ u (cid:8) θ ε x∂ x u + ( xθ ε ) ′ u (cid:9) dx = − γ | u (0) | + 2 Z Γ xθ ′ ε | ∂ x u | dx + Z Γ (2 θ ′ ε + xθ ′′ ε ) Re( u∂ x u ) dx + 2 Z Γ θ ε | ∂ x u | dx. (2.14)Noting thatRe (cid:0) V ( x ) u (cid:8) θ ε x∂ x u + ( xθ ε ) ′ u (cid:9)(cid:1) = ∂ x (cid:0) xV ( x ) θ ε | u | (cid:1) − xV ′ ( x ) θ ε | u | , we get(2.15) Re Z Γ V ( x ) u (cid:8) θ ε x∂ x u + ( xθ ε ) ′ u (cid:9) dx = − Z Γ xV ′ ( x ) θ ε | u | dx. NSTABILITY OF GROUND STATES 11
Moreover, Re Z Γ −| u | p − u (cid:8) θ ε x∂ x u + ( xθ ε ) ′ u (cid:9) dx = − Z Γ | u | p +1 θ ε dx − Z Γ | u | p +1 xθ ′ ε dx − Z Γ ( | u | ) p − ∂ x ( | u | ) xθ ε dx = − p − p + 1 Z Γ | u | p +1 θ ε dx − p − p + 1 Z Γ | u | p +1 xθ ′ ε dx. (2.16)Finally, from (2.13)-(2.16) we get h ′ ε ( t ) = Z Γ θ ε | ∂ x u | dx − Z Γ xV ′ ( x ) θ ε | u | dx − γ | u (0) | − p − p + 1 Z Γ | u | p +1 θ ε dx + Z Γ xθ ′ ε | ∂ x u | dx + Z Γ (2 θ ′ ε + xθ ′′ ε ) Re( u∂ x u ) dx − p − p + 1 Z Γ | u | p +1 xθ ′ ε dx. Since θ ε , θ ′ ε , xθ ′ ε , xθ ′′ ε are bounded with respect to x and ε , and θ ε → , θ ′ ε → , xθ ′ ε → , xθ ′′ ε → ε ↓ , by the Dominated Convergence Theorem we havelim ε ↓ h ′ ε ( t ) = 2 k ∂ x u k − Z Γ xV ′ ( x ) | u | dx − γ | u (0) | − p − p + 1 k u k p +1 p +1 =: g ( t ) . Moreover, again by the Dominated Convergence Theorem,lim ε ↓ h ε ( t ) = Im Z Γ xu∂ x u dx =: h ( t ) . Using continuity of g ( t ) and the fact that the operator A = ddt in the space C [0 , T ] withdom( A ) = C [0 , T ] is closed, we arrive at h ′ ( t ) = g ( t ) , t ∈ [0 , T ], i.e. h ′ ( t ) = 2 k ∂ x u k − Z Γ xV ′ ( x ) | u | dx − γ | u (0) | − p − p + 1 k u k p +1 p +1 , and h ( t ) is C function. Finally, (2.4) holds for u ∈ dom( H γ,V ). Step 3.
To conclude the proof consider { u n } n ∈ N ⊂ dom( H γ,V ) such that u n → u in H (Γ) and xu n → xu in L (Γ) as n → ∞ . Let u n ( t ) be the maximal solutions of thecorresponding Cauchy problem associated with (1.1). From (2.3) and (2.4) we obtain k xu n ( t ) k = k xu n k + 4 t Im Z Γ xu n ∂ x u n dx + t Z s Z P ( u n ( y )) dy ds. Using continuous dependence and repeating the arguments from [11, Corollary 6.5.3], weobtain as n → ∞k xu ( t ) k = k xu k + 4 t Im Z Γ xu ∂ x u dx + t Z s Z P ( u ( y )) dy ds, that is (2.4) holds for u ∈ H (Γ). (cid:3) existence of ground states In this section, we prove Proposition 1.1. We begin with two technical lemmas. Through-out this section we assume that ω > ω . Lemma 3.1. If I ω ( v ) < , then d ω < p − p + 1) k v k p +1 p +1 , and d ω < p − p + 1) (cid:0) F γ,V ( v ) + ω k v k (cid:1) . Moreover, d ω = inf (cid:26) p − p + 1) k v k p +1 p +1 : v ∈ H (Γ) \{ } , I ω ( v ) ≤ (cid:27) = inf (cid:26) p − p + 1) (cid:0) F γ,V ( v ) + ω k v k (cid:1) : v ∈ H (Γ) \{ } , I ω ( v ) ≤ (cid:27) . (3.1) Proof.
Noting that S ω ( v ) = 12 I ω ( v ) + p − p + 1) k v k p +1 p +1 , v ∈ H (Γ) , (3.2)we get d ω = inf (cid:26) p − p + 1) k v k p +1 p +1 : v ∈ H (Γ) \{ } , I ω ( v ) = 0 (cid:27) . Set d ∗ ω := inf (cid:26) p − p + 1) k v k p +1 p +1 : v ∈ H (Γ) \{ } , I ω ( v ) ≤ (cid:27) . It is clear that d ∗ ω ≤ d ω . Let v ∈ H (Γ) \{ } and I ω ( v ) <
0. Put λ := F γ,V ( v ) + ω k v k k v k p +1 p +1 ! p − . Then, since I ω ( λv ) = λ (cid:0) F γ,V ( v ) + ω k v k (cid:1) − λ p +1 k v k p +1 p +1 =: f ( λ ), we obtain I ω ( λ v ) = 0and 0 < λ < f (1) < , f (0) = 0, and f ′ ( λ ) > λ ). Hence we have d ω ≤ p − p + 1) k λ v k p +1 p +1 = p − p + 1) λ p +11 k v k p +1 p +1 < p − p + 1) k v k p +1 p +1 . NSTABILITY OF GROUND STATES 13
Thus, we obtain d ω ≤ d ∗ ω . Similarly we can show d ω < p − p +1) (cid:0) F γ,V ( v ) + ω k v k (cid:1) and thesecond part of (3.1) since we can rewrite d ω = inf (cid:26) p − p + 1) (cid:0) F γ,V ( v ) + ω k v k (cid:1) : u ∈ H (Γ) \{ } , I ω ( v ) = 0 (cid:27) . (cid:3) To get the existence of the minimizers of d ω , one has at a certain point to compare theaction S ω for γ > S ω of the nonpotential case ( V ( x ) ≡ γ > S ω ( v ) = 12 k v ′ k + ω k v k − γ | v (0) | − p + 1 k v k p +1 p +1 ,I ω ( v ) = k v ′ k + ω k v k − γ | v (0) | − k v k p +1 p +1 ,d ω = inf (cid:8) S ω ( v ) : v ∈ H (Γ) \{ } , I ω ( v ) = 0 (cid:9) = inf (cid:26) p − p + 1) k v k p +1 p +1 : v ∈ H (Γ) \{ } , I ω ( v ) = 0 (cid:27) , and M ω := (cid:8) φ ∈ H (Γ) \{ } : I ω ( φ ) = 0 , S ω ( φ ) = d ω (cid:9) . It is known that for γ > γ ∗ , where γ ∗ is defined by (1.6), the set M ω is not empty (see [2]).Throughout this section we assume γ > γ ∗ . Lemma 3.2. d ω > d ω > .Proof. First, we show that d ω >
0. Let v ∈ H (Γ) \{ } satisfy I ω ( v ) = 0. Then k v k p +1 p +1 = F γ,V ( v ) + ω k v k . Since ω > ω , by the Sobolev embedding and Lemma 4.13-( ii ), we have k v k p +1 ≤ C k v k H (Γ) ≤ C (cid:0) F γ,V ( v ) + ω k v k (cid:1) = C k v k p +1 p +1 . Hence we obtain C − p − ≤ k v k p +1 . Taking the infimum over v , we get d ω >
0. Next, weprove d ω > d ω . Since M ω is not empty, we can take φ ∈ M ω . By Assumption 3 , I ω ( φ ) = ( V φ, φ ) < . Then, by Lemma 3.1, we obtain d ω < p − p + 1) k φ k p +1 p +1 = d ω . (cid:3) Lemma 3.3.
Let { v n } ⊂ H (Γ) \{ } be a minimizing sequence for d ω , i.e. I ω ( v n ) = 0 and lim n →∞ S ω ( v n ) = d ω . Then there exist a subsequence { v n k } of { v n } and v ∈ H (Γ) \{ } such that lim k →∞ k v n k − v k H (Γ) = 0 , I ω ( v ) = 0 and S ω ( v ) = d ω . Therefore, M ω is notempty. Proof.
Since ω > ω and S ω ( v n ) = p − p + 1) (cid:0) F γ,V ( v n ) + ω k v n k (cid:1) = p − p + 1) k v n k p +1 p +1 −→ n →∞ d ω , (3.3)the sequence { v n } is bounded in H (Γ) (see Lemma 4.13-( ii )). Hence there exist a sub-sequence { v n k } of { v n } and v ∈ H (Γ) such that { v n k } converges weakly to v in H (Γ).We may assume that v n k = 0 and define λ k = (cid:13)(cid:13) v ′ n k (cid:13)(cid:13) + ω k v n k k − γ | v n k , (0) | k v n k k p +1 p +1 ! p − . Notice that λ k > I ω ( λ k v n k ) = 0. Therefore, by Lemma 3.2 and the definition of d ω ,we obtain d ω < d ω ≤ p − p + 1) k λ k v n k k p +1 p +1 = λ p +1 k p − p + 1) k v n k k p +1 p +1 , for all k ∈ N . (3.4)Furthermore, by I ω ( v n k ) = 0, (3.3) and the weak continuity of ( V v, v ) = Z Γ V ( x ) | v ( x ) | dx (see [23, Theorem 11.4]), we getlim k →∞ λ k = lim k →∞ k v n k k p +1 p +1 − ( V v n k , v n k ) k v n k k p +1 p +1 ! p − = d ω − p − p +1) ( V v , v ) d ω ! p − . Taking the limit in (3.4), we obtain d ω < lim k →∞ λ p +1 k d ω . Since d ω >
0, we arrive atlim k →∞ λ k >
1, and consequently − ( V v , v ) >
0. Thus, v = 0.By the weak convergence, we obtainlim k →∞ (cid:8) ( F γ,V ( v n k ) − F γ,V ( v n k − v )) + ω (cid:0) k v n k k − k v n k − v k (cid:1)(cid:9) (3.5) = F γ,V ( v ) + ω k v k . Next, passing to a subsequence of { v n k } if necessary, we may assume that v n k −→ k →∞ v a.e. on Γ. Therefore, by the Brezis-Leib lemma [8],lim k →∞ I ω ( v n k ) − I ω ( v n k − v ) = lim k →∞ − I ω ( v n k − v ) = I ω ( v ) . Since v = 0, then the right-hand side of (3.5) is positive. It follows from (3.3) and(3.5) that p − p + 1) lim k →∞ (cid:0) F γ,V ( v n k − v ) + ω k v n k − v k (cid:1) < p − p + 1) lim k →∞ (cid:0) F γ,V ( v n k ) + ω k v n k k (cid:1) = d ω . Hence, by (3.1), we have I ω ( v n k − v ) > k large enough. Thus, since − I ω ( v n k − v ) −→ k →∞ I ω ( v ), we obtain I ω ( v ) ≤
0. Then, by (3.1) and the weak lower semicontinuity of norms,
NSTABILITY OF GROUND STATES 15 we see that d ω ≤ p − p + 1) (cid:0) F γ,V ( v ) + ω k v k (cid:1) ≤ p − p + 1) lim k →∞ (cid:0) F γ,V ( v n k ) + ω k v n k k (cid:1) = d ω . Therefore, from (3.5) we getlim k →∞ F γ,V ( v n k − v ) + ω k v n k − v k = 0 , and consequently, by Lemma 4.13-( ii ), we have v n k −→ k →∞ v in H (Γ) and I ω ( v ) = 0. Thisconcludes the proof. (cid:3) Proof of Proposition 1.1.
Step 1.
We prove that G ω = M ω . Let ϕ ∈ M ω . Since I ω ( ϕ ) = 0, we have h I ′ ω ( ϕ ) , ϕ i = 2 (cid:0) F γ,V ( ϕ ) + ω k ϕ k (cid:1) − ( p + 1) k ϕ k p +1 p +1 = − ( p − k ϕ k p +1 p +1 < . (3.6)There exists a Lagrange multiplier µ ∈ R such that S ′ ω ( ϕ ) = µI ′ ω ( ϕ ). Furthermore, since µ h I ′ ω ( ϕ ) , ϕ i = h S ′ ω ( ϕ ) , ϕ i = I ω ( ϕ ) = 0 , then, by (3.6), µ = 0. Hence S ′ ω ( ϕ ) = 0. Moreover, for v ∈ H (Γ) \{ } satisfying S ′ ω ( v ) = 0, we have I ω ( v ) = h S ′ ω ( v ) , v i = 0. Then, from the definition of M ω , weget S ω ( ϕ ) ≤ S ω ( v ). Hence, we obtain ϕ ∈ G ω . Now, let φ ∈ G ω . Since M ω is notempty, we take ϕ ∈ M ω . By the first part of the proof, we have ϕ ∈ G ω , therefore S ω ( φ ) = S ω ( ϕ ) = d ω . This implies φ ∈ M ω . Step 2.
Let ϕ ∈ G ω . Below we show that ϕ has the form ϕ ( x ) = e iθ φ ( x ) with positive φ ( x ) ∈ dom( H γ,V ) . Set φ := | ϕ | , then k φ ′ k ≤ k ϕ ′ k and S ω ( φ ) ≤ S ω ( ϕ ) = d ω . Using G ω = M ω , we obtain I ω ( ϕ ) = 0, then I ω ( φ ) ≤
0. It follows from Lemma 3.1 that φ ∈ M ω and S ω ( ϕ ) = S ω ( φ ). Observe that this implies k φ ′ k = N X e =1 ∞ Z | φ ′ e ( x ) | dx = N X e =1 ∞ Z | ϕ ′ e ( x ) | dx = k ϕ ′ k . (3.7)From S ′ ω ( φ ) = 0, repeating the proof of [2, Theorem 4] (see also [5, Lemma 4.1]), onegets φ ∈ dom( H γ,V ) and H γ,V φ + ωφ − φ p = 0 , therefore − φ ′′ e + ωφ e + V e ( x ) φ e − φ pe = 0 , x ∈ (0 , ∞ ) , e = 1 , . . . , N. Recalling that V ( x ) ≤ φ e is either trivial or strictly positive on (0 , ∞ ). Indeed, to prove that, we need toset β ( s ) := ωs − s p and observe that β ( s ) ∈ C [0 , ∞ ) is nondecreasing for s small, and β (0) = β ( ω p − ) = 0.Now assume φ e (0) = φ ′ e (0) = 0 and put e φ e ( x ) = ( φ e ( x ) , x ∈ [0 , ∞ )0 , x ∈ ( − δ, . Then, by the Sobolev extension theorem, we have e φ e ∈ H ( − δ, ∞ ). Moreover, − e φ ′′ e + ω e φ e + V e ( x ) e φ e − e φ pe = 0 , on ( − δ, ∞ ) . Therefore, by [28, Theorem 1], arguing as above, we find that e φ e = 0 on ( − δ, ∞ ).Next assume φ (0) = 0, i.e. φ (0) = . . . = φ N (0). Since φ e ∈ C (0 , ∞ ), φ e ≥ φ e (0) = 0, then φ ′ e (0) ≥
0. By N P e =1 φ ′ e (0) = − γφ (0) = 0, we get φ e (0) = φ ′ e (0) = 0. Then φ e = 0 on (0 , ∞ ) for all e = 1 , . . . , N , and by continuity φ = 0 on Γ, which is absurdsince φ ∈ M ω . Hence φ e (0) > e = 1 , . . . , N , therefore φ e > , ∞ ) for all e = 1 , . . . , N , i.e. φ > Step 3.
Now, we can write ϕ e ( x ) = φ e ( x ) τ e ( x ) , where τ e ∈ C (0 , ∞ ), | τ e | = 1. Then ϕ ′ e = φ ′ e τ e + φ e τ ′ e = τ e ( φ ′ e + φ e τ e τ ′ e ) . Using Re( τ e τ ′ e ) = 0, we have | ϕ ′ e | = | φ ′ e | + | φ e τ ′ e | . Therefore, from (3.7) we obtain N X e =1 ∞ Z | φ ′ e | dx = N X e =1 ∞ Z | ϕ ′ e | dx = N X e =1 ∞ Z | φ ′ e | dx + N X e =1 ∞ Z | φ e τ ′ e | dx. So far as φ e >
0, we have τ ′ e = 0 for all e = 1 , . . . , N . Since τ e ∈ C (0 , ∞ ), there existsa constant θ e ∈ R such that τ e ( x ) = e iθ e on (0 , ∞ ). By the continuity at the vertex, weobtain θ e = θ = const for all e = 1 , . . . , N . This ends the proof.Re( τ e τ ′ e ) (cid:3) instability of standing waves In this section, we prove Theorem 1.4 and Corollary 1.5.4.1.
Proof of the main result.
We begin with the following lemma.
Lemma 4.1.
Let ϕ ω ∈ M ω . Then ( i ) k ϕ ω k p +1 p +1 = inf n k v k p +1 p +1 : v ∈ H (Γ) \{ } , I ω ( v ) = 0 o = inf n k v k p +1 p +1 : v ∈ H (Γ) \{ } , I ω ( v ) ≤ o , ( ii ) S ω ( ϕ ω ) = inf { S ω ( v ) : v ∈ H (Γ) , k v k p +1 p +1 = k ϕ ω k p +1 p +1 } . Proof. ( i ) This is an immediate consequence of Lemma 3.1( ii ) Set d ∗∗ ω := inf { S ω ( v ) : v ∈ H (Γ) , k v k p +1 p +1 = k ϕ ω k p +1 p +1 } . As far as d ∗∗ ω ≤ S ω ( ϕ ω ), itsuffices to prove S ω ( ϕ ω ) ≤ d ∗∗ ω . If v ∈ H (Γ) satisfies k v k p +1 p +1 = k ϕ ω k p +1 p +1 , then, by item ( i )and (3.2), we have I ω ( v ) ≥
0. Hence, by (3.2), S ω ( ϕ ω ) = p − p + 1) k ϕ ω k p +1 p +1 = p − p + 1) k v k p +1 p +1 ≤ S ω ( v ) . Thus, we obtain S ω ( ϕ ω ) ≤ d ∗∗ ω . (cid:3) NSTABILITY OF GROUND STATES 17
Recall that P ( v ) = k v ′ k − Z Γ xV ′ ( x ) | v ( x ) | dx − γ | v (0) | − p − p + 1) k v k p +1 p +1 . Lemma 4.2. If ∂ λ E ( ϕ λω ) | λ =1 < , then there exist δ > and ε > such that the followingholds: for any v ∈ N ε ( ϕ ω ) satisfying k v k ≤ k ϕ ω k , there exists λ ( v ) ∈ (1 − δ, δ ) suchthat E ( ϕ ω ) ≤ E ( v ) + ( λ ( v ) − P ( v ) , where N ε ( ϕ ω ) is defined by (1.7) .Proof. Since ∂ λ E ( ϕ λω ) | λ =1 < ∂ λ E ( v λ ) is continuous in v (we mean ”orbit”-continuity)and λ , there exist positive constants ε and δ such that ∂ λ E ( v λ ) < v ∈ N ε ( ϕ ω )and λ ∈ (1 − δ, δ ). Using P ( v ) = ∂ λ E ( v λ ) | λ =1 , the Taylor expansion at λ = 1 gives E ( v λ ) ≤ E ( v ) + ( λ − P ( v ) , λ ∈ (1 − δ, δ ) , v ∈ N ε ( ϕ ω ) . (4.1)Let v ∈ N ε ( ϕ ω ) satisfy k v k ≤ k ϕ ω k . We define λ ( v ) := k ϕ ω k p +1 p +1 k v k p +1 p +1 ! p − . Then, (cid:13)(cid:13) v λ ( v ) (cid:13)(cid:13) p +1 p +1 = k ϕ ω k p +1 p +1 and we can take ε small enough to guarantee λ ( v ) ∈ (1 − δ, δ ). Since (cid:13)(cid:13) v λ ( v ) (cid:13)(cid:13) = k v k ≤ k ϕ ω k , by Lemma 4.1-( ii ), we have E ( v λ ( v ) ) = S ω ( v λ ( v ) ) − ω (cid:13)(cid:13) v λ ( v ) (cid:13)(cid:13) ≥ S ω ( ϕ ω ) − ω k ϕ ω k = E ( ϕ ω ) , which together with (4.1) implies that E ( ϕ ω ) ≤ E ( v ) + ( λ ( v ) − P ( v ). (cid:3) To prove Theorem 1.4, we introduce the following definition.
Definition 4.3.
Let ε be the positive constant given by Lemma 4.2. Set Z ε ( ϕ ω ) := { v ∈ N ε ( ϕ ω ) : E ( v ) < E ( ϕ ω ) , k v k ≤ k ϕ ω k , P ( v ) < } , and for any u ∈ N ε ( ϕ ω ) , we define the exit time from N ε ( ϕ ω ) by T ε ( u ) = sup { T > u ( t ) ∈ N ε ( ϕ ω ) , ≤ t ≤ T } , with u ( t ) being a solution of (1.1) . Lemma 4.4.
Assume ∂ λ E ( ϕ λω ) | λ =1 < , then for any u ∈ Z ε ( ϕ ω ) , there exists b = b ( u ) > such that P ( u ( t )) ≤ − b for ≤ t < T ε ( u ) .Proof. Set b := E ( ϕ ω ) − E ( u ) >
0, with u ∈ Z ε ( ϕ ω ). From the conservation of energyand Lemma 4.2, we have b ≤ ( λ ( u ( t )) − P ( u ( t )) , ≤ t < T ε ( u ) . (4.2)Therefore, for 0 ≤ t < T ε ( u ) we get P ( u ( t )) = 0. Indeed, if P ( u ( t )) = 0 for some t ∈ [0 , T ( u )) , then from (4.2) it follows b ≤
0, which contradicts the definition of b .Since P ( u ) < t P ( u ( t )) is continuous, we see that P ( u ( t )) < ≤ t < T ε ( u ) and hence λ ( u ( t )) − < ≤ t < T ε ( u ). Thus, from Lemma 4.2 and(4.2), we have P ( u ( t )) ≤ b λ ( u ( t )) − ≤ − b δ , ≤ t < T ε ( u ) . Hence, taking b = b δ , we arrive at P ( u ( t )) ≤ − b for 0 ≤ t < T ε ( u ). (cid:3) Now we are ready to prove Theorem 1.4.
Proof of Theorem 1.4.
Observe that P ( v ) = ∂ λ S ω ( v λ ) | λ =1 = (cid:10) S ′ ω ( v ) , ∂ λ v λ | λ =1 (cid:11) . Since S ′ ω ( ϕ ω ) = 0, we obtain P ( ϕ ω ) = ∂ λ S ω ( ϕ λω ) | λ =1 = 0. Moreover, by P ( ϕ λω ) = λ∂ λ E ( ϕ λω ), wehave ∂ λ E ( ϕ λω ) | λ =1 = 0. Then, from the assumption ∂ λ E ( ϕ λω ) | λ =1 <
0, we get E ( ϕ λω )
0, we define χ a ∈ C ∞ c (Γ) by( χ a ) e ( x ) = χ (cid:16) xa (cid:17) , x ∈ R + , e = 1 , . . . , N. Then we have lim a →∞ (cid:13)(cid:13) χ a ϕ λ ω − ϕ λ ω (cid:13)(cid:13) H (Γ) = 0 and (cid:13)(cid:13) χ a ϕ λ ω (cid:13)(cid:13) ≤ (cid:13)(cid:13) ϕ λ ω (cid:13)(cid:13) = k ϕ ω k for all a >
0. Thus, by continuity of E and P , for any δ ≤ ε there exists a > χ a ϕ λ ω ∈ Z δ ( ϕ λ ω ), therefore χ a ϕ λ ω ∈ Z δ ( ϕ ω ) ⊆ Z ε ( ϕ ω ).Observe that χ a ϕ λ ω ∈ Σ(Γ) (see Proposition 2.6 for the definition of Σ(Γ)), and byvirial identity (2.4), we see that d dt k xu ( t ) k = 8 P ( u ( t )) , ≤ t ≤ T ε ( χ a ϕ λ ω ) , (4.3)where u ( t ) is the solution to (1.1) with u (0) = χ a ϕ λ ω . From Lemma 4.4, there exists b = b ( λ , a ) > P ( u ( t )) ≤ − b, ≤ t < T ε ( χ a ϕ λ ω ) . (4.4)Then, from (4.4) and (4.3), we can see that T ε ( χ a ϕ λ ω ) < ∞ .Summarizing the above, we affirm: there exists ε > δ > u = χ a ϕ λ ω ∈ N δ ( ϕ ω ) and t > u ( t ) of (1.1) satisfies u ( t ) / ∈ N ε ( ϕ ω ) . Hence, the standing wave solution e iωt ϕ ω of (1.1) is orbitally unstable. (cid:3) Rescaled variational problem and proof of Corollary 1.5.
Assume that V ( x ) = − βx α , β > , < α <
1. Recall that v λ ( x ) = λ / v ( λx ) for λ >
0. By simple computations,we have E ( v λ ) = λ k v ′ k + λ α V v, v ) − λ γ | v (0) | − λ p − p + 1 k v k p +1 p +1 , NSTABILITY OF GROUND STATES 19 ∂ λ E ( v λ ) | λ =1 = k v ′ k + α ( α − V v, v ) − ( p − p − p + 1) k v k p +1 p +1 . Since P ( ϕ ω ) = ∂ λ S ω ( ϕ λω ) | λ =1 = 0, then we get ∂ λ E ( ϕ λω ) | λ =1 = − α (2 − α )2 ( V ϕ ω , ϕ ω ) + γ | ϕ ω, (0) | − ( p − p − p + 1) k ϕ ω k p +1 p +1 , and ∂ λ E ( ϕ λω ) | λ =1 < − α (2 − α )( V ϕ ω , ϕ ω ) + γ | ϕ ω, (0) | k ϕ ω k p +1 p +1 < ( p − p − p + 1) . (4.5)Below we prove that the left-hand side of (4.5) converges to 0 as ω → ∞ . To this end,we consider the following rescaling of ϕ ω ∈ M ω : ϕ ω ( x ) = ω p − e ϕ ω ( √ ωx ) , ω ∈ ( ω , ∞ ) , (4.6)and observe − ω − − α α (2 − α )( V e ϕ ω , e ϕ ω ) + ω − γ | e ϕ ω, (0) | k e ϕ ω k p +1 p +1 = − α (2 − α )( V ϕ ω , ϕ ω ) + γ | ϕ ω, (0) | k ϕ ω k p +1 p +1 . (4.7)Put e I ω ( v ) : = k v ′ k + k v k − ω − − α β Z Γ | v ( x ) | x α dx − ω − γ | v (0) | − k v k p +1 p +1 , e I ( v ) : = k v ′ k + k v k − k v k p +1 p +1 . Consider the minimization problem e d := inf n k v k p +1 p +1 : v ∈ H (Γ) \{ } , e I ( v ) ≤ o . (4.8)In [2, Theorem 3] it was shown that e d >
0. The following lemma is the key result toprove Corollary 1.5.
Lemma 4.5.
Assume γ > , β > , < α < and p > . Let ϕ ω ∈ M ω , and e ϕ ω ( x ) bethe rescaled function given in (4.6) . Then ( i ) lim ω →∞ k e ϕ ω k p +1 p +1 = e d , ( ii ) lim ω →∞ e I ( e ϕ ω ) = 0 , ( iii ) lim ω →∞ k e ϕ ω k H (Γ) = e d . Proof.
Notice that k e ϕ ω k p +1 p +1 = inf n k v k p +1 p +1 : v ∈ H (Γ) \{ } , e I ω ( v ) = 0 o = inf n k v k p +1 p +1 : v ∈ H (Γ) \{ } , e I ω ( v ) ≤ o := e d ω . (4.9) By definition we have(4.10) e I ( v ) = e I ω ( v ) − ω − − α ( V v, v ) + ω − γ | v (0) | , and(4.11) e I ( v ) = λ − e I ( λv ) + ( λ p − − k v k p +1 p +1 . Using, (4.10), (4.11), e I ω ( e ϕ ω ) = 0, estimate (4.18), and the Sobolev embedding, for any λ > λ − e I ( λ e ϕ ω ) = − ω − − α ( V e ϕ ω , e ϕ ω ) + ω − γ | e ϕ ω, (0) | − ( λ p − − k e ϕ ω k p +1 p +1 ≤ C ω − − α k e ϕ ω k H (Γ) + C ω − γ k e ϕ ω k H (Γ) − ( λ p − − k e ϕ ω k p +1 p +1 . (4.12)Moreover, from e I ω ( e ϕ ω ) = 0, we deduce k e ϕ ω k H (Γ) = − ω − − α ( V e ϕ ω , e ϕ ω ) + ω − γ | e ϕ ω, (0) | + k e ϕ ω k p +1 p +1 ≤ C ω − − α k e ϕ ω k H (Γ) + C ω − γ k e ϕ ω k H (Γ) + k e ϕ ω k p +1 p +1 . This implies (cid:16) − C ω − − α − C ω − γ (cid:17) k e ϕ ω k H (Γ) ≤ k e ϕ ω k p +1 p +1 . Since for ω sufficiently large (cid:16) − C ω − − α − C ω − γ (cid:17) >
0, from (4.12) we get λ − e I ( λ e ϕ ω ) ≤ − λ p − − − C ω − − α + C ω − γ − C ω − − α − C ω − γ ! k e ϕ ω k p +1 p +1 . (4.13)Hence for any λ >
1, there exists ω = ω ( λ ) ∈ ( ω , ∞ ) such that e I ( λ e ϕ ω ) < ω ∈ ( ω , ∞ ). Thus, by (4.8), e d ≤ λ p +1 k e ϕ ω k p +1 p +1 for ω ∈ ( ω , ∞ ). Observe that e I ( v ) ≤ e I ω ( v ) ≤
0, then from (4.9) we obtain e d ω = k e ϕ ω k p +1 p +1 ≤ e d . Therefore,(4.14) λ − ( p +1) e d ≤ k e ϕ ω k p +1 p +1 ≤ e d , ω ∈ ( ω , ∞ ) . Letting λ ↓
1, we get that ω → ∞ , and from (4.14) it follows ( i ).Now, assume that λ = 1 in (4.13), then using ( i ), we deducelim sup ω →∞ e I ( e ϕ ω ) ≤ . (4.15)Furthermore, define λ ( ω ) = k e ϕ ′ ω k + k e ϕ ω k k e ϕ ω k p +1 p +1 ! p − > , then e I ( λ ( ω ) e ϕ ω ) = 0. Therefore, we have e d ≤ λ ( ω ) p +1 k e ϕ ω k p +1 p +1 . (4.16)Thus, by ( i ) and (4.16), we arrive atlim inf ω →∞ λ ( ω ) ≥ lim inf ω →∞ e d k e ϕ ω k p +1 p +1 ! p +1 = 1 . NSTABILITY OF GROUND STATES 21
Moreover, by (4.11), e I ( λ ( ω ) e ϕ ω ) = 0 and ( i ), we havelim inf ω →∞ e I ( e ϕ ω ) = lim inf ω →∞ ( λ ( ω ) p − − k e ϕ ω k p +1 p +1 ≥ , which together with (4.15) implies ( ii ). Finally, from ( i ) and ( ii ), we obtain e d = lim ω →∞ k e ϕ ω k p +1 p +1 = lim ω →∞ k e ϕ ω k H (Γ) , which shows ( iii ). (cid:3) Proof of Corollary 1.5.
Recall that, by Theorem 1.4, if ∂ λ E (cid:0) ϕ λω (cid:1) | λ =1 <
0, then e it ϕ ω ( x )is orbitally unstable. Since ∂ λ E (cid:0) ϕ λω (cid:1) | λ =1 < ⇐⇒ − α (2 − α ) ( V ϕ ω , ϕ ω ) + γ | ϕ ω, (0) | k ϕ ω k p +1 p +1 < ( p − p − p + 1) , by (4.7), it suffices to provelim ω →∞ − ω − − α α (2 − α )( V e ϕ ω , e ϕ ω ) + ω − γ | e ϕ ω, (0) | k e ϕ ω k p +1 p +1 = 0 . (4.17)We have 0 ≤ − ω − − α α (2 − α )( V e ϕ ω , e ϕ ω ) + ω − γ | e ϕ ω, (0) | ≤ (cid:16) C ω − − α + C ω − γ (cid:17) k e ϕ ω k H (Γ) . Hence, by Lemma 4.5-( i ),( iii ), we obtain (4.17). This concludes the proof. (cid:3) Instability results in H (Γ) . We discuss counterparts of Proposition 1.1, Theorem1.4, Corollary 1.5 for arbitrary γ ∈ R and symmetric V ( x ), i.e. V ( x ) = . . . = V N ( x ), inthe space H (Γ) = { v ∈ H (Γ) : v ( x ) = . . . = v N ( x ) , x > } . The well-posedness in H (Γ) follows analogously to [17, Lemma 2.6]. We use index · eq todenote counterparts of the objects for the space H (Γ).It is known that d ω, eq = S ω ( φ γ ) (see page 12 in [18]) for any γ ∈ R , where φ γ ( x ) = (cid:18)n ( p +1) ω sech (cid:16) ( p − √ ω x + arctanh( γN √ ω ) (cid:17)o p − (cid:19) Ne =1 . Then for 0 < ω , eq < ω (observe that ω , eq ≤ ω ) one can repeat all the proofs in Section3 and Subsections 4.1 and 4.2 with H (Γ) instead of H (Γ) . Thus, we get the followingresults.
Proposition 4.6.
Let p > , γ ∈ R , ω > ω , eq . If V ( x ) = V ( x ) is symmetric andsatisfies Assumptions 1-3, then the set of ground states G ω, eq is not empty, in particular, G ω, eq = M ω, eq . If ϕ ω ∈ G ω, eq , then there exist θ ∈ R and a positive function φ ∈ H (Γ) such that ϕ ω ( x ) = e iθ φ ( x ) . Theorem 4.7.
Let p > , γ ∈ R , ω > ω , eq . If V ( x ) = V ( x ) is symmetric and satisfiesAssumptions 1-4, ϕ ω ( x ) ∈ G ω, eq , and ∂ λ E ( ϕ λω ) | λ =1 < , then the standing wave solution e iωt ϕ ω ( x ) of (1.1) is orbitally unstable in H (Γ) and therefore in H (Γ) . Corollary 4.8.
Assume that V ( x ) = − βx α , β > , < α < , γ ∈ R . Let p > and ϕ ω ( x ) ∈ G ω, eq . Then there exists ω ∗ eq ∈ ( ω , eq , ∞ ) such that for any ω ∈ ( ω ∗ eq , ∞ ) thestanding wave solution e iωt ϕ ω ( x ) of (1.1) is orbitally unstable in H (Γ) . Remark 4.9. ( i ) Observe that when dealing with H (Γ) , no restriction on γ appears.This is due to the fact that the corresponding constrained variational problem is closelyrelated to the one on R , which in turn admits a minimizer for any γ (see [18, Remark3.1]). ( ii ) Consider i∂ t u = − ∂ x u − γδ ( x ) u + V ( x ) u − | u | p − u, ( t, x ) ∈ R × R ,γ ∈ R . Notice that the above results are valid with H (Γ) substituted by H ( R ) = { f ∈ H ( R ) : f ( x ) = f ( − x ) } and analogous assumptions on V ( x ) . One onlyneeds to recall that d ω, rad = S ω ( φ γ ) (see [14, Theorem 1]), where φ γ ( x ) = n ( p +1) ω sech (cid:16) ( p − √ ω | x | + arctanh( γ √ ω ) (cid:17)o p − . Appendix
Below we show some properties of the operator H γ,V introduced by (1.2). Lemma 4.10.
Let γ ∈ R and V ( x ) = V ( x ) ∈ L (Γ) + L ∞ (Γ) . The quadratic form F γ,V given by (1.3) is semibounded and closed, and the operator H γ,V defined by ( H γ,V v ) e = − v ′′ e + V e v e , dom( H γ,V ) = ( v ∈ H (Γ) : − v ′′ e + V e v e ∈ L ( R + ) , N X e =1 v ′ e (0) = − γv (0) ) . is the self-adjoint operator associated with F γ,V in L (Γ) .Proof. We can write V ( x ) = V ( x ) + V ( x ), with V ∈ L (Γ) and V ∈ L ∞ (Γ). Thus, usingthe Gagliardo-Nirenberg inequality (see formula (2.1) in [10]) and the Young inequality,we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Γ V ( x ) | v ( x ) | dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k V k k v k ∞ + k V k ∞ k v k ≤ C k V k k v ′ k k v k + k V k ∞ k v k ≤ ε k v ′ k + C ε k v k , ε > . (4.18)Similarly, by the Sobolev embedding, we obtain (cid:12)(cid:12) γ | v (0) | (cid:12)(cid:12) ≤ | γ |k v k ∞ ≤ C k v ′ k k v k ≤ ε k v ′ k + C ε k v k . Therefore,(4.19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ | v (0) | + Z Γ V ( x ) | v ( x ) | dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε k v ′ k + C ε k v k , for every ε > . NSTABILITY OF GROUND STATES 23
Then, by the KLMN theorem [26, Theorem X.17], we infer that the quadratic form F γ,V is associated with a semibounded self-adjoint operator T γ,V defined by (observe that A = H , in [26, Theorem X.17], i.e. V ≡ , γ = 0)dom( T γ,V ) = (cid:8) u ∈ H (Γ) : ∃ y ∈ L (Γ) s.t. ∀ v ∈ H (Γ) , F γ,V ( u, v ) = ( y, v ) (cid:9) ,T γ,V u = y. It is easily seen that dom( H γ,V ) ⊆ dom( T γ,V ) and T γ,V u = H γ,V u, u ∈ dom( H γ,V ). Henceit is sufficient to prove that dom( T γ,V ) ⊆ dom( H γ,V ).Let ˜ u ∈ dom( T γ,V ) and ˜ v ∈ H (Γ), then there exists ˜ y ∈ L (Γ) such that(4.20) F γ,V (˜ u, ˜ v ) = Z Γ (˜ u ′ ˜ v ′ + V ˜ u ˜ v ) dx − γ ˜ u (0)˜ v (0) = (˜ y, ˜ v ) . Observe that ˜ y − V ˜ u ∈ L loc (Γ) and set z = ( z e ) Ne =1 , z e ( x ) = x Z (˜ y e ( t ) − V e ( t )˜ u e ( t )) dt. Suppose now additionally that ˜ v has a compact support, then(4.21) Z Γ (˜ y − V ˜ u )˜ vdx = Z Γ z ′ ˜ vdx = − ˜ v (0) N X e =1 z e (0) − Z Γ z ˜ v ′ dx. From (4.20) we deduce(4.22) Z Γ (˜ y − V ˜ u )˜ vdx = Z Γ ˜ u ′ ˜ v ′ dx − γ ˜ u (0)˜ v (0) . Combining (4.21) and (4.22) we get(4.23) Z Γ (˜ u ′ + z )˜ v ′ dx + ˜ v (0) − γ ˜ u (0) + N X e =1 z e (0) ! = 0 . Choose ˜ v = (˜ v e ) Ne =1 such that ˜ v ( x ) ∈ C ∞ ( R + ) and ˜ v ( x ) ≡ . . . ≡ ˜ v N ( x ) ≡ . Then weobtain ∞ Z (˜ u ′ + z )˜ v ′ dx = 0 , therefore ˜ u ′ + z ≡ const ≡ c . We have used that ˜ u ′ + z ∈ Ran( A ) ⊥ , where Av = v ′ with dom( A ) = C ∞ ( R + ) in L ( R + ). Analogously ˜ u ′ e + z e ≡ const ≡ c e , e = 2 , . . . , N. Finally, from (4.23) we deduce˜ v (0) − γ ˜ u (0) − N X e =1 (˜ u ′ e (0) + z e (0)) + N X e =1 z e (0) ! = 0 . Assuming that ˜ v (0) = 0 , we arrive at N P e =1 ˜ u ′ e (0) = − γ ˜ u (0) . Moreover, − ˜ u ′′ + V ˜ u = z ′ + V ˜ u = ˜ y − V ˜ u + V ˜ u = ˜ y ∈ L (Γ) . Hence ˜ u ∈ dom( H γ,V ) and dom( T γ,V ) ⊆ dom( H γ,V ) . (cid:3) Lemma 4.11.
Suppose that V ( x ) = V ( x ) ∈ L ε (Γ) + L ∞ (Γ) , i.e. for any ε > and V ∈ L ε (Γ) + L ∞ (Γ) there exists a representation V = V + V , V ∈ L (Γ) , V ∈ L ∞ (Γ) ,with k V k ≤ ε . Then we have (4.24) dom( H γ,V ) = ( v ∈ H (Γ) : v e ∈ H ( R + ) , N X e =1 v ′ e (0) = − γv (0) ) := D H . Moreover, for m sufficiently large, H γ,V -norm k ( H γ,V + m ) · k is equivalent to H -normon Γ . Proof.
Observe that, by V ( x ) ∈ L ε (Γ) + L ∞ (Γ), the Sobolev and the Young inequalitieswe get(4.25) k V v k ≤ k V k k v k ∞ + k V k ∞ k v k ≤ ε k v k H (Γ) + C k v k and, | ( v ′′ , V v ) | ≤ k v ′′ k k V v k ≤ k v ′′ k k V k k v k ∞ + k v ′′ k k V k ∞ k v k ≤ C k v ′′ k k V k k v k H (Γ) + C k v ′′ k k v k ≤ ε k v k H (Γ) + ε k v ′′ k + C ε k v k ≤ ε k v k H (Γ) + C ε k v k . (4.26)It is immediate from (4.25), (4.26) that(4.27) k H γ,V v k = k v ′′ k + 2 Re( v ′′ , V v ) + k V v k ≤ C k v k H (Γ) . And for m sufficiently large, inequalities (4.25) and (4.26) imply(4.28) k H γ,V v k + m k v k = k v ′′ k + 2 Re( v ′′ , V v ) + k V v k + m k v k ≥ C k v k H (Γ) . Thus, we get (4.24).The second assertion follows from (4.27),(4.28), and k ( H γ,V + m ) v k = k H γ,V v k + m k v k + 2 m ( H γ,V v, v ) , | ( H γ,V v, v ) | ≤ k H γ,V v k k v k ≤ ε k H γ,V v k + C ε k v k . (cid:3) Remark 4.12.
Observe that there exists potential V ( x ) satisfying Assumptions 1-4 suchthat dom( H γ,V ) = D H . For example, consider V ( x ) = − /x α , / ≤ α < , and N = γ = 2 , then v = ( e − x , e − x ) ∈ D H , but k H γ,V v k = 2 k − v ′′ − v x α k > e − ε ε Z dxx α = ∞ . Lemma 4.13.
Let γ > and V ( x ) = V ( x ) satisfy Assumptions 1 e 3. Then the followingassertions hold. ( i ) The number − ω defined by (1.5) is negative. ( ii ) Let also m > ω , then p F γ,V ( v ) + m k v k defines a norm equivalent to the H -norm. ( iii ) The number − ω is the first eigenvalue of H γ,V . Moreover, it is simple, and thereexists the corresponding positive eigenfunction ψ ∈ dom( H γ,V ) , i.e. H γ,V ψ = − ω ψ . NSTABILITY OF GROUND STATES 25
Proof. ( i ) To show − ω <
0, observe that − ω = inf σ ( H γ,V ) = inf (cid:8) F γ,V ( v ) : v ∈ H (Γ) , k v k = 1 (cid:9) . (4.29)Consider v λ ( x ) = λ v ( λx ) with λ >
0. Hence F γ,V ( v λ ) = λ k v ′ k − λγ | v (0) | + ( V v λ , v λ ) . For λ small enough, we have F γ,V ( v λ ) <
0. Finally, − ω is finite since F γ,V ( v ) is lowersemibounded.( ii ) Let ε >
0. Firstly, notice that from (4.19) one easily gets F γ,V ( v ) + m k v k ≤ (1 + 2 ε ) k v ′ k + ( C ε + m ) k v k ≤ C k v k H (Γ) . Secondly, for ε and δ sufficiently small, F γ,V ( v ) + m k v k = δ k v ′ k + (1 − δ ) (cid:18) k v ′ k + 11 − δ ( V v, v ) − γ − δ | v (0) | (cid:19) + m k v k ≥ δ k v ′ k − (1 + ε )(1 − δ ) ω k v k + m k v k ≥ C k v k H (Γ) . Indeed, the family of sesquilinear formst( κ )[ u, v ] = ( u ′ , v ′ ) + 11 − κ ( V u, v ) − γ − κ ( u (0) v (0))is holomorphic of type (a) in the sense of Kato in the complex neighborhood of zero(see [21, Chapter VII, §
4] for the definition and [21, Chapter VI, §
1, Example 1.7] for theproof of sectoriality). Using inequality (4 .
7) in [21, Chapter VII] with κ = κ = 0 , κ = δ ,we obtain | t( δ )[ v ] − t(0)[ v ] | ≤ ε | t(0)[ v ] | . Hencet( δ )[ v ] ≥ t(0)[ v ] − ε | t(0)[ v ] | = F γ,V ( v ) − ε | F γ,V ( v ) | ≥ − (1 + ε ) ω k v k . ( iii ) Step 1.
Let { v n } be a minimizing sequence, that is, F γ,V ( v n ) −→ n →∞ − ω , k v n k = 1for all n ∈ N . From ( ii ), we deduce that { v n } is bounded in H (Γ). Then there exista subsequence { v n k } of { v n } and v ∈ H (Γ) such that { v n k } converges weakly to v in H (Γ). Observe that, by the weak lower semicontinuity of L -norm and F γ,V ( · ), we get k v k ≤ F γ,V ( v ) ≤ lim k →∞ F γ,V ( v n k ) = − ω < . We have k v k = 1, since, otherwise, there would exist λ > k λv k = 1 and F γ,V ( λv ) = λ F γ,V ( v ) < − ω , which is a contradiction. Consequently v is a minimizerfor (4.29).Let ψ = | v | , then ψ ≥ k ψ k = k v k = 1. Notice that k ψ ′ k ≤ k v ′ k , therefore F γ,V ( ψ ) ≤ F γ,V ( v ). Then ψ is a minimizer of (4.29). This implies the existenceof the Lagrange multiplier − µ such that F ′ γ,V ( ψ ) = − µQ ′ ( ψ ) , Q ( v ) = k v k . Repeating the arguments from the proof of [2, Theorem 4], we get ψ ∈ dom( H γ,V ) and H γ,V ψ = − µψ . Multiplying the above equation by ψ and integrating we conclude µ = ω . Recalling that V ( x ) ≤ ψ > . Notice that one needs to apply [28, Theorem 1] with β ( s ) = ω s. Step 2.
Suppose that u is a nonnegative solution of(4.30) H γ,V u = − ω u . Let us show that there exists
C > u ( x ) = Cψ ( x ). Assume that this isfalse. Then there exists C > e u ( x ) = u ( x ) − Cψ ( x ) takes both positive andnegative values. We have H γ,V e u = − ω e u , consequently e v = e u / k e u k is the minimizerof (4.29). Arguing as in Step 1 , one can show that | e v | is also a minimizer and | e v | > e u ( x ) has a constant sign. This is a contradiction.Suppose now that u is an arbitrary solution to (4.30) such that k u k = 1 (that is u is a minimizer of (4.29)). Define w = | Re u | + i | Im u | , then | w | = | u | and | w ′ | = | u ′ | ,consequently F γ,V ( u ) = F γ,V ( w ) and k w k = 1 . Therefore, w is a minimizer of (4.29).This implies that w satisfies (4.30), and, in particular, | Re u | and | Im u | satisfy (4.30).Thus, | Re u | = C ψ and | Im u | = C ψ , C , C >
0, consequently Re u = e C ψ and Im u = e C ψ , e C , e C ∈ R , since Re u and Im u do not change the sign. Finally, u = e C ψ + i e C ψ = e Cψ , e C ∈ C , and therefore , − ω is simple. (cid:3) Acknowledgments.
The authors are kindly grateful to Prof. Gl´aucio Terra for the proofof Remark 1.2.
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