Nonlinear Schrödinger equation with growing potential on infinite metric graphs

Nonlinear Analysis 184 (2019) 258–272

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Nonlinear Analysis www.elsevier.com/locate/na

Nonlinear Schrödinger equation with growing potential on infinite metric graphs Setenay Akduman a , Alexander Pankov b,c ,∗ a

Department of Mathematics, Izmir Democracy University, Izmir, 35140, Turkey Department of Mathematics, Morgan State University, Baltimore, MD 21251, USA Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation b c

article

info

Article history: Received 21 December 2018 Accepted 19 February 2019 Communicated by V.D. Radulescu MSC: primary 35Q55 35R02 secondary 49J40 58E30

abstract The paper deals with nonlinear Schrödinger equations on infinite metric graphs. We assume that the linear potential is infinitely growing. We prove an existence and multiplicity result that covers both self-focusing and defocusing cases. Furthermore, under some additional assumptions we show that solutions obtained bifurcate from trivial ones. We prove that these solutions are superexponentially localized. Our approach is variational and based on generalized Nehari manifold. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Metric graph Nonlinear Schrödinger equation Generalized Nehari manifold Bifurcation Exponential localization

1. Introduction Differential equations on metric graphs, or networks, form a new research area that demonstrates a great progress in last decades. In particular, the spectral theory of Schr¨odinger operators on such graphs received a substantial attention in mathematics and physics literature in which such objects are called quantum graphs. Any survey of the field is not in the scope of the present paper, and we refer to [6,9,10,17,18,22,23,28,29] and references therein. Also we mention that nonlinear equations on combinatorial graphs are studied in recent years (see, e.g., [13–15,20,33] and references therein). In the present paper we deal with a class of nonlinear Schr¨odinger equations (NLS) on metric graphs. Let us start with time dependent NLS iψt = −∆ψ + V (x)ψ − g(x)|ψ|

p−2

ψ,

p > 2,

∗ Corresponding author at: Department of Mathematics, Morgan State University, Baltimore, MD 21251, USA. E-mail addresses: [email protected] (S. Akduman), [email protected] (A. Pankov).

https://doi.org/10.1016/j.na.2019.02.020 0362-546X/© 2019 Elsevier Ltd. All rights reserved.

(1.1)

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on edges of a metric graph Γ subject to Kirchhoff vertex conditions (see below for details). Power nonlinearity in (1.1) is chosen for the sake of simplicity. Looking for standing wave solutions of the form ψ(t, x) = u(x) exp(−iλt), we arrive at the following stationary NLS − ∆u + V (x)u − λu = g(x)|u|

p−2

u

(1.2)

for the (real valued) profile function u(x) that satisfies the same vertex conditions. In general, there are two ways to construct NLS standing waves. The first one consists of finding profile functions u(x) with prescribed L2 norm and unknown frequency λ. More precisely, such a function u is a critical point (for example, minimizers) of the functional ∫ ( ∫ ) 1 1 p ′2 2 J0 (u) = u + V (x)u dx − g(x)|u| dx 2 Γ p Γ restricted to the mass constraint ∥u∥2L2 = µ > 0. In this approach the frequency λ appears as the Lagrange multiplier. In the context of NLS on noncompact metric graphs this approach is used to study standing solitary waves in the case when V = 0, g = 1 (self-focusing) and 2 < p < 6 (subcriticality) [1–5] (see also [11] for nonzero linear potentials). In these papers topological properties of Γ come into play, and a number of existence, nonexistence and multiplicity results are obtained. A detailed analysis of such solutions in the case of (compact) dumbbell graph is given in [21]. Another approach is based on solving Eq. (1.2) with prescribed frequency λ. In [24,27], perturbation analysis is used to find a solution of (1.2), still in the case of zero linear and constant positive (self-focusing) nonlinear potentials. In these papers the tadpole and a special periodic graphs are considered, respectively. In both cases the existence of standing waves is obtained provided that λ < 0 is close to zero. On the other hand, Eq. (1.2) with given λ can be studied by means of critical point theory applied to the functional ∫ λ u2 dx. Jλ (u) = J0 (u) − 2 Γ This approach is more flexible and applies to sufficiently general nonconstant linear and nonlinear potentials, as well as to the defocusing case (negative nonlinear potential). To the best of our knowledge, [26] is the first paper that employs this approach in the case of NLS on metric graphs. In that paper NLS with periodic linear and nonlinear potentials on a general periodic metric graph is considered. The main assumption is that λ does not belong to the linear spectrum in the self-focusing case, or belongs to a finite spectral gap in the defocusing case. The existence of both solitary waves and periodic waves with arbitrarily large periods is proven. Furthermore, periodic waves converge to solitary ones in the large wave length limit. The techniques make use of generalized Nehari manifold in its original form introduced in [25]. Also we mention paper [19] in which the classical Nehari manifold is used to study self-focusing NLS with constant potentials on star graphs. In the present paper we consider NLS on sufficiently general metric graphs assuming that the linear potential is infinitely growing in certain integral sense, while the nonlinear potential is bounded and either everywhere self-focusing or everywhere defocusing. In particular, the linear spectrum is discrete. Making use of the version of generalized Nehari manifold introduced in [31,32], we prove the existence and multiplicity of solitary waves for all λ in self-focusing case and for all λ above the bottom of linear spectrum in the defocusing case. Furthermore, under some additional assumptions we show that if λ is close to an eigenvalue of the linear part from an appropriate side, depending on whether the problem is self-focusing or defocusing, then the solution obtained bifurcates from the trivial solution, and provides an estimate of its norm. In addition, we prove that the solutions obtained decay superexponentially fast at infinity.

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The paper is organized as follows. Section 2 contains a necessary background on metric graphs. In Section 3 we provide the statement of problem and its variational setting. Generalized Nehari manifold is discussed in Section 4. Section 5 contains the proof of main existence result. Section 6 is devoted to bifurcation of solutions. Finally, in Section 7 we prove that the solutions are superexponentially localized. 2. Metric graphs We consider a graph Γ = (E, V) with countably infinite sets of edges E and vertices V. In our study, the loops and multiple edges are allowed, and it is assumed that the graph is connected in the sense that any two vertices are terminal vertices of a path of edges. Recall that the degree deg(v) of a vertex v ∈ V is the number of edges emanating from v. We assume that all vertices of Γ have finite degrees which are positive due to the connectedness of Γ . For any vertex v ∈ V we denote by Ev the set of edges adjacent to v. By definition, an interval in Γ is a subinterval of any edge. The graph Γ is called a metric graph if each edge e is identified with an interval [0, le ] of real line. We always assume that there exist two positive constants l and l such that l ≤ le ≤ l

(2.1)

for all e ∈ E. For each edge e ∈ E, xe denotes the induced coordinate of e. We often skip the index e in this notation. We use the same symbol x for a point on Γ . Notice that the first inequality in (2.1) is an essential assumption. On the other hand, the second bound in (2.1) is imposed for the sake of convenience. Actually, if Γ contains edges of arbitrarily large length, or even edges that extend to infinity, then, adding new vertices of degree 2, one can split any such edge into a finite or infinite number of edges with uniformly bounded lengths. For any two points x and y in Γ , the distance d(x, y) is defined as the length of a shortest path that connects these points. Since the graph is connected, the distance is well-defined. Fixing an arbitrary vertex o ∈ V considered as an origin, let us set d(x) = d(x, o) . (2.2) By B(r), r > 0, we denote the ball {x ∈ Γ : d(x) ≤ r}. Parametrizations of edges induce a natural measure, dx, on Γ which coincides with the Lebesgue measure on each edge. By |S| we denote the measure of a set S ⊂ Γ . We say that the graph has at most exponential growth if there exist positive constants a and C such that |B(r)| ≤ C exp(ar) (2.3) for all r > 0. For the sake of completeness recall that the graph is of subexponential growth if (2.3) holds for all a > 0 with C = Ca > 0. Graphs of power growth are defined in a similar way. Graphs with at most exponential, but not subexponential growth, are of exponential growth. The following are typical examples. Example 2.1. Vertices of periodic hypercubic graph in RN form the subgroup ZN ⊂ RN . Every vertex is connected to each of its 2N nearest neighbors by an edge of length 1. This graph has power (hence, subexponential) growth. Example 2.2. Let Γ be a regular tree such that all vertices have the same degree d ≥ 3, and all edges are of the length 1. This is an example of metric graph with exponential growth. The notion of growth of graph will be needed in Sections 6 and 7. We utilize the standard notation Lp (Γ ), 1 ≤ p ≤ ∞, for Lebesgue spaces on Γ with respect to the measure dx. The norm in a Banach space E is denoted by ∥ · ∥E .

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Now let us introduce basic functional spaces on metric graphs. We denote by Lploc (Γ ), 1 ≤ p ≤ ∞, the space of all measurable functions f on Γ such that f |e ∈ Lp (e) for all e ∈ E. The Sobolev space H 1 (Γ ) consists of all continuous functions f on Γ such that f |e ∈ H 1 (e) for all edges e ∈ E and ∑ ∥f ∥2H 1 = ∥f ∥2H 1 (e) < ∞. e∈E

There is a continuous, dense embedding H (Γ ) ⊂ Lp (Γ ) if p ≥ 2. Finally we introduce the space BS(Γ ) of Stepanov bounded functions (known also under the name uniform L1 space [30]) that consists of all functions f ∈ L1loc (Γ ) such that 1

∥f ∥BS = sup ∥f ∥L1 (e) < ∞. e∈E

3. Statement of problem Non-linear Schr¨ odinger (NLS) equation is − u′′ (x) + V (x)u(x) − λu(x) = κf (x, u(x)),

x ∈ Γ,

(3.1)

together with the Kirchhoff vertex conditions, where κ = +1 or κ = −1 (two cases). Here λ is a real constant. Throughout the paper we impose the following assumptions. (V1 ) V ∈ L1loc (Γ ), V − ∈ BS(Γ ), where V − (x) = min[0, V (x)] (V2 ) For every α ∈ (0, l) ∫ V (x)dx → ∞ S

as the interval S of length α escapes to infinity, i.e., for every M > 0 there exists rM > 0 such that ∫ V (x)dx ≥ M S

for all intervals S of length α such that S ⊂ Γ \ B(rM ). (f1 ) The function f (x, s) satisfies the Caratheodory condition, i.e. for every s ∈ R the function f (x, s) is measurable with respect to x ∈ Γ , and f (x, s) is continuous in s ∈ R for almost all x ∈ Γ . (f2 ) For almost all x ∈ Γ , |f (x, s)| ≤ µ(R)|s| (3.2) whenever |s| ≤ R, where µ(R) is non-decreasing, µ(R) > 0 if R > 0 and µ(R) → 0 as R → 0. Obviously, µ(0) = 0. (f3 ) The function f (x, s)/|s| of u ∈ R, extended to s = 0 by 0, is strictly increasing. (f4 ) F (x, s)/s2 → ∞ as |s| → ∞ for almost all x ∈ Γ , where ∫ s F (x, s) = f (x, τ )dτ. 0

Example 3.1.

The power nonlinearity p−2

f (x, s) = g(x)|s|

s,

where p > 2, g ∈ L∞ (Γ ) and g(x) ≥ α> 0, satisfies Assumptions (f1 )–(f4 ). From the point of view of applications, this is the most important nonlinearity.

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Remark 3.1. There is a simple sufficient condition for the validity of the assumptions on the potential V listed above. Assume that V ∈ L∞ loc (Γ ) and V (x) → ∞ as x → ∞ in the sense that lim ess inf{V (x) : d(x) ≥ r} = +∞.

r→∞

Then both (V1 ) and (V2 ) are satisfied. An example of such potential is [d(x)]α , α > 0. Under Kirchhoff conditions, u is a solution of (3.1) if u|e is differentiable, (u|e )′ is absolutely continuous, (3.1) holds a.e. on Γ and Kirchhoff conditions are satisfied. We consider the Schr¨ odinger operator L associated to the differential expression L=−

d2 + V (x) dx2

together with certain vertex conditions. The domain D(L) of L consists of all u ∈ L2 (Γ ) such that u and u′ are absolutely continuous on each edge of Γ (hence, u′′ ∈ L1loc (Γ )), u is continuous at all vertices of Γ ,

(3.3)

∑ du (v) = 0 dne

(3.4)

e∈Ev

for all vertices v ∈ V , where dnd e stands for the outward derivatives at the endpoints of the edge e, and Lu ∈ L2 (Γ ). Then the action of L is defined by Lu = Lu for all u ∈ D(L). Assumption (V1 ) guarantees that L is a densely defined, self-adjoint operator in L2 (Γ ) [7]. Furthermore, L is bounded below and D(L) ⊂ H 1 (Γ ). Notice that conditions (3.3) and (3.4) are called Kirchhoff vertex conditions. Alternatively, the operator L can be defined in terms of quadratic forms, as in [7]. Consider the quadratic form ∫ q(u) = (u′ (x)2 + V (x)u2 (x))dx Γ

on L2 (Γ ) with the form domain { D(q) =

1

∫

u ∈ H (Γ ) :

} |V (x)||u(x)| dx < ∞ . 2

Γ

Then q is a closed, bounded below quadratic form and generates a self-adjoint operator that coincides with L (see [7]). Also we consider the associated bilinear form ∫ q(u, v) = (u′ (x)v ′ (x) + V (x)u(x)v(x))dx Γ

so that q(u) = q(u, u). Notice that q(u, v) = (Lu, v)L2 whenever u ∈ D(L) and v ∈ L2 (Γ ). Without loss of generality, we assume that the bottom of the spectrum σ(L) of L is equal to 1 and, hence, q(u) ≥ ∥u∥2L2 (Γ )

(3.5)

for all u ∈ D(q). Endowed with the inner product q(·, ·), the form domain D(q) becomes a Hilbert space denoted by E. The norm in E is denoted by ∥ · ∥. Note that E ⊂ H 1 (Γ ) continuously and densely. It is convenient to introduce the following bilinear and quadratic forms qλ (u, v) = q(u, v) − λ(u, v)L2 and qλ (u) = qλ (u, u) = ∥u∥2 − λ∥u∥2L2 , respectively defined on the space E, where λ ∈ R.

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As shown in [7], (V2 ) is a necessary and sufficient condition for the discreteness of σ(L), as well as for the compactness of embedding E ⊂ L2 (Γ ). This is the so-called Molchanov’s type criterion. As consequence, σ(L) consists of distinct eigenvalues 1 = λ1 < λ2 < · · · < λn < · · · → ∞, all of finite multiplicity. Let Ek ⊂ E be the eigensubspace of L that corresponds to the eigenvalue λk , k ∈ N. These are finite dimensional subspaces. Furthermore, Ek and Ej are mutually orthogonal with respect to both inner products q(u, v) and (u, v)L2 whenever j ̸= k. We associate with problem (3.1), (3.3), (3.4) the energy functional Jλ (u) =

1 qλ (u) − κΨ (u) 2

where

(3.6)

∫ Ψ (u) =

F (x, u)dx. Γ

The functional Jλ is well defined on the space E and is of class C 1 . Its derivative Jλ′ (u), u ∈ E, as a linear functional on E is given by ∫ Jλ′ (u)h = qλ (u, h) − κ f (x, u)hdx , h ∈ E. Γ

Since the quadratic part of Jλ is bounded and, hence, is of class C 1 on E, this follows from Lemma 3.1. Lemma 3.1. Under Assumptions (f1 ) and (f2 ), the functional Ψ on E belongs to the class C 1 , is weakly continuous, its derivative is given by ∫ ′ Ψ (u)h = f (x, u)hdx , h ∈ E , (3.7) Γ

and the mapping Ψ ′ : E → E ′ , where E ′ is the dual space to E, is completely continuous. Proof . Let BR ⊂ E be the open ball of radius R > 0 centered at the origin. Since the embeddings E ⊂ H 1 (Γ ) ⊂ L∞ (Γ ) are continuous, there exists R1 = R1 (R) such that ∥u∥L∞ < R1 for all u ∈ BR . It is easily seen that there exists a Carath´eodory function f˜R (x, u) on Γ × R such that f˜R (x, u) = f (x, u) if |u| ≤ R1 and coincides with a linear function when |u| ≥ R1 . Consider the functional ∫ ˜ R (u) = Ψ F˜R (x, u)dx, Γ

where F˜R (x, u) =

∫

u

f˜R (x, s)ds.

0

˜ R is continuously By classical results on the Nemytski operators (see, e.g., [12, Section 3.2]), the functional Ψ 2 2 differentiable on L (Γ ). Due to the continuity of embedding E ⊂ L (Γ ), the same holds on E. Furthermore, ˜ ′ is given by the equation of the form (3.7), with f replaced by fR . The compactness of embedding Ψ R ˜ R and complete continuity of Ψ ˜ ′ . Since Ψ = Ψ ˜ R on BR and R E ⊂ L2 (Γ ) implies the weak continuity of Ψ R is an arbitrary positive number, the proof is complete. □ Remark 3.2. The operator Ψ ′ is the composition of the Nemytski operator u ↦→ f (x, u) as an operator from E into L2 (Γ ) and the continuous embedding L2 (Γ ) ⊂ E ′ .

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It is not difficult to verify that critical points of the functional Jλ are solutions of problem (3.1), (3.3), (3.4) in the space E. Indeed, testing equation Jλ′ (u)h = 0 with a smooth, compactly supported functions h on an arbitrary edge e ∈ E and integrating by parts, we see that Eq. (3.1) holds true on e. Condition (3.3) is trivial because u ∈ E ⊂ H 1 (Γ ). To verify (3.4) at a vertex v ∈ V, it is enough to take as h a smooth function that vanishes outside the ball of radius, say, l/2 centered at v and is equal to 1 in a concentric ball of radius l/4, and make use of integration by parts. Eq. (3.1) complemented with Kirchhoff vertex conditions (3.3) and (3.4) makes sense on finite, i.e., compact graphs as well. Assume that V ∈ L∞ (Γ ). Then the linear Schr¨odinger operator L possesses properties stated above in the case of infinite graph, and potential satisfying (V1 ) and (V2 ). In particular, the spectrum of L is discrete. The conclusions of Theorems 5.1 and 6.1 hold true in this case under the same assumptions on the nonlinearity. Notice that all eigenfunctions of L belong to Lr (Γ ) for all r ∈ [1, ∞]. 4. Generalized Nehari manifold In this section E stands for an abstract Hilbert space. Let Q(u, v) be a bounded, symmetric bilinear form on E and let Q(u) = Q(u, u) be the associated quadratic form. We assume that the space E splits as E = E− ⊕ E0 ⊕ E+, where the subspaces E − , E 0 and E + ̸= {0} are mutually orthogonal with respect to the bilinear form Q, the form Q is negative definite on E − , positive definite on E + and Q = 0 on E 0 . Then for every u ∈ E we have a unique decomposition u = u− + u0 + u+ , where u− ∈ E − , u0 ∈ E 0 and u+ ∈ E + . Notice that Q(u, v) = Q(u+ , v + ) + Q(u− , v − ) for all u, v ∈ E. Also we set F = E − ⊕ E 0 , E(u) = Ru ⊕ F = Ru+ ⊕ F and ˆ E(u) = {tu + v : t ≥ 0, v ∈ F } = {tu+ + v : t ≥ 0, v ∈ F }. On the space E we consider a functional J of the form J(u) =

1 Q(u) − Φ(u), 2

where Φ is a C 1 functional on E such that Φ(0) = 0. The derivative of J at u is given by J ′ (u)v = Q(u, v) − Φ ′ (u)v for all v ∈ E. In this section we make the following assumptions: (i) The functional Φ is weakly lower semicontinuous and 1 ′ Φ (u)u > Φ(u) > 0 2 for all u ̸= 0.

(4.1)

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ˆ (ii) For each w ∈ E \ F the functional Φ|E(w) has a unique nonzero critical point m(w) ∈ E(w) which is the ˆ global maximum point. The generalized Nehari manifold of the functional J is defined by N = N (J) = {u ∈ E \ F : J ′ (u)u = 0 and J ′ (u)v = 0 for all v ∈ F }. If u ̸= 0 is a critical point of J, then for the critical value we have 1 1 J(u) = J(u) − J ′ (u)u = Φ ′ (u)u − Φ(u) > 0. 2 2 On the other hand, J(u) ≤ 0 for all u ∈ F . Hence, N contains all nonzero critical points. Furthermore, if ˆ w ∈ E \ F , then N ∩ E(w) consists of exactly one point m(w). Actually, N = {m(w) : w ∈ E \ F }. Remark 4.1. We assume that the subspace E + is nonzero because otherwise Assumption (i) implies that the functional does not possess nontrivial critical points. The following result is borrowed from [32] (see Theorem 35). Notice that in [32] it is assumed implicitly that the subspace E + is infinite dimensional. The case of finite dimension is easily covered by the proof given there. Theorem 4.1.

In addition to Assumptions (i) and (ii), we suppose that

(iii) Φ ′ (u) = o(∥u∥) as u → 0; (iv) Φ(tu)/t2 → ∞ as s → ∞ uniformly on weakly compact subsets of E \ {0}; (v) Φ ′ is completely continuous. Then c = inf J(u) > 0 u∈N

is a nontrivial critical point of J. Furthermore, if the functional Φ is even, then J has at least dim E + pairs of nontrivial critical points. Remark 4.2.

The critical value c has the following minimax characterization c=

inf

max J(u) =

ˆ w∈E\F u∈E(w)

inf

max J(u).

ˆ w∈E + ,∥w∥=1 u∈E(w)

The critical value c and corresponding critical points are called ground level and ground critical points of the functional, respectively. 5. Existence and multiplicity In this section we apply Theorem 4.1 to problem (3.1), (3.3), (3.4). Theorem 5.1.

Assume that the potential and nonlinearity satisfy Assumptions (V1 ), (V2 ) and (f1 )–(f4 ).

(a) Let κ = +1. Then problem (3.1), (3.3), (3.4) has a nontrivial solution u ∈ E. If, in addition, f (x, s) is odd with respect to s, then there exist infinitely many pairs of nontrivial solutions. (b) Let κ = −1. If λ > λ1 , then there exists a nontrivial solution in E. If, in addition, the nonlinearity is odd and λ > λn , then the problem has at least N pairs of nontrivial solutions, where n ⨁ N = dim Ek . k=1

(c) Let κ = −1. If λ ≤ λ1 , then the problem has no nontrivial solution in E.

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Proof . We apply Theorem 4.1 to the functional κJλ (u) =

κ qλ (u) − Ψ (u). 2

The first part of Assumption (i) and Assumption (v) follow from Lemma 3.1. Since f (x, s) > 0 (respectively, f (x, s) < 0) if s > 0 (respectively, s < 0), we see that F (x, s) > 0 for all s ̸= 0. Making use of Assumption (f3 ), we easily obtain the following estimate ∫ s 1 −1 |τ ||s| f (x, s)dτ = f (x, s)s, F (x, s) < 2 0 which implies (4.1). Assumption (f2 ), and the embeddings E ⊂ L∞ (Γ ) and E ⊂ L2 (Γ ) imply that if u ∈ E, then ∥f (x, u)∥2L2 ≤ µ2 (R)∥u∥2L2 ≤ Cµ2 (R)∥u∥2 , where R = ∥u∥L∞ ≤ C1 ∥u∥. Since µ(R) → 0 as R → 0, we see, due to Remark 3.2, that Assumption (iii) holds true. To verify (iv) we assume the contrary. Then there exist a weakly compact set W ⊂ E \ {0}, a sequence un ∈ W and a sequence τn → ∞ such that τn−2 Ψ (τn un ) is bounded. Passing to a subsequence, we can assume that un → u ̸= 0 weakly in E and, by the compactness of embedding E ⊂ L2 (Γ ), strongly in L2 (Γ ). Hence, passing to a further subsequence, un (x) → u(x) a.e. on Γ . Since F ≥ 0, and |τn un (x)| → ∞ if u(x) ̸= 0, the Fatou Lemma yields ∫ Ψ (τn un ) F (x, τn un ) 2 = 2 u dx → ∞, τn 2 (τn un ) Γ which is a contradiction. Now we verify (ii). Given λ ∈ R, we denote by E − , E 0 and E + the (closed) subspaces generated by eigenvectors with eigenvalues < λ, = λ and > λ, respectively. The form qλ is positive (respectively, negative) definite on E + (respectively, on E − ), i.e., there exists β = β(λ) > 0 such that ± qλ (u) ≥ β∥u∥2 ,

u ∈ E± .

(5.1)

Notice that E − and E 0 are finite dimensional subspaces, while E + has infinite dimension. If λn < λ ≤ ⨁n λn+1 , then E − = k=1 Ek and N is the dimension of this space. These subspaces serve the functional κJλ , with κ = 1, and then F = E − ⊕ E 0 . If κ = −1, then E − and E + are positive and negative subspaces, respectively, for the form −qλ and, hence, we set F = E + ⊕ E 0 in this case. We consider the case κ = −1, the other one being similar and simpler. ˆ First we prove that E(w) ∩N = ̸ ∅ for all w ∈ E \ F = E \ (E 0 ⊕ E + ). Without loss we may assume that − ˆ w ∈ E and ∥w∥ = 1. We show that there exists R > 0 such that −Jλ (u) ≤ 0 for all u ∈ E(w) with ∥u∥ ≥ R. Indeed, assume the contrary. Then there exists a sequence un such that −Jλ (un ) ≥ 0 and ∥un ∥ → ∞. Let vn = ∥un ∥−1 un . Passing to a subsequence, we may assume that vn → v weakly in E. Then 0≤

qλ (vn− ) qλ (vn+ ) Ψ (∥un ∥vn ) −Jλ (un ) =− − − . 2 ∥un ∥ 2 2 ∥un ∥2

(5.2)

The first two terms in (5.2) are bounded. If v ̸= 0, then, by (iv), the third term tends to −∞, a contradiction. Hence, v = 0. As consequence, vn± → 0 and vn0 → 0 weakly. Then vn− → 0 and vn0 → 0 strongly because vn− and vn0 belong to finite dimensional subspaces Rw and E 0 , respectively. Since Ψ ≥ 0, inequalities (5.1) and (5.2) imply that ∥vn+ ∥ ≤ ∥vn− ∥ → 0. On the other hand, 1 = ∥vn ∥2 = ∥vn− ∥2 + ∥vn0 ∥2 + ∥vn+ ∥2 → 0, a contradiction.

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In particular, the boundedness of Ψ on bounded subsets of E implies that supu∈E(w) (−Jλ (u)) < ∞. ˆ 2 2 Since, by (iii), −Jλ (tw) = γt + o(t ) as t → 0, where γ = −qλ (w)/2 > 0, the supremum is positive. Let us ˆ show that −Jλ attains its (positive) maximum on E(w). For, we need to prove that −Jλ is upper weakly semicontinuous on E(w). Since Ψ is weakly continuous, it is enough to prove that the quadratic form qλ is weakly low semicontinuous on E(w). Let un = tn w + u0n + u+ n ∈ E(w) converges weakly in E. Then + + tn → t, u+ → u weakly, and q (u ) = q (t w) + q (u ). Since qλ |E(w) is a positive definite continuous λ n λ n λ n n quadratic form, it is a convex continuous function on E(w), hence, weakly low semicontinuous. This implies ˆ the required because qλ (tn w) → qλ (tw). Thus, E(w) ∩ N ̸= ∅. The uniqueness of global maximum of −Jλ |E(w) can be obtained exactly as in [32, Proposition 39] (see ˆ also [31]). The proof is complete. □ 6. Bifurcation In this section we consider the case when λ ̸∈ σ(L) and study the dependence of solution on λ as λ approaches one of the closest eigenvalues (depending on κ). We denote by δ(λ) the distance between λ and the spectrum of L. We start with the following simple lemma which is well known in the folklore and improves inequality (5.1). For convenience, we set λ0 = −∞. The subspaces E ± ⊂ E are introduced in the proof of Theorem 5.1. Note that in our case E 0 = {0}. We denote by P + and P − the orthogonal projectors in E onto the subspaces E + and E − , respectively. Obviously, P − = I − P + , where I stands for the identity operator. Notice that P + and P − extend to orthogonal projectors in L2 (Γ ) Lemma 6.1. Assume that (V1 ) and (V2 ) hold true. Let λ ∈ (λk−1 , λk ). Then qλ (u) ≥ and qλ (u) ≤

λk − λ ∥u∥2 , λk

λk−1 − λ ∥u∥2 , λk−1

u ∈ E+,

u ∈ E − (k > 1).

Proof . For every n ∈ N, let {en,j }j=1,...,mn be an L2 -orthonormal system of eigenfunctions of L with eigenvalue λn . If ∑ u= an,j en,j ∈ E + , n≥k,1≤j≤mn

then ∑

∥u∥2L2 =

(an,j )2 ,

n≥k,1≤j≤mn

while ∑

∥u∥2 =

λn (an,j )2 .

n≥k,1≤j≤mn 2

Since qλ = ∥u∥ −

λ∥u∥2L2 ,

we obtain easily that qλ (u) =

∑ n≥k,1≤j≤mn

The case of E − is similar.

(1 −

λk − λ λ )λn (an,j )2 ≥ ∥u∥2 . λn λk

□

In the following lemma we need an extra assumption on the graph Γ .

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Lemma 6.2. Assume that (V1 ) and (V2 ) hold, λ ∈ (λk−1 , λk ), with k > 1, and Γ has at most exponential growth. Then, for every p ≥ 2, the projectors P + and P − are bounded operators with respect to Lp norm. Proof . Since P + = I − P − , it is enough to consider the projector P − only. Recall that the subspace E − is finite dimensional. According to [8, Corollary 5.2], eigenfunctions of L belong to Lr (Γ ) for all r ∈ [1, ∞] and, hence, E − ⊂ Lr (Γ ) for all r ∈ [1, ∞]. Let {e1 , e2 , . . . , eN }, N = dim E − , be an L2 -orthonormal basis in E − . For any u ∈ E, we express P − u in the form P −u =

N ∑

ξj ej .

j=1

Since all norms on a finite dimensional space are equivalent, to prove the lemma it is enough to show that |ξj | ≤ C∥u∥Lp ,

j = 1, 2, . . . , N,

with a constant C > 0 independent of j and u. It is easily seen that ξj = (u, ej )L2

1, 2, . . . , N.

′

Since u ∈ E ⊂ Lp (Γ ) and ej ∈ Lp (Γ ), j = 1, 2, . . . , N , H¨older’s inequality yields the required. □ The next lemma provides estimates for norms of a critical point in terms of its critical value. Lemma 6.3. In addition to (V1 ), (V2 ) and (f1 )–(f4 ) assume that the nonlinearity satisfies the following Ambrosetti–Rabinowitz condition (AR) there exists a constant q > 2 such that 0 < qF (x, s) ≤ f (x, s)s ,

s ∈ R \ {0},

and p

F (x, s) ≥ a0 |s| ,

(6.1)

with p ≥ 2 and a0 > 0. Let λ ∈ (λk−1 , λk ), with k > 1 if κ = −1. Then there exists a constant C > 0 such that for any critical point u ∈ E of κJλ ∥u∥pLp ≤ CκJλ (u) . (6.2) Assume , in addition, that |f (x, s)| ≤ a1 |s|

p−1

,

(6.3)

where a1 > 0, and Γ has at most exponential growth if k > 1. Then δ(λ)∥u∥2 ≤ CκJλ (u) .

(6.4)

Proof . Let us consider the case κ = 1 only, the other one being similar. Making use of (AR) condition, we have that ∫ Jλ (u) ≥ (2−1 − q −1 ) f (x, u)udx . (6.5) Γ

Then (AR) and (6.1) imply (6.2). Now we prove (6.4). Assume that k > 1. Since 0 = Jλ′ (u)u+ = qλ (u+ ) −

∫ Γ

f (x, u)u+ dx,

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Lemma 6.1 implies that δ(λ) + 2 ∥u ∥ ≤ λk

(

λk − λ λk

)

λ − λk−1 λk−1

)

+ 2

∫

f (x, u)u+ dx .

∥u ∥ ≤ Γ

Similarly, δ(λ) − 2 ∥u ∥ ≤ λk

(

∫

− 2

∥u ∥ ≤ −

f (x, u)u− dx.

Γ

Adding these two inequalities and making use of (6.3), we obtain that ∫ ∫ δ(λ) ∥u∥2 ≤ |f (x, u)||u+ |dx + |f (x, u)||u− |dx λk Γ (∫ Γ ) ∫ p−1 + p−1 ≤ a1 |u| |u |dx + |u| |u− |dx . Γ

Γ

We estimate the integrals in the right-hand side by means of H¨older’s inequality with mutually conjugate exponents p′ and p, i.e., 1/p + 1/p′ = 1. We obtain that ) δ(λ) p/p′ ( ∥u∥2 ≤ a1 ∥u∥Lp ∥u+ ∥Lp + ∥u− ∥Lp . λk Since Γ has at most exponential growth, Lemma 6.2 implies that the projectors P + and P − are Lp bounded. Hence, ∥u+ ∥Lp + ∥u− ∥Lp ≤ C∥u∥Lp , and we obtain (6.4). The case k = 1 is simpler because the subspace E − becomes trivial, and Lemma 6.2 is not needed.

□

Theorem 6.1. Under the assumptions of Theorem 5.1, suppose that Γ has at most exponential growth, and λ ∈ (λk−1 , λk ) with k > 1. Furthermore, we assume that the nonlinearity satisfies condition (AR) and inequalities (6.1) and (6.3). Let uλ ∈ E be a ground critical point of the functional κJλ . Then there exists a constant C > 0 such that 1 ∥uλ ∥ ≤ C(λk − λ) p−2 if κ = 1 and λ is sufficiently close to λk , and 1

∥uλ ∥ ≤ C(λ − λk−1 ) p−2 if κ = −1 and λ is sufficiently close to λk−1 . Proof . Let κ = 1. We assume that λ is sufficiently close to λk and, in particular, δ(λ) = λk − λ. Let e ∈ D(L) be any eigenvector of L with eigenvalue λk such that ∥e∥L2 = 1. By Remark 4.2, the critical value c = cλ = Jλ (uλ ) can be estimated as follows ) ( 2 ∫ t qλ (e) qλ (w) c ≤ max Jλ (v) = max + − F (x, te + w)dx . ˆ 2 2 t>0,w∈E − v∈E(e) Γ Since qλ (e) = (λk − λ) and qλ (w) ≤ 0, making use of inequality (6.1), we obtain that ( ) (λk − λ)t2 c ≤ max − a0 ∥te + w∥pLp . 2 t>0,w∈E −

(6.6)

To estimate the right-hand side, consider the (continuous) projector P : te + w ↦→ te in the subspace E(e). Since E(e) is finite dimensional, all norms on it are equivalent and, hence, P is bounded with respect to Lp -norm and ∥te + w∥pLp ≥ b1 ∥e∥pLp tp .

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As consequence, ( c ≤ max t>0

) (λk − λ)t2 − b2 tp . 2

By means of elementary calculus we obtain that p

c ≤ C(λk − λ) p−2 ,

(6.7)

with a constant C > 0 that can be expressed in terms of p and b2 . If λ is sufficiently close to λk , then c = cλ = Jλ (uλ ) is sufficiently small. Combining inequalities (6.4) and (6.7), we obtain that p (λk − λ)∥uλ ∥2 ≤ Jλ (uλ ) = c ≤ C(λk − λ) p−2 . C Hence, 2

∥uλ ∥2 ≤ C 2 (λk − λ) p−2 , and we obtain the required. The case when κ = −1 is similar to the previous one. We assume that λ ∈ (λk−1 , λk ) is close to λk−1 so that δ(λ) = λ − λk−1 . The only difference is in the estimate for critical value c = cλ = −Jλ (uλ ) of the functional −Jλ . In this case we use the subspace E + instead of E − and choose as e an eigenfunction with ∥e∥L2 = 1 and eigenvalue λk−1 . Arguing as in the proof of (6.6), we obtain that ( ) (λ − λk−1 )t2 p c ≤ max − a0 ∥te + w∥Lp . 2 t>0,w∈E + In this case the projector P : te + w, where w ∈ E + , acts in the subspace E(e) = Re ⊕ E + which is infinite dimensional now. However, P is the restriction of P − to E(e). Hence, by Lemma 6.2, P is bounded with respect to Lp -norm, and we conclude exactly as in the case κ = 1. The proof is complete. □ Note that the proofs of inequality (6.2), and inequality (6.4) in the case when κ = 1 and k = 1, are independent of the exponential growth assumption on Γ . Also, the proof of estimate (6.7) does not use that assumption. As consequence, we obtain the following results. Corollary 6.1. Under the assumptions of Theorem 5.1, let κ = 1 and λ < λ1 be sufficiently close to λ1 . Assume that the nonlinearity satisfies condition (AR), and inequalities (6.1) and (6.3). Then the solution uλ satisfies 1

∥uλ ∥ ≤ C(λ1 − λ) p−2 , where C > 0 is independent of λ. Corollary 6.2. Under the assumptions of Theorem 5.1, let κ = 1 and λ ∈ (λk−1 , λk ), k > 1, be sufficiently close to λk . Assume that the nonlinearity satisfies condition (AR), and inequalities (6.1) and (6.3). Then the solution uλ satisfies 1

∥uλ ∥Lp ≤ C(λ1 − λ) p−2 , where C > 0 is independent of λ. Remark 6.1.

The results of this section apply obviously to the case of power nonlinearity.

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7. Localization of solutions Now we show that solutions obtained in Theorem 5.1 decay at infinity superexponentially fast. In fact, we have the following more general result. Theorem 7.1. Assume that Assumptions (V1 ), (V2 ) and (f2 ) hold true. If u ∈ L2 (Γ ) ∩ L∞ (Γ ) is a solution of (3.1) and satisfies vertex conditions (3.3) and (3.4), then u decays at infinity superexponentially fast, i.e., for every α > 0 there exists a constant Cα > 0 such that |u(x)| ≤ Cα exp(−αd(x)). Proof . Assumption (f2 ) implies easily that f (x, u) ∈ L2 (Γ ). Hence, −u′′ + V u ∈ L2 (Γ ). Since u satisfies the vertex conditions, we see that u ∈ D(L). Now we define the function V0 (x) on Γ as follows: V0 (x) = f (x, u(x))/u(x) if u(x) ̸= 0 at the point x ∈ Γ and V0 (x) = 0 otherwise. Due to Assumption (f2 ), V0 ∈ L∞ (Γ ). Hence, the multiplication operator by V0 is bounded in L2 . On the other hand, the resolvent of L is compact. Then it is easily seen that V0 is a relatively compact perturbation of the operator L. Therefore, the operator L0 = L + V0 , with the domain D(L0 ) = D(L) is self-adjoint, and its essential spectrum is empty (see, e.g., [16]), that is, σ(L0 ) is purely discrete. By the construction of L0 , u is an eigenfunction of L0 associated to the eigenvalue λ of finite multiplicity. Now the result follows from [8, Corollary 4.2]. □ Making use of [8, Corollary 5.2], we obtain the following result. Corollary 7.1. Under the assumptions of Theorem 7.1, suppose that Γ has at most exponential growth. If u ∈ L2 (Γ ) ∩ L∞ (Γ ) is a solution of (3.1) and satisfies vertex conditions (3.3) and (3.4), then u ∈ Lp (Γ ) for all p ∈ [1, ∞]. Acknowledgments The work of second author is supported by Simons Foundation USA, Award 410289, and by the RUDN University Program 5-100 Russian Federation. The authors thank anonymous referees for remarks and comments. References [1] R. Adami, C. Cacciapuoti, D. Finco, D. Noja, Constrained energy minimization and orbital stability for the NLS equation on a star graph, Ann. Inst. H. Poincar´ e, Anal. non Lin´ eaire 31 (2014) 1289–1310. [2] R. Adami, C. Cacciapuoti, D. Finco, D. Noja, Variational properties and orbital stability of standing waves for NLS equation on a star graph, J. Differential Equations 257 (2014) 3738–3777. [3] R. Adami, E. Serra, P. Tilli, NLS Ground states on graphs, Calc. Var. 54 (1) (2015) 743–761. [4] R. Adami, E. Serra, P. Tilli, Threshold phenomena and existence results for NLS ground states on metric graphs, J. Funct. Anal. 271 (2016) 201–223. [5] R. Adami, E. Serra, P. Tilli, Negative energy ground states for the L2 -critical NLSE on metric graphs, Comm. Math. Phys. 352 (2017) 387–406.

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