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BY 4.0 license Open Access Published by De Gruyter March 8, 2019

An Abstract Linking Theorem Applied to Indefinite Problems Via Spectral Properties

Liliane A. Maia EMAIL logo and Mayra Soares

Abstract

An abstract linking result for Cerami sequences is proved without the Cerami condition. It is applied directly in order to prove the existence of critical points for a class of indefinite problems in infinite-dimensional Hilbert Spaces. The applications are given to Schrödinger equations. Here spectral properties inherited by the potential features are exploited in order to establish a linking structure, and hence hypotheses of monotonicity on the nonlinearities are discarded.

1 Introduction

In this work, the groundbreaking paper [3] by V. Benci and P. H. Rabinowitz is revisited. The aim is to prove an abstract linking theorem for Cerami sequences [6], which will complement related works found in the literature and make possible to extend for many applications. Our interest in applications is twofold: On one hand, extending results for existence of solutions to nonlinear Schrödinger equations, elliptic systems or even Hamiltonian systems, with very general potentials which make the problems strongly indefinite. On the other hand working with nonlinear terms which do not satisfy any monotonicity condition such as those required to perform projections on the so called Nehari manifold as in, for instance, [20, 29]. In this purpose, spectral properties of self-adjoint operators are going to be exploited in order to get the geometry of a linking structure and then apply an abstract result to obtain a critical point to the functional associated to the nonlinear equation, namely a solution to the problem. Furthermore, a compactness structure given by Cerami sequences, (C)c sequences for short, is faced here since asymptotically linear problems at infinity are studied. Thus, inspired by the theory developed in [3], in this work a more general version of its main result [3, Theorem 1.29] is provided for (C)c sequences. To do so, a deformation lemma adapted for Cerami sequences is proved and then the abstract results obtained by V. Benci and P. Rabinowitz are extended.

Our main result, developed throughout this paper, is the following.

Theorem (Linking Theorem for Cerami Sequences).

Let E be a real Hilbert space with inner product (,), let E1 be a closed subspace of E and E2=E1. Let IC1(E,R) satisfy the following conditions:

  1. I ( u ) = 1 2 ( L u , u ) + B ( u ) for all u E , where u = u 1 + u 2 E 1 E 2 , Lu=L1u1+L2u2 and Li:EiEi, i=1,2, is a bounded linear self-adjoint mapping.

  2. B is weakly continuous and uniformly differentiable on bounded subsets of E.

  3. There exist Hilbert manifolds S , Q E such that Q is bounded and has boundary Q , and constants α > ω and v E 2 such that

    1. S v + E 1 and I α on S,

    2. I ω on Q ,

    3. S and Q link.

  4. If I ( u n ) , for a sequence ( u n ) , is bounded and ( 1 + u n ) I ( u n ) 0 as n + , then ( u n ) is bounded.

Then I possesses a critical value cα.

It is important to highlight that (I2) implies that B(u) maps weakly convergent to strongly convergent sequences, which gives a kind of partial compactness for I. Moreover, (I4) is a weakened version of the Cerami condition, (C)c condition for short, once the boundedness of any (C)c sequence will be enough to look for a nontrivial critical point without wondering whether it has a convergent subsequence (see the first paragraph of the proof of Theorem 2.4). Together, hypotheses (I2) and (I4) ensure the existence of a critical value c which can be characterized as a minimax level. Furthermore, hypotheses (I1) and (I3) produce a quite general linking geometry for the functional I so that both subspaces in the Hilbert decomposition are allowed to be infinite-dimensional. The conjunction of these hypotheses reproduces sufficient tools to obtain a nontrivial critical point for I in the desired applications scenario. In fact, our approach of constructing a linking structure by means of a sharp study on the spectral properties of the Schrödinger operator is a methodological novelty. This idea extremely enhances the current references since it lights up the fundamental relation between the asymptotic behavior of the nonlinear term and the spectrum features. Therefore, the purpose of revisiting the core of linking structures was precisely to understand this interaction and discard any unnecessary hypothesis.

The pioneering work in this direction is [1] by P. Bartolo, V. Benci and D. Fortunato, where a deformation lemma for Cerami sequences was developed assuming the (C)c condition, as a qualitative deformation lemma with the purpose of extending previous critical point results to non-super-quadratic problems. Thereafter, D. Costa and C. Magalhães [8] proved abstract linking results for strongly indefinite non-quadratic problems on bounded domains, making use of the deformation lemma introduced in [1] and proving that under their assumptions the associated functional satisfied the (C)c condition. Alternatively, as aforementioned, the same lines as in [3] are followed, and hence a deformation lemma without the (C)c condition is proved. Furthermore, our version of the linking theorem only requires the boundedness of Cerami sequences.

In the literature one also finds a paper by G. Li and C. Wang [18] which presented a similar argument, introducing a new kind of deformation lemma, without the (C)c condition, but subsequently used in a linking theorem under the (C)c condition. Moreover, in the abstract result they required that one of the subspaces in the linking decomposition is of finite dimension, while in our result both subspaces in the decomposition may be of infinite dimension. Their construction was inspired by ideas of M. Willem [31] for the quantitative deformation lemma. Although a kind of deformation lemma is developed, it is deeply different from theirs since nonstandard ideas in [3] are closely followed. In fact, the mapping η in deformation is in general determined by solving an appropriate initial value problem involving I(η). However, this is not suitable for our purposes because is necessary to construct an η satisfying special properties, which will be fundamental in attaining the critical minimax level.

It is also important to mention the theory developed by W. Kryszewski and A. Szulkin in [16], where they solved a more general class of superlinear problems, with assumptions of periodicity. Developing a new degree theory and a weaker topology, they generalized abstract linking theorems introduced in [3] also working with Palais–Smale sequences. Following the same idea, G. Li and A. Szulkin [17] extended the results in [16] obtaining a (C)c sequence for the asymptotically linear case. Nevertheless, so as to get a nontrivial solution, without the (C)c condition, these authors required extra assumptions, including a monotonicity condition on the nonlinearity in an auxiliary problem solved in [30], which had been treated by adapting techniques in [16, 14].

Posteriorly, T. Bartsch and Y. Ding [2] complemented the results in [16] considering both, Palais–Smale and Cerami sequences, in order to apply their abstract results to a Dirac equation where the nonlinearity could be asymptotically linear or superlinear at infinity. Similarly, in [11], Y. Ding and B. Ruf worked with an asymptotically linear problem with a Dirac operator, but without periodicity conditions. Their operator satisfies that the essential spectrum is (-a,a) and that the discrete spectrum intersects the interval (0,q0) for some positive q0. Then they could make use of discrete and positive eigenvalues in the linking structure and apply a particular case of the result in [2], so as to obtain a Cerami sequence. Under their assumptions they were able to prove that their functional satisfied the (C)c condition, which yielded their results.

It is worth highlighting that [11] adapted assumptions and arguments introduced in another very inspiring paper [10], where L. Jeanjean and Y. Ding worked with Hamiltonian Systems, looking for homoclinic orbits, without any periodicity condition. These authors also applied the abstract critical point theory developed in [2], and in order to recover the desired compactness they imposed hypotheses controlling the size of the nonlinearity with respect to the behavior of the potential at infinity. Thus, their assumptions yielded the linking geometry and provided the (C)c condition. In contrast to the argument presented by these authors, our approach does not require the guarantee of the (C)c condition, the necessary compactness to solve the problem is embedded inside the four conditions assumed in the abstract result.

Still referring to abstract results involving linking structure, it is well known that M. Schechter and W. Zou have developed many relevant papers in this spirit; see especially [24, 23, 26, 25] among other works by these authors. In our understanding, their results are different from ours in the sense that, roughly speaking, they usually work with weaker linking geometries in order to get either a Palais–Smale or a Cerami bounded sequence. Then they apply widely alternative arguments to find a solution to the proposed application. On the other side, our idea is to obtain a result which could ensure the existence of a nontrivial critical point directly, without stressing either on geometry or on compactness of the associated functional, separately. Notwithstanding, it is worth pointing out clever abstract results obtained in [25, Theorem 2.1] and [26, Theorem 2.1], where Schechter and Zou also made use of the “Monotonicity Trick” developed by L. Jeanjean in [14], for the purpose of getting critical points for a family of functionals, converging to a critical point the functional associated to the initial problem. These results have been applied to solve asymptotically linear problems with spectral properties similar to those presented in this paper; see, for instance, [7].

Here we present two applications for Schrödinger equations for our abstract result. Other applications can be found in [27], where M. Soares proves the existence of solutions to Hamiltonian and elliptic systems by applying this abstract result.

For our applications we consider problem (P),

- Δ u + V ( x ) u = g ( x , u ) in  N , N 3 ,

in the case where g(x,s)=h(x)f(s), and h satisfies

  1. h L ( N ) L q ( N ) , q=2*2*-p for some p(2,2*) and h>0 almost everywhere.

Furthermore, f is asymptotically linear satisfying the following conditions:

  1. f C ( , ) and lims0f(s)s=0.

  2. There exists a>0 such that lims+F(s)s2=a2, where F(s):=0sf(t)𝑑t, and F(s)0.

  3. Setting Q(s):=12f(s)s-F(s)>0 for all s/{0}, we have

    lim s + Q ( s ) = + .

Moreover, for the first application, we assume that V satisfies the following conditions:

  1. V C ( N , ) and lim|x|+V(x)=V>0.

  2. Setting A:=-Δ+V(x) as an operator of L2(N) and denoting by σ(A) the spectrum of A, we have

    sup [ σ ( A ) ( - , 0 ) ] = σ - < 0 < σ + = inf [ σ ( A ) ( 0 , + ) ] .

This application was inspired by [19], where L. Maia, J. Oliveira Junior and R. Ruviaro solved problem (P) with potential V satisfying (V1)–(V2). In addition, they required that 0σ(A) and assumed some more hypotheses of decay and compactness on V. About the nonlinearity they set h1, fC3(N,) with some growth hypotheses on its derivatives, and assumed f(s)/s being increasing as well. Since this kind of potential ensures that the subspace where A is negative definite is finite-dimensional, they could apply the aforementioned version of the linking theorem introduced by G. Li and C. Wang in [18] to get the (C)c sequence. They used the associated problem at infinity and a splitting lemma to compare the levels of both problems and get the necessary compactness. Trying to improve their result, our abstract linking theorem for (C)c sequences is applied and a nontrivial critical point is obtained straightway, avoiding such monotonicity assumptions on f.

Staring at our hypotheses, it is also possible to say that our second application complements the work by L. Jeanjean and K. Tanaka in [15]. In fact, they assumed V(x)α>0, and so they worked with Ekeland’s principle to get a (C)c sequence and due to the geometry of their functional they applied the Mountain Pass Theorem to get a critical point. They also worked with an asymptotically linear problem where f(s)/s is not necessarily increasing. In addition, they assumed h1 and f(s)/sa>infσ(A)>0 as |s|+. Differently, in our case V changes sign and infσ(A)<0, which implies a linking geometry and prevents us to use the same argument. However, by considering the positive spectrum, a similar hypothesis is assumed:

a > inf u 1 E 1 , u 0 N ( | u 1 ( x ) | 2 + V ( x ) u 1 2 ( x ) ) 𝑑 x N h ( x ) u 1 2 ( x ) 𝑑 x 1 h inf [ σ ( A ) ( 0 , + ) ] = σ + h > 0 ,

where hL(N):=h and E1 is the subspace of H1(N) on which the operator A is positive definite. This hypothesis allows to develop a linking structure.

On the other hand, the work [9] by D. Costa and H. Tehrani can be cited since the same V as theirs is presented here. More specifically, they required (V1), and our hypothesis (V2) implies that either 0 is an isolated point of σ(A) or it is in a gap of the spectrum, which is also required by them. However, they did not work with an asymptotically linear problem, but they assumed the well-known Ambrosetti–Rabinowitz condition and required a nondecreasing nonlinearity. In their assumptions, h=a is a sign-changing function in C1(N,) and such that lim|x|+a(x)=a<0, differently from lim|x|+h(x)=0 in our case. Moreover, instead of using an abstract linking theorem, they applied a method of approximations to solve their problem.

It is also worth mentioning the paper [12] by A. Edelson and C. Stuart since assumptions close to theirs are considered here. However, they required f(s)/s strictly increasing, which is removed here. Moreover, they applied the method of sub- and super-solution and bifurcation to get a solution to their problem.

Finally, for the second application we keep all assumptions on h and f, but on V we assume (V2) and replace assumption (V1) by the following:

  1. V C ( N , ) is (2π)N-periodic in xN.

This application is motivated by the fact that in virtue of (V1) the subspace in which the operator A is negative definite is finite-dimensional. Since this is irrelevant for applying our abstract result, we sought for a problem where both subspaces, in which the operator A is positive and negative definite, are infinite-dimensional. In fact, (V1) combined with (V2) ensures the desired. Although all other hypotheses are kept, this replacement changes completely the spectral properties of A, which are fundamental to determine the linking geometry. It would be interesting to note that we only require V being a periodic function in order to explore spectral properties, we do not need a periodic nonlinearity. Hence unlike most of the works in the literature (see [16, 2]), we do not make use of periodicity to translate a (C)c sequence and ensure the existence of a critical point.

2 The Notion of Linking and Some Definitions

In this section, the notion of link is presented and a new version of an abstract linking theorem is proved based on [3], however with (C)c sequences. Throughout this section, E always denotes a Hilbert space, E=E1E2 and if uE, writing u=u1+u2 with uiEi, i=1,2, then set Piu:=ui, where Pi:EEi is the projector on Ei, i=1,2. Furthermore, mappings h:[0,1]×EE will be denoted by ht(u), and the closed ball of E centered in zero with radius r will be denoted by Br. Furthermore, let τ=(BτE1)(BτE2).

Let Σ denote the class of mappings ΦC([0,1]×E,E) for which P2Φt(u)=u2-Wt(u), with Wt compact for t[0,1] and Φ0(u)=u. Let S and Q be Hilbert manifolds, Q having boundary Q. We say S and Q “link” if whenever ΦΣ and Φt(Q)S= for all t[0,1], then Φt(Q)S for all t[0,1].

Remark 2.1.

A geometric understanding of this definition is that S and Q link if every Hilbert manifold modeled on Q and sharing the same boundary intersects S (see [3]).

A useful example of linking sets is provided in [3] and stated below.

Lemma 2.2 (See [Lemma 1.3][3]).

Let eB1E1 and r1>ρ>0. If

S = B ρ E 1 𝑎𝑛𝑑 Q = { r e : r [ 0 , r 1 ] } ( B r 2 E 2 ) ,

then S and Q link.

First of all, some definitions and notations introduced in [3] are required.

Let B:E be a functional. We call B uniformly differentiable in bounded subsets of E if for any R,ε>0 there exists δ=δ(R,ε)>0, independent of u, such that

| B ( u + v ) - B ( u ) - B ( u ) v | ε v

for all u,u+vBR and vδ.

Lemma 2.3.

Let B:ER be a functional which is weakly continuous and uniformly differentiable in bounded subsets of E. Then B:EE is completely continuous.

The proof follows from elementary arguments.

Let Γ denote the set of mappings hC([0,1]×E,E) satisfying the following conditions:

  1. h t ( u ) = U t ( u ) + K t ( u ) , where U,KC([0,1]×E,E),Ut is a homeomorphism of E onto E and Kt is compact for each t[0,1].

  2. U 0 ( u ) = u , K0(u)=0.

  3. P i U t ( u ) = U t ( P i ( u ) ) , i=1,2.

  4. h t maps bounded sets to bounded sets.

In addition, for hΓ, let htj(u) denote the j-fold composite of h with itself, i.e. ht1(u)=ht(u), ht2(u)=ht(ht(u)) and htj(u)=ht(htj-1(u)) for j>1.

Now, it is convenient to state the Linking Theorem for Cerami sequences from Section 1.

Theorem 2.4 (Abstract Linking Theorem).

Let E be a real Hilbert space with inner product (,), let E1 be a closed subspace of E and let E2=E1. Let IC1(E,R) satisfy the following conditions:

  1. I ( u ) = 1 2 ( L u , u ) + B ( u ) for all u E , where u = u 1 + u 2 E 1 E 2 , Lu=L1u1+L2u2 and Li:EiEi, i=1,2, is a bounded linear self-adjoint mapping.

  2. B is weakly continuous and uniformly differentiable on bounded subsets of E.

  3. There exist Hilbert manifolds S , Q E such that Q is bounded and has boundary Q , and constants α > ω and v E 2 such that

    1. S v + E 1 and I α on S,

    2. I ω on Q ,

    3. S and Q link, that is, they satisfy the linking definition in [ 3 ].

  4. Set c by

    (2.1) c := inf h Λ sup u Q ¯ I ( h 1 ( u ) ) ,

    where Q ¯ is the closure of Q,

    Λ = { h C ( [ 0 , 1 ] × E , E ) : h = h ( 1 ) h ( m ) , h ( 1 ) , , h ( m ) Γ , m , h t ( Q ) I α - ω 2 - β , β ( 0 , α - ω 2 ) } ,

    and I λ = { u E : I ( u ) λ } for all λ . If for a sequence ( u n ) there exists a constant b > 0 such that ( u n ) I - 1 ( [ c - b , c + b ] ) and ( 1 + u n ) I ( u n ) 0 as n + , then ( u n ) is bounded.

Then cα, and c is a critical value of I.

Let IC1(E,) be a functional. A sequence (un)E is said to be a Cerami sequence, or a (C) sequence for short, if it satisfies

sup n | I ( u n ) | < and I ( u n ) ( 1 + u n ) 0

as n+. Given c, then (un)E is said to be a Cerami sequence on level c, or a (C)c sequence for short, if it satisfies

I ( u n ) c and I ( u n ) ( 1 + u n ) 0

as n+.

3 A Quantitative Deformation Lemma and Proof of the Main Result

Inspired by [3], we find it suitable to state a variant of the standard quantitative deformation lemma for Cerami sequences, without Cerami condition, which is necessary to prove Theorem 2.4.

Lemma 3.1 (Deformation Lemma).

Let IC1(E,R) satisfying (I1)(I2) as in Theorem 2.4. Then for any RN, ϱ>0 and ε(0,110), if s:=(R+2)2, there exist kN and ηΓ, such that the following assertions hold:

  1. I ( η t k s ( u ) ) I ( u ) + ϱ for all u R + 2 and t [ 0 , 1 ] .

  2. If c and I ( w ) ( 1 + w ) 2 ε for all w R + 1 I - 1 ( [ c - ε , c + ε ] ) , then I ( η k s ( u ) ) c - ε 2 whenever u R / 2 I - 1 ( [ c - ε , c + ε ] ) .

Remark 3.2.

The mapping η is usually determined by solving an appropriate differential equation involving I(η). Such an approach seems to fail here since it does not give an η satisfying (Γ1)–(Γ3) which are crucial for the purpose of proving Theorem 2.4. Hence, in order to prove Lemma 3.1 it is necessary to argue similarly to the proof of [3, Theorem 1.5].

Proof.

First, choose χC(,) such that

χ ( t ) = 1 for  s R + 1 ,
χ ( t ) = 0 for  s R + 2 ,
χ ( t ) < 0 for  t ( R + 1 , R + 2 ) ,
χ ( t ) ( R + 2 - t ) 2 for  t [ R + 3 2 , R + 2 ] .

For u=u1+u2E1E2, set Vi(u):=χ(ui)PiI(u), i=1,2, and V(u):=V1(u)+V2(u). Now note that V(u)I(u) in R+1. Moreover, by (I1)–(I2), there is a constant M=M(R) such that I(u)M for uR+2. With this, set

(3.1) ε ¯ := 1 M s min ( ϱ , ε 2 ) .

Since I is uniformly differentiable on bounded sets, there is a δ=δ(ε¯,R)>0 such that

(3.2) | I ( u + v ) - I ( u ) - I ( u ) v | ε ¯ v

for u,u+vR+2 and vδ. Assume δ1 and choose k such that

(3.3) 1 k < min ( δ 2 M , 1 8 ( R + 2 ) ( 1 + max | χ ( s ) | ( L 1 + L 2 ) ) .

Now define ηt(u):=u-tkV(u).

Claim.

We have ηΓ.

By assuming this claim and postponing its proof, it is necessary to check that R+2 is an invariant set for ηt for the purpose of proving (i)–(ii). In fact, for u=u1+u2R+2, by the definitions of η and V it follows that

(3.4) P i η t ( u ) - u i = ( u i - t k V i ( u ) ) - u i = t k V i ( u ) M k χ ( u i ) .

Note that, for uiBR+3/2, the choice of χ implies that

M k χ ( u i ) 1 2 R + 2 - u i

via (3.3), while for uiR+32 it follows that

1 2 R + 2 - u i ( R + 2 - u i ) 2 M k χ ( u i )

by the choice of χ and k. Hence, the right-hand side of (3.4) does not exceed R+2-ui, i=1,2, which implies that R+2 is invariant for ηt. Indeed, from (3.4) and the triangular inequality, this yields

(3.5) P i η t ( u ) M k χ ( u i ) + u i R + 2 .

Since dist(ηt(u),R+2), the distance from ηt(u) to R+2 satisfies

dist ( η t ( u ) , R + 2 ) = min i = 1 , 2 ( R + 2 - P i ( η t ( u ) ) ) .

Thus (3.5) implies that ηt(u)R+2.

In order to prove (i), observe that by (3.3) and the definitions of η, M and k it follows that

(3.6) η t ( u ) - u = - t k V ( u ) = t k V ( u ) M k < δ

for all uR+2. Hence, fixing uR+2 and using (3.2) with v=-tkV(u), we obtain

(3.7) I ( u + v ) = I ( η t ( u ) ) I ( u ) - t k I ( u ) V ( u ) + ε ¯ t k V ( u ) .

Since E2=E1, we obtain P1(I(u))P2(I(u)), and by the definition of V it follows that

(3.8) I ( u ) V ( u ) = χ ( u 1 ) P 1 ( I ( u ) ) 2 + χ ( u 2 ) P 2 ( I ( u ) ) 2 0 .

Thus, by (3.1), (3.6)–(3.8) and the definition of ε¯ it follows that

(3.9) I ( η t ( u ) ) I ( u ) + ε ¯ M k I ( u ) + ϱ M s M k = I ( u ) + ϱ k s .

Provided that R+2 is invariant under ηt, (i) holds by iterating (3.9) ks times. In fact, iterating twice means using ηt(u) instead of u in (3.9), and after that using again (3.9), but for u it yields

I ( η t 2 ( u ) ) = I ( η t ( η t ( u ) ) ) I ( η t ( u ) ) + ϱ k s I ( u ) + 2 ϱ k s .

Then, after ks iterations, it yields

I ( η t k s ( u ) ) I ( u ) + k s ϱ k s I ( u ) + ϱ ,

and (i) is proved.

In order to prove (ii), take uR/2I-1([c-ε,c+ε]). Three cases are considered.

Case I: η1j(u)BR+1I-1([c-ε,c+ε]) for 1jks. By definition, V(u)=I(u) in R+1. Then, by fixing j, 1jks, the definition of η1 yields

η 1 j ( u ) - η 1 j - 1 ( u ) = - 1 k V ( η 1 j - 1 ( u ) ) = - 1 k I ( η 1 j - 1 ( u ) ) .

Then, by using (3.7) for η1j(u) and η1j-1(u) instead of u+v and u, by the definition of M, and due to (3.1), it yields

(3.10) I ( η 1 j ( u ) ) - I ( η 1 j - 1 ( u ) ) - 1 k I ( η 1 j - 1 ( u ) ) 2 + 1 k ε ¯ M - 1 k I ( η 1 j - 1 ( u ) ) 2 + ε 2 k s .

Then, by the telescopic sum, and using (3.10) for all 1jks, it follows that

(3.11) I ( η 1 k s ( u ) ) - I ( u ) = j = 1 k s [ I ( η 1 j ( u ) ) - I ( η 1 j - 1 ( u ) ) ] j = 1 k s [ - 1 k I ( η 1 j - 1 ( u ) ) 2 + ε 2 k s ] .

Provided that

( R + 2 ) I ( u ) ( 1 + u ) I ( u ) 2 ε

from the assumption, by setting εs:=εs, it follows that I(u)22εs for all uR+1. Thus, (3.11) yields

(3.12) I ( η 1 k s ( u ) ) - I ( u ) j = 1 k s [ - 2 ε s k + ε s 2 k ] = - 3 ε 2 .

Therefore, since I(u)c+ε, inequality (3.12) implies that I(η1ks(u))I(u)-3ε2c-ε2, and the result holds for the first case.

Case II: η1j(u)BR+1I-1([c-ε,c+ε]) for 1jm-1 but η1m(u)I-1([c-ε,c+ε]) for some 1mks. From (3.10) and since I(u)22εs for all uR+1, it follows that

I ( η 1 j ( u ) ) - I ( η 1 j - 1 ( u ) ) - 2 ε s k + ε s 2 k = - 3 ε s 2 k ,

and hence I(η1j(u))<I(η1j-1(u)) for 1jm. Then η1m(u)I-1([c-ε,c+ε]) and I(η1m(u))<I(η1m-1(u)) imply that I(η1m(u))<c-ε. Thus, by using again a telescopic sum, it follows that

(3.13) I ( η 1 k s ( u ) ) = I ( η 1 m ( u ) ) + j = m + 1 k s [ I ( η 1 j ( u ) ) - I ( η 1 j - 1 ( u ) ) ] c - ε + j = m + 1 k s [ I ( η 1 j ( u ) ) - I ( η 1 j - 1 ( u ) ) ] .

By fixing j, m+1jks, and using (3.7) as in (3.10), it yields

(3.14) I ( η 1 j ( u ) ) - I ( η 1 j - 1 ( u ) ) - 1 k I ( η 1 j - 1 ( u ) ) 2 + ε 2 k s ε 2 k s .

By replacing (3.14) in (3.13) for all m+1jks, it follows that

I ( η 1 k s ( u ) ) c - ε + j = m + 1 k s [ ε 2 k s ] = c - ε + ( k s - m k s ) ε 2 c - ε 2 .

Therefore, the result also holds in this case.

Case III: η1j(u)BR+1I-1([c-ε,c+ε]) for 1jm-1 but η1m(u)BR+1 for some 1mks. Since uR/2, it follows that

u + R 2 + 1 R + 1 η 1 m ( u ) ,

and hence, by the triangular inequality and the telescopic sum, it follows that

(3.15) R + 2 2 η 1 m ( u ) - u η 1 m ( u ) - u j = 1 m η 1 j ( u ) - η 1 j - 1 ( u ) .

By the definition of η1 and since V(u)=I(u) in R+1, it follows that

(3.16) η 1 j ( u ) - η 1 j - 1 ( u ) = 1 k V ( η 1 j - 1 ( u ) ) = 1 k I ( η 1 j - 1 ( u ) )

for all 1jm because η1j(u)R+1 for all 1jm-1. Hence, by replacing (3.16) in (3.15) and applying Hölder’s inequality for finite sums, it yields

(3.17) R + 2 2 1 k j = 1 m I ( η 1 j - 1 ( u ) ) m 1 2 k [ j = 1 m I ( η 1 j - 1 ( u ) ) 2 ] 1 2 .

From (3.10) it follows that

1 k I ( η 1 j - 1 ( u ) ) 2 - [ I ( η 1 j ( u ) ) - I ( η 1 j - 1 ( u ) ) ] + ε 2 k s ,

which yields

(3.18) I ( η 1 j - 1 ( u ) ) 2 k ( I ( η 1 j - 1 ( u ) ) - I ( η 1 j ( u ) ) ) + ε s 2

for all 1jm. Thus, by using (3.18) in (3.17), it yields

R + 2 2 m 1 2 k [ j = 1 m k ( I ( η 1 j - 1 ( u ) ) - I ( η 1 j ( u ) ) ) + ε s 2 ] 1 2 = m 1 2 k [ k ( I ( u ) - I ( η 1 m ( u ) ) ) + m ε s 2 ] 1 2 .

By squaring both sides, it follows that

s 4 = ( R + 2 2 ) 2
m k 2 [ k ( I ( u ) - I ( η 1 m ( u ) ) ) + m ε s 2 ]
= m k [ I ( u ) - I ( η 1 m ( u ) ) + m ε 2 k s ]
(3.19) m k [ c + ε - I ( η 1 m ( u ) ) + ε 2 ] .

By multiplying both sides of (3.19) by km, it yields

1 4 k s 4 m c + 3 ε 2 - I ( η 1 m ( u ) ) ,

which implies I(η1m(u))c+3ε2-10ε4=c-ε since 0<ε<110 by assumption. Now, as in Case II, by using a telescopic sum and (3.14), it yields

I ( η 1 k s ( u ) ) = I ( η 1 m ( u ) ) + j = m + 1 k s [ I ( η 1 j ( u ) ) - I ( η 1 j - 1 ( u ) ) ]
c - ε + j = m + 1 k s [ ε 2 k s ]
= c - ε + ( k s - m k s ) ε 2
c - ε 2 .

Therefore, (ii) also holds for the third case.

Finally, to finish the proof of this lemma, it is left to prove the claim. Denoting by Id:EE the identity map in E, since PiV(u)=Vi(u)=χ(ui)PiI(u)=χ(ui)(Liui+PiB(u)), we obtain that

P i η t ( u ) = P i ( u - t k V ( u ) ) = ( Id - t k χ ( u i ) L i ) u i - t k χ ( u i ) P i B ( u ) .

Hence, it is suitable to set

U t ( u ) := i = 1 , 2 ( Id - t k χ ( u i ) L i ) u i

and

K t ( u ) := - t k i = 1 , 2 χ ( u i ) P i B ( u ) .

In fact, observe that given unu in E, (I2) and Lemma 2.3 imply that B(un)B(u) in E. Since Pi is continuous, it follows that Kt(un)Kt(u). Thus Kt is completely continuous, and therefore is compact for all t[0,1]. Moreover, by the definitions of Ut and Kt, it is clear that η satisfies (Γ2)–(Γ3). Via (I1)–(I2) there is a constant C=C(r) such that I(u)C for all uBr. Hence by the definition of ηt,

η t ( u ) r + I ( u ) r + C ,

and so ηt maps bounded sets to bounded sets, namely η satisfies (Γ4). Lastly, it suffices to show that Ut is a homeomorphism of E onto E, in order to complete the verification of (Γ1). By (Γ3), it suffices to show that PiUt is a homeomorphism of Ei onto Ei, i=1,2. Let u,vEi. For any t[0,1], by the definition of χ, if u,vR+2, then

t k χ ( u ) L i u - χ ( v ) L i v = 0 1 2 u - v .

Hence, without loss of generality, suppose that vR+2. By the definitions of χ and k, and via the Mean Value Theorem, for any t[0,1] it yields

t k χ ( u ) L i u - χ ( v ) L i v 1 k L i u - v + 1 k L i | χ ( u ) - χ ( v ) | v
1 k L i ( R + 2 ) [ 1 + max | χ ( s ) | ] u - v
(3.20) 1 2 u - v .

Thus, (3.20) holds for all u,vEi. Now, for each wEi fixed, set w(u):EiEi given by

w ( u ) := t k χ ( u ) L i u + w .

Note that due to (3.20), w(u) is a contraction on Ei. Then it follows from the contracting mapping theorem that w(u) has a unique fixed point uw. Therefore, uwEi is the unique function such that w(uw)=uw, which implies that PiUt(uw)=w is an one-to-one correspondence, and hence PiUt is a bijection. Furthermore, (3.20) yields

P i U t ( u ) - P i U t ( v ) = ( u - v ) - t k ( χ ( u ) L i u - χ ( v ) L i v )
u - v - 1 2 u - v
= 1 2 u - v ,

which implies that (PiUt)-1 is continuous. Since PiUt is continuous by definition, it ensures that PiUt is a homeomorphism of Ei onto Ei. Consequently, η satisfies (Γ1), and finally the claim is proved. ∎

The following lemma gives a significant information about the level c.

Lemma 3.3 (See [3, Proposition 1.17]).

If I satisfies (I3), then cα.

Proof.

For the sake of completeness, this proof is included here. In fact, it suffices to show that

(3.21) h 1 ( Q ¯ ) S

for all hΛ. In fact, if (3.21) holds, there is a yh1(Q¯)S, and hence

(3.22) sup u Q ¯ I ( h 1 ( u ) ) I ( y ) inf w S I ( w ) α

due to (I3) (i). Since (3.22) holds for all hΛ, the definition of c in (I4) yields cα. The proof of (3.21) follows from the stronger assertion that

(3.23) h t ( Q ¯ ) S

for all hΛ and t[0,1]. Since S-vE1 via (I3) (i), inequality (3.23) is equivalent to finding, for each t[0,1], a uQ¯ such that

(3.24) { P 1 h t ( u ) S - v , P 2 h t ( u ) = v .

In order to solve (3.24), it is necessary to convert it into an equivalent problem to which the linking geometry hypotheses can be applied. Suppose first that hΛ with the corresponding m=1. Let u=u1+u2E1E2 as usual. By (Γ1) and (Γ3), (3.24) becomes

(3.25) { P 1 h t ( u ) S - v , P 2 h t ( u ) = U t ( u 2 ) + P 2 K t ( u ) = v .

More generally, suppose the second equation of (3.25) replaced by

(3.26) P 2 h t ( u ) = P 2 Z t ( u ) ,

where Zt(u) is an arbitrary compact operator with Z0(u)=v. Note that in (3.25), Zt(u)=v is the compact constant operator for all t[0,1]. Again via (Γ1) and (Γ3), equation (3.26) is equivalent to

U t ( u 2 ) = - P 2 ( K t ( u ) - Z t ( u ) ) ,

which is equivalent to

u 2 = U t - 1 ( - P 2 ( K t ( u ) - Z t ( u ) ) ) P 2 Y t ( u ) ,

where Yt is compact due to the compactness of Kt and Zt, and Y0(u)=v since

U 0 - 1 ( - P 2 ( K 0 ( u ) - Z 0 ( u ) ) ) = - P 2 ( - Z 0 ( u ) ) = v .

Now, suppose by induction that (3.26) is equivalent to

(3.27) u 2 = P 2 Y t ( u ) ,

with Yt compact and Y0(u)=v, whenever hΛ with the corresponding m=n-1. Then let hΛ with m=n so that h=h(1)h(m), and let h^=h(2)h(m). Hence h=h(1)h^ and, again by (Γ1) and (Γ3), the equation

v = P 2 h t ( u ) = P 2 ( U t ( 1 ) + K t ( 1 ) ) h ^ t ( u ) = U t ( 1 ) P 2 h ^ t ( u ) + P 2 K t ( 1 ) h ^ t ( u )

is equivalent to

(3.28) P 2 h ^ t ( u ) = ( U t ( 1 ) ) - 1 ( - P 2 K t ( 1 ) ( h ^ t ( u ) ) + v ) = : P 2 Z ^ t ( u ) ,

where h(1)=U(1)+K(1). Since Kt(1) is compact, Z^t given by the right-hand side of (3.28) is compact and P2Z^0(u)=v since K0=0. Thus, by the induction hypothesis there is a compact Yt such that (3.28) is equivalent to solving (3.27).

Now set Φt(u)=P1ht(u)+u2-P2Yt(u)+v, and note that ΦΣ since

P 2 Φ t = u 2 - ( P 2 Y t - v )

and

Φ 0 ( u ) = P 1 h 0 ( u ) + u 2 - P 2 Y 0 ( u ) + v = P 1 u + u 2 - v + v = u .

In addition, P1Φt=P1ht, and provided that P2ht=v is equivalent to (3.27), due to all remarks above, it follows that P2Φt=v is equivalent to P2ht=v by the definition of Φt. Therefore, Φt(u)S if and only if ht(u)S, and hence to obtain (3.23) and complete the proof it is only necessary to show that

(3.29) Φ t ( Q ) S

for all t[0,1]. Since ΦΣ and via (I3) (iii), S and Q link. Then (3.29) holds if

(3.30) Φ t ( Q ) S = .

Suppose the contrary, so there is a uQ and t[0,1] such that Φt(u)S. Then ht(u)S, but

h t ( Q ) I α + ω 2 - β

since hΛ. On the other hand,

S I α + ω 2 - β =

due to (I3) (i), and provided that β(0,α-ω2), it hence yields a contradiction. Thus (3.30) is satisfied and the proof of the lemma is complete. ∎

Now, under the knowledge of all results in the previous sections, Theorem 2.4 can be finally proved.

Proof of Theorem 2.4.

First, since the identity map h(u)=u is in Λ, we have c<+ in view of (I2) and (I4). Moreover, cα by Lemma 3.3. Now suppose that c is not a critical value of I. Then I(u)0, for all uI-1(c), and hence there exists ε>0 such that

(3.31) ( 1 + u ) I ( u ) 2 ε for all  u I - 1 ( [ c - ε , c + ε ] ) .

If not, for all n there exists a sequence of positive εn0 and unI-1([c-εn,c+εn]) such that

( 1 + u n ) I ( u n ) < 2 ε n .

From (I4) this sequence is bounded, and thus it possesses a weakly convergent subsequence, still denoted by (un), namely unu as n+ for some uE. By (I2) and Lemma 2.3, one has B(un)B(u) along this subsequence and by assumption, I(un)0. Then it follows that Lun=I(un)-B(un)-B(u) as n+. On the other hand, Lun also converges weakly to Lu along this subsequence. Hence Lu=-B(u) and LunLu strongly. Then I(u)=Lu+B(u)=0. Since I(un)c, again by (I2) it follows that

I ( u n ) = 1 2 ( L u n , u n ) + B ( u n ) I ( u ) = c .

But it means that c is a critical value of I, contrary to the assumption. Thus there exists an ε as desired in (3.31). Further, ε<110 can be assumed. By the definition of the infimum, choose an hΛ with corresponding β such that

(3.32) c sup u Q ¯ I ( h 1 ( u ) ) c + ε and h t ( Q ) I α - ω 2 - β .

Since hΛ, we have that ht maps bounded sets on bounded sets due to the definition of Γ, and hence h1(Q¯) is bounded. Therefore, there is an R such that h1(Q¯)BR/2. By Lemma 3.1, with ϱ=12min(β,ε), there exist ηΓ and k such that ηtks satisfies (i) and (ii) of that lemma. Let gt(u)=ηtks(ht(u)). Provided that

h t ( Q ) I α - ω 2 - β ,

in view of Lemma 3.1 (i),

g t ( Q ) I α - ω 2 - β 2 .

Hence gΛ and from (2.1) it follows that

(3.33) c sup u Q ¯ I ( g 1 ( u ) ) .

From (3.32), I(h1(u))c+ε for all uQ¯. Thus, if h1(u)I-1([c-ε,c+ε]), by (3.31) it is possible to apply Lemma 3.1 (ii) to conclude that g1(u)Ic-ε/2. On the other hand, if h1(u)Ic-ε, then Lemma 3.1 (i) yields

I ( g 1 ( u ) ) = I ( η 1 k s ( h 1 ( u ) ) ) I ( h 1 ( u ) ) + ϱ c - ε + ε 2 = c - ε 2 ,

and thus g1(u)Ic-ε/2 by the choice of ϱ. Consequently, it follows that

sup u Q ¯ I ( g 1 ( u ) ) c - ε 2 ,

which contradicts (3.33) and the theorem is proved. ∎

4 Application to Asymptotically Linear Schrödinger Equations in N

This section introduce two applications for the abstract critical point theorem developed previously. The main difference between them is how to obtain the linking geometry, based on their spectra, which are very different to each other.

First, consider problem (P),

(4.1) - Δ u + V ( x ) u = g ( x , u ) in  N

for N3, where the potential V satisfies (V1)–(V2) as stated in Section 1.

Remark 4.1.

In view of hypothesis (V1), V(x) is bounded and σess(A)=[V,+) (see [28, Theorem 3.15, p. 44]), and hence σ(A)(-,V)=σd(A)(-,V). Furthermore, hypothesis (V2) implies that either 0σ(A) or 0σd(A) since 0(σ-,σ+) is an isolated point and the essential spectrum [V,+) does not have isolated points. Hence 0σess(A)=[V,+), which implies that V>0, namely V is positive, and therefore assumption V>0 in (V1) is redundant. With this in hand, it is possible to introduce by means of the operator A a norm equivalent to the usual norm H1(N) in H1(N) (see [9, Lemma 1.2]). Thus, E=(H1(N),) will be the Hilbert space used in order to apply Theorem 2.4. Finally, from (V2) we have σ(A)(-,0)=σd(A)(-,0), i.e., the operator A has negative eigenvalues. Furthermore, this set is finite (see [13, Theorem 30, p. 150]). An example satisfying (V1)–(V2) is given by a continuous V(x) such that

V ( x ) = { - V 0 , | x | < R , V , | x | > 2 R ,

where V0>λ1(1)R2>0 is a constant and λ1(1) is the first eigenvalue of the operator (-Δ,H01(B1(0))).

Henceforth, consider the case where g(x,s)=h(x)f(s) and h satisfies (h1). Furthermore, f is asymptotically linear satisfying (f1)–(f3).

Remark 4.2.

Setting hL(N):=h, we observe that assumption (h1) implies that 0<h<+. In addition, (f2) implies that lims+F(s)=+ and lims+f(s)s=a. Moreover, due to assumptions (f1)–(f2) there exists κ>0 such that |f(s)|κ|s| for all s and aκ. An example satisfying (f1)–(f3) but not with f(s)s increasing, is a continuous f(s) such that

f ( s ) = { s 7 - 3 2 s 5 + 2 s 3 1 + s 6 , | s | < 5 , s 3 1 + s 2 , | s | > 10 .

Let I:E be the energy functional associated with problem (P) in (4.1), which is given by

I ( u ) = 1 2 N ( | u | 2 + V ( x ) u 2 ) - N h ( x ) F ( u ) 𝑑 x

for all uE. As observed in Remark 4.1, the set of eigenvalues σd(A)(-,0) is finite. Then one can denote it by {λi}i=1j for some j, counting multiplicities, and in addition denote by φiE the eigenfunction associated with λi for i=1,,j and then set E-:=span{φi}i=1j. Moreover, by setting E0:=ker(A), if 0σ(A), then E0={0}, if not, then 0σd(A). Hence E0 is finite-dimensional. Thus, E-E0 is a finite-dimensional subspace of E and, by setting E+:=(E-E0), it is the subspace of E in which the operator A is positive definite. With this, E=E+E-E0 and every function uE can be uniquely written as u=u++u-+u0, with u+E+, u0E0 and u-E-. Furthermore, as in [9], the operator A induces a norm equivalent to the standard H1(N)-norm and a corresponding inner product (,) in E given by

u 2 := ( A u + , u + ) 2 - ( A u - , u - ) 2 + u 0 2 2

and

( u , v ) = { N ( u ( x ) v ( x ) + V ( x ) u ( x ) v ( x ) ) 𝑑 x = ( A u , v ) 2 if  u , v E + , - N ( u ( x ) v ( x ) + V ( x ) u ( x ) v ( x ) ) 𝑑 x = - ( A u , v ) 2 if  u , v E - , ( u , v ) 2 if  u , v E 0 , 0 if  u E j , v E k , j k ,

for j,k{+,-,0}. Henceforth, the Hilbert space used in this application is E=(H1(N),) and in addition it is possible to write

I ( u ) = 1 2 ( u + 2 - u - 2 ) - N h ( x ) F ( u ( x ) ) 𝑑 x

for all u=u++u-+u0E. Note that the orthogonality among E+, E-, E0 ensures that u+, u-, u0 are also orthogonal in L2(). As usual, u22=(u,u)2 denotes the norm and inner product in L2(). Then (uj,uk)2=0, k and j,k{+,-,0}.

Denote by {(λ)} the spectral family of the operator A. In view of the spectral theory (see [4, Supplement S1.1], see also [21, Chapter 3]) it is possible to define E2:=(0)E=E-E0 and E1:=(I-(0))E. Furthermore, (0)=(λ) for all 0<λ<σ+ by the definition of σ+ in (V2). Then E1=(I-(λ))E for all 0<λ<σ+. Hence, by [4, Theorem 1.1’, p. 394] it follows that σ+u122u12 for all u1E1. Therefore,

inf u 1 E 1 , u 1 0 u 1 2 u 1 2 2 σ + ,

and thus, by setting

(4.2) a 0 := inf u 1 E 1 , u 1 0 u 1 2 h 1 2 u 1 2 2 1 h inf u 1 E 1 , u 1 0 u 1 2 u 1 2 2 σ + h > 0 ,

it follows that

u 1 2 a 0 N h ( x ) u 1 2 ( x ) 𝑑 x

for all u1E1.

Under all the previous assumptions and notations, it is possible to state the first main result of this section.

Theorem 4.3.

Assume V satisfying (V1)(V2), h satisfying (h1) and f satisfying (f1)(f3), with a>a0. Then problem (P) has a nontrivial weak solution uH1(RN).

In order to prove Theorem 4.3 it is necessary to check that I satisfies hypotheses (I1)–(I4) in Theorem 2.4, and then it is possible to ensure the result by applying this theorem. First of all, see that IC1(E,) due to the hypotheses assumed about h and f. Moreover, on one hand,

( L u , u ) = ( L 1 u 1 + L 2 u 2 , u 1 + u 2 ) = ( L 1 u 1 , u 1 ) + ( L 2 u 2 , u 2 ) ,

and on the other hand, denoting by I1:E1E1 the identity operator in E1, and by P-:E2E2 the projector operator of E2 on E-, we note that u2E2 is such that u2=u-+u0. Hence,

u + 2 - u - 2 = u 1 2 - u 2 - u 0 2 = ( I 1 ( u 1 ) , u 1 ) + ( - P - ( u 2 ) , u 2 ) .

Thus, by setting L1:=I1 and L2:=-P-, it follows that Li:EiEi are bounded, linear and self-adjoint operators for i=1,2. Therefore, I(u)=12(Lu,u)+B(u), where

(4.3) B ( u ) = - N h ( x ) F ( u ( x ) ) 𝑑 x ,

and this gives (I1).

In order to prove (I2) the following lemma is needed.

Lemma 4.4.

Assume that (h1) and (f1)(f2) hold for I. Then B given in (4.3) is weakly continuous and uniformly differentiable on bounded subsets.

Proof.

Let unu be a sequence in E. Then un(x)u(x) almost everywhere in N, and F(un)(x)F(u)(x) almost everywhere in N since F(s) is a continuous function. Moreover, (4.11) yields

( | F ( u n ) | 2 * p ) L 1 ( N )

since 2<22*p<2* and (un)ELs(N) for 2s2*. Thus

( F ( u n ( ) ) ) L 2 * p ( N )

is bounded, provided that (un) is bounded in E, and hence it is bounded in L22*p(N) and in L2*(N). Since F(un)(x)F(u)(x) almost everywhere in N and

F ( u n ) L 2 * p ( N ) C for all  n ,

by Brezis–Lieb’s Lemma in [5], F(un)F(u) in L2*p(N). Provided that (h1) implies that hLq(N), where q is the conjugate exponent of 2*p, then

N h ( x ) F ( u n ( x ) ) 𝑑 x N h ( x ) F ( u ( x ) ) 𝑑 x

as n+. Therefore, B is weakly continuous.

Showing that B is uniformly differentiable on bounded subsets of E means that given ε>0 and BRE, there exists δ>0 such that

| B ( u + v ) - B ( u ) - B ( u ) v | < ε v

for all u+vBR with v<δ. First, note that B satisfies

(4.4) | B ( u + v ) - B ( u ) - B ( u ) v | h 1 2 N ( h ( x ) ) 1 2 | f ( z ( x ) ) - f ( u ( x ) ) | | v ( x ) | 𝑑 x

since for ψ(t):=F(u+tv) it yields ψ(t)=f(u+tv)v. Hence, the Mean Value Theorem implies there exists some function θ(x) such that 0<θ(x)<1 almost everywhere in N and, by writing z=u+θv, it follows that

F ( u + v ) - F ( u ) = ψ ( 1 ) - ψ ( 0 ) = ψ ( θ ) = f ( z ) v ,

almost everywhere in N. Moreover, provided that hL(N), from (4.4) we have

(4.5) | B ( u + v ) - B ( u ) - B ( u ) v | h 1 2 ξ L 2 ( N ) v L 2 ( N ) h 1 2 C 2 ξ L 2 ( N ) v ,

where C2>0 is the constant given by the continuous embedding EL2(N) and ξ:=h12()|f(z())-f(u())| belongs to L2(N). Indeed, since 2*p is the conjugate exponent of q, by applying Hölder’s inequality for q and 2*p, it follows that

N | ξ | 2 𝑑 x = N h ( x ) | f ( z ( x ) ) - f ( u ( x ) ) | 2 𝑑 x h L q ( N ) f ( z ( x ) ) - f ( u ( x ) ) L 2 2 * p ( N ) 2 < +

since

2 < 2 2 * p < 2 * and | f ( u ) | 2 2 * p κ 2 2 * p | u | 2 2 * p L 1 ( N )

due to assumption (h1). Observe that by (4.5) it is sufficient to show that, given ε>0 and BRE, there exists δ>0 such that

ξ L 2 ( N ) ε h 1 / 2 C 2

for all u+vBR with v<δ.

In order to prove this indirectly, suppose there exist ε0>0 and BR0E fixed such that for all δ>0 it is possible to obtain uδ+vδBR0 with

v δ < δ and ξ δ L 2 ( N ) > ε 0 h 1 / 2 C 2 ,

where

ξ δ = h 1 2 ( ) | f ( z δ ( ) ) - f ( u δ ( ) ) | and z δ = u δ + θ v δ .

Choosing δn=1n, for each n there exist un+vnBR0 such that vn1n and

ξ n L 2 ( N ) > ε 0 h 1 / 2 C 2 .

Hence, vn0 in E and unu in E, up to subsequences as n+. In addition, znu in E and zn(x),un(x)u(x) almost everywhere in N, up to subsequences. Thus,

| f ( z n ( x ) ) - f ( u n ( x ) ) | 2 0

almost everywhere in N as n+. Moreover, (zn) and (un) are bounded in E and the Sobolev embedding L22*p(N)E holds. Then

| f ( z n ) - f ( u n ) | 2 L 2 * p ( N ) 2 * p C ( f ( z n ) L 2 2 * p ( N ) 2 2 * p + f ( u n ) L 2 2 * p ( N ) 2 2 * p )
C κ 2 2 * p ( z n L 2 2 * p ( N ) 2 2 * p + u n L 2 2 * p ( N ) 2 2 * p )
2 C ( C 2 2 * p κ R 0 ) 2 2 * p ,

where C22*p>0 is the constant given by the continuous embedding EL22*p(N). Therefore, the sequence (|f(zn)-f(un)|2) is bounded in L2*p(N). Applying Brezis–Lieb’s Lemma again, it yields

| f ( z n ) - f ( u n ) | 2 0

in L2*p(N) as n+. Since hLq(N), which is the dual space of L2*p(N), by weak convergence it yields

ξ n L 2 ( N ) 2 = N h ( x ) | f ( z n ( x ) ) - f ( u n ( x ) ) | 2 𝑑 x 0

as n+, which contradicts

ξ n L 2 ( N ) > ε 0 h 1 / 2 C 2

and completes the proof. ∎

In order to prove (I3), choose Q={re:r[0,r1]}(E2Br2) and S=BρE1, where 0<ρ<r1<r2 are constants and eE1, e=1, must be a suitable vector. Hence, observe that if a as in (f2) is such that a>a0, then for ε>0 small enough and aε:=a-ε it follows that a>aε>a0, and by the definition of a0 in (4.2), there exists some e0E1 such that

a 0 N h ( x ) e 0 2 ( x ) 𝑑 x e 0 2 a ε N h ( x ) e 0 2 ( x ) 𝑑 x .

By normalizing e0 it follows that e=e0e0E1 is such that

(4.6) 1 = e 2 = N ( | e ( x ) | 2 + V ( x ) e 2 ( x ) ) 𝑑 x a ε N h ( x ) e 2 ( x ) 𝑑 x .

Therefore, choose such e for the structure of Q. Furthermore, by Lemma 2.2, S and Q “link”, where Q can be written as Q=Q1Q2Q3, with Q1={0}(E2Br2), Q2={re:r[0,r1]}(E2Br2) and Q3={r1e}(E2Br2). The following lemma shows that I satisfies (I3) (i)–(ii) in Theorem 2.4 for some α>0, ω=0 and arbitrary vE2.

Lemma 4.5.

Assume that (V1)(V2), (h1) and (f1)(f2) hold for I. For Q and S as above, and for sufficiently large r1>0, the inequalities I|Sα>0 and I|Q0 hold for some α>0.

Proof.

By definition, SE1, and hence for all u1S it yields

(4.7) I ( u 1 ) 1 2 ρ 2 - h N ( ε 2 | u 1 ( x ) | 2 + C ε p | u 1 ( x ) | p ) 𝑑 x = ρ 2 ( 1 2 ( 1 - ε h C 2 2 ) - C ε p h C p p ρ p - 2 ) .

Thus, if ε,ρ are sufficiently small, from (4.7) we obtain I(u1)α>0.

Now, for the purpose of checking that I|Q0<α, consider the three cases as follows.

Case I: uQ1E2. Thus

I ( u ) = - 1 2 u 2 - N h ( x ) F ( u ( x ) ) 𝑑 x 0

since h(x)F(u(x))0 for all xN.

Case II: uQ2. Thus u=u1+u2, where u1=re, with 0u1=rr1 and u2=r2>r1, and therefore

I ( u ) = 1 2 ( u 1 2 - r 2 2 ) - N h ( x ) F ( u ( x ) ) 𝑑 x 1 2 ( r 1 2 - r 2 2 ) < 0 .

Case III: uQ3. Thus u=r1e+u2, where 0u2r2. If r1u2r2, then

I ( u ) = 1 2 ( r 1 2 - u 2 2 ) - N h ( x ) F ( u ( x ) ) 𝑑 x 1 2 ( r 1 2 - r 1 2 ) 0 .

If 0u2<r1, put u2=r1v2, where v2B1E2. Thus,

(4.8) I ( u ) = 1 2 r 1 2 ( 1 - v 2 2 ) - N h ( x ) F ( u ( x ) ) 𝑑 x 1 2 r 1 2