## Abstract

We demonstrate the existence of solutions in the discrete nonlinear Schrödinger equation (DNLS) with saturable nonlinearity. We consider two types of solutions to DNLS periodic and vanishing at infinity. Calculus of variations and the Nehari manifolds are employed to establish the existence of these solutions. We present some extensions of our results, combining the Nehari manifold approach and the Mountain Pass argument.

## 1. Introduction

In recent years, several authors have studied the properties of solitons in photorefractive media (Krolikowski *et al*. 2003). Owing to the small optical power required for their generation, it is very easy to obtain them experimentally even with continuous-wave lasers and standard optical equipment, and an almost full control of the relevant parameters can be obtained in the experiment. Moreover, the magnitude of the saturable nonlinearity of photorefractive crystals can be easily driven by adjusting the applied external electrical field. The nonlinear waveguide array was introduced in soliton theory 15 years ago. It is suggested that these arrays possess a great potential for various applications such as optical interconnects, beam deflectors and modulators as well as nonlinear all-optical switches and amplifiers. The first experimental observation of discrete spatial solitons in nonlinear waveguide arrays with Kerr nonlinearity was reported 10 years ago (Eisenberg *et al*. 1998). Soon thereafter, waveguides with a negative diffraction were obtained, which enabled defocusing of light and paved the way to the discovery of the discrete diffraction-managed spatial solitons.

In this case, the equation describing these media is a modification of the original nonlinear Schrödinger (NLS), which consists of substituting the Kerr nonlinearity term with another one of saturable type. This saturable NLS (SNLS) equation is non-integrable and the soliton collision processes are inelastic, leading to annihilation, fusion or creation of solitons (Cowan *et al*. 1986; Snyder & Sheppard 1993; Jakubowski *et al*. 1997; Królikowski & Holmstrom 1997). The evolution equation of bright one-dimensional optical spatial solitons in bulk photorefractive media, based on the Vinetskii–Kukhtarev model (with the neglected diffusion term), can be written as(1.1)where *u* is a normalized slowly varying envelope of the electric field of the light wave.

This last phenomenon consists of the appearance of three solitons after the collision of only two of them. Another important feature of the SNLS is that the behaviour of the solutions is quite robust, being independent of the details of the mathematical model. The discrete version of the NLS equation can be used to describe nonlinear waveguide arrays within the tight binding approximation (Christodoulides *et al*. 2003). The existence and properties of mobile discrete breathers/solitons in discrete nonlinear Schrödinger lattices have been considered in a number of studies.

The optical pulse propagation in one-dimensional equidistant nonlinear waveguide arrays with saturable nonlinearity can be modelled, within the nearest-neighbour approximation and with neglected influence of diffusion of charge carriers, by virtue of the following discrete version of the Vinetskii–Kukhtarev equation (discrete NLS with saturable nonlinearity),(1.2)where *μ*>0 and *ν*≠0.

We study standing wave solutions of (1.2), i.e. solutions of the form(1.3)where the amplitude *u*_{n} is supposed to be real. The equation for the amplitude is(1.4)where is the discrete Laplacian and(1.5)In this paper, we consider two types of solutions to (1.4) as follows: (a) *k*-periodic, i.e. *u*_{n+k}=*u*_{n} and (b) vanishing at infinity, i.e. . Actually, in case (b), we look for solutions in the space *l*^{2} of square summable sequences, which certainly vanish at infinity. Equation (1.4) has a trivial solution *u*_{n}≡0. We are looking for non-trivial solutions. Our main result on equation (1.4) with nonlinearity (1.5) is the following.

*Suppose that either ω*<0 *and ω*+*ν*/*μ*>0 *or ω*>4 *and ω*+*ν*/*μ*<4. *Then for every k*≥2*, there exist two non-trivial k-periodic solutions* ±*u*^{(k)} *as well as two non-trivial solutions* ±*u*∈*l*^{2} *of equation* (*1.4*). *If ω*<0*, then u*^{(k)} *and u are strictly positive. Moreover, the solution u decays exponentially at infinity, i.e.**with C*>0 *and a*>0.

Note, both periodic and solitary solutions possessing the temporal frequency *ω*=2(1−*ν*/2*μ*) to equation (1.4) with nonlinearity (1.5) are obtained explicitly in Khare *et al*. (2005).

Let us point out that the spectrum of −*Δ* in *l*^{2} coincides with the interval [0,4]. The assumption of theorem 1.1 means that *ω* does not belong to the spectrum, while *ω*+*ν*/*μ* is on the opposite side of the spectrum endpoint closest to *ω*.

We prove theorem 1.1 by means of a variational approach. More precisely, we introduce functionals *J*_{k} and *J* on the spaces of *k*-periodic sequences and *l*^{2}, respectively, whose critical points are solutions of equation (1.4). To produce non-trivial critical points, we use the Nehari manifold approach suggested by Nehari (1960). We consider the so-called Nehari manifolds *N*_{k} and *N*. These are *C*^{1} submanifolds of corresponding spaces located away from the origin. All non-trivial solutions (if exist) belong to these manifolds. Being restricted to the Nehari manifolds, the functionals *J*_{k} and *J* are bounded below by positive constants, while on whole spaces the functionals are not bounded both from above and below. Therefore, we can minimize them over these manifolds. The key point is that on *N*_{k} and *N* the derivatives and *J*′, respectively, vanish along transverse directions to these manifolds. Hence, if we find a minimum point of the functional over the Nehari manifold, this minimum point is automatically a solution of equation (1.4).

First, we consider the periodic problem. In this case, *N*_{k} is finite dimensional. The second part of theorem 1.1 concerning *l*^{2} solutions is more involved because the functional *J* does not satisfy the Palais–Smale condition. Our idea is to pass to the limit as *k*→∞. This idea was employed in Pankov (2005*a*,*b*, 2006); however, he considered the case of superlinear nonlinearity and periodic coefficients. The key point is to show that the norms of *u*^{(k)} are bounded. The proof of this is based on an indirect argument that uses some concentration compactness. Once this is done, passing to a subsequence, we can assume that *u*^{(k)}→*u* pointwise. It is not difficult to see that *u* is a solution of (1.4) and that *u*∈*l*^{2}. Finally, we show that *u* is a minimum point of *J* on *N*. An exponential decay estimate for *u* follows exactly as in Pankov (2006).

Actually, we give the proofs only in the case when *ω*<0. The other case is similar, except for the positivity property. The only change we need is to replace the functionals *J*_{k} and *J* with −*J*_{k} and −*J*, respectively.

Let us mention that the solutions obtained this way are the so-called ground-state solutions. This means that they have minimum possible action (±*J*_{k} or ±*J*) among all non-trivial solutions.

Actually, in what follows we consider equation (1.4) for a wider class of nonlinearities. Here, according to Stuart (1993), saturable means asymptotically linear at infinity. Theorem 1.1 is a consequence of theorems 2.2 and 2.3.

The paper is organized as follows: in §2, we present the main results and in §3, we introduce the Nehari manifolds. In §§4 and 5, we prove the existence of periodic and decaying solutions, respectively. Section 6 is devoted to the global convergence. Finally, in §7, we give some extensions of main results, combining the Nehari manifold approach with the Mountain Pass argument.

Let us point out that all our results, as well as the techniques, extend straightforwardly to the case of NLS on multidimensional lattices. The only change is that in dimension *d* the spectrum of −*Δ* is the interval [0,4*d*]. We consider the one-dimensional case only to simplify the notation.

## 2. Main results

We consider the following equation:(2.1)where *σ*=±1 and *ω*<0. Throughout the paper, we suppose that the nonlinearity *f*(*t*) satisfies the following assumptions, in whichis the primitive function of *f*(*t*)

It is easily verified that under assumptions (*h*1)–(*h*3), the function is strictly increasing, while the function strictly increases for *t*≥0 and strictly decreases for *t*≤0.

Let us introduce some notation. Throughout the paper, *k*>1 stands for an integer andwhere [.] is the integer part.

We denote by *X*_{k} the space of all *k*-periodic sequences. This is a finite-dimensional space endowed with the Euclidean normSometimes, we will consider *l*^{p} norm on *X*_{k}with the corresponding change when *p*=∞. This is an equivalent norm on *X*_{k}. The symbol *X* stands for the space with the normWe also consider the spaces *l*^{p}=*l*^{p}(), 1≤*p*≤∞ with the normwith corresponding change when *p*=∞. We mention that(2.2)whenever 1≤*p*≤*q*≤∞.

Also, we denote by (.,.)_{k} and (.,.) the natural inner products on the spaces *X*_{k} and *X*, respectively. For the sake of simplicity, we denote by the same letter *L* the operatoracting either on *X*_{k} or *X*. The operator *L* is bounded and self-adjoint in both the spaces *X*_{k} and *X*. Now we introduce the action functionals(2.3)and(2.4)on the spaces *X*_{k} and *X*, respectively. It is readily verified that these are *C*^{1} functionals and the derivatives are given by(2.5)and(2.6)for any *v*∈*X*_{k} and *v*∈*X*, respectively. Here and thereafter 〈*h*,*v*〉 stands for the value of a linear functional *h* on an element *v*.

Hence, critical points of *J*_{k} and *J* are *k*-periodic and *l*^{2} solutions of (2.1), respectively. In case *σ*=1, we say that a solution of (2.1) is a ground-state solution if it minimizes the action among all solutions of the same type (*k*-periodic or *l*^{2}, respectively). In case *σ*=−1, ground states are solutions that maximize the action.

Now we are ready to formulate our main results.

*Assume that the nonlinearity satisfies* (*h*1)–(*h*4) *and either σ*=1*, ω*<0 *and l*+*ω*>0*, or σ*=−1*, ω*>4 *and* −*l*+*ω*<4. *Then for every k*>1*, equation* (*2.1*) *possesses a non-trivial k-periodic ground-state solution u*^{(k)}∈*X*_{k}. *Moreover, in the case when f is odd, i.e. f*(−*u*)=−*f*(*u*)*, there are two non-trivial ground states* ±*u*^{(k)} *and one of them is positive, provided σ*=1.

*Under the assumptions of* *theorem 2.2**,* *there exists a non-trivial ground-state solution u*∈*l*^{2} *and u decays exponentially fast**for some α*>0 *and C*>0. *Moreover, if f is odd, there are two non-trivial ground states* ±*u*^{(k)} *and one of them is positive, provided σ*=1.

Theorem 1.1 follows from theorems 2.2 and 2.3 because the nonlinearity given by equation (1.5) satisfies assumptions (*h*1)–(*h*4).

*Under the assumptions of* *theorem 2.2**, let u*^{(k)}∈*X*_{k} *be the solution obtained in that theorem. Then, there exists a ground-state solution u*∈*l*^{2} *and b*_{k}∈ *such that**as k*→∞.

We complement theorems 2.2–2.4 with the following results.

*Suppose that either ω*<0 *and l*+*ω*<0*,* *or ω*>4 *and l*+*ω*>4. *Then, equation* (*2.1*) *has no non-trivial solutions in X*_{t} *and X*.

*Suppose that ω*∈[0,4] *and* *,* *where C*>0 *and p*>0. *Then equation* (*2.1*) *has no non-trivial solution such that* .

The proofs of propositions 2.5 and 2.6 are essentially the same as those in Pankov (2006), proposition 2.1 and theorem 6.2.

In the proofs of theorems 2.2–2.4 given below, we consider the case *σ*=1 only, the other case being similar to the functionals *J*_{k} and *J* replaced by −*J*_{k} and −*J*, respectively.

## 3. The Nehari manifolds

In this section, we study the main properties of the Nehari manifolds associated with the functionals *J*_{k} and *J*. These manifolds are defined as follows:

Letwhere *u*∈*X*_{k} and *u*∈*X*, respectively. These are *C*^{1} functionals and their derivatives are given by(3.1)and(3.2)

*Under the assumptions of* *theorems 1.1–2.3**,* *the sets N*_{k} *and N are non-empty closed C*^{1} *submanifolds in X*_{k} *and X, respectively*. *The derivatives* *and I*′ *are non-zero on the corresponding Nehari manifolds. Moreover, there exists β*_{0}>0 *such that* , *u*∈*N*_{k} *and* , *u*∈*N*.

We provide the proof in case of *N*_{k}, the other case being similar.

First we show that . Let *δ*∈(−*ω*, *l*) and *E*_{δ} be the spectral subspace of the operator *L*_{k}=−*Δ*−*ω* in that it corresponds to [0,*δ*]. Since −*ω*∈*σ*(*L*_{k}), we see that *E*_{δ}≠{0}. Let *v*∈*E*_{δ} and *v*≠0. By (*h*1),for *t*>0 small. On the other hand,By (*h*2), the sum above tends to be and, hence, for *t*>0 large enough. As a consequence, there exists *t*^{*}>0 such that , and *t*^{*}*v*∈*N*_{k}.

Let *u*∈*N*_{k}. By equations (2.5) and (3.1), and the definition of *N*_{k}, we haveBy assumption (*h*3), this quantity is negative. Hence, , and the implicit function theorem implies that *N*_{k} is a *C*^{1} submanifold of .

Now let us prove the last statement of the lemma. LetThis is an increasing function of *r*≥0, and, by assumption (*h*1), *φ*(*r*)→0, as *r*→0. Let *u*∈*N*_{k}. Note that the operator *L*_{k} is positive definite. Then, by the definition of the Nehari manifold and equation (2.2),Hence, *φ*(‖*u*‖_{k})≥|*ω*|, and this implies what is required.

Closedness of *N*_{k} is obvious. ▪

Actually, the proof of lemma 3.1 shows that if *I*_{k}(*v*)≤0 (respectively, *I*(*v*)≤0), then there exists a unique *t*^{*}∈(0,1] such that *t*^{*}*v*∈*N*_{k} (respectively, *t*^{*}*v*∈*N*). Also the same argument as in the proof of lemma 3.1 gives us the existence of 0≠*v*∈*X*_{k} (respectively, 0≠*v*∈*X*), such that *J*_{k}(*v*)≤0 (respectively, *J*(*v*)<0).

From (2.3) and (2.4), it follows that on *N*_{k}:(3.3)By (*h*3) . Similarly,(3.4)and . Thus, being restricted to their Nehari manifolds, the functionals *J*_{k} and *J* are bounded below. Actually, we have the following simple result.

*There exists α*_{0}=*α*_{0}(*k*)>0 *such that**for all u*∈*N*_{k}.

On *N*_{k}, we haveBy lemma 3.1, . Hence, there exist *n*_{0}∈*Q*_{k} (depending on *u*) and (but independent of *u*), such that . By remark 2.1, we obtain what is required, with ▪

In fact, the functional *J* on *N* is also bounded below by a positive constant. But this fact is less easy and will be obtained in the proof of theorem 2.2.

From (the proof of) lemma 3.1, it follows that the tangent spaces *T*_{u}*N*_{k} and *T*_{u}*N* at *u*∈*N*_{k} or *u*∈*N*, respectively, areandand the line is a transverse line. The functionals *J*_{k} and *J* have a remarkable behaviour along the transverse line as stated in the following.

*For u*∈*N*_{k}*, the function J*_{k}(*tu*)*, t*>0*, has a unique critical point at t*=1*, which is, actually, a global maximum. The same statement holds for N and J*.

Let . Computing the derivative of *φ*, we have(3.5)This shows that *t*=1 is a critical point. Its uniqueness follows from the strict monotonicity of *f*(*t*)/*t* stated in remark 2.1. The other case is similar. ▪

By lemma 3.5, minimum points of *J*_{k} and *J* on the corresponding Nehari manifolds are solutions of equation (2.1). Therefore, to prove theorems 2.2 and 2.3, we consider the following minimization problems:(3.6)and(3.7)We shall prove theorems 2.2 and 2.3 by solving the minimization problems (3.6) and (3.7).

## 4. Existence of periodic solutions

We start with the following.

*Under the assumptions of* *theorem 2.2**, the minimum value in problem* (3.6) *is attained*.

Let *u*^{j}∈*N*_{k} be a minimizing sequence for *J*_{k}, i.e. . By equation (3.3),Now assumption (*h*4) implies that is bounded. Since the space *X*_{k} is finite dimensional, the *l*^{∞} norm is equivalent to the Euclidean norm on *X*_{k}, and the sequence *u*^{j} is bounded. Passing to a subsequence, we can assume that *u*^{j} converges to *u*∈*X*_{k}. Since the set *N*_{k} is closed and the functional *J*_{k} is continuous, we obtain that *u*∈*N*_{k} and *J*_{k}(*u*)=*m*_{k}. ▪

The solution *u* of problem (3.6) is a non-trivial solution of equation (2.1). By construction, this is a ground state.

Suppose now that the nonlinearity *f* is odd. Hence, *F* is even. It is easy to check whetherAlso and . This implies thatOn the other hand, (note that the left-hand part of this equation is not )By remark 3.2, there exists , such that . Then, by remark 2.1 and equation (3.3),Hence, *J*_{k}(*u*^{*})=*m*_{k} and *u*^{*} is a non-negative ground state and we can assume that *u*=*u*^{*}.

Finally, let us prove that the solution *u* is strictly positive. Let *G*(*n*,*m*) be the Green function of −*Δ*−ω. Since *ω*<0, we have that *G*(*n*,*m*)>0 for all *n*,*m*∈ (see, Teschl 2000). From equation (2.1), we obtain that (note that we consider case *ω*<0 and *σ*=1)Since *u* is non-negative and not equal to zero, we see that *u*_{n}>0 for all *n*∈, and the proof is complete. ▪

## 5. Existence of decaying solutions

To obtain a localized ground state, we pass to the limit as *k*→∞. To do that we need the following.

*Let u*_{k} *be a k-periodic ground state, i.e. a solution of problem* (*3.6*). *Then the sequences* *and* *are bounded*.

First, we prove that *m*_{k} is bounded. Let *w*∈*X* be a non-zero vector that belongs to the spectral subspace of the operator *L*=−*Δ*−*ω* in *X* that corresponds to [0,*δ*], with *δ*∈(−*ω*,*l*). As in the beginning of the proof of lemma 3.1, there exists *t*>0 such that *I*(*tw*)<0. Since finitely supported sequences are dense in *X*, we can approximate *tw* by a finitely supported element such that . By remark 3.2, there exists *t*^{*}∈(0,1) such that *I*(*v*)=0, where . For all sufficiently large *k*, we have that . Now for any such *k*, let be a unique element such that whenever . It is easy to see that and . Hence, is bounded.

Assume now that is unbounded. Passing to a subsequence (still denoted by *u*^{k}), we can assume that . For , one of the following alternatives holds:

either

the sequence

*v*^{k}is vanishing, i.e. orthe sequence

*v*^{k}is non-vanishing, i.e. (after passage to a further subsequence) there exist*δ*>0 and*b*_{k}∈, such that for all*k*.

First, we rule out vanishing (case (i)). We have thatHence,(5.1)By assumption (*h*1), there exists *t*_{0}>0 such that whenever . Let and . We have thatTogether with equation (5.1), this implies that(5.2)On the other hand, , with the certain constant *c*_{0}>0, and by the Hölder inequality,(5.3)for any *p*>2. But it is easy to verify thatSince , equations (5.2) and (5.3) show that . LetBy equation (3.3) and remark 2.1,a contradiction.

Now we rule out non-vanishing (case (ii)). First, because of discrete translation invariance, we can assume that *b*_{k}=0. Since , passing to a further subsequence, we can also assume that there exists *v*=(*v*_{n}), such that for all *n*∈. Moreover, it is obvious that *v*∈*X*, with and |*v*_{0}|≥*δ*. Hence, *v*≠0.

Now since *u*^{k} is a (*k*-periodic) solution of equation (2.1), with *σ*=1, we have(5.4)where and, by assumption (*h*2), . Given *n*∈, if *v*_{n}≠0, then . Hence, passing to the limit in equation (5.4), we obtain thati.e. *v*∈*X* is a (non-zero) eigenvector of the operator −*Δ*, with the eigenvalue *ω*+*l*. But the spectrum of −*Δ* in *X* is absolutely continuous (see Teschl 2000), a contradiction.

The proof is complete. ▪

Let be a ground-state solution. By lemma 5.1, the sequence is bounded and, therefore, *u*^{k} is either vanishing or non-vanishing. In case of vanishing, as in the proof of lemma 5.1, we have that as *k*→∞ for any *p*>2. By assumption (*h*1), for every *ϵ*>0, there exists *C*_{ϵ}> such thatSince *u*^{k} is a *k*-periodic solution, we have thatLetting *ϵ*=|*ω*|/2, we obtain thatThis contradicts the conclusion of lemma 3.1 and, hence, vanishing is not possible.

Thus, the sequence *u*_{k} is non-vanishing, and making use of passage to a subsequence and discrete translation invariance, we can assume that , with some *δ*>0. Passing to a further subsequence, we can also assume that there exists a sequence *u*=(*u*_{n}) such that for all *n*∈. It is easily seen that *u*∈*X* and *u*≠0. Moreover, equation (2.1) possesses point-wise limits and, hence, *u* is a non-trivial solution of that equation.

Now we prove that the solution *u* just constructed is a ground state, i.e. *m*=*J*(*u*). Actually, we have proven that, for every sequence *k*_{j}→∞, passing to a subsequence still denoted by *k*_{j} and making appropriate shifts, we can suppose that point-wise, where *u*∈*X* is a non-trivial solution. Let *N*>0 be an integer. Then, by remark 2.1, we have thatLetting *N*→∞, we obtain thatand, hence,(5.5)

Let us prove thathence,(5.6)Given *ϵ*>0, let *w*∈*N* be such thatChoose *t*_{1}>1 sufficiently close to 1 such thatWe also have that *I*(*t*_{1}*w*)<0. By density argument, we can find a finitely supported sequence *v* sufficiently close to *t*_{1}*w* in *X* such that *I*(*v*)<0 andThen there exists *t*_{2}∈(0,1) such that *I*(*t*_{2}*v*)=0. By remark 2.1,Let be such that if *n*∈*Q*_{k}. If *k* is large enough, then andThis implies (5.6). Hence, the solution constructed above is a ground state.

Positivity of ground state can be proven as in the proof of theorem 2.2. ▪

Since *m* is attained, we see that *m*>0.

## 6. Global convergence

In this section, we prove theorem 2.4 that concerns the convergence of periodic ground states to a solitary ground state globally, i.e. on the expanding family of sets *Q*_{k}.

Let *u*_{k}∈*X*_{k} be a ground state. We can assume without loss that for every *n*∈, where *u*∈*X* is a solitary ground state. (This means that *b*_{k}=0.) Denote by a unique element of *X*_{k} such that whenever *n*∈*Q*_{k} and let . We have to show that as *k*→∞.

Let us first prove that(6.1)and(6.2)as *k*→∞.

To prove (6.1), we start with the following identity:(6.3)which is easy to verify. Note that . Next,Since , with *C*>0, and *u*∈*X*=*l*^{2}, the second term on the right converges to as *k*→∞. As for the first term, we have thatHere, for *n*∈*Q*_{k}, except in the case when is an endpoint of *Q*_{k}. In the last case, does not exceed , which tends to be 0 as *k*→∞. Hence,andSimilarly, .

Now we examine the last term, *s*_{k}, in the right-hand side of (6.3). Given an integer *N*>0 and a real number *ϵ*>0, we represent this term as follows:By the Lagrange mean value theorem,where . Since and , with *C*>0, we have thatSince is bounded and *u*∈*X*, we can find *N* such that for all sufficiently large *k*. Now, since for all *n*∈, we see that for all sufficiently large *k*. Hence, for all *k* large enough and, therefore, *s*_{k}→0 as *k*→∞. Thus, we have proven (6.1).

Similar arguments prove (6.2).

Now, by (6.1) and (6.2),By remark 2.1, this implies that . Since is bounded, we have that(6.4)for any *p*>2. But for every *ϵ*>0, there exists *C*_{ϵ}>0 such thatHence,Taking *ϵ*=|*ω*|/4 and using (6.2) and (6.4), we obtain that as *k*→∞. The proof is complete. ▪

## 7. Some extensions of main results

Theorems 2.2–2.4 do not apply to the nonlinearity given by(7.1)with *p*≠2. Another important nonlinearity not covered by those results is given by(7.2)where *Χ*>0, *p*>0 and *a*>0. This nonlinearity appears in certain waveguide problems (see, Stuart 1993). In this section, we present some extensions of our results that cover the examples just mentioned. For the above-mentioned nonlinearities, the assumption (*h*4) breaks down, and we replace it by

The function is bounded.

At the same time, we keep the assumptions (*h*1)–(*h*3).

Let *σ*_{k} be the spectrum of the operator −*Δ* in the space *X*_{k} that consists of eigenvalues , *j*=0, 1, …, *k*−1. The union ∪_{k}*σ*_{k} and the complement are dense in the spectrum [0,4] of −*Δ* in the space *X*.

We begin with the following analogue of theorem 2.2:

*Assume that the nonlinearity satisfies* (*h*1)–(*h*3) *and* (*h*5)*, and either σ*=1*, ω*<0*, l*+*ω*>0 *and l*+*ω*∉*σ*_{k}*, or σ*=−1*, ω*>4*,* −*l*+*ω*<4 *and* −*l*+*ω*∉*σ*_{k}. *Then for every k*>1*, equation* (*2.1*) *possesses a non-trivial k-periodic ground-state solution u*^{(k)}∈*X*_{k}. *Moreover, in the case when f is odd, i.e. f*(−*u*)=−*f*(*u*), *and σ*=1*, there are two non-trivial ground states,* ±*u*^{(k)} *and u*^{(k)}>0.

To prove theorem 7.1, we use the Mountain Pass theorem combined with the Nehari manifold approach. Therefore, we need the following:

*Under the assumptions of* *theorem 7.1**, the functional J*_{k} *satisfies the Palais–Smale condition, i.e. every sequence* *such that* *is bounded and* (*a Palais–Smale sequence*) *contains a convergent subsequence*.

Since the space *X*_{k} is finite dimensional, it is enough to show that every Palais–Smale sequence is bounded.

Let and be spectral subspaces of the operator *L*_{k} that correspond to eigenvalues *λ*>*l* and *λ*<*l*, respectively. Since *l*+*ω* is not an eigenvalue of −*Δ*, *l* is not an eigenvalue of *L*_{k} and we have the orthogonal decomposition . For any *u*∈*X* we write *u*=*u*^{+}+*u*^{−}, where .

Now let us consider a Palais–Smale sequence *u*^{j}. It splits as . We haveTaking *u*=*u*^{j} and *v*=*u*^{j+}, and using orthogonality of and , we obtainSince on , with *α*>0, for *j* large enough and all norms on a finite-dimensional space being equivalent, we get, using assumption (*h*5),which implies that the sequence *u*^{j+} is bounded.

Similarly, since on , we obtain thatHence, *u*^{j−} is bounded. Thus, is bounded and the proof is complete.

We prove theorem 7.1. LetandAs in the proof of lemma 3.3, we see that if (at this point, we only use assumption (*h*1)). By remark 3.2, there exists an element *v*^{0}∈*X*_{k} such that . Since, by lemma 7.2, *J*_{k} satisfies the Palais–Smale condition, the standard Mountain Pass theorem (see, Rabinowitz 1986; Willem 1996) implies that *c*_{k}>0 and *c*_{k} is a critical value of *J*_{k}. Therefore, there exists a solution *u*^{k}∈*X*_{k} of equation (2.1) such that and .

To prove that *u*^{k} is a ground state, it is enough to show that *c*_{k}=*m*_{k}. Let *u*∈*N*_{k} and . (We use here the notation from the proof of lemma 3.5.) Then, by lemma 3.5, *t*=1 is the maximum point and the only non-zero critical point of *φ*(*t*). Let us fix *t*_{0}>1 sufficiently close to 1. Then, . Equation (3.5) and monotonicity of *f*(*t*)/*t* imply thatwhenever *t*≥*t*_{0}. Hence, as *t*→∞. Therefore, for *t* large enough. This means that, after a reparameterization, the ray produces a path and . Hence, *c*_{k}≤*m*_{k}.

Now let . Since , remark 2.1 implies that . By (*h*1), whenever *t*>0 is small enough. Hence, there exists such that . This implies that *m*_{k}≤*c*_{k}, hence, *c*_{k}=*m*_{k} and the solution *u*^{k} is a ground state.

Positivity of *u*^{k} follows exactly as in the proof of theorem 2.2, provided *σ*=1 and the nonlinearity is even.

*Assume that the nonlinearity satisfies* (*h*1)–(*h*3) *and* (*h*5)*, and either σ*=1*, ω*<0 *and l*+*ω*>0*, or σ*=−1*, ω*>4 *and* −*l*+*ω*<4. *Then there exists a non-trivial ground-state solution u*∈*l*^{2} *of equation* (*2.1*)*, and u decays exponentially fast**for some α*>0 *and C*>0. *Moreover, if f is odd and σ*=1, *then* −*u is also a ground-state solution and* ±*u*>0.

For every *k*, there exists *ω*_{k}, such that *ω*_{k}→*ω* and *l*+*ω*_{k}∉*σ*_{k}. By theorem 7.1, there exists a ground-state solution of equation (2.1), with *ω* being replaced by *ω*_{k}. Now we can use the same arguments as in the proof of theorem 2.3 to complete the proof. ▪

As in the proof of theorem 2.4, we can show that the conclusion of that theorem holds for the solutions constructed in the proof of theorem 7.3.

Note that theorems 7.1 and 7.3 cover nonlinearities (7.1), with *p*>2, and (7.2). The case of nonlinearity (7.1) with *p*∈(0,2) remains open.

## Footnotes

- Received June 18, 2008.
- Accepted July 16, 2008.

- © 2008 The Royal Society