Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 181369 | 12 pages | https://doi.org/10.1155/2011/181369

On the Travelling Waves for the Generalized Nonlinear Schrödinger Equation

Academic Editor: Ondřej Došlý
Received25 Feb 2011
Accepted05 May 2011
Published25 Jul 2011


This paper is devoted to the analysis of the travelling waves for a class of generalized nonlinear Schrödinger equations in a cylindric domain. Searching for travelling waves reduces the problem to the multiparameter eigenvalue problems for a class of perturbed p-Laplacians. We study dispersion relations between the eigenparameters, quantitative analysis of eigenfunctions and discuss some variational principles for eigenvalues of perturbed p-Laplacians. In this paper we analyze the Dirichlet, Neumann, No-flux, Robin and Steklov boundary value problems. Particularly, a “duality principle” between the Robin and the Steklov problems is presented.

1. Introduction

The main concerns of the paper are the travelling waves for the generalized nonlinear Schrödinger (NLS) equation with the free initial condition in the following form (see [1] for generalized NLS):𝑖𝑣𝑡||||div𝑣𝑝2𝑣=𝜈|𝑣|𝑞2𝑣,𝑝>1,𝑣|𝜕𝑄=0,(1.1) where 𝑣=𝑣(𝑡,𝑥1,𝑥2,,𝑥𝑛+1) and 𝜈 is a parameter. 𝑄=×Ω is a cylinder, 𝜕𝑄 is the lateral boundary of 𝑄, 𝑡>0, 𝑥1, and (𝑥2,𝑥3,,𝑥𝑛+1)Ω. Assume that Ω is a bounded domain in 𝑛 with the smooth boundary. Particularly, in the case of 𝑝=2 we get 𝑖𝑣𝑡Δ𝑣=𝜈|𝑣|𝑞2𝑣,(1.2) which is a nonlinear Scrödinger equation (see [1]). On the other hand problem (1.1) can be considered as an evolution 𝑝𝑞-Laplacian equation. Different aspects of such kind of problems, with some initial conditions, have been studied in [2]. Thus problem (1.1) models the linear Schrödinger equations (𝑝=𝑞=2), NLS (𝑝=2,𝑞2), evolution 𝑝𝑞-Laplacians, and generalized NLS. This definitely means that we have a good motivation for problem (1.1).

In this paper by the travelling waves we mean the solutions of (1.1) in the form 𝑣=𝑒𝑖(𝑤𝑡𝑘𝑥)𝑢(𝑥2,,𝑥𝑛+1), where 𝑥=𝑥1 and 𝑢 is a real-valued function. A simple computation yields 𝑣𝑡=𝑖𝑤𝑒𝑖(𝑤𝑡𝑘𝑥)𝑢, 𝑣𝑥=𝑖𝑘𝑒𝑖(𝑤𝑡𝑘𝑥)𝑢, and 𝑣𝑥𝑖=𝑒𝑖(𝑤𝑡𝑘𝑥)𝑢𝑥𝑖, 𝑖=2,3,,𝑛+1. Hence, 𝑣=(𝑣𝑥,𝑣𝑥2,,𝑣𝑥𝑛+1)=(𝑖𝑘𝑢,𝑢)𝑒𝑖(𝑤𝑡𝑘𝑥) and |𝑣|=(𝑘2𝑢2+𝑢2𝑥2++𝑢2𝑥𝑛+1)1/2. By using the notation 𝑘𝑢(𝑥2,𝑥3,,𝑥𝑛+1)=(𝑘𝑢,𝑢𝑥2,,𝑢𝑥𝑛+1) we obtain |𝑣|=|𝑘𝑢|. Finally, by setting all of these into (1.1) we can obtain the following nonstandard multiparameter eigenvalue problems for perturbed 𝑝-Laplacians: 𝑤𝑢+𝑘2||𝑘𝑢||𝑝2||𝑢div𝑘𝑢||𝑝2𝑢=𝜈|𝑢|𝑞2𝑢,𝑝>1,𝑢|𝜕Ω=0.(1.3) In what follows, by shifting the variables 𝑥2,,𝑥𝑛+1, we have used the following notations: 𝑢=𝑢(𝑥1,,𝑥𝑛), |𝑘𝑢|=(𝑘2𝑢2+𝑢2𝑥1++𝑢2𝑥𝑛)1/2, and 𝑢=(𝑢𝑥1,𝑢𝑥2,,𝑢𝑥𝑛).

At this point we have to note that by searching for the standing waves 𝑣=𝑒𝑖𝑤𝑡𝑢(𝑥1,𝑥2,,𝑥𝑛+1) for the NLS equation 𝑖𝑣𝑡Δ𝑣=𝜈|𝑣|𝑞2𝑣(1.4) we obtain the following eigenvalue problem: 𝑤𝑢Δ𝑢=𝜈|𝑢|𝑞2𝑢.(1.5) On the other hand by setting the travelling wave solutions of the form 𝑣=𝑒𝑖(𝑤𝑡𝑘𝑥)𝑢(𝑥2,,𝑥𝑛+1) into the NLS equation we obtain 𝑤+𝑘2𝑢Δ𝑢=𝜈|𝑢|𝑞2𝑢.(1.6) Thus we obtain the same type eigenvalue problems for both standing and travelling waves for the NLS equation (see [3] and references therein for standing waves for NLS). However, standing and travelling waves for generalized NLS are associated with quite different type of eigenvalue problems. Particularly, the eigenvalue problem associated to the travelling wave solutions is problem (1.3), which is clearly a nonstandard multiparameter eigenvalue problem in the nonlinear analysis, and we are not aware of any known result for this problem.

A solution of (1.3) is a weak solution, defined in the following way.

Definition 1.1. 0𝑢𝑊01,𝑝(Ω) is a solution of (1.3) if and only if 𝑤Ω𝑢𝑣𝑑𝑥+𝑘2Ω||𝑘𝑢||𝑝2𝑢𝑣𝑑𝑥+Ω||𝑘𝑢||𝑝2𝑢𝑣𝑑𝑥=𝜈Ω|𝑢|𝑞2𝑢𝑣𝑑𝑥(1.7)

holds for all 𝑣𝑊01,𝑝(Ω), where 𝑊01,𝑝(Ω) is the Sobolev space (for Sobolev spaces, see [4]). In this case, we say that 𝑢 is an eigenfunction, corresponding to the eigenpair (𝑤,𝑘) and 𝜈, where 𝑤 is a frequency, 𝑘 is a wave number, and the parameter 𝜈 comes from the initial equation (1.1). We prefer to denote the test functions in (1.7) by 𝑣, which is clearly different from the notation that is used in (1.1). Let us define 𝑤𝐹(𝑢)=2Ω𝑢21𝑑𝑥+𝑝Ω||𝑘𝑢||𝑝𝜈𝑑𝑥𝑞Ω|𝑢|𝑞𝐺𝑑𝑥,𝑘1(𝑢)=𝑝Ω||𝑘𝑢||𝑝𝑑𝑥.(1.8) We set 𝑋=𝑊01,𝑝(Ω). Then, 𝑢𝑋 is a solution of (1.7) if and only if 𝑢 is a free critical point for 𝐹(𝑢), that is, 𝐹(𝑢),𝑣=0,forall𝑣𝑋, where 𝐹𝑋𝑋 is the Fréchet derivative of 𝐹, 𝑋 is the dual space, and 𝐹(𝑢),𝑣 denotes the value of the functional 𝐹(𝑢) at 𝑣𝑋. Indeed, the existence of Fréschet derivative implies the existence of directional (Gateaux) derivative. Using the definition of Gateaux derivative, we can obtain 𝐺𝑘(=𝑑𝑢),𝑣𝐺𝑑𝑡𝑘(|||𝑢+𝑡𝑣)𝑡=0=𝑘2Ω||𝑘𝑢||𝑝2𝑢𝑣𝑑𝑥+Ω||𝑘𝑢||𝑝2𝑢𝑣𝑑𝑥,(1.9) which is enough to see that (1.7) is the variational equation for the functional 𝐹(𝑢).

As 𝑢𝑊01,𝑝(Ω), by Sobolev embedding theorems (see [4]) the functional 𝐹(𝑢) can be well defined if(i)𝑝𝑛 or(ii)1<𝑝<𝑛 and max{2,𝑞}𝑛𝑝/(𝑛𝑝).

In the next section two cases 𝜈=0 and 𝜈0 will be studied separately. If 𝜈=0, then we may rewrite (1.7) in the form𝑘2Ω||𝑘𝑢||𝑝2𝑢𝑣𝑑𝑥+Ω||𝑘𝑢||𝑝2𝑢𝑣𝑑𝑥=𝑤Ω𝑢𝑣𝑑𝑥,(1.10) which is the equation for free critical points of the functional 1𝐹(𝑢)=𝑝Ω||𝑘𝑢||𝑝𝑤𝑑𝑥2Ω𝑢2𝑑𝑥.(1.11) Evidently, there are not nontrivial solutions of (1.10) in the case of 𝑤0. Thus, 𝑤>0, and by the scaling property, we obtain that if 𝑝2 and (1.10) has a nontrivial solution for some 𝑤>0, then it has nontrivial solutions for all 𝑤>0.

In what follows, 𝑢=[Ω|𝑢|𝑝𝑑𝑥]1/𝑝 denotes the standard norm in 𝑊01,𝑝(Ω) and 𝑢𝑘=[Ω|𝑘𝑢|𝑝𝑑𝑥]1/𝑝, which is equivalent to the norm 𝑢.

This paper consists of an introduction (Section 1) and two sections. In Section 2 we study the structure of the eigenparameters 𝜈, 𝑘, 𝑤 and the eigenfunctions for problem (1.3), including the dispersion relations between 𝑤, 𝑘, and 𝜈 and variational principles in some special cases. We consider separately two cases: 𝜈=0 and𝜈0. In the case of 𝜈=0 we have estimated bounds for the set of eigenfunctions, proved the existence of infinitely many eigenfunctions, corresponding to an eigenpair (𝑤,𝑘), 𝑤>0, and demonstrated that, in the case of 𝑝>2, the set crit𝐹 is compact. For the general case 𝜈0 we study the existence of positive solutions and variational principles in some special cases. The proofs are based on the Sobolev imbedding theorems, the Palais-Smale condition, variational techniques, and the Ljusternik-Schnirelman critical point theory. Various boundary problems and some relations between them are studied in Section 3.

As mentioned above the problem of the existence of travelling waves and a quantitative analysis for travelling waves is a multiparameter eigenvalue problem given by (1.7) or (1.10). This section is devoted to these problems, and the techniques we use in this section are partially close to that used in [5].

We study separately two cases: 𝜈=0 and𝜈0.

Case 1 (𝜈=0). This subsection is devoted to the quantitative analysis of solutions of (1.10). We assume that(i)𝑝𝑛 or (ii)1<𝑝<𝑛 and 2<𝑛𝑝/(𝑛𝑝).

Our first observation for eigenvalue problem (1.10) is given in the following proposition.

Proposition 2.1. (a) Let 𝑝=2. In this case, all the eigenpairs (𝑤,𝑘) of problem (1.10) lie in the parabola 𝜆1+𝑘2𝑤, where 𝜆1 is the first eigenvalue of Δ in 𝐿2(Ω) and 𝜆1>0. Moreover, for a fixed 𝑤, there is a finite number of 𝑘 and for a fixed 𝑘, there is a countable number of 𝑤𝑛(𝑘), such that 𝑤𝑛(𝑘)+ as 𝑛,
(b) If 𝑝>2, then for a fixed (𝑤,𝑘)×, the solutions of (1.10) are bounded and 𝑢𝑘𝐶𝑝,𝑘(Ω)𝑤1/(𝑝2)(2.1) holds for some 𝐶𝑝,𝑘(Ω)>0,
(c) Let 𝑝<2. In this case, one has 𝑢𝑘1𝑤𝐶𝑝,𝑘(Ω)1/(𝑝2)>0.(2.2)

Proof. The proof easily follows from (1.10), by using the Courant-Weyl variational principle 0<𝜆1=inf𝑢0Ω||||𝑢2𝑑𝑥Ω|𝑢|2,𝑑𝑥(2.3) the bounded embedding 𝑊01,𝑝(Ω)𝐿2(Ω), and the equivalence of the norms and 𝑘.

In the following theorem, we have proved some properties of the functional 𝐹(𝑢)=(1/𝑝)Ω|𝑘𝑢|𝑝𝑑𝑥(𝑤/2)Ω𝑢2𝑑𝑥 in 𝑋, which guarantee the existence of nontrivial solutions of (1.10) for a fixed (𝑤,𝑘)×, 𝑤>0.

Theorem 2.2. Let 𝑝>2 and (𝑤,𝑘)×, 𝑤>0. Then (i)1𝑝12𝑤𝐶𝑝,𝑘(Ω)𝑝/(𝑝2)inf𝑋𝐹(𝑢)<0(2.4)for some 𝐶𝑝,𝑘(Ω)>0.(ii)𝐹 attains its infimum at a nontrivial vector 𝑢0𝑋.

Proof. One has 𝐹(𝑢)=(1/𝑝)Ω|𝑘𝑢|𝑝𝑑𝑥𝑤/2Ω𝑢2𝑑𝑥. As 𝑝>2, the Poincaré inequality yields Ω𝑢2𝑑𝑥𝐶𝑝,𝑘(Ω)𝑢2𝑘. Hence, 1𝐹(𝑢)=𝑝𝑢𝑝𝑘𝑤2Ω𝑢21𝑑𝑥𝑝𝑢𝑝𝑘𝑤2𝐶𝑝,𝑘(Ω)𝑢2𝑘.(2.5) Let 𝑓(𝑥)=(1/𝑝)𝑥𝑝(𝑤/2)𝐶𝑝,𝑘(Ω)𝑥2. We have 𝑓(0)=0 and 𝑓(𝑥)+ as 𝑛. This indicates that 𝑓 has a global minimum point, and clearly, this point is 𝑥=[𝑤𝐶𝑝,𝑘(Ω)]1/(𝑝2). Now, by putting the vectors 𝑢 with 𝑢𝑘=[𝑤𝐶𝑝,𝑘(Ω)]1/(𝑝2) into 𝐹(𝑢), we obtain 1𝑝12𝑤𝐶𝑝,𝑘(Ω)𝑝/(𝑝2)inf𝑋𝐹(𝑢).(2.6) Now, we will show that inf𝑋𝐹(𝑢)<0. Let 𝑢 be a vector such that 𝑢𝑘=1. Subsequently, by setting 𝑡𝑢 in 𝐹(𝑢), we obtain 𝐹(𝑡𝑢)=(𝑡𝑝/𝑝)(𝑤/2)𝑡2𝑐, where 𝑐=Ω𝑢2𝑑𝑥. Thus 𝐹(𝑡𝑢)<0 if 0<𝑡<[(𝑝/2)𝑤𝑐]1/(𝑝2).
(ii) We have to show that inf𝑋𝐹(𝑢) is attained. Let inf𝑋𝐹(𝑢)=𝛼. Then, there exists a sequence 𝑢𝑛𝑋 such that 𝐹(𝑢𝑛)𝛼 as 𝑛. The sequence 𝑢𝑛 should be bounded, because 1𝐹(𝑢)𝑝𝑢𝑝𝑘𝑤2𝐶𝑝,𝑘(Ω)𝑢2𝑘=𝑢2𝑘1𝑝𝑢𝑘𝑝2𝑤2𝐶𝑝,𝑘(Ω)(2.7) and 𝐹(𝑢)+ as 𝑢𝑘+, which means that 𝐹 is coercive. However, 𝑋=𝑊01,𝑝(Ω) is a reflexive Banach space. Consequently, 𝑢𝑛𝑢0 for some 𝑢0𝑋, where “” denotes the weak convergence in 𝑋. Evidently, 𝐹(𝑢) is sequentially lower semicontinuous, that is, 𝑢𝑛𝑢0𝑢implies𝐹0lim𝑛𝑢inf𝐹𝑛.(2.8)
Indeed, Ω|𝑘𝑢|𝑝𝑑𝑥=𝑢𝑝𝑘, and it is known that the norm is sequentially lower semicontinuous. The second term of 𝐹(𝑢) is Ω𝑢2𝑑𝑥, and this term is sequentially continuous, because the embedding 𝑊01,𝑝(Ω)𝐿2(Ω) is compact. Now, it follows from (2.8) that 𝐹(𝑢0)𝛼, which means inf𝑋𝐹(𝑢)=𝐹(𝑢0). Finally, 𝑢00 because inf𝑋𝐹(𝑢)<0 by (i) and 𝐹(0)=0.

Corollary 2.3. In the case of 𝑝>2, all pairs (𝑤,𝑘)×, 𝑤>0, are eigenpairs of problem (1.10).

Proof. This immediately follows from (ii) of Theorem 2.2.

Now, we will prove that there are infinitely many solutions of (1.10) for all (𝑤,𝑘)×, 𝑤>0. For this, our main component will be Proposition 2.5 ([6], p. 324 Proposition  44.18) about free critical points of a functional, that is, about the solutions of the operator equation𝐹(𝑢)=0,𝑢𝑋.(2.9)

First, we will give some definitions, including the Palais-Smale (PS-condition) which are crucial in the theory of nonlinear eigenvalue problems (see [6, 7]).

Definition 2.4. Let 𝐹𝐶(𝑋,). 𝐹 satisfies the PS-condition at a point 𝑐 if each sequence 𝑢𝑛𝑋, such that 𝐹(𝑢𝑛)𝑐 and 𝐹(𝑢𝑛)0 in 𝑋 has a convergent subsequence.
Particularly, 𝐹 satisfies (PS) if and only if it satisfies the PS-condition for all 𝑐<0.

Let us denote by 𝒦𝑚 the class of all compact, symmetric, and zero-free subsets 𝐾 of 𝑋, such that gen𝐾𝑚. Here, gen𝐾 is defined as the smallest natural number 𝑛1 for which there exists an odd and continuous function 𝑓𝐾𝑛{0}. Let 𝑐𝑚=inf𝐾𝒦𝑚sup𝑢𝐾𝐹(𝑢),𝑚=1,2,.(2.10) Suppose that(𝐻1)𝑋 is a real 𝐵-space,(𝐻2)𝐹 is an even functional with 𝐹𝐶(𝑋,),(𝐻3)𝐹 satisfies (PS) with respect to 𝑋 and 𝐹(0)=0.

As mentioned earlier, our main component will be the following proposition.

Proposition 2.5. If (𝐻1), (𝐻2), and (𝐻3) hold and <𝑐𝑚<0, then 𝐹 has a pair of critical points (𝑢,𝑢) on 𝑋 such that 𝐹(±𝑢)=𝑐𝑚, to which solutions of (2.9) correspond. Moreover, if <𝑐𝑚=𝑐𝑚+1==𝑐𝑚+𝑝<0,𝑝1, then gen(crit𝑋,𝑐𝑚𝐹)𝑝+1, where crit𝑋,𝑐𝑚𝐹={𝑢𝑋𝐹(𝑢)=0,𝐹(𝑢)=𝑐𝑚}.

Now, we are ready to prove the following theorem.

Theorem 2.6. Let 𝑝>2. (a) For each (𝑤,𝑘),𝑤>0, problem (1.10) has an infinite number of nontrivial solutions.
(b) The set crit𝐹 is compact, where crit𝐹={𝑢𝑋𝐹(𝑢)=0}.

Proof. Our proof is based on the previous proposition. Clearly, conditions (𝐻1) and (𝐻2) are satisfied. The fact that 𝐹 satisfies (PS) is standard (see [5]). In our case 𝐹 satisfies PS-condition for all 𝑐. It needs to be demonstrated that 𝑐𝑚<0,𝑚=1,2,. By the definition of “𝑖𝑛𝑓𝑠𝑢𝑝”, it is adequate to show the existence of a set 𝐾𝒦𝑚 such that sup𝐾𝐹(𝑢)<0. We have 1𝐹(𝑢)=𝑝Ω||𝑘𝑢||𝑝𝑤𝑑𝑥2Ω𝑢2𝑑𝑥.(2.11) Let 𝑋𝑚 be an 𝑚-dimensional subspace of 𝑋 and 𝑆1 the unit sphere in 𝑋. We can choose 𝑢𝑋𝑚𝑆1 and define 𝐹(𝑡𝑢)=(𝑡𝑝/𝑝)(𝑤/2)𝑡2Ω𝑢2𝑑𝑥. As 𝑋𝑚𝑆1 is compact, inf𝑢𝑋𝑚𝑆1Ω𝑢2𝑑𝑥=𝛼(𝑚)>0. Hence, 𝑡𝐹(𝑡𝑢)𝑝𝑝𝑤2𝑡2𝛼(𝑚),𝑢𝑋𝑚𝑆1.(2.12) Moreover, lim𝑡0𝐹(𝑡𝑢)=0 and 𝐹(𝑡𝑢)<0 provided 0<𝑡<[(𝑝/2)𝑤𝛼(𝑚)]1/(𝑝2). Using this fact we obtain that for every 𝑚 there exist 𝜀𝑚>0 and 𝑡𝑚>0 such that 𝐹(𝑡𝑚𝑢)<𝜀𝑚 for all 𝑢𝑋𝑚𝑆1. Clearly 𝑡𝑚𝑢𝑆𝑡𝑚 and gen(𝑋𝑚𝑆𝑡𝑚)=𝑚. Now, set 𝐾=𝑋𝑚𝑆𝑡𝑚. Then sup𝐾𝐹(𝑢)𝜀𝑚<0. Consequently, inf𝐾𝒦𝑚sup𝑢𝐾𝐹(𝑢)<0. By Theorem 2.2  𝐹 is bounded below. Hence, <𝑐𝑚=inf𝐾𝒦𝑚sup𝑢𝐾𝐹(𝑢)<0,(2.13) and the statements of the theorem in (a) follow from Proposition 2.5.
(b) Let us prove that the set crit𝐹 is compact. Let 𝑢𝑛crit𝐹 be a sequence. Then 𝐹𝑢𝑛,𝑢𝑛=Ω||𝑘𝑢𝑛||𝑝𝑤𝑑𝑥2Ω𝑢2𝑛𝑑𝑥=0.(2.14) However, 𝐹𝑢𝑛=1𝑝Ω||𝑘𝑢𝑛||𝑝𝑤𝑑𝑥2Ω𝑢2𝑛1𝑑𝑥2𝐹𝑢𝑛,𝑢𝑛=0,(2.15) and by Theorem 2.2,𝐹 is bounded below. Thus, 𝐹(𝑢𝑛) is bounded, and consequently, it has a convergent subsequence (denoted again by 𝐹(𝑢𝑛)). We have 𝐹(𝑢𝑛)𝑐 and 𝐹(𝑢𝑛)=0. Hence, by PS-condition, 𝑢𝑛 has a convergent subsequence. The limit points of 𝑢𝑛 belong to crit𝐹 because, by PS-condition, the set crit𝐹 is closed.

The case 𝑝<2. This case is standard, and by similar methods that are given in [5] and earlier for the case 𝑝>2, one can establish the existence of nontrivial solutions of (1.10) for all (𝑤,𝑘), 𝑤>0.

Case 2 (𝜈0). We first look at the following problem: 𝑘2Ω||𝑘𝑢||𝑝2𝑢𝑣𝑑𝑥+Ω||𝑘𝑢||𝑝2𝑢𝑣𝑑𝑥𝜈Ω|𝑢|𝑝2𝑢𝑣𝑑𝑥=𝑤Ω𝑢𝑣𝑑𝑥,𝑢>0inΩ,(2.16) where 𝑢𝑊01,𝑝(Ω) and (2.16) holds for all 𝑣𝑊01,𝑝(Ω). The main result is as follows.

Proposition 2.7. Let 2<𝑛𝑝/(𝑛𝑝). Then (a) for all 𝜈<𝜈1(𝑘) there is a positive solution to problem (2.16), where 0<𝜈1(𝑘)=inf𝑢𝑊01,𝑝(Ω)𝑢0Ω|𝑘𝑢|𝑝/Ω|𝑢|𝑝𝑑𝑥,
(b) for each 𝜈 there is a number 𝑘 such that for all 𝑘>𝑘 problem (2.16) has a positive solution.

Proof. (a) As a result of the compact imbedding 𝑊01,𝑝(Ω)𝐿2(Ω) and the fact that Ω|𝑘𝑢|𝑝𝑑𝑥 is a norm in 𝑊01,𝑝(Ω), the functional 1𝐹(𝑢)=𝑝Ω||𝑘𝑢||𝑝𝜈𝑑𝑥𝑝Ω|𝑢|𝑝𝑑𝑥,(2.17) is coercive and lower semicontinuous on the weakly closed set 𝑀={𝑢Ω𝑢2=1}. From these properties, by using the condition 𝜈<𝜈1(𝑘) we obtain the existence of a nonnegative solution. The positivity follows from the maximum principle.(b) This fact follows from (a) and the relation 𝑘𝑝<𝜈1(𝑘) as 𝑘.

Note 1. The case 2=𝑛𝑝/(𝑛𝑝) is the critical case: lack of compactness, which is a subject that deserves a separate study.

Now, our concern is the following typical eigenvalue problem:𝑘2Ω||𝑘𝑢||𝑝2𝑢𝑣𝑑𝑥+Ω||𝑘𝑢||𝑝2𝑢𝑣𝑑𝑥=𝜈Ω|𝑢|𝑝2𝑢𝑣𝑑𝑥.(2.18)

Let us look at problem (2.18) with respect to 𝜈 for a fixed 𝑘. This problem is a typical eigenvalue problem. If 𝑘=0, then we get the 𝑝-Laplacian eigenvalue problem, and these questions have been studied by many authors (see [8, 9] and the references therein). Particularly, it has been shown in [8] that there is a sequence of “variational eigenvalues” which can be described by the Ljusternik-Schnirelman type variational principles. Our aim is to get the similar results for perturbed 𝑝-Laplacian eigenvalue problem (2.18). In our case 𝑘0, and we can apply two methods.

Method 1. For the Diriclet problem the norms: 𝑢=[Ω|𝑢|𝑝𝑑𝑥]1/𝑝, which is the standard norm in 𝑊01,𝑝(Ω), and 𝑢𝑘=[Ω|𝑘𝑢|𝑝𝑑𝑥]1/𝑝 are equivalent. Then it is enough to replace 𝑋,𝑢 by the Banach space 𝑋,𝑢𝑘 and follow the methods of [8, 9] to get the needed results.

Method 2. One can construct a Ljusternik-Schnirelman deformation (see [6, 7]) and check Palais-Smale condition for the functional 1𝐹(𝑢)=𝑝Ω|𝑢|𝑝𝑑𝑥(2.19) on the manifold 𝐺𝑘=𝑢Ω||𝑘𝑢||𝑝𝑑𝑥=1.(2.20) A such construction was given in our previous paper (see [10]). We follow our construction and just give the final result.

Theorem 2.8. For a fixed 𝑘, there exists a sequence of eigenvalues of problem (2.18), depending on 𝑘, which is given by 1𝜈𝑛(𝑘)=sup𝐾𝒦𝑛(𝑘)inf𝑢𝐾𝐹(𝑢).Moreover,𝜈𝑛(𝑘),as𝑛,(2.21) where one denotes by 𝒦𝑛(𝑘) the class of all compact, symmetric subsets 𝐾 of 𝐺𝑘, such that gen𝐾𝑛.

3. On the Neumann, No-Flux, Robin, and Steklov Boundary Value Problems

At the end of the paper we briefly discuss the other boundary problems, such as Neumman, No-flux, Robin, and Steklov. We note that all of the above given results are related to problem (1.3) with the Diriclet boundary condition; however the similar results are valid for the following boundary conditions too:

Neumann problem: 𝑤𝑢+𝑘2||𝑘𝑢||𝑝2||𝑢div𝑘𝑢||𝑝2𝑢=𝜈|𝑢|𝑞2𝑢,𝑝>1,𝜕𝑢|||𝜕𝑛𝜕Ω=0.(3.1)

No-flux problem:𝑤𝑢+𝑘2||𝑘𝑢||𝑝2||𝑢div𝑘𝑢||𝑝2𝑢=𝜈|𝑢|𝑞2𝑢|𝑢,𝑝>1,𝜕Ω=constant,𝜕Ω||𝑘𝑢||𝑝2𝜕𝑢𝜕𝑛𝑑𝑠=0.(3.2)

Robin problem: 𝑤𝑢+𝑘2||𝑘𝑢||𝑝2||𝑢div𝑘𝑢||𝑝2𝑢=𝜈|𝑢|𝑞2||𝑢,𝑝>1,𝑘𝑢||𝑝2𝜕𝑢𝜕𝑛+𝛽(𝑥)|𝑢|𝑝2𝑢|||𝜕Ω=0.(3.3)

Steklov problem:𝑤𝑢+𝑘2||𝑘𝑢||𝑝2||𝑢div𝑘𝑢||𝑝2𝑢=|𝑢|𝑞2||𝑢,𝑝>1,𝑘𝑢||𝑝2𝜕𝑢𝜕𝑛=𝜈|𝑢|𝑝2𝑢on𝜕Ω.(3.4)

Evidently, the energy space 𝑋 (the Banach space, we use in the critical point theory) for Dirichlet, Neuman, No-flux, Robin, and Steklov problems is 𝑊01,2(Ω), 𝑊1,2(Ω), 𝑊01,2(Ω), 𝑊1,2(Ω), and 𝑊1,2(Ω), respectively. In the case of 𝑤=0,𝑘=0,and 𝑝=𝑞 we obtain the standard eigenvalue problems for 𝑝-Laplacians, which have been studied in detail in [8] for all of the above given boundary value problems. Many results for standard 𝑝-Laplacians, including the regularity results, may be extended to the perturbed 𝑝-Laplacians by the similar techniques that are used in [8]. However, we omit these questions in this paper.

Our simple observation between Robin and Steklov problems is as follows: (𝑤,𝑘,𝜈) is an eigentriple for Steklov problem if and only if (𝑤,𝑘,1) is an eigentriple for Robin problem at 𝛽=𝜈.

Finally, we use a similar connection between the typical Robin and Steklov eigenvalue problems to prove the existence of negative eigenvalues for the Robin problem. For sake of simplicity we choose 𝑘=0 and consider the following standard eigenvalue problems for 𝑝-Laplacians:||||Robinproblem:div𝑢𝑝2𝑢=𝜈|𝑢|𝑝2||||𝑢,𝑝>1,𝑢𝑝2𝜕𝑢𝜕𝑛+𝛽|𝑢|𝑝2𝑢|||𝜕Ω||||=0,Steklovproblem:div𝑢𝑝2𝑢=|𝑢|𝑝2||||𝑢,𝑝>1,𝑢𝑝2𝜕𝑢𝜕𝑛=𝜈|𝑢|𝑝2𝑢on𝜕Ω.(3.5) Problems (3.5) can be rewritten in the following variational forms:Ω||||𝑢𝑝2𝑢𝑣𝑑𝑥+𝛽𝜕Ω|𝑢|𝑝2𝑢𝑣𝑑𝑠=𝜈Ω|𝑢|𝑝2𝑢𝑣,𝑢𝑊1,2(Ω),𝑣𝑊1,2(Ω),(3.6)Ω||||𝑢𝑝2𝑢𝑣𝑑𝑥+Ω|𝑢|𝑝2𝑢𝑣=𝜈𝜕Ω|𝑢|𝑝2𝑢𝑣𝑑𝑠,𝑢𝑊1,2(Ω),𝑣𝑊1,2(Ω),(3.7) respectively.

It is known that (see [8])(I)if 𝛽0 then the Robin problem has a sequence of positive eigenvalues 𝜈𝑛(𝛽) such that 𝜈𝑛(𝛽)+ as 𝑛;(II)the Steklov problem also has a sequence of positive eigenvalues 𝜈𝑛 such that 𝜈𝑛+ as 𝑛.

An Inverse Problem
Now let us be given 𝜈<0. Our question is as follows: for what values of 𝛽 the given number 𝜈<0 will be an eigenvalue for Robin problem (3.6). To answer this question we use a “duality principle” between Robin and Steklov problems and give the final result in the following theorem.

Theorem 3.1. For a given 𝜈<0 there exists a sequence 𝛽𝑛, such that the number 𝜈<0 will be an eigenvalue for the Robin problem at 𝛽=𝛽𝑛, 𝑛=1,2,. Moreover, 𝛽𝑛=𝜈𝑛 and 𝜈𝑛 are the eigenvalues of the Steklov problem.

Proof. The proof is based on the relations between the Robin and Steklov problems. To answer this question we consider the Steklov problem in the form ||||div𝑢𝑝2𝑢=𝛽|𝑢|𝑝2||||𝑢,𝑝>1,𝑢𝑝2𝜕𝑢𝜕𝑛=𝜈|𝑢|𝑝2𝑢on𝜕Ω.(3.8) Then the variational problem (3.7) is replaced by Ω||||𝑢𝑝2𝑢𝑣𝑑𝑥+𝛽Ω|𝑢|𝑝2𝑢𝑣=𝜈𝜕Ω|𝑢|𝑝2𝑢𝑣𝑑𝑠.(3.9) By comparing (3.6) and (3.9) we obtain that (𝛽,𝜈) is an eigenpair for the Steklov problem if and only if (𝜈,𝛽) is an eigenpair for the Robin problem. We know that (see [8]) for a positive number 𝛽 Steklov problem (3.9) has a sequence of positive eigenvalues 𝜈𝑛 such that 𝜈𝑛+ as 𝑛. Thus (𝛽,𝜈𝑛), 𝑛=1,2, are eigenpairs for (3.9). Then it follows that (𝜈𝑛,𝛽), 𝑛=1,2, are eigenpairs for the Robin problem. To end the proof we notice that 𝜈=𝛽 and 𝛽𝑛=𝜈𝑛.


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Copyright © 2011 Mahir Hasanov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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