On the Travelling Waves for the Generalized Nonlinear Schrödinger Equation

Abstract

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):𝑖𝑣_𝑡−div||∇𝑣||𝑣^𝑝−2∇|_𝜕𝑄𝑣=𝜈|𝑣|^𝑞−2𝑣,𝑝>1,=0,(1.1) where 𝑣∶=𝑣(𝑡,𝑥₁,𝑥₂,…,𝑥_𝑛+1) and 𝜈 is a parameter. 𝑄=ℝ×Ω is a cylinder, 𝜕𝑄 is the lateral boundary of 𝑄, 𝑡>0, 𝑥₁∈ℝ, and (𝑥₂,𝑥₃,…,𝑥_𝑛+1)∈Ω. Assume that Ω is a bounded domain in ℝ^𝑛 with the smooth boundary. Particularly, in the case of 𝑝=2 we get 𝑖𝑣_𝑡−Δ𝑣=𝜈|𝑣|^𝑞−2𝑣,(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,𝑞≠2), 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=𝑒^{𝑖(𝑤𝑡−𝑘𝑥)}𝑢(𝑥₂,…,𝑥_𝑛+1), where 𝑥∶=𝑥₁ and 𝑢 is a real-valued function. A simple computation yields 𝑣_𝑡=𝑖𝑤𝑒^{𝑖(𝑤𝑡−𝑘𝑥)}𝑢, 𝑣_𝑥=−𝑖𝑘𝑒^{𝑖(𝑤𝑡−𝑘𝑥)}𝑢, and 𝑣_𝑥𝑖=𝑒^{𝑖(𝑤𝑡−𝑘𝑥)}𝑢_𝑥𝑖, 𝑖=2,3,…,𝑛+1. Hence, ∇𝑣∶=(𝑣_𝑥,𝑣_𝑥2,…,𝑣_𝑥𝑛+1)=(−𝑖𝑘𝑢,∇𝑢)𝑒^{𝑖(𝑤𝑡−𝑘𝑥)} and |∇𝑣|=(𝑘²𝑢²+𝑢^2𝑥₂+⋯+𝑢^2𝑥_𝑛+1)^1/2. By using the notation ∇_𝑘𝑢(𝑥₂,𝑥₃,…,𝑥_𝑛+1)∶=(𝑘𝑢,𝑢_𝑥2,…,𝑢_𝑥𝑛+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||^𝑝−2𝑢−div𝑢|||∇_𝑘𝜕Ω𝑢||^𝑝−2∇𝑢=𝜈|𝑢|^𝑞−2𝑢,𝑝>1,=0.(1.3) In what follows, by shifting the variables 𝑥₂,…,𝑥_𝑛+1, we have used the following notations: 𝑢∶=𝑢(𝑥₁,…,𝑥_𝑛), |∇_𝑘𝑢|=(𝑘²𝑢²+𝑢^2𝑥₁+⋯+𝑢^2𝑥_𝑛)^1/2, and ∇𝑢=(𝑢_𝑥1,𝑢_𝑥2,…,𝑢_𝑥𝑛).

At this point we have to note that by searching for the standing waves 𝑣=𝑒^𝑖𝑤𝑡𝑢(𝑥₁,𝑥₂,…,𝑥_𝑛+1) for the NLS equation 𝑖𝑣_𝑡−Δ𝑣=𝜈|𝑣|^𝑞−2𝑣(1.4) we obtain the following eigenvalue problem: −𝑤𝑢−Δ𝑢=𝜈|𝑢|^𝑞−2𝑢.(1.5) On the other hand by setting the travelling wave solutions of the form 𝑣=𝑒^{𝑖(𝑤𝑡−𝑘𝑥)}𝑢(𝑥₂,…,𝑥_𝑛+1) into the NLS equation we obtain −𝑤+𝑘²𝑢−Δ𝑢=𝜈|𝑢|^𝑞−2𝑢.(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. 0≠𝑢∈𝑊^01,𝑝(Ω) is a solution of (1.3) if and only if −𝑤_Ω𝑢𝑣𝑑𝑥+𝑘²_Ω||∇_𝑘𝑢||^𝑝−2𝑢𝑣𝑑𝑥+_Ω||∇_𝑘𝑢||^𝑝−2∇𝑢⋅∇𝑣𝑑𝑥=𝜈_Ω|𝑢|^𝑞−2𝑢𝑣𝑑𝑥(1.7)

holds for all 𝑣∈𝑊^01,𝑝(Ω), where 𝑊^01,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 𝐹(𝑢)=−𝑤2_Ω𝑢²𝑑𝑥+1𝑝_Ω||∇_𝑘𝑢||^𝑝𝑑𝑥−𝜈𝑞_Ω𝐺|𝑢|^𝑞𝑑𝑥,_𝑘(𝑢)=1𝑝_Ω||∇_𝑘𝑢||^𝑝𝑑𝑥.(1.8) We set 𝑋∶=𝑊^01,𝑝(Ω). Then, 𝑢∈𝑋 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,forall𝑣∈𝑋, where 𝐹^∶𝑋→𝑋^∗ is the Fréchet derivative of 𝐹, 𝑋^∗ is the dual space, and ⟨𝐹^(𝑢),𝑣⟩ denotes the value of the functional 𝐹^(𝑢) at 𝑣∈𝑋. Indeed, the existence of Fréschet derivative implies the existence of directional (Gateaux) derivative. Using the definition of Gateaux derivative, we can obtain 𝐺^𝑘(𝑢),𝑣=𝑑𝑑𝑡𝐺^𝑘(𝑢+𝑡𝑣)|||_𝑡=0=𝑘²_Ω||∇_𝑘𝑢||^𝑝−2𝑢𝑣𝑑𝑥+_Ω||∇_𝑘𝑢||^𝑝−2∇𝑢⋅∇𝑣𝑑𝑥,(1.9) which is enough to see that (1.7) is the variational equation for the functional 𝐹(𝑢).

As 𝑢∈𝑊^01,𝑝(Ω), by Sobolev embedding theorems (see [4]) the functional 𝐹(𝑢) can be well defined if(i)𝑝≥𝑛 or(ii)1<𝑝<𝑛 and max{2,𝑞}≤𝑛𝑝/(𝑛−𝑝).

In the next section two cases 𝜈=0 and 𝜈≠0 will be studied separately. If 𝜈=0, then we may rewrite (1.7) in the form𝑘²_Ω||∇_𝑘𝑢||^𝑝−2𝑢𝑣𝑑𝑥+_Ω||∇_𝑘𝑢||^𝑝−2∇𝑢⋅∇𝑣𝑑𝑥=𝑤_Ω𝑢𝑣𝑑𝑥,(1.10) which is the equation for free critical points of the functional 𝐹(𝑢)=1𝑝_Ω||∇_𝑘𝑢||^𝑝𝑑𝑥−𝑤2_Ω𝑢²𝑑𝑥.(1.11) Evidently, there are not nontrivial solutions of (1.10) in the case of 𝑤≤0. Thus, 𝑤>0, and by the scaling property, we obtain that if 𝑝≠2 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 𝑊^01,𝑝(Ω) and ‖𝑢‖_𝑘∶=[∫_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 and𝜈≠0. 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 𝜈≠0 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.

2. The Structure of Eigenparameters 𝑤, 𝑘, 𝜈 and Related Eigenfunctions

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 and𝜈≠0.

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

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+𝑘²≤𝑤, 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 𝑤_𝑛(𝑘), such that 𝑤_𝑛(𝑘)→+∞ as 𝑛→∞,
(b) If 𝑝>2, then for a fixed (𝑤,𝑘)∈ℝ×ℝ, the solutions of (1.10) are bounded and ‖𝑢‖_𝑘≤𝐶_𝑝,𝑘(Ω)𝑤^1/(𝑝−2)(2.1) holds for some 𝐶_𝑝,𝑘(Ω)>0,
(c) Let 𝑝<2. In this case, one has ‖𝑢‖_𝑘≥1𝑤𝐶_𝑝,𝑘(Ω)^1/(𝑝−2)>0.(2.2)

Proof. The proof easily follows from (1.10), by using the Courant-Weyl variational principle 0<𝜆₁=inf_𝑢≠0∫_Ω||∇𝑢||²𝑑𝑥∫_Ω|𝑢|²𝑑𝑥,(2.3) the bounded embedding 𝑊^01,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)∈ℝ×ℝ, 𝑤>0.

Theorem 2.2. Let 𝑝>2 and (𝑤,𝑘)∈ℝ×ℝ, 𝑤>0. Then (i)1𝑝−21𝑤𝐶^𝑝,𝑘(Ω)^{𝑝/(𝑝−2)}≤inf_𝑋𝐹(𝑢)<0(2.4)for some 𝐶_𝑝,𝑘(Ω)>0.(ii)𝐹 attains its infimum at a nontrivial vector 𝑢₀∈𝑋.

Proof. One has 𝐹(𝑢)=(1/𝑝)∫_Ω|∇_𝑘𝑢|^𝑝𝑑𝑥−𝑤/2∫_Ω𝑢²𝑑𝑥. As 𝑝>2, the Poincaré inequality yields ∫_Ω𝑢²𝑑𝑥≤𝐶_𝑝,𝑘(Ω)‖𝑢‖^2𝑘. Hence, 𝐹(𝑢)=1𝑝‖𝑢‖^𝑝𝑘−𝑤2_Ω𝑢²𝑑𝑥≥1𝑝‖𝑢‖^𝑝𝑘−𝑤2𝐶^𝑝,𝑘(Ω)‖𝑢‖^2𝑘.(2.5) Let 𝑓(𝑥)=(1/𝑝)𝑥^𝑝−(𝑤/2)𝐶_𝑝,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=[𝑤𝐶_𝑝,𝑘(Ω)]^1/(𝑝−2). Now, by putting the vectors 𝑢 with ‖𝑢‖_𝑘=[𝑤𝐶_𝑝,𝑘(Ω)]^1/(𝑝−2) into 𝐹(𝑢), we obtain 1𝑝−21𝑤𝐶^𝑝,𝑘(Ω)^{𝑝/(𝑝−2)}≤inf_𝑋𝐹(𝑢).(2.6) Now, we will show that inf_𝑋𝐹(𝑢)<0. Let 𝑢 be a vector such that ‖𝑢‖_𝑘=1. Subsequently, by setting 𝑡𝑢 in 𝐹(𝑢), we obtain 𝐹(𝑡𝑢)=(𝑡^𝑝/𝑝)−(𝑤/2)𝑡²𝑐, where 𝑐=∫_Ω𝑢²𝑑𝑥. Thus 𝐹(𝑡𝑢)<0 if 0<𝑡<[(𝑝/2)𝑤𝑐]^1/(𝑝−2).
(ii) We have to show that inf_𝑋𝐹(𝑢) is attained. Let inf_𝑋𝐹(𝑢)=𝛼. Then, there exists a sequence 𝑢_𝑛∈𝑋 such that 𝐹(𝑢_𝑛)→𝛼 as 𝑛→∞. The sequence 𝑢_𝑛 should be bounded, because 𝐹(𝑢)≥1𝑝‖𝑢‖^𝑝𝑘−𝑤2𝐶^𝑝,𝑘(Ω)‖𝑢‖^2𝑘=‖𝑢‖^2𝑘1𝑝‖𝑢‖^𝑘𝑝−2−𝑤2𝐶^𝑝,𝑘(Ω)(2.7) and 𝐹(𝑢)→+∞ as ‖𝑢‖_𝑘→+∞, which means that 𝐹 is coercive. However, 𝑋=𝑊^01,𝑝(Ω) is a reflexive Banach space. Consequently, 𝑢_𝑛⇀𝑢₀ for some 𝑢₀∈𝑋, where “⇀” denotes the weak convergence in 𝑋. Evidently, 𝐹(𝑢) is sequentially lower semicontinuous, that is, 𝑢_𝑛⇀𝑢₀implies𝐹𝑢₀≤lim_𝑛→∞inf𝐹𝑢_𝑛.(2.8)
Indeed, ∫_Ω|∇_𝑘𝑢|^𝑝𝑑𝑥=‖𝑢‖^𝑝𝑘, 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 𝑊^01,𝑝(Ω)↪𝐿₂(Ω) is compact. Now, it follows from (2.8) that 𝐹(𝑢₀)≤𝛼, which means inf_𝑋𝐹(𝑢)=𝐹(𝑢₀). Finally, 𝑢₀≠0 because inf_𝑋𝐹(𝑢)<0 by (i) and 𝐹(0)=0.

Corollary 2.3. In the case of 𝑝>2, all pairs (𝑤,𝑘)∈ℝ×ℝ, 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)∈ℝ×ℝ, 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,𝑢∈𝑋.(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 𝐹∈𝐶^(𝑋,ℝ). 𝐹 satisfies the PS-condition at a point 𝑐∈ℝ if each sequence 𝑢_𝑛∈𝑋, such that 𝐹(𝑢_𝑛)→𝑐 and 𝐹^(𝑢_𝑛)→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 𝒦_𝑚 the class of all compact, symmetric, and zero-free subsets 𝐾 of 𝑋, such that gen𝐾≥𝑚. Here, gen𝐾 is defined as the smallest natural number 𝑛≥1 for which there exists an odd and continuous function 𝑓∶𝐾→ℝ^𝑛⧵{0}. Let 𝑐_𝑚=inf_{𝐾⊂𝒦𝑚}sup_𝑢∈𝐾𝐹(𝑢),𝑚=1,2,….(2.10) Suppose that(𝐻1)𝑋 is a real 𝐵-space,(𝐻2)𝐹 is an even functional with 𝐹∈𝐶^(𝑋,ℝ),(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 −∞<𝑐_𝑚<0, then 𝐹 has a pair of critical points (𝑢,−𝑢) on 𝑋 such that 𝐹(±𝑢)=𝑐_𝑚, to which solutions of (2.9) correspond. Moreover, if −∞<𝑐_𝑚=𝑐_𝑚+1=⋯=𝑐_𝑚+𝑝<0,𝑝≥1, then gen(crit_{𝑋,𝑐𝑚}𝐹)≥𝑝+1, where crit_{𝑋,𝑐𝑚}𝐹={𝑢∈𝑋∣𝐹^(𝑢)=0,𝐹(𝑢)=𝑐_𝑚}.

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∶={𝑢∈𝑋∣𝐹^(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∈ℝ. It needs to be demonstrated that 𝑐_𝑚<0,𝑚=1,2,…. By the definition of “𝑖𝑛𝑓𝑠𝑢𝑝”, it is adequate to show the existence of a set 𝐾∈𝒦_𝑚 such that sup_𝐾𝐹(𝑢)<0. We have 𝐹(𝑢)=1𝑝_Ω||∇_𝑘𝑢||^𝑝𝑑𝑥−𝑤2_Ω𝑢²𝑑𝑥.(2.11) Let 𝑋_𝑚 be an 𝑚-dimensional subspace of 𝑋 and 𝑆₁ the unit sphere in 𝑋. We can choose 𝑢∈𝑋_𝑚∩𝑆₁ and define 𝐹(𝑡𝑢)=(𝑡^𝑝/𝑝)−(𝑤/2)𝑡²∫_Ω𝑢²𝑑𝑥. As 𝑋_𝑚∩𝑆₁ is compact, inf_{𝑢∈𝑋𝑚∩𝑆1}∫_Ω𝑢²𝑑𝑥∶=𝛼(𝑚)>0. Hence, 𝐹(𝑡𝑢)≤𝑡^𝑝𝑝−𝑤2𝑡²𝛼(𝑚),∀𝑢∈𝑋_𝑚∩𝑆₁.(2.12) Moreover, lim_𝑡→0𝐹(𝑡𝑢)=0 and 𝐹(𝑡𝑢)<0 provided 0<𝑡<[(𝑝/2)𝑤𝛼(𝑚)]^1/(𝑝−2). Using this fact we obtain that for every 𝑚 there exist 𝜀_𝑚>0 and 𝑡_𝑚>0 such that 𝐹(𝑡_𝑚𝑢)<−𝜀_𝑚 for all 𝑢∈𝑋_𝑚∩𝑆₁. Clearly 𝑡_𝑚𝑢∈𝑆_𝑡𝑚 and gen(𝑋_𝑚∩𝑆_𝑡𝑚)=𝑚. Now, set 𝐾∶=𝑋_𝑚∩𝑆_𝑡𝑚. Then sup_𝐾𝐹(𝑢)≤−𝜀_𝑚<0. Consequently, inf_{𝐾⊂𝒦𝑚}sup_𝑢∈𝐾𝐹(𝑢)<0. By Theorem 2.2  𝐹 is bounded below. Hence, −∞<𝑐_𝑚=inf_{𝐾⊂𝒦𝑚}sup_𝑢∈𝐾𝐹(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 𝑢_𝑛∈crit𝐹 be a sequence. Then 𝐹^𝑢_𝑛,𝑢_𝑛=_Ω||∇_𝑘𝑢_𝑛||^𝑝𝑑𝑥−𝑤2_Ω𝑢^2𝑛𝑑𝑥=0.(2.14) However, 𝐹𝑢_𝑛=1𝑝_Ω||∇_𝑘𝑢_𝑛||^𝑝𝑑𝑥−𝑤2_Ω𝑢^2𝑛𝑑𝑥≤12𝐹^𝑢_𝑛,𝑢_𝑛=0,(2.15) and by Theorem 2.2,𝐹 is bounded below. Thus, 𝐹(𝑢_𝑛) is bounded, and consequently, it has a convergent subsequence (denoted again by 𝐹(𝑢_𝑛)). We have 𝐹(𝑢_𝑛)→𝑐 and 𝐹^(𝑢_𝑛)=0. Hence, by PS-condition, 𝑢_𝑛 has a convergent subsequence. The limit points of 𝑢_𝑛 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 (𝜈≠0). We first look at the following problem: 𝑘²_Ω||∇_𝑘𝑢||^𝑝−2𝑢𝑣𝑑𝑥+_Ω||∇_𝑘𝑢||^𝑝−2∇𝑢⋅∇𝑣𝑑𝑥−𝜈_Ω|𝑢|^𝑝−2𝑢𝑣𝑑𝑥𝑢=>𝑤0in_ΩΩ𝑢𝑣,𝑑𝑥,(2.16) where 𝑢∈𝑊^01,𝑝(Ω) and (2.16) holds for all 𝑣∈𝑊^01,𝑝(Ω). The main result is as follows.

Proposition 2.7. Let 2<𝑛𝑝/(𝑛−𝑝). Then (a) for all 𝜈<𝜈₁(𝑘) there is a positive solution to problem (2.16), where 0<𝜈₁(𝑘)=inf_𝑢∈𝑊^01,𝑝_(Ω)𝑢≠0∫_Ω|∇_𝑘𝑢|^𝑝/∫_Ω|𝑢|^𝑝𝑑𝑥,
(b) for each 𝜈∈ℝ 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 𝑊^01,𝑝(Ω)↪𝐿₂(Ω) and the fact that ∫_Ω|∇_𝑘𝑢|^𝑝𝑑𝑥 is a norm in 𝑊^01,𝑝(Ω), the functional 𝐹(𝑢)=1𝑝_Ω||∇_𝑘𝑢||^𝑝𝑑𝑥−𝜈𝑝_Ω|𝑢|^𝑝𝑑𝑥,(2.17) is coercive and lower semicontinuous on the weakly closed set 𝑀∶={𝑢∣∫_Ω𝑢²=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 𝑘^𝑝<𝜈₁(𝑘)→∞ as 𝑘→∞.

Note 1. The case 2=𝑛𝑝/(𝑛−𝑝) 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:𝑘²_Ω||∇_𝑘𝑢||^𝑝−2𝑢𝑣𝑑𝑥+_Ω||∇_𝑘𝑢||^𝑝−2∇𝑢⋅∇𝑣𝑑𝑥=𝜈_Ω|𝑢|^𝑝−2𝑢𝑣𝑑𝑥.(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 𝑘≠0, 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 𝑊^01,𝑝(Ω), and ‖𝑢‖_𝑘∶=[∫_Ω|∇_𝑘𝑢|^𝑝𝑑𝑥]^1/𝑝 are equivalent. Then it is enough to replace ⟨𝑋,‖𝑢‖⟩ by the Banach space ⟨𝑋,‖𝑢‖_𝑘⟩ 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 𝐹(𝑢)=1𝑝_Ω|𝑢|^𝑝𝑑𝑥(2.19) on the manifold 𝐺_𝑘=𝑢∣_Ω||∇_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∈ℝ, there exists a sequence of eigenvalues of problem (2.18), depending on 𝑘, which is given by 1𝜈_𝑛(𝑘)=sup_{𝐾⊂𝒦𝑛(𝑘)}inf_𝑢∈𝐾𝐹(𝑢).Moreover,𝜈_𝑛(𝑘)⟶∞,as𝑛⟶∞,(2.21) where one denotes by 𝒦_𝑛(𝑘) the class of all compact, symmetric subsets 𝐾 of 𝐺_𝑘, such that gen𝐾≥𝑛.

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||^𝑝−2𝑢−div𝜕𝑢||∇_𝑘𝑢||^𝑝−2∇𝑢=𝜈|𝑢|^𝑞−2𝑢,𝑝>1,𝜕𝑛|||_𝜕Ω=0.(3.1)

No-flux problem:−𝑤𝑢+𝑘𝑢²|||∇_𝑘𝑢||^𝑝−2𝑢−div||∇_𝑘𝑢||^𝑝−2∇𝑢=𝜈|𝑢|^𝑞−2𝑢,𝑝>1,_𝜕Ω=constant,_𝜕Ω||∇_𝑘𝑢||^𝑝−2𝜕𝑢𝜕𝑛𝑑𝑠=0.(3.2)

Robin problem: −𝑤𝑢+𝑘²||∇_𝑘𝑢||∇||^𝑝−2𝑢−div||∇_𝑘𝑢||^𝑝−2∇𝑢=𝜈|𝑢|^𝑞−2𝑢,𝑝>1,_𝑘𝑢||^𝑝−2𝜕𝑢𝜕𝑛+𝛽(𝑥)|𝑢|^𝑝−2𝑢|||_𝜕Ω=0.(3.3)

Steklov problem:−𝑤𝑢+𝑘²||∇_𝑘𝑢||∇||^𝑝−2𝑢−div||∇_𝑘𝑢||^𝑝−2∇𝑢=|𝑢|^𝑞−2𝑢,𝑝>1,_𝑘𝑢||^𝑝−2𝜕𝑢𝜕𝑛=𝜈|𝑢|^𝑝−2𝑢on𝜕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 𝑊^01,2(Ω), 𝑊^1,2(Ω), 𝑊^01,2(Ω)⊕ℝ, 𝑊^1,2(Ω), and 𝑊^1,2(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:Robinproblem||∇:𝑢−||d_𝑝−2iv||∇𝑢||^𝑝−2∇𝑢=𝜈|𝑢|^𝑝−2𝑢,𝑝>1,𝜕𝑢𝜕𝑛+𝛽|𝑢|^𝑝−2Steklovproblem:div||∇𝑢||_𝑝−2𝑢|||_𝜕Ω=0,||∇𝑢||_𝑝−2∇𝑢=|𝑢|^𝑝−2𝑢,𝑝>1,𝜕𝑢𝜕𝑛=𝜈|𝑢|^𝑝−2𝑢on𝜕Ω.(3.5) Problems (3.5) can be rewritten in the following variational forms:_Ω||∇𝑢||^𝑝−2∇𝑢𝑢∈⋅∇𝑊𝑣𝑑𝑥_1,2+𝛽_𝜕Ω|𝑢|^𝑝−2𝑢𝑣𝑑𝑠=𝜈_Ω|𝑢|^𝑝−2𝑢𝑣,(Ω),∀𝑣∈𝑊^1,2(Ω),(3.6)_Ω||∇𝑢||^𝑝−2∇𝑢𝑢∈⋅∇𝑊𝑣𝑑_1,2𝑥+_Ω|𝑢|^𝑝−2𝑢𝑣=𝜈_𝜕Ω|𝑢|^𝑝−2𝑢𝑣𝑑𝑠,(Ω),∀𝑣∈𝑊^1,2(Ω),(3.7) respectively.

It is known that (see [8])(I)if 𝛽≥0 then the Robin problem has a sequence of positive eigenvalues 𝜈_𝑛(𝛽) such that 𝜈_𝑛(𝛽)→+∞ as 𝑛→∞;(II)the Steklov problem also has a sequence of positive eigenvalues 𝜈_𝑛 such that 𝜈_𝑛→+∞ 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 𝛽_𝑛→−∞, such that the number 𝜈<0 will be an eigenvalue for the Robin problem at 𝛽=𝛽_𝑛, 𝑛=1,2,…. Moreover, 𝛽_𝑛=−𝜈_𝑛 and 𝜈_𝑛 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 div||∇||𝑢∇||𝑢||^𝑝−2_𝑝−2∇𝑢=𝛽|𝑢|^𝑝−2𝑢,𝑝>1,𝜕𝑢𝜕𝑛=𝜈|𝑢|^𝑝−2𝑢on𝜕Ω.(3.8) Then the variational problem (3.7) is replaced by _Ω||∇𝑢||^𝑝−2∇𝑢⋅∇𝑣𝑑𝑥+𝛽_Ω|𝑢|^𝑝−2𝑢𝑣=𝜈_𝜕Ω|𝑢|^𝑝−2𝑢𝑣𝑑𝑠.(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 𝜈_𝑛 such that 𝜈_𝑛→+∞ as 𝑛→∞. Thus (𝛽,𝜈_𝑛), 𝑛=1,2,… are eigenpairs for (3.9). Then it follows that (−𝜈_𝑛,−𝛽), 𝑛=1,2,… are eigenpairs for the Robin problem. To end the proof we notice that 𝜈=−𝛽 and 𝛽_𝑛=−𝜈_𝑛.