BY 4.0 license Open Access Published by De Gruyter December 20, 2016

On sign-changing solutions for (p,q)-Laplace equations with two parameters

Vladimir Bobkov ORCID logo and Mieko Tanaka

Abstract

We investigate the existence of nodal (sign-changing) solutions to the Dirichlet problem for a two-parametric family of partially homogeneous ( p , q ) -Laplace equations - Δ p u - Δ q u = α | u | p - 2 u + β | u | q - 2 u where p q . By virtue of the Nehari manifolds, the linking theorem, and descending flow, we explicitly characterize subsets of the ( α , β ) -plane which correspond to the existence of nodal solutions. In each subset the obtained solutions have prescribed signs of energy and, in some cases, exactly two nodal domains. The nonexistence of nodal solutions is also studied. Additionally, we explore several relations between eigenvalues and eigenfunctions of the p- and q-Laplacians in one dimension.

MSC 2010: 35J62; 35J20; 35P30

1 Introduction

In this article, we study the existence and nonexistence of sign-changing solutions for the problem

(${\mathrm{GEV};\alpha,\beta}$) { - Δ p u - Δ q u = α | u | p - 2 u + β | u | q - 2 u in  Ω , u = 0 on  Ω ,

where Ω N , N 1 , is a bounded domain with a sufficiently smooth boundary Ω , and α , β are parameters. The operator Δ r u := div ( | u | r - 2 u ) is the classical r-Laplacian, r = { q , p } > 1 , and without loss of generality we assume that q < p .

Boundary value problems with a combination of several differential operators of different nature (in particular, as in ((${\mathrm{GEV};\alpha,\beta}$))) arise mainly as mathematical models of physical processes and phenomena, and have been extensively studied in the last two decades; see, e.g., [15, 30, 19, 13] and the references below. Among the historically first examples one can mention the Cahn–Hilliard equation [12] describing the process of separation of binary alloys, and the Zakharov equation [33, (1.8)] which describes the behavior of plasma oscillations. Elliptic equations with the ( 2 , 6 ) - and ( 2 , p ) -Laplacians were considered explicitly in [8, 7] with the aim of obtaining soliton-type solutions (in particular, as a model for elementary particles).

The considered problem ((${\mathrm{GEV};\alpha,\beta}$)) attracts special attention due to its symmetric and partially homogeneous structure; cf. [31, 32, 28, 20, 34, 10, 4]. By developing the results of [31, 28, 20], the authors of the present article obtained in [10] a reasonably complete description of the subsets of the ( α , β ) -plane which correspond to the existence/nonexistence of positive solutions to problem ((${\mathrm{GEV};\alpha,\beta}$)). At the same time, to the best of our knowledge, analogous results for sign-changing solutions have not been obtained circumstantially so far, although a particular information on the existence can be extracted from [24, 32, 1]. The main reason for this is a crucial dependence of the structure of the solution set to problem ((${\mathrm{GEV};\alpha,\beta}$)) on parameters α and β. As a consequence, the existence can not be treated by a unique approach, and various tools have to be used for different parts of the ( α , β ) -plane.

The aim of the present article is to allocate and characterize the sets of parameters α and β for which problem ((${\mathrm{GEV};\alpha,\beta}$)) possesses or does not possess sign-changing solutions (see Figure 1). In this sense, this work can be seen as the second part of the article [10].

1.1 Notations and preliminaries

Before formulating the main results, we introduce several notations. In what follows, L r ( Ω ) with r ( 1 , + ) and L ( Ω ) stand for the Lebesgue spaces with the norms

u r := ( Ω | u | r 𝑑 x ) 1 / r and u := ess sup x Ω | u ( x ) | ,

respectively, and W 0 1 , r := W 0 1 , r ( Ω ) denotes the Sobolev space with the norm u r . For u W 0 1 , r we define u ± := max { ± u , 0 } . Note that u ± W 0 1 , r and u = u + - u - .

By a (weak) solution of ((${\mathrm{GEV};\alpha,\beta}$)) we mean function u W 0 1 , p which satisfies

(1.1) Ω | u | p - 2 u φ d x + Ω | u | q - 2 u φ d x = α Ω | u | p - 2 u φ 𝑑 x + β Ω | u | q - 2 u φ 𝑑 x

for all φ W 0 1 , p . If u is a solution of ((${\mathrm{GEV};\alpha,\beta}$)) and u ± 0 (a.e. in Ω), then u is called nodal or sign-changing solution. It is not hard to see that any solution of ((${\mathrm{GEV};\alpha,\beta}$)) is a critical point of the energy functional E α , β C 1 ( W 0 1 , p , ) defined by

E α , β ( u ) := 1 p H α ( u ) + 1 q G β ( u ) ,

where

H α ( u ) := Ω | u | p 𝑑 x - α Ω | u | p 𝑑 x and G β ( u ) := Ω | u | q 𝑑 x - β Ω | u | q 𝑑 x .

Notice that the supports of u + and u - are disjoint for any u W 0 1 , p . This fact, together with evenness of the functionals H α and G β , easily implies that

H α ( u + ) + H α ( u - ) = H α ( u ) and G β ( u + ) + G β ( u - ) = G β ( u ) .

Remark 1.

Any solution u W 0 1 , p of problem ((${\mathrm{GEV};\alpha,\beta}$)) belongs to C 0 1 , γ ( Ω ¯ ) for some γ ( 0 , 1 ) . In fact, u L ( Ω ) by the Moser iteration process; cf. [25, Appendix A]. Furthermore, the regularity up to the boundary in [21, Theorem 1] and [22, p. 320] provides u C 0 1 , γ ( Ω ¯ ) , γ ( 0 , 1 ) .

Next, we recall several facts related to the eigenvalue problem for the Dirichlet r-Laplacian, r > 1 . We say that λ is an eigenvalue of - Δ r , if the problem

(${\mathrm{EV};r,\lambda}$) { - Δ r u = λ | u | r - 2 u in  Ω , u = 0 on  Ω

has a nontrivial (weak) solution. Analogously to the linear case, the set of all eigenvalues of ((${\mathrm{EV};r,\lambda}$)) will be denoted as σ ( - Δ r ) . It is well known that the lowest positive eigenvalue λ 1 ( r ) can be obtained through the nonlinear Rayleigh quotient as (cf. [2])

(1.2) λ 1 ( r ) := inf { Ω | u | r 𝑑 x Ω | u | r 𝑑 x : u W 0 1 , r , u 0 } .

The eigenvalue λ 1 ( r ) is simple and isolated, and the corresponding eigenfunction φ r W 0 1 , p (defined up to an arbitrary multiplier) is strictly positive (or strictly negative) in Ω. Moreover, λ 1 ( r ) is the unique eigenvalue with a corresponding sign-constant eigenfunction [2]. Note also that any eigenfunction φ of - Δ r belongs to C 0 1 , γ ( Ω ¯ ) for some γ ( 0 , 1 ) .

The following lemma directly follows from the definition of λ 1 ( r ) and its simplicity.

Lemma 2.

Assume that u W 0 1 , p { 0 } . Then we have the following results:

  1. (i)

    Let α λ 1 ( p ) . Then H α ( u ) 0 , and H α ( u ) = 0 if and only if α = λ 1 ( p ) and u = t φ p for some t { 0 } .

  2. (ii)

    Let β λ 1 ( q ) . Then G β ( u ) 0 , and G β ( u ) = 0 if and only if β = λ 1 ( q ) and u = t φ q for some t { 0 } .

Although the structure of σ ( - Δ r ) is not completely known except for the case r = 2 or N = 1 (see, e.g., [17, Theorem 3.1]), several unbounded sequences of eigenvalues can be introduced by virtue of minimax variational principles. In what follows, by { λ k ( r ) } k we denote a sequence of eigenvalues for ((${\mathrm{EV};r,\lambda}$)) introduced in [18]. It can be described variationally as

(1.3) λ k ( r ) := inf h k ( r ) max z S k - 1 h ( z ) r r ,

where S k - 1 is the unit sphere in k and

(1.4) k ( r ) := { h C ( S k - 1 , S ( r ) ) : h  is odd } ,
S ( r ) := { u W 0 1 , r : u r = 1 } .

It is known [18] that λ k ( r ) + as k + . Moreover, λ 2 ( r ) coincides with the second eigenvalue of - Δ r , i.e.,

λ 2 ( r ) = inf { λ σ ( - Δ r ) : λ > λ 1 ( r ) } ,

and it can be alternatively characterized as in [16]:

(1.5) λ 2 ( r ) = inf γ Γ max s [ 0 , 1 ] γ ( s ) r r ,
Γ := { γ C ( [ 0 , 1 ] , S ( r ) ) : γ ( 0 ) = φ r , γ ( 1 ) = - φ r } ,

where the first eigenfunction φ r is normalized such that φ r S ( r ) . We denote any eigenfunction corresponding to λ 2 ( r ) as φ 2 , r . Notice that λ 2 ( r ) > λ 1 ( r ) . Furthermore, in the one-dimensional case the sequence (1.3) describes the whole σ ( - Δ r ) (cf. [17, Theorem 4.1], where this result is proved for the Krasnosel’skii-type eigenvalues).

Finally, we introduce the notation for the eigenspace of - Δ r at λ :

(1.6) ES ( r ; λ ) := { v W 0 1 , r : v  is a solution of (EV;  r λ ) } .

It is clear that ES ( r ; λ ) { 0 } if and only if λ σ ( - Δ r ) .

1.2 Main results

Let us state the main results of this article. We begin with the nonexistence of nodal solutions for ((${\mathrm{GEV};\alpha,\beta}$)).

Theorem 3.

Assume that

( α , β ) ( - , λ 2 ( p ) ] × ( - , λ 1 ( q ) ] ( - , λ 1 ( p ) ] × ( - , λ 2 ( q ) ] .

Then ((${\mathrm{GEV};\alpha,\beta}$)) has no nodal solutions.

In the one-dimensional case Theorem 3 can be refined as follows.

Theorem 4.

Let N = 1 . If ( α , β ) ( - , λ 2 ( p ) ] × ( - , λ 2 ( q ) ] , then ((${\mathrm{GEV};\alpha,\beta}$)) has no nodal solutions.

In the case of general dimensions an additional information on hypothetical nodal solutions to ((${\mathrm{GEV};\alpha,\beta}$)) for α ( λ 1 ( p ) , λ 2 ( p ) ] and β ( λ 1 ( q ) , λ 2 ( q ) ] is given in Lemma 5 below.

Figure 1 
                  The case 
                        
                           
                              
                                 
                                    
                                       λ
                                       2
                                    
                                    ⁢
                                    
                                       (
                                       q
                                       )
                                    
                                 
                                 <
                                 
                                    
                                       λ
                                       3
                                    
                                    ⁢
                                    
                                       (
                                       q
                                       )
                                    
                                 
                              
                           
                           
                           {\lambda_{2}(q)<\lambda_{3}(q)}
                        
                     , 
                        
                           
                              
                                 
                                    
                                       λ
                                       2
                                    
                                    ⁢
                                    
                                       (
                                       p
                                       )
                                    
                                 
                                 <
                                 
                                    
                                       λ
                                       3
                                    
                                    ⁢
                                    
                                       (
                                       p
                                       )
                                    
                                 
                              
                           
                           
                           {\lambda_{2}(p)<\lambda_{3}(p)}
                        
                     , and 
                        
                           
                              
                                 
                                    
                                       (
                                       
                                          
                                             λ
                                             2
                                          
                                          ⁢
                                          
                                             (
                                             p
                                             )
                                          
                                       
                                       ,
                                       
                                          
                                             λ
                                             3
                                          
                                          ⁢
                                          
                                             (
                                             p
                                             )
                                          
                                       
                                       )
                                    
                                    ∩
                                    
                                       σ
                                       ⁢
                                       
                                          (
                                          
                                             -
                                             
                                                Δ
                                                p
                                             
                                          
                                          )
                                       
                                    
                                 
                                 =
                                 ∅
                              
                           
                           
                           {(\lambda_{2}(p),\lambda_{3}(p))\cap\sigma(-\Delta_{p})=\emptyset}
                        
                     . Existence (light gray, solid lines),nonexistence (dark gray, zigzag lines), unknown (white, dashed lines).

Figure 1

The case λ 2 ( q ) < λ 3 ( q ) , λ 2 ( p ) < λ 3 ( p ) , and ( λ 2 ( p ) , λ 3 ( p ) ) σ ( - Δ p ) = . Existence (light gray, solid lines),nonexistence (dark gray, zigzag lines), unknown (white, dashed lines).

Now we formulate the existence result for nodal solutions with a positive energy. Let us define the following “lower” critical value depending on α :

(1.7) β ( α ) := inf { min { Ω | u + | q 𝑑 x Ω | u + | q 𝑑 x , Ω | u - | q 𝑑 x Ω | u - | q 𝑑 x } : u ( α ) } ,

where

(1.8) ( α ) := { u W 0 1 , p : u ± 0 , max { Ω | u + | p 𝑑 x Ω | u + | p 𝑑 x , Ω | u - | p 𝑑 x Ω | u - | p 𝑑 x } α } ,

and put β ( α ) = + whenever the admissible set ( α ) is empty.

Theorem 5.

Let α > λ 2 ( p ) . Then for all β < β L ( α ) problem ((${\mathrm{GEV};\alpha,\beta}$)) has a nodal solution u with E α , β ( u ) > 0 and precisely two nodal domains.

Several main properties of the function β ( α ) are collected in Lemma 11 below. Let us remark that the parametrization by α in (1.7) is different from the parametrization by s of the form ( α , β ) = ( λ + s , λ ) which was used in [10] in order to construct a critical curve for the existence of positive solutions. In the context of the present article, the parametrization by α makes problem ((${\mathrm{GEV};\alpha,\beta}$)) easier to analyze. We also note that (1.7) is conceptually similar to the characterization of the first nontrivial curve of the Fučík spectrum given in [26]. In Section 2 below, we introduce and study several other critical points besides (1.7), which although are not directly used in the proofs of the main results, increase the understanding of the construction of the ( α , β ) -plane, and could be employed in further investigations.

Next, we state the existence of negative energy nodal solutions for ((${\mathrm{GEV};\alpha,\beta}$)). Consider the “upper” critical value

(1.9) β 𝒰 * ( α ) := sup { Ω | φ | q 𝑑 x Ω | φ | q 𝑑 x : φ ES ( p ; α ) { 0 } } ,

where α , and set β 𝒰 * ( α ) = - provided α σ ( - Δ p ) . Several lower and upper bounds for β 𝒰 * ( α ) are given in Lemmas 6 and 7 below. Define k α := min { k : α < λ k + 1 ( p ) } and notice that λ k α + 1 ( q ) λ 2 ( q ) for all α .

Theorem 6.

Let α R σ ( - Δ p ) ¯ . Then for all β > max { β U * ( α ) , λ k α + 1 ( q ) } problem ((${\mathrm{GEV};\alpha,\beta}$)) has a nodal solution u satisfying E α , β ( u ) < 0 .

Evidently, if σ ( - Δ p ) is a discreet set (as it is for p = 2 or N = 1 ), then σ ( - Δ p ) ¯ = . Moreover, λ 1 ( p ) and λ 2 ( p ) belong to σ ( - Δ p ) ¯ for all p > 1 and N 1 since λ 1 ( p ) is isolated and there are no eigenvalues between λ 1 ( p ) and λ 2 ( p ) (see Section 1.1).

One of the main ingredients for the proof of Theorem 6 is the result on the existence of three nontrivial solutions (positive, negative and sign-changing) to the problem with the ( p , q ) -Laplacian and a nonlinearity in the general form given by Theorem 13 below. This result is of independent interest.

Theorem 6 can be refined as follows.

Theorem 7.

Assume that

( α , β ) ( - , λ 2 ( p ) ) × ( λ 2 ( q ) , + ) { ( λ 1 ( p ) , φ p q q / φ p q q ) } .

Then ((${\mathrm{GEV};\alpha,\beta}$)) has a nodal solution u satisfying E α , β ( u ) < 0 .

Remark 8.

In the one-dimensional case we have φ p q q / φ p q q < λ 2 ( q ) (see Lemma 2 in Appendix A), and hence the assertion of Theorem 7 holds for all ( α , β ) ( - , λ 2 ( p ) ) × ( λ 2 ( q ) , + ) .

Let us note that unlike the case of positive solutions, the structure of the set of nodal solutions for problem ((${\mathrm{GEV};\alpha,\beta}$)) is more complicated, and we are not aware of the maximality of the regions obtained in Theorems 5 and 6.

The article is organized as follows: In Section 2, we apply the method of the Nehari manifold in order to prove Theorem 5. In Section 3, by means of linking arguments and the descending flow method, we provide two general existence results which yield, in particular, Theorems 6 and 7. For the convenience of the reader we collect the proofs of the main theorems in Section 4. In Appendix A, we prove several additional facts on the relation between eigenvalues and eigenfunctions of the p- and q-Laplacians in the one-dimensional case. Finally, in Appendix B, we give a sketch of the proof of Theorem 13.

2 Nodal solutions with positive energy

The classical Nehari manifold for problem ((${\mathrm{GEV};\alpha,\beta}$)) is defined by

𝒩 α , β := { u W 0 1 , p { 0 } : E α , β ( u ) , u = H α ( u ) + G β ( u ) = 0 } .

It can be readily seen that 𝒩 α , β contains all nontrivial solutions of ((${\mathrm{GEV};\alpha,\beta}$)). On the other hand, if u W 0 1 , p is a sign-changing solution of ((${\mathrm{GEV};\alpha,\beta}$)), then

0 = E α , β ( u ) , u + = E α , β ( u + ) , u + = H α ( u + ) + G β ( u + ) ,
0 = - E α , β ( u ) , u - = E α , β ( u - ) , u - = H α ( u - ) + G β ( u - ) .

These equalities bring us to the definition of the so-called nodal Nehari set for ((${\mathrm{GEV};\alpha,\beta}$)):

(2.1) α , β := { u W 0 1 , p : u ± 0 , H α ( u ± ) + G β ( u ± ) = 0 } = { u W 0 1 , p : u ± 𝒩 α , β } .

By construction, α , β contains all sign-changing solutions of ((${\mathrm{GEV};\alpha,\beta}$)), and hence α , β 𝒩 α , β .

Let us divide α , β into the following three subsets:

α , β 1 := { u α , β : H α ( u + ) < 0 , H α ( u - ) < 0 } ,
α , β 2 := { u α , β : H α ( u + ) > 0 , H α ( u - ) > 0 } ,
α , β 3 := { u α , β : H α ( u + ) H α ( u - ) 0 } .

Evidently, α , β = α , β 1 α , β 2 α , β 3 and all α , β i are mutually disjoint. The main aim of this section is to prove the existence of nodal solutions for ((${\mathrm{GEV};\alpha,\beta}$)) through minimization of E α , β over α , β 1 in an appropriate subset of the ( α , β ) -plane.

2.1 Preliminary analysis

In this subsection, we mainly study the properties of the sets α , β 1 , α , β 2 , and α , β 3 . First of all, we give the following auxiliary lemma, which is in fact analogous to [10, Proposition 6] and can be proved in the same manner.

Lemma 1.

Let u W 0 1 , p . If H α ( u ) G β ( u ) < 0 , then there exists a unique critical point t ( u ) > 0 of E α , β ( t u ) with respect to t > 0 and t ( u ) u N α , β . In particular, if

H α ( u ) < 0 < G β ( u ) ,

then t ( u ) is the unique maximum point of E α , β ( t u ) with respect to t > 0 and E α , β ( t ( u ) u ) > 0 .

We start our consideration of the sets α , β i with several simple facts.

Lemma 2.

Let α , β R . The following hold:

  1. (i)

    If β λ 1 ( q ) , then α , β 1 = α , β and, consequently, α , β 2 , α , β 3 = .

  2. (ii)

    If α λ 1 ( p ) , then α , β 2 = α , β and, consequently, α , β 1 , α , β 3 = .

Proof.

Let us first prove assertion (i). Assume that β λ 1 ( q ) and w α , β . Then Lemma 2 implies that G β ( w ± ) 0 and in fact G β ( w ± ) > 0 , since otherwise w ± = φ q , which is impossible in view of the strict positivity of φ q in Ω. Thus, the Nehari constraints H α ( w ± ) + G β ( w ± ) = 0 yield H α ( w ± ) < 0 , whence w α , β 1 . Assertion (ii) can be shown by the same arguments. ∎

Let us introduce the following sets:

(2.2) 1 ( α ) := { u W 0 1 , p : H α ( u + ) < 0 , H α ( u - ) < 0 } ,
(2.3) 2 ( α ) := { u W 0 1 , p : H α ( u + ) > 0 , H α ( u - ) > 0 } .

Obviously, α , β 1 1 ( α ) and α , β 2 2 ( α ) . Moreover, we have the following result.

Lemma 3.

Let α , β R . The following hold:

  1. (i)

    If α λ 2 ( p ) , then 1 ( α ) = and, consequently, α , β 1 = .

  2. (ii)

    If β λ 2 ( q ) , then 2 ( α ) = and, consequently, α , β 2 = .

Proof.

We give the proof of assertion (i). The second part can be proved analogously. Suppose, by contradiction, that α λ 2 ( p ) and there exists w 1 ( α ) . These assumptions read as

(2.4) max { Ω | w + | p 𝑑 x Ω | w + | p 𝑑 x , Ω | w - | p 𝑑 x Ω | w - | p 𝑑 x } < α λ 2 ( p ) .

On the other hand, it is shown in [9, Proposition 4.2] that the second eigenvalue λ 2 ( r ) , r > 1 , can be characterized as follows:

(2.5) λ 2 ( r ) = inf { max { Ω | u + | r 𝑑 x Ω | u + | r 𝑑 x , Ω | u - | r 𝑑 x Ω | u - | r 𝑑 x } : u W 0 1 , r , u ± 0 } .

Comparing (2.4) and (2.5) (with r = p ), we obtain a contradiction. ∎

Lemmas 2 and 3 readily entail the following information about the emptiness of α , β and, consequently, the nonexistence of nodal solutions for ((${\mathrm{GEV};\alpha,\beta}$)).

Lemma 4.

If α λ 2 ( p ) and β λ 1 ( q ) , or α λ 1 ( p ) and β λ 2 ( q ) , then M α , β = .

Lemma 5.

Let α λ 2 ( p ) and β λ 2 ( q ) . If u is a nodal solution of ((${\mathrm{GEV};\alpha,\beta}$)), then

α > λ 1 ( p ) , β > λ 1 ( q ) , u α , β 3 .

Let us now subsequently treat the emptiness and nonemptiness of α , β 1 and α , β 2 . First we consider α , β 1 . Introduce the critical value

β 1 ( α ) := sup { min { Ω | u + | q 𝑑 x Ω | u + | q 𝑑 x , Ω | u - | q 𝑑 x Ω | u - | q 𝑑 x } : u 1 ( α ) }

for each α , where the admissible set 1 ( α ) is defined by (2.2), or, equivalently,

1 ( α ) = { u W 0 1 , p : u ± 0 , max { Ω | u + | p 𝑑 x Ω | u + | p 𝑑 x , Ω | u - | p 𝑑 x Ω | u - | p 𝑑 x } < α } .

We assume that β 1 ( α ) = - whenever 1 ( α ) is empty. Consider also

(2.6) β 1 * := sup { min { Ω | φ + | q 𝑑 x Ω | φ + | q 𝑑 x , Ω | φ - | q 𝑑 x Ω | φ - | q 𝑑 x } : φ ES ( p , λ 2 ( p ) ) { 0 } } ,

where ES ( p , λ 2 ( p ) ) is the eigenspace of the second eigenvalue λ 2 ( p ) defined by (1.6).

The main properties of β 1 ( α ) are collected in the following lemma.

Lemma 6.

The following assertions hold:

  1. (i)

    β 1 ( α ) = - for any α λ 2 ( p ) , and β 1 ( α ) [ β 1 * , + ) for all α > λ 2 ( p ) .

  2. (ii)

    β 1 ( α ) is nondecreasing for α ( λ 2 ( p ) , + ) .

  3. (iii)

    β 1 ( α ) is left-continuous for α ( λ 2 ( p ) , + ) .

  4. (iv)

    β 1 ( α ) + as α + .

  5. (v)

    α , β 1 if and only if α > λ 2 ( p ) and β < β 1 ( α ) .

Proof.

(i) If α λ 2 ( p ) , then 1 ( α ) = in view of Lemma 3, and hence β 1 ( α ) = - . On the other hand, if α > λ 2 ( p ) , then any second eigenfunction φ 2 , p satisfies H α ( φ 2 , p ± ) < 0 and, in consequence, it belongs to 1 ( α ) . This implies that ES ( p , λ 2 ( p ) ) { 0 } 1 ( α ) and β 1 ( α ) β 1 * .

Consider the set

(2.7) X ( α ) := { v W 0 1 , p : v p p α v p p } .

It is known that for any α there exists C ( α ) > 0 such that v p C ( α ) v q for all v X ( α ) ; see [31, Lemma 9]. Therefore, since u ± X ( α ) for any u 1 ( α ) , the Hölder inequality yields the existence of a constant C 1 > 0 such that

C 1 u ± q u ± p C ( α ) u ± q for all  u 1 ( α ) ,

which gives the boundedness of β 1 ( α ) from above.

(ii) If λ 2 ( p ) < α 1 α 2 , then 1 ( α 1 ) 1 ( α 2 ) , which implies the desired monotonicity.

(iii) Let us fix an arbitrary α 0 > λ 2 ( p ) . Since assertion (ii) readily leads to lim α α 0 - 0 β 1 ( α ) β 1 ( α 0 ) , it is enough to show that lim α α 0 - 0 β 1 ( α ) β 1 ( α 0 ) . By the definition of β 1 ( α 0 ) , for any ε > 0 there exists u ε 1 ( α 0 ) such that

(2.8) β 1 ( α 0 ) - ε min { Ω | u ε + | q 𝑑 x Ω | u ε + | q 𝑑 x , Ω | u ε - | q 𝑑 x Ω | u ε - | q 𝑑 x } .

Recalling that H α 0 ( u ε ± ) < 0 , we can find δ = δ ( ε ) > 0 such that H α ( u ε ± ) < 0 for any α ( α 0 - δ , α 0 ] . Therefore, u ε 1 ( α ) , and for all α ( α 0 - δ , α 0 ] the definition of β 1 ( α ) leads to

(2.9) min { Ω | u ε + | q 𝑑 x Ω | u ε + | q 𝑑 x , Ω | u ε - | q 𝑑 x Ω | u ε - | q 𝑑 x } β 1 ( α ) .

Combining (2.8) and (2.9), we obtain the inequality lim α α 0 - 0 β 1 ( α ) β 1 ( α 0 ) , since ε > 0 is arbitrary.

(iv) Let L > λ 1 ( q ) be an arbitrary positive constant. Recalling that for the variational eigenvalues λ k ( q ) there holds λ k ( q ) + as k + , we can find k L 2 such that λ k L ( q ) > L . Take an eigenfunction φ corresponding to λ k L ( q ) . Since φ C 0 1 , γ ( Ω ¯ ) and φ changes its sign in Ω (see Section 1.1), there exists α L satisfying

max { Ω | φ + | p 𝑑 x Ω | φ + | p 𝑑 x , Ω | φ - | p 𝑑 x Ω | φ - | p 𝑑 x } < α L .

Therefore, φ 1 ( α L ) , and from the definition of β 1 ( α L ) and its monotonicity it follows that

β 1 ( α ) β 1 ( α L ) min { Ω | φ + | q 𝑑 x Ω | φ + | q 𝑑 x , Ω | φ - | q 𝑑 x Ω | φ - | q 𝑑 x } = λ k L ( q ) > L

provided α α L . Since L can be chosen arbitrary large, we conclude that lim α + β 1 ( α ) = + .

(v) If α > λ 2 ( p ) and β < β 1 ( α ) , then, by the definition of β 1 ( α ) , there exists u 1 ( α ) such that

(2.10) β < min { Ω | u + | q 𝑑 x Ω | u + | q 𝑑 x , Ω | u - | q 𝑑 x Ω | u - | q 𝑑 x } β 1 ( α ) .

This means that H α ( u ± ) < 0 and G β ( u ± ) > 0 . Hence, by Lemma 1 we obtain t ± > 0 such that t ± u ± 𝒩 α , β , whence t + u + - t - u - α , β 1 .

Suppose now that there exists u α , β 1 for some α , β . Lemma 3 implies that α > λ 2 ( p ) . On the other hand, u α , β 1 1 ( α ) . Hence, from the Nehari constraints it follows that G β ( u ± ) > 0 , and we arrive to (2.10). ∎

Consider now the set α , β 2 . The corresponding critical value, parametrized again by α , appears to be the following:

β 2 ( α ) := inf { max { Ω | u + | q 𝑑 x Ω | u + | q 𝑑 x , Ω | u - | q 𝑑 x Ω | u - | q 𝑑 x } : u 2 ( α ) } ,

where the admissible set 2 ( α ) is defined by (2.3).

The main properties of β 2 ( α ) are similar to those for β 1 ( α ) and collected in the following lemma.

Lemma 7.

The following assertions hold:

  1. (i)

    β 2 ( α ) [ λ 2 ( q ) , + ) for any α .

  2. (ii)

    β 2 ( α ) is nondecreasing for α , and β 2 ( α ) = β 2 ( λ 1 ( p ) ) = λ 2 ( q ) for α λ 1 ( p ) .

  3. (iii)

    β 2 ( α ) is right-continuous for α .

  4. (iv)

    α , β 2 if and only if α and β > β 2 ( α ) .

Proof.

(i) It is easy to see that for any α the admissible set 2 ( α ) is nonempty. For example, any eigenfunction corresponding to λ σ ( - Δ p ) belongs to 2 ( α ) provided λ > max { α , λ 1 ( p ) } . Hence, β 2 ( α ) < + . On the other hand, the definition of β 2 ( α ) and characterization (2.5) with r = q directly imply that β 2 ( α ) λ 2 ( q ) for any α since 2 ( α ) W 0 1 , p W 0 1 , q .

(ii) If α 1 α 2 , then 2 ( α 2 ) 2 ( α 1 ) , which leads to the desired monotonicity. Since any sign-changing function w W 0 1 , p satisfies H λ 1 ( p ) ( w ± ) > 0 (see Lemma 2), we get 2 ( α ) = 2 ( λ 1 ( p ) ) = { u W 0 1 , p : u ± 0 } for all α λ 1 ( p ) , and hence β 2 ( α ) = β 2 ( λ 1 ( p ) ) for all α λ 1 ( p ) . In order to show that β 2 ( λ 1 ( p ) ) = λ 2 ( q ) , let us recall that any eigenfunction φ 2 , q corresponding to λ 2 ( q ) belongs to C 0 1 , γ ( Ω ¯ ) (see Section 1.1). Hence, φ 2 , q 2 ( λ 1 ( p ) ) and, consequently,

λ 2 ( q ) = max { Ω | φ 2 , q + | q 𝑑 x Ω | φ 2 , q + | q 𝑑 x , Ω | φ 2 , q - | q 𝑑 x Ω | φ 2 , q - | q 𝑑 x } β 2 ( λ 1 ( p ) ) λ 2 ( q ) ,

where the equality follows from (2.5) with r = q , and the last inequality is given by assertion (i).

Assertions (iii) and (iv) can be proved in much the same way as in Lemma 6. ∎

For the further proof of the existence of nodal solutions to ((${\mathrm{GEV};\alpha,\beta}$)) in α , β 1 , let us study the properties of the critical value (1.7) defined as

β ( α ) := inf { min { Ω | u + | q 𝑑 x Ω | u + | q 𝑑 x , Ω | u - | q 𝑑 x Ω | u - | q 𝑑 x } : u ( α ) } ,

where the admissible set ( α ) is given by (1.8), or, equivalently,

( α ) = { u W 0 1 , p : u ± 0 , H α ( u + ) 0 , H α ( u - ) 0 } .

We put β ( α ) = + whenever ( α ) = . Arguing as in the proof of Lemma 3, it can be shown that ( α ) = if and only if α < λ 2 ( p ) . Note that α , β 1 1 ( α ) ( α ) .

First we give two auxiliary results.

Lemma 8.

Let α > 0 , β R , and { u n } n N be an arbitrary sequence in B L ( α ) (or in M α , β 1 ). Denote by { v n } n N a sequence normalized as follows:

(2.11) v n := u n + u n + p - u n - u n - p , n .

Then the following assertions hold:

  1. (i)

    v n ( α ) (or v n α , β 1 ) for all n .

  2. (ii)

    v n converges, up to a subsequence, to some v 0 W 0 1 , p weakly in W 0 1 , p and strongly in L p ( Ω ) .

  3. (iii)

    v 0 ± 0 and H α ( v 0 ± ) 0 , that is, v 0 ( α ) .

Proof.

Obviously, v n ± = u n ± / u n ± p , and hence assertion (i) follows from the p-homogeneity of H α . Assertion (ii) is a consequence of the boundedness of { v n } n in W 0 1 , p . Since H α ( v n ± ) 0 for all n , we get v n ± p p 1 / α , whence v 0 ± 0 a.e. in Ω, due to the strong convergence of v n in L p ( Ω ) . Moreover, using the weak lower semicontinuity of the W 0 1 , p -norm, we conclude that H α ( v 0 ± ) lim inf n + H α ( v n ± ) 0 . This is assertion (iii). ∎

Proposition 9.

For any α λ 2 ( p ) there exists a minimizer u α B L ( α ) of β L ( α ) .

Proof.

If α λ 2 ( p ) , then ( α ) is nonempty, since H α ( φ 2 , p ± ) 0 for any second eigenfunction φ 2 , p corresponding to λ 2 ( p ) . Thus, there exists a minimizing sequence { u n } n ( α ) for β ( α ) . Consider the corresponding normalized sequence { v n } n ( α ) given by (2.11). Lemma 8 implies that the limit point v 0 ( α ) , and hence

β ( α ) min { Ω | v 0 + | q 𝑑 x Ω | v 0 + | q 𝑑 x , Ω | v 0 - | q 𝑑 x Ω | v 0 - | q 𝑑 x } lim inf n + min { Ω | v n + | q 𝑑 x Ω | v n + | q 𝑑 x , Ω | v n - | q 𝑑 x Ω | v n - | q 𝑑 x } = β ( α ) ,

which means that v 0 is a minimizer of β ( α ) . ∎

Remark 10.

The definition (1.7) of β ( α ) is equivalent to

(2.12) β ( α ) := inf { Ω | u + | q 𝑑 x Ω | u + | q 𝑑 x : u ( α ) } .

This can be seen by testing β ( α ) either with the corresponding minimizer u α or with - u α .

Consider now the critical value

(2.13) β * := inf { Ω | φ + | q 𝑑 x Ω | φ + | q 𝑑 x : φ ES ( p , λ 2 ( p ) ) { 0 } } .

The following lemma contains the main properties of β ( α ) .

Lemma 11.

The following assertions hold:

  1. (i)

    β ( α ) = + for any α < λ 2 ( p ) , and β ( α ) ( λ 1 ( q ) , β * ] for any α λ 2 ( p ) .

  2. (ii)

    β ( α ) is nonincreasing for α [ λ 2 ( p ) , + ) .

  3. (iii)

    β ( α ) is right-continuous for α [ λ 2 ( p ) , + ) .

  4. (iv)

    𝒦 α , β if and only if α λ 2 ( p ) and β β ( α ) , where 𝒦 α , β is defined by

    (2.14) 𝒦 α , β := { u W 0 1 , p : u ± 0 , H α ( u + ) 0 , H α ( u - ) 0 , G β ( u + ) 0 }
    = ( α ) { u W 0 1 , p : G β ( u + ) 0 } .

Proof.

(i) As stated in the proof of Lemma 3, we easily see that ( α ) = for all α < λ 2 ( p ) , and hence β ( α ) = + . If α λ 2 ( p ) , then ES ( p , λ 2 ( p ) ) { 0 } ( α ) , and using (2.12), we obtain that β ( α ) β * . Since any sign-changing function w W 0 1 , p satisfies w ± q q > λ 1 ( q ) w ± q q (see Lemma 2), taking a minimizer u α of β ( α ) (see Proposition 9), we conclude that

β ( α ) = u α + q q / u α + q q > λ 1 ( q ) for all  α λ 2 ( p ) .

Assertion (ii) can be proved as in Lemma 6.

(iii) Due to assertion (ii), it is sufficient to show that β ( α 0 ) lim α α 0 + 0 β ( α ) for all α 0 λ 2 ( p ) . Since β ( α ) is monotone and bounded in a right neighborhood of α 0 , for any decreasing sequence { α n } n such that α n α 0 + 0 as n + there holds

lim n + β ( α n ) = lim α α 0 + 0 β ( α ) .

According to Proposition 9, for each n there exists a minimizer u n ( α n ) of β ( α n ) , and we can assume that u n ± p = 1 . Thus, passing to an appropriate subsequence, u n converges to some u 0 W 0 1 , p weakly in W 0 1 , p and strongly in L p ( Ω ) . Moreover, u 0 ± 0 in Ω since H α n ( u n ± ) 0 implies that u n ± p p 1 / α n . Furthermore, due to the weak lower semicontinuity of the W 0 1 , p -norm, we have H α 0 ( u 0 ± ) 0 , and hence u 0 ( α 0 ) . Consequently, using (2.12), we conclude that

β ( α 0 ) Ω | u 0 + | q 𝑑 x Ω | u 0 + | q 𝑑 x lim inf n + Ω | u n + | q 𝑑 x Ω | u n + | q 𝑑 x = lim inf n + β ( α n ) = lim α α 0 + 0 β ( α ) .

(iv) Assume that α λ 2 ( p ) and β β ( α ) . Let u ( α ) be a minimizer of β ( α ) . Then H α ( u ± ) 0 and, in view of (2.12), we may suppose that G β ( α ) ( u + ) = 0 . Therefore, G β ( u + ) G β ( α ) ( u + ) = 0 and hence u 𝒦 α , β .

Suppose now that there exists u 𝒦 α , β for some α , β . Since 𝒦 α , β ( α ) , assertion (i) implies that α λ 2 ( p ) . Moreover, since G β ( u + ) 0 , the definition of β ( α ) leads to

β ( α ) min { Ω | u + | q 𝑑 x Ω | u + | q 𝑑 x , Ω | u - | q 𝑑 x Ω | u - | q 𝑑 x } Ω | u + | q 𝑑 x Ω | u + | q 𝑑 x β ,

which completes the proof. ∎

In the sequel, it will be convenient to use the notation

(2.15) Σ := { ( α , β ) 2 : α > λ 2 ( p )