On the Fredholm Alternative for the Fučík Spectrum

Abstract

We consider resonance problems for the one-dimensional 𝑝-Laplacian assuming Dirichlet boundary conditions. In particular, we consider resonance problems associated with the first three curves of the Fučík Spectrum. Using variational arguments based on linking theorems, we prove sufficient conditions for the existence of at least one solution. Our results are related to the classical Fredholm Alternative for linear operators. We also provide a new variational characterization for points on the third Fučík curve.

1. Introduction

In this paper, we study the solvability of the problem −𝑢^′𝑝−2𝑢^′′−𝛼𝑢^+𝑝−1+𝛽(𝑢⁻)^𝑝−1(),𝑢(0)(𝑇)=𝑓in0,𝑇=𝑢=0, (1.1) where 𝑝>1, 𝑇>0, (𝛼,𝛽)∈ℝ², and 𝑓∈𝐿¹(0,𝑇). A solution of (1.1) is defined as a function 𝑢∈𝐶¹₀[0,𝑇], such that |𝑢^′|^𝑝−2𝑢^′ is absolutely continuous and satisfies (1.1) a.e. in (0,𝑇).

It is helpful to restate the given problem as an operator equation. Let 𝑋∶=𝑊^1,𝑝(0,𝑇) and let ⟨⋅,⋅⟩ represent the duality pairing between 𝑋 and 𝑋^∗. Define the operators 𝐽,𝑆∶𝑋→𝑋^∗ by ⟨𝐽(𝑢),𝑣⟩=^𝑇₀𝑢^′𝑝−2𝑢^′𝑣^′,⟨𝑆(𝑢),𝑣⟩=^𝑇₀|𝑢|^𝑝−2𝑢𝑣,for𝑢,𝑣∈𝑋. (1.2) In [1, page 306], it is proved that 𝐽 is an isomorphism (in particular, 𝐽⁻¹ exists and is continuous) and that 𝑆 is continuous and compact. Using an integration by parts argument, it is easy to verify [2, page 120] that solutions of (1.1) are in a one-to-one correspondence with solutions of the operator equation 𝐽(𝑢)−𝛼𝑆𝑢⁺+𝛽𝑆(𝑢⁻)=𝐹, (1.3) where 𝐹∈𝑋^∗ is defined by ⟨𝐹,𝑣⟩=^𝑇₀𝑓𝑣,𝑣∈𝑋. (1.4)

We are interested in the case where (𝛼,𝛽)∈Σ_𝑝 which represents the Fučík Spectrum associated with (1.1), that is, the set of all (𝛼,𝛽)∈ℝ² such that 𝐽(𝑢)−𝛼𝑆𝑢⁺+𝛽𝑆(𝑢⁻)=0 (1.5) has a nontrivial solution. An explicit form of Σ_𝑝 is given in [2, page 132]. For convenience, we recall the first parts of it. First we note that for 𝛼=𝛽=𝜆, we have (𝜆,𝜆)∈Σ_𝑝 if and only if 𝜆 is an eigenvalue of 𝐽(𝑢)−𝜆𝑆(𝑢)=0. (1.6) These eigenvalues can be expressed explicitly as𝜆=𝜆_𝑘=𝑝−1𝑘𝜋_𝑝𝑇^𝑝,𝑘=1,2,…, (1.7) where 𝜋_𝑝∶=2𝜋/𝑝sin(𝜋/𝑝). The associated eigenfunctions are scalar multiples of𝑢_𝑘(𝑡)=sin_𝑝𝑘𝜋_𝑝𝑡𝑇,𝑘=1,2,…, (1.8) where sin_𝑝 is defined by the implicit formula𝜏=^sin𝑝₀^(𝜏)𝑑𝑠(1−𝑠^𝑝)^1/𝑝𝜋,𝜏∈0,^𝑝2, (1.9) which is extended to [0,𝜋_𝑝] and then [0,2𝜋_𝑝] by symmetry, and then to all of ℝ as a 2𝜋_𝑝 periodic function. See, for example, [3, 4]. Note that, we have 𝑢₁>0 in (0,𝑇), and 𝑢₁ is a nontrivial solution of (1.5) for (𝛼,𝛽)=(𝜆₁,𝛽), with arbitrary 𝛽∈ℝ. Obviously, this implies that 𝜆₁×ℝ⊂Σ_𝑝. Similarly, ℝ×𝜆₁⊂Σ_𝑝 with a corresponding nontrivial solution, −𝑢₁<0 in (0,𝑇). It is helpful to separate this so-called trivial part of the Fučík Spectrum into𝒞⁻₁∶=𝜆₁,𝛽∈ℝ²∶𝛽≤𝜆₁∪𝛼,𝜆₁∈ℝ²∶𝛼≥𝜆₁,𝒞⁺₁∶=𝜆₁,𝛽∈ℝ²∶𝛽≥𝜆₁∪𝛼,𝜆₁∈ℝ²∶𝛼≤𝜆₁. (1.10) We set 𝒞₁=𝒞⁻₁∪𝒞⁺₁. The set 𝒞₁ is the component, that is, maximal connected subset, of Σ_𝑝 which contains (𝜆₁,𝜆₁). The other components of Σ_𝑝 lie in the first quadrant. Hence, from now on we assume 𝛼>0, 𝛽>0.

The next component of Σ_𝑝, which contains (𝜆₂,𝜆₂), is called 𝒞₂. It is a curve (𝑝-hyperbola) which passes through (𝜆₂,𝜆₂) and has the asymptotes 𝛼=𝜆₁ and 𝛽=𝜆₁. More precisely,𝒞₂∶=𝛼,𝛽∈ℝ²∶1𝛼^1/𝑝+1𝛽^1/𝑝=𝑝−1^−1/𝑝𝑇𝜋_𝑝. (1.11) For (𝛼,𝛽)∈𝒞₂, a corresponding nontrivial solution, 𝑢_𝛼𝛽, of (1.5) is a two-bump function in (0,𝑇) which can be constructed in a piecewise fashion using appropriately shifted and scaled sin_𝑝 functions as the pieces. In particular, for 𝛼>𝛽, 𝑢_𝛼𝛽 is a positive multiple of either 𝜙₂₁ or 𝜙₂₂, both 𝐿^𝑝 normalized, as shown in Figure 1. We note that 𝜙₂₂(𝑡)=𝜙₂₁(𝑇−𝑡), 𝑡∈[0,𝑇]. For 𝛼<𝛽, the situation is similar but the positive bump is now bigger than the negative one. For further reference, we denote𝒞⁻₂∶=𝛼,𝛽∈𝒞₂,𝒞∶𝛼≥𝛽₊₂∶=𝛼,𝛽∈𝒞₂∶𝛼≤𝛽. (1.12) In the special case where 𝛼=𝛽=𝜆₂, we have 𝜙₂₁=𝑐𝑢₂ and 𝜙₂₂=−𝑐𝑢₂, where 𝑐>0 is an 𝐿^𝑝 normalizing constant. 𝒞₂ is called the first nontrivial part of the Fučík Spectrum.

(a) 𝜙21

(b) 𝜙22

Figure 1 

Eigenfunctions for (𝛼,𝛽)∈𝒞2 with 𝛼>𝛽.

The component of Σ_𝑝 containing (𝜆₃,𝜆₃) is denoted by 𝒞₃ and consists of two 𝑝-hyperbolas which intersect at (𝜆₃,𝜆₃).𝒞¹₃∶=𝛼,𝛽∈ℝ²∶2𝛼^1/𝑝+1𝛽^1/𝑝=𝑝−1^−1/𝑝𝑇𝜋_𝑝,𝒞²₃∶=𝛼,𝛽∈ℝ²∶1𝛼^1/𝑝+2𝛽^1/𝑝=𝑝−1^−1/𝑝𝑇𝜋_𝑝. (1.13) Note that the asymptote for 𝒞¹₃ as 𝛼→∞ is 𝛽=𝜆₁, while as 𝛽→∞ the asymptote is 𝛼=𝜆₂. The set 𝒞²₃ is just the reflection of 𝒞¹₃ with respect to the diagonal line 𝛼=𝛽. In Theorem 1.4 below we will consider (𝛼,𝛽)∈𝒞¹₃ with 𝛼>𝛽, with a corresponding normalized eigenfunction, 𝜙₃₁, as depicted in Figure 2.

Figure 2 

Eigenfunction for (𝛼,𝛽)∈𝐶13 with 𝛼>𝛽, that is, 𝜙31.

Components 𝒞_𝑛 of Σ_𝑝 containing the points (𝜆_𝑛,𝜆_𝑛), 𝑛>3, are obtained similarly (see [2]). Note that a nontrivial solution of (1.5) associated with (𝛼,𝛽)∈𝒞_𝑛 consists of 𝑛 bumps in the interval (0,𝑇), and can be expressed explicitly in terms of the sin_𝑝 function.

We adopt a convention used in [5] to identify the Fučík eigenvalues and eigenfunctions referred to in our theorems and proofs. Fix a positive constant 𝑠 and consider the intersection of the line (𝜆+𝑠,𝜆) with Σ_𝑝. This produces a sequence 0<𝜆₁₁<𝜆₁₂<𝜆₂₁=𝜆₂₂<𝜆₃₁<𝜆₃₂<⋯ with associated normalized eigenfunctions 𝜙₁₁,𝜙₁₂,𝜙₂₁,𝜙₂₂,𝜙₃₁,𝜙₃₂,…. See Figure 3.

Figure 3 

First few components in the Fučík Spectrum.

Figure 4 

𝜕𝑄.

There is an interesting literature developing for problems such as (1.1). There are two different cases which have to be distinguished: (𝛼,𝛽)∉Σ_𝑝, the so-called nonresonance case; (𝛼,𝛽)∈Σ_𝑝, the so-called resonance case.

The nonresonance case is well understood. If (𝛼,𝛽) belongs to a component of ℝ²⧵Σ_𝑝 containing a point (𝜆,𝜆), then (1.1) has a solution for arbitrary 𝑓∈𝐿¹(0,𝑇). (The proof is based on the homotopy invariance property of Leray-Schauder degree.) On the other hand, if the component of ℝ²⧵Σ_𝑝 does not contain a point (𝜆,𝜆), then there exists 𝑓∈𝐿¹(0,𝑇) for which (1.1) does not have a solution (see, e.g., [2]).

The resonance case is much more involved. The case where 𝑝≠2 and (𝛼,𝛽)=(𝜆₁,𝜆₁) was studied in detail in [4] and generalized to the PDE case by [6–10]. The case where (𝛼,𝛽)=(𝜆₂,𝜆₂) was studied in [11] and there are no generalizations of their results to the PDE case available so far. It is particularly interesting to note that these papers show that in some respects the semilinear (𝑝=2) and quasilinear (𝑝≠2) cases can be quite different.

Our work is also related to results in [12–15], where solvability conditions in the spirit of Landesman and Lazer, [16], or Ahmad et al. [17], are derived for nonlinear perturbations of (1.1). Theorem 6.1 on page 334 of [12] is identical to our Theorem 1.2. In this case the novelty of our theorem is that it is proved using variational techniques rather than the method of upper and lower solutions. Problem (1.1) represents a subcase of the problems considered in [13–15]. We note that [13, 15] deal with the semilinear case, 𝑝=2, and [14] deals with the general quasilinear case, 1<𝑝<∞. Moreover, the solvability conditions in these papers reduce to the solvability conditions in our theorems. However, the given papers make significant use of the additional assumption that (𝛼,𝛽)∈(𝜆_𝑘−1,𝜆_𝑘+1)² for some 𝜆_𝑘−1<𝜆_𝑘<𝜆_𝑘+1. Our arguments do not rely on this local restriction and thus allow us to consider any point on the first three curves of Σ_𝑝. More specifically, we can consider 𝜆=𝜆₁₁,𝜆₁₂,𝜆₂₁, and 𝜆₃₁ with no further restriction.

We proceed with the statements of our main theorems, assuming throughout that (𝛼,𝛽)=(𝜆+𝑠,𝜆)∈Σ_𝑝, where 𝑠>0 is fixed. The theorems do not apply if 𝑠=0. Our assumption that 𝑠>0 restricts attention to those points of Σ_𝑝 which are below the diagonal 𝛼=𝛽. If we consider 𝑠<0, then the part of Σ_𝑝 above the diagonal is considered and (1.14), (1.15), and (1.18) are reversed by the symmetry of the problem. The assumption (1.17) remains unchanged for both cases.

Theorem 1.1. Suppose that 𝜆=𝜆₁₁ (𝜆₁₂) and assume ^𝑇₀𝑓𝜙₁₁<0⋅^𝑇₀𝑓𝜙₁₂>0. (1.14) Then (1.1) has at least one solution.

In the special case 𝑝=2, this theorem can be significantly strengthened.

Theorem 1.2. Suppose that 𝑝=2 and that 𝜆=𝜆₁₁ (𝜆₁₂). Then (1.1) has a solution if and only if ^𝑇₀𝑓𝜙₁₁≤0⋅^𝑇₀𝑓𝜙₁₂≥0. (1.15)

The statements above are written so as to keep notation consistent. Notice that 𝜙₂₁=−𝜙₁₁, so the given inequalities in (1.14) are actually the same and can be restated as^𝑇₀𝑓𝑢₁<0. (1.16) The inequalities in (1.15) are also identical.

Theorem 1.3. Suppose that 𝜆=𝜆₂₁ and assume ^𝑇₀𝑓𝜙₂₁⋅^𝑇₀𝑓𝜙₂₂>0. (1.17) Then (1.1) has at least one solution.

Notice that this theorem contains two cases, one where both integrals in (1.17) are positive and one where both integrals are negative. These cases correspond in some sense to resonance above and below the Fučík Spectrum.

Our last theorem is the following.

Theorem 1.4. Suppose that 𝜆=𝜆₃₁ and assume ^𝑇₀𝑓𝜙₃₁<0. (1.18) Then (1.1) has at least one solution.

The proofs of our theorems will be variational. Note that (1.1) is the Euler-Lagrange equation associated with the functional 𝐸(𝑢)1∶=𝑝^𝑇₀𝑢^′𝑝−𝛼𝑝^𝑇₀𝑢^+𝑝−𝛽𝑝^𝑇₀(𝑢⁻)^𝑝−^𝑇₀=1𝑓𝑢𝑝^𝑇₀𝑢^′𝑝−𝜆𝑝^𝑇₀|𝑢|^𝑝−𝑠𝑝^𝑇₀𝑢^+𝑝−^𝑇₀𝑓𝑢,𝑢∈𝑋. (1.19) Solutions of (1.1) are critical points of 𝐸. One part of the proof of Theorem 1.1 will locate a critical point via minimization. All other arguments will characterize critical points as saddle points. This requires us to establish the appropriate compactness and geometric properties of the functional. In particular, saddle point arguments require the appropriate geometry on linked sets.

In order to understand the functional 𝐸 it is important to first understand the functional 𝐼(𝑢)∶=^𝑇₀𝑢^′𝑝−𝑠^𝑇₀𝑢^+𝑝,𝑢∈𝑋. (1.20) It is well known that 𝜆₁₁,𝜆₁₂,𝜆₂₁,… are critical values of 𝐼 restricted to the 𝐿^𝑝 unit sphere 𝒮∶={𝑢∈𝑋∶∫^𝑇₀|𝑢|^𝑝=1}. The corresponding critical points are 𝜙₁₁,𝜙₁₂,𝜙₂₁,…. In particular, 𝜙₁₁ is a global minimum, 𝜙₁₂ is a local minimum, and 𝜙₂₁ is a saddle point with 𝜆₂₁ characterized as the minimax over continuous curves on 𝒮 connecting 𝜙₁₁ and 𝜙₁₂. See [5] for details. Throughout the paper we will make use of the notations 𝐼^𝑐∶={𝑢∈𝑋∶𝐼(𝑢)≥𝑐∫^𝑇₀|𝑢|^𝑝} and 𝐼_𝑐∶={𝑢∈𝑋∶𝐼(𝑢)≤𝑐∫^𝑇₀|𝑢|^𝑝}, to describe super- and sub-level sets.

The linking property that is crucial for the proofs of Theorems 1.3 and 1.4 also leads to a new variational characterization of 𝜆₃₁, which we discuss in the last section. A similar characterization was proved in [18], but we note that this was assuming (𝜆₃₁+𝑠,𝜆₃₁)∈(𝜆₂,𝜆₄)², that is, 𝜆₃₁+𝑠<𝜆₄. Let us emphasize that our characterization does not restrict the value of 𝑠.

Finally, since our arguments rely on variational structures that are also available in the related PDE case, it should be clear that the given theorems generalize under appropriate circumstances. For example, Theorems 1.1 and 1.2 generalize to the PDE case with no further assumptions. Generalizing Theorem 1.3 is more complicated. We would need for 𝜆₂₁ to be isolated, and to have an associated set of solutions, that is, an eigenspace, composed of positive multiples of a pair of two-bump solutions such as 𝜙₂₁,𝜙₂₂. We note that it is known that eigenfunctions associated with 𝜆₂₁ have only two nodes, see [5] or [19], but it is not true in general that the solution set can be spanned by just the two given solutions. It is not known if 𝜆₂₁ is necessarily isolated in the PDE case.

2. A Discussion of the Semilinear Case

We provide a brief discussion of the special properties of the semilinear case, that is, 𝑝=2, with (𝛼,𝛽)∈𝒞₁. In this case (1.1) becomes−𝑢^′′−𝛼𝑢⁺+𝛽𝑢⁻(),𝑢(0)(𝑇)=𝑓in0,𝑇=𝑢=0. (2.1) For the first case we assume that 𝜆=𝜆₁₁, so 𝛼=𝜆₁ and 𝛽=𝜆₁−𝑠 for some 𝑠>0. If we multiply equation (2.1) by the positive eigenfunction 𝜙₁₁ and integrate by parts, we obtain−𝑠^𝑇₀𝑢⁻𝜙₁₁=^𝑇₀𝑓𝜙₁₁, (2.2) and the necessity of (1.15) follows. A similar argument applies if 𝜆=𝜆₁₂. In this case it is important to recall that 𝜙₁₂=−𝜙₁₁<0.

On the other hand let us assume that ^𝑇₀𝑓𝜙₁₁=^𝑇₀𝑓𝑢₁=0. (2.3) Then the linear problem−𝑢^′′−𝜆₁(),𝑢(0)(𝑇)𝑢=𝑓in0,𝑇=𝑢=0, (2.4) has a solution 𝑢. If we choose 𝑐>0 large enough then 𝑢=𝑢+𝑐𝜙₁₁ is a positive solution of this linear problem. It follows that 𝑢 is a solution of−𝑢^′′−𝜆₁𝑢⁺+𝛽𝑢⁻(),𝑢(0)(𝑇)=𝑓in0,𝑇=𝑢=0, (2.5) for arbitrary 𝛽∈ℝ. Similarly, we can choose 𝑐<0 large enough so that 𝑢=𝑢+𝑐𝜙₁₁<0 in (0,𝑇), and 𝑢 is then a solution of−𝑢^′′−𝛼𝑢⁺+𝜆₁𝑢⁻(),𝑢(0)(𝑇)=𝑓in0,𝑇=𝑢=0 (2.6) for arbitrary 𝛼∈ℝ. Hence (2.3) is a sufficient condition. The sufficiency of the strict inequalities is postponed until a later section where it will be proved for the more general case, 𝑝>1. We note that our proof of Theorem 1.1, in a later section, will also complete the proof of Theorem 1.2.

3. The Palais-Smale Condition on Σ_𝑝

In this section, we prove that under conditions (1.14), (1.17), and (1.18) the energy functional 𝐸 satisfies the Palais-Smale condition (PS), that is, if {𝑢_𝑛}⊂𝑋 such that {𝐸(𝑢_𝑛)} is bounded and 𝐸′(𝑢_𝑛)→0 in 𝑋^∗, then {𝑢_𝑛} contains a converging subsequence. In fact, we prove the more general result stated below.

Proposition 3.1. Let (𝛼,𝛽)∈Σ_𝑝 and let 𝑓∈𝐿¹(0,𝑇) such that ^𝑇₀𝑢_𝛼𝛽𝑓≠0, (3.1) for any 𝑢_𝛼𝛽 which is a nontrivial solution of (1.5). Then 𝐸 satisfies (PS).

Proof. Let {𝑢_𝑛}⊂𝑋 be a sequence such that 𝐸(𝑢_𝑛) is bounded and 𝐸′(𝑢_𝑛)→0 in 𝑋^∗. (We will refer to such a sequence as a PS-sequence.) We proceed in two steps. First we will show that {𝑢_𝑛} is bounded in 𝑋. Then we will select a converging subsequence.
We argue by contradiction, so assume ‖𝑢_𝑛‖_𝑋→∞. Let 𝑤_𝑛∶=𝑢_𝑛/‖𝑢_𝑛‖_𝑋, and, without loss of generality, assume that 𝑤_𝑛⇀𝑤 in 𝑋. Our assumptions imply0=lim_𝑛→∞𝐸′𝑢_𝑛𝑢_𝑛𝑋^𝑝−1=lim_𝑛→∞𝐽𝑤_𝑛−𝛼𝑆𝑤⁺_𝑛+𝛽𝑆𝑤⁻_𝑛−𝐹𝑢_𝑛𝑋^𝑝−1. (3.2) The compactness of 𝑆 and (3.2) yield lim_𝑛→∞𝐽𝑤_𝑛=𝛼𝑆𝑤⁺−𝛽𝑆(𝑤⁻), (3.3) and the continuity of 𝐽⁻¹ subsequently leads to 𝑤=lim_𝑛→∞𝑤_𝑛=𝐽⁻¹𝛼𝑆𝑤⁺−𝛽𝑆(𝑤⁻), (3.4) that is, 𝐽(𝑤)−𝛼𝑆𝑤⁺+𝛽𝑆(𝑤⁻)=0. (3.5)
In order to rule out the case 𝑤=0, we observe in (3.2) thatlim_𝑛→∞𝐽𝑤_𝑛−𝛼𝑆𝑤⁺_𝑛+𝛽𝑆𝑤⁻_𝑛=0. (3.6) This together with 𝑤=0 would imply 0=lim_𝑛→∞𝐽𝑤_𝑛,𝑤_𝑛=lim_𝑛→∞^𝑇₀𝑤^′_𝑛^𝑝=1, (3.7) a contradiction. Thus 𝑤 is a nontrivial solution of (3.5), that is, 𝑤=𝑢_𝛼𝛽. Our assumptions also imply that ^𝑇₀𝑓𝑢_𝑛=𝑝𝐸𝑢_𝑛−𝐸′𝑢_𝑛,𝑢_𝑛=𝑜𝑢_𝑛𝑋 (3.8) as 𝑛→∞. Dividing through by ‖𝑢_𝑛‖_𝑋 we get that 0=lim_𝑛→∞^𝑇₀𝑓𝑤_𝑛=^𝑇₀𝑓𝑢_𝛼𝛽≠0, (3.9) a contradiction. We have proved that {𝑢_𝑛} is bounded.
Without loss of generality we assume that there is a 𝑢∈𝑋 such that 𝑢_𝑛⇀𝑢 in 𝑋 and 𝑢_𝑛→𝑢 in 𝐿^𝑝(0,𝑇). Our assumptions imply that0=lim_𝑛→∞𝐸^′𝑢_𝑛=lim_𝑛→∞𝐽𝑢_𝑛−𝛼𝑆𝑢⁺_𝑛+𝛽𝑆𝑢⁻_𝑛−𝐹. (3.10) Using the continuity of 𝐽⁻¹ and the compactness of 𝑆 we now have 𝑢=lim_𝑛→∞𝑢_𝑛=𝐽⁻¹𝛼𝑆𝑢⁺−𝛽𝑆(𝑢⁻)+𝐹. (3.11) Hence every PS-sequence has a converging subsequence and the proof is finished.

4. Proof of Theorems 1.1 and 1.2

We will treat two cases.

Case 1 (𝜆=𝜆₁₁=𝜆₁−𝑠). If 𝜆=𝜆₁₁, then 𝐸(𝑢)=1𝑝^𝑇₀𝑢^′𝑝−𝜆₁𝑝^𝑇₀|𝑢|^𝑝+𝑠𝑝^𝑇₀(𝑢⁻)^𝑝−^𝑇₀𝑓𝑢. (4.1)Lemma 4.1. 𝐸 is bounded below.Proof. Assume the contrary, that is, there exists a sequence {𝑢_𝑛}⊂𝑋 such that lim_𝑛→∞𝐸𝑢_𝑛=−∞. (4.2) As a consequence, ‖𝑢_𝑛‖_𝑋→∞. Let 𝑤_𝑛=𝑢_𝑛/‖𝑢_𝑛‖_𝑋. Then 0≥liminf_𝑛→∞𝐸𝑢_𝑛𝑢_𝑛^𝑝_𝑋=liminf_𝑛→∞1𝑝^𝑇₀𝑤^′_𝑛^𝑝−𝜆₁𝑝^𝑇₀𝑤_𝑛^𝑝+𝑠𝑝^𝑇₀𝑤⁻_𝑛^𝑝−1𝑢_𝑛^𝑝_𝑋^𝑇0𝑓𝑢_𝑛=liminf_𝑛→∞1𝑝^𝑇₀𝑤^′_𝑛^𝑝−𝜆₁𝑝^𝑇₀𝑤_𝑛^𝑝+𝑠𝑝^𝑇₀𝑤⁻_𝑛^𝑝. (4.3) Without loss of generality we assume that there exists 𝑤∈𝑋 such that 𝑤_𝑛⇀𝑤 in 𝑋 and 𝑤_𝑛→𝑤 in 𝐿^𝑝(0,𝑇). Thus 0≥liminf_𝑛→∞1𝑝^𝑇₀𝑤^′_𝑛^𝑝−𝜆₁𝑝^𝑇₀|𝑤|^𝑝+𝑠𝑝^𝑇₀(𝑤⁻)^𝑝. (4.4) If 𝑤=0, then (4.4) yields lim_𝑛→∞‖𝑤_𝑛‖_𝑋=0, a contradiction. Hence 𝑤≠0. By the weak lower semicontinuity of the norm we deduce from (4.4) that 10≥𝑝^𝑇₀𝑤^′𝑝−𝜆₁𝑝^𝑇₀|𝑤|^𝑝+𝑠𝑝^𝑇₀(𝑤⁻)^𝑝. (4.5)
The variational characterization and simplicity of 𝜆₁ yield that 𝑤 must be a positive multiple of 𝑢₁, and thus it follows that 𝑤=𝜙₁₁. Sincelim_𝑛→∞^𝑇₀𝑓𝑤_𝑛=^𝑇₀𝑓𝜙₁₁<0, (4.6) there exists 𝑛₀∈ℕ such that for 𝑛≥𝑛₀, we have ∫^𝑇₀𝑓𝑢_𝑛<0. Using this and the variational characterization of 𝜆₁, we see that 1𝑝^𝑇₀𝑢^′_𝑛^𝑝−𝜆₁𝑝^𝑇₀𝑢_𝑛^𝑝+𝑠𝑝^𝑇₀𝑢⁻_𝑛^𝑝−^𝑇₀𝑓𝑢_𝑛≥0, (4.7) for 𝑛≥𝑛₀, and thus we arrive at a contradiction with (4.2).
The boundedness of 𝐸 from below together with the (PS) condition yields the existence of a critical point, that is, a minimizer, of 𝐸, and thus of a solution to (1.1). The proof of the first case is finished.

Case 2 (𝜆=𝜆₁₂=𝜆₁). If 𝜆=𝜆₁₂, then we can write (1.19) as 𝐸(𝑢)=1𝑝^𝑇₀𝑢^′𝑝−𝜆₁𝑝^𝑇₀|𝑢|^𝑝−𝑠𝑝^𝑇₀𝑢^+𝑝−^𝑇₀𝑓𝑢. (4.8) We will show that 𝐸 has a saddle point.
For 𝑡>0, we have 𝐸𝑡𝑢₁=−𝑠𝑡^𝑝𝑝^𝑇₀𝑢₁^𝑝−𝑡^𝑇₀𝑓𝑢₁, (4.9) and for 𝑡<0, we have 𝐸𝑡𝑢₁=−𝑡^𝑇₀𝑓𝑢₁=−𝑡𝑐^𝑇₀𝑓𝜙₁₁, (4.10) for some 𝑐>0, so it is clear that lim_|𝑡|→∞𝐸𝑡𝑢₁=−∞. (4.11)
Observe that for 𝑢∈𝐼^𝜆21, we have 𝐸(𝑢)≥𝜆₂₁−𝜆₁𝑝^𝑇₀|𝑢|^𝑝−^𝑇₀𝑓𝑢. (4.12) Since 𝜆₂₁>𝜆₁, the functional 𝐸 is bounded below by some 𝐾 on 𝐼^𝜆21. Hence, we obtain from (4.11) that, for 𝑅>0 large enough, max𝐸𝑅𝑢₁,𝐸−𝑅𝑢₁<𝐾≤inf_{𝑢∈𝐼𝜆21}𝐸(𝑢). (4.13)
Let Γ∶={ℎ∶[−1,1]→𝑋∶ℎiscontinuous,andℎ(±1)=±𝑅𝑢₁}. We claim that if ℎ∈Γ, then ℎ([−1,1])∩𝐼^𝜆21 is nonempty. If ℎ(𝑡)=0 for some 𝑡, then the conclusion is immediate. If ℎ(𝑡)≠0 for all 𝑡, then it follows from the variational characterization of 𝜆₂₁ in [5] that max_{𝑡∈[−1,1]}𝐼ℎ(𝑡)‖ℎ(𝑡)‖_𝐿^𝑝≥𝜆₂₁, (4.14) and so our claim is established. It follows that max_{[]𝑡∈−1,1}𝐸(ℎ(𝑡))≥𝐾∀ℎ∈Γ, (4.15) and so 𝑐=inf_ℎ∈Γmax_{[]𝑡∈−1,1}𝐸(ℎ(𝑡))≥𝐾. (4.16) Since the (PS) condition has already been established, we can conclude that 𝑐 is a critical value and thus (1.1) has a solution.
The proof of the second case is finished, and thus both Theorems 1.1 and 1.2 have been proved.

5. Proof of Theorem 1.3

Consider two cases.

Case 1 (∫^𝑇₀𝑓𝜙₂₁>0 and ∫^𝑇₀𝑓𝜙₂₂>0). This is the most interesting case. The plan is to construct a copy of 𝑆¹, the unit circle in ℝ², in 𝑋 that links with 𝐼^𝜆31. We will then argue that these sets enjoy the appropriate saddle-point geometry. Since the (PS) condition has already been established for our functional, we will have proved the existence of a saddle point.
The construction is accomplished via the union of six curves. Let 𝛾₁∶=𝛼𝜙⁺₂₁−𝛽𝜙⁻₂₁∶𝛼≥0,𝛽≥0,𝛼^𝑝𝑇₀𝜙⁺₂₁^𝑝+𝛽^𝑝𝑇₀𝜙⁻₂₁^𝑝=1, (5.1) a curve in 𝒮 connecting 𝜙⁺₂₁/‖𝜙⁺₂₁‖_𝐿𝑝 and −𝜙⁻₂₁/‖𝜙⁻₂₁‖_𝐿𝑝. A straight-forward calculation shows that 𝐼≡𝜆₂₁ on 𝛾₁. (See [19].)
Let 𝛾₂𝜙∶=(1−𝑡)⁺²¹𝜙⁺_21𝐿𝑝^𝑝+𝑡𝜙^𝑝₁₁^1/𝑝∶0≤𝑡≤1, (5.2) a curve in 𝒮 connecting 𝜙⁺₂₁/‖𝜙⁺₂₁‖_𝐿𝑝 to 𝜙₁₁. Another straight-forward calculation shows that 𝐼|_𝛾2 decreases from 𝜆₂₁ to 𝜆₁₁ as 𝑡 goes from 0 to 1. Similarly, we construct 𝛾₃ connecting −𝜙⁻₂₁/‖𝜙⁻₂₁‖_𝐿𝑝 to 𝜙₁₂=−𝜙₁₁.
The curves 𝛾₄, 𝛾₅ and 𝛾₆ are constructed similarly to 𝛾₁, 𝛾₂, and 𝛾₃ using 𝜙₂₂ in place of 𝜙₂₁. Let 𝜕𝑄∶=𝛾₁∪⋯∪𝛾₆. It is clear that 𝜕𝑄 is homeomorphic to 𝑆¹, and for convenience we name a homeomorphism ℎ₀∶𝑆¹→𝜕𝑄 such that ℎ₀(0,−1)=𝜙₁₁, ℎ₀(0,1)=𝜙₁₂, ℎ₀(−1,0)=𝜙₂₁, and ℎ₀(1,0)=𝜙₂₂.
Before proceeding it is useful to modify 𝜕𝑄 by lowering its 𝐼-energy in a way that will help with later estimates. The construction so far shows that 𝐼≤𝜆₂₁ on 𝜕𝑄, but it will be convenient to have a strict inequality on most of the curve 𝜕𝑄. We begin with the following lemma whose proof is clear.
Lemma 5.1. There exists an 𝜖>0 and a 𝛿>0 such that ^𝑇₀𝑓𝑢>𝜖, (5.3) for any 𝑢∈𝐵_𝛿(𝜙₂₁)∪𝐵_𝛿(𝜙₂₂).Now let 𝐶∶=𝜕𝑄⧵(𝐵_𝛿/2(𝜙₂₁)∪𝐵_𝛿/2(𝜙₂₂)), let 𝐵∶=∅, and apply the basic deformation lemma for 𝐶¹-manifolds from [20, Lemma 3.7, page 55], where 𝐼 is the 𝐶¹ functional in question and 𝒮 is the Finsler manifold. Let 𝛼(𝑡,𝑥) be the deformation guaranteed by the lemma and let 𝜕𝑄′∶=𝛼(𝑡,𝜕𝑄) for any small fixed 𝑡>0. We have 𝐼(𝛼(𝑡,𝑢))≤𝐼(𝑢) for all 𝑢, so 𝐼≤𝜆₂₁ on all of 𝜕𝑄′. It also follows that there is a 𝜅>0 such that 𝐼(𝑢)≤𝜆₂₁−𝜅 on 𝜕𝑄′⧵(𝐵_𝛿(𝜙₂₁)∪𝐵_𝛿(𝜙₂₂)). For convenience, we rename 𝜕𝑄′ as 𝜕𝑄 again.
Recall that 𝐸(𝑢)=1𝑝^𝑇₀𝑢^′𝑝−𝑠𝑝^𝑇₀𝑢^+𝑝−𝜆₂₁𝑝^𝑇₀|𝑢|^𝑝−^𝑇₀𝑓𝑢,𝑢∈𝑋. (5.4)
The following lemmas establish the desired saddle geometry.
Lemma 5.2. 𝐸 is bounded below on 𝐼^𝜆31 by some constant 𝐾.Proof. Observe that for 𝑢∈𝐼^𝜆31, we have 𝐸(𝑢)≥𝜆₃₁−𝜆₂₁𝑝^𝑇₀|𝑢|^𝑝−^𝑇₀𝑓𝑢. (5.5) The lemma follows.Lemma 5.3. There is an 𝑅>0 such that max_𝑅𝜕𝑄𝐸(𝑢)<𝐾, where 𝑅𝜕𝑄∶={𝑅𝑢∶𝑢∈𝜕𝑄}.Proof. For 𝑢∈𝜕𝑄∩(𝐵_𝛿(𝜙₂₁)∪𝐵_𝛿(𝜙₂₂)) and 𝑡>0, we have 𝐸(𝑡𝑢)≤−𝑡^𝑇₀𝑓𝑢<−𝑡𝜖. (5.6) For 𝑢∈𝜕𝑄⧵(𝐵_𝛿(𝜙₂₁)∪𝐵_𝛿(𝜙₂₂)) and 𝑡>0, we have 𝐸(𝑡𝑢)≤−𝑡^𝑝𝜅𝑝−𝑡^𝑇₀𝑓𝑢. (5.7) The lemma follows.Now we prove a linking lemma. It is clear that 𝜕𝑄∩𝐼^𝜆31=∅, so what remains to be proved is the following. For convenience we set 𝐷∶={(𝑥,𝑦)∈ℝ²∶𝑥²+𝑦²≤1}.Lemma 5.4. Given any continuous ℎ∶𝐷→𝑋 such that ℎ|_𝑆¹≡ℎ₀, we have ℎ(𝐷)∩𝐼^𝜆31≠∅. Proof. Suppose not. If 0∈ℎ(𝐷), then we are done. If 0∉ℎ(𝐷), then without loss of generality we may replace ℎ by its projection onto 𝒮 and restrict the remainder of the argument to that surface. If ℎ(𝐷)∩𝐼^𝜆31=∅ then there is a 𝛿>0 such that max_𝐷𝐼(ℎ(𝑥))≤𝜆₃₁−𝛿. Using the results of Corvellec, [21], there exists a strong deformation retract 𝜂∶𝐼_{𝜆31−𝛿/2}×[0,1]→𝐼_𝜆21. (See the proof of Theorem 4 in [21] where the existence of a strong deformation retract is proved as one step towards proving the existence of a weak deformation retract.) Replacing ℎ with 𝜂(ℎ(⋅),1), and calling it ℎ again, we may now assume that max_𝐷𝐼(ℎ(𝑥))≤𝜆₂₁. Observe that 𝜂 has not changed the values of ℎ on 𝑆¹, and so ℎ≡ℎ₀ on 𝑆¹.
We will now establish the existence of a path, 𝛾, in 𝐷 connecting (0,±1). The image, via ℎ, of this path will connect 𝜙₁₁ to 𝜙₁₂ in (𝐼_𝜆21∩𝒮)⧵{𝜙₂₁,𝜙₂₂}. This will be a contradiction of the variational characterization of 𝜆₂₁ established in [5], and so the lemma will be proved.
Our tool for constructing 𝛾 is the Hex Theorem as stated in [22]. Let 𝐸∶={(𝑎,𝑏)∈𝑆¹∶𝑎>0,−√2/2≤𝑏≤√2/2}, and 𝑊∶={(𝑎,𝑏)∈𝑆¹∶𝑎<0,−√2/2≤𝑏≤√2/2}. The other two components of 𝑆¹ are 𝑁, the component containing (0,1), and 𝑆, the component containing (0,−1). It is clear that given any 𝑛 there is a homeomorphism from 𝐷 to the standard 𝑛×𝑛 Hex board with edges corresponding to 𝐸, 𝑊, 𝑁, and 𝑆 in a natural way. We will refer to such a homeomorphism as an 𝑛×𝑛 hexagonal tiling of 𝐷. It is also clear that given any 𝑑>0 there is an 𝑛×𝑛 hexagonal tiling of 𝐷 such that the maximum diameter of the hexagons is less than 𝑑.
Since ℎ(𝐸)∩ℎ(𝑊)=∅, we have 𝑟∶=dist(ℎ(𝐸),ℎ(𝑊))>0. Choose 𝛿>0 such that |𝑥−𝑦|<𝛿 implies ‖ℎ(𝑥)−ℎ(𝑦)‖_𝑋<𝑟. Now tile 𝐷 with an 𝑛×𝑛 collection of hexagons with maximum diameter 𝛿/3. If a hexagon contains a point in ℎ⁻¹(𝜙₂₁)∪ℎ⁻¹(𝜙₂₂), then color it red. Otherwise color the hexagon blue. By the Hex Theorem, we either have a path of red hexagons connecting 𝐸 and 𝑊, or a path of blue hexagons connecting 𝑁 and 𝑆. Suppose that there is a path of red hexagons connecting 𝐸 and 𝑊. Consider a hexagon at the beginning of the path so that it is touching 𝐸. Since all of the points of this hexagon are within distance 𝛿/3 of 𝐸, the image of this hexagon via ℎ cannot intersect ℎ(𝑊), and in particular cannot contain 𝜙₂₁. Hence this first red hexagon contains a point in ℎ⁻¹(𝜙₂₂). Similarly, the last hexagon in the path contains a point of ℎ⁻¹(𝜙₂₁). Somewhere along this path there must be a pair of hexagons that share an edge and where one hexagon contains a point 𝑥∈ℎ⁻¹(𝜙₂₂) and the other a point 𝑦∈ℎ⁻¹(𝜙₂₁). However, this implies that |𝑥−𝑦|<𝛿 with ‖ℎ(𝑥)−ℎ(𝑦)‖_𝑋=‖𝜙₂₂−𝜙₂₁‖_𝑋>𝑟, a contradiction. Hence there must be a path of blue hexagons connecting 𝑁 and 𝑆. This clearly implies that there is a continuous path in 𝐷 connecting (0,1) to (0,−1) where no point on the path is in ℎ⁻¹(𝜙₂₂)∪ℎ⁻¹(𝜙₂₁). The image of this path via ℎ is the 𝛾 that we were looking for. The lemma is proved, and hence the theorem is proved. It is important to observe that a similar argument establishes the linking of 𝑅𝜕𝑄 and 𝐼^𝜆31 for any 𝑅≥1.

Case 2 (∫^𝑇₀𝑓𝜙₂₁<0 and ∫^𝑇₀𝑓𝜙₂₂<0). Lemma 5.5. 𝐸 is bounded below by some 𝐾 on 𝐼^𝜆21.Proof. Let 𝜖>0 and 𝛿>0 such that for 𝑢∈(𝐵_𝛿(𝜙₂₁)∪𝐵_𝛿(𝜙₂₂)), we have ∫^𝑇₀𝑓𝑢<−𝜖. There is a 𝜅>0 such that 𝐼(𝑢)≥𝜆₂₁+𝜅 for 𝑢∈(𝒮∩𝐼^𝜆21)⧵(𝐵_𝛿(𝜙₂₁)∪𝐵_𝛿(𝜙₂₂)). Therefore 𝐸(𝑡𝑢)≥𝜖𝑡 for 𝑢∈(𝒮∩𝐼^𝜆21)∩(𝐵_𝛿(𝜙₂₁)∪𝐵_𝛿(𝜙₂₂)) and 𝐸(𝑡𝑢)≥(𝜅/𝑝)𝑡^𝑝−𝑡∫^𝑇₀𝑓𝑢 for 𝑢∈(𝒮∩𝐼^𝜆21)⧵(𝐵_𝛿(𝜙₂₁)∪𝐵_𝛿(𝜙₂₂)). The lemma follows.Lemma 5.6. There is an 𝑅>0 such that max{𝐸(𝑅𝜙₁₁),𝐸(𝑅𝜙₁₂)}<𝐾.Proof. For 𝑡>0, we have 𝐸(𝑡𝜙₁₂)=((𝜆₁₂−𝜆₂₁)/𝑝)𝑡^𝑝−𝑡∫^𝑇₀𝜙₁₂𝑓 and 𝐸(𝑡𝜙₁₁)=((𝜆₁₁−𝜆₂₁)/𝑝)𝑡^𝑝−𝑡∫^𝑇₀𝜙₁₁𝑓, so the lemma follows easily.Lemma 5.7. Every continuous path connecting 𝑅𝜙₁₁ to 𝑅𝜙₂₁ must intersect 𝐼^𝜆21.Proof. This is a direct consequence of the variational characterization of 𝜆₂₁ proved in [5]. Hence the theorem is proved.

6. Proof of Theorem 1.4

In this section we assume that 𝜆=𝜆₃₁. Once again we must establish the appropriate saddle geometry and linking. Most of the arguments follow from work done in previous sections. The one exception is the following lemma.

Lemma 6.1. 𝐸 is bounded below by some 𝐾 on 𝐼^𝜆31.

Proof. There is an 𝜖>0 and a 𝛿>0 such that if 𝑢∈𝐵_𝛿(𝜙₃₁), then ∫^𝑇₀𝑓𝑢<−𝜖. For 𝑢∈(𝒮∩𝐼^𝜆31)⧵𝐵_𝛿(𝜙₃₁) there is a 𝜅>0 such that 𝐼(𝑢)≥𝜆₃₁+𝜅. It follows that 𝐸(𝑡𝑢)≥𝜖𝑡 for 𝑢∈𝐵_𝛿(𝜙₃₁)∩𝒮∩𝐼^𝜆31 and that 𝐸(𝑡𝑢)≥(𝜅/𝑝)𝑡^𝑝−𝑡∫^𝑇₀𝑢𝑓 for 𝑢∈(𝒮∩𝐼^𝜆31)⧵𝐵_𝛿(𝜙₃₁). The lemma follows.

Once again we consider 𝜕𝑄 as constructed in the previous section. We can employ the 𝜕𝑄 with or without the modifying flow.

Lemma 6.2. There is an 𝑅>0 such that max_𝑅𝜕𝑄𝐸(𝑢)<𝐾, where 𝑅𝜕𝑄∶={𝑅𝑢∶𝑢∈𝜕𝑄}.

Proof. This follows easily from the fact that for 𝑢∈𝜕𝑄, we have 𝐼(𝑢)≤𝜆₂₁ and so 𝐸(𝑡𝑢)≤((𝜆₂₁−𝜆₃₁)/𝑝)𝑡^𝑝−𝑡∫^𝑇₀𝑓𝑢.
The linking of 𝐼^𝜆31 and 𝑅𝜕𝑄 has already been established in Lemma 5.4. Hence the theorem is proved.

7. Variational Characterization of 𝜆₃₁

Let 𝒬∶={𝑄⊂𝒮∶ there exists a continuous ℎ∶𝐷→𝒮 such that ℎ(𝐷)=𝑄,ℎ∶𝑆¹→ℎ(𝑆¹) is a homeomorphism into 𝐼_𝜆21, and ℎ(𝑆¹) contains 𝜙₁₁,𝜙₂₁,𝜙₁₂,𝜙₂₂ in that order}. For convenience we refer to ℎ(𝑆¹) as 𝜕𝑄.

Lemma 5.4 shows that𝑐∶=inf_𝑄∈𝒬max_𝑢∈𝑄𝐼(𝑢)≥𝜆₃₁. (7.1) The set 𝒬 is clearly preserved by pseudogradient flows on 𝒮 associated with 𝐼, and 𝐼 satisfies the (PS) condition. Hence 𝑐 is a critical value of 𝐼. To show that 𝑐=𝜆₃₁, and thus that 𝜆₃₁ has the given variational characterization, it suffices to construct an element 𝑄∈𝒬 such that max_𝑄𝐼(𝑢)=𝜆₃₁.

We begin by constructing a triangle using the positive and negative parts of 𝜙₃₁. For convenience we write𝜙₃₁𝑢(𝑥)[],𝑣(𝑥)[],𝑤(𝑥)[],∶=,𝑥∈0,𝑎,𝑥∈𝑎,𝑏,𝑥∈𝑏,𝑇 (7.2) where 𝑢, 𝑤 are the positive parts of 𝜙₃₁ and 𝑣 is the negative part. Each of 𝑢, 𝑣, 𝑤 is an appropriately scaled multiple of sin_𝑝 on the appropriate interval, and is 0 everywhere else. Now let 𝑃={𝛼𝑢+𝛽𝑣+𝛾𝑤∶𝛼,𝛽,𝛾≥0,𝛼^𝑝∫^𝑇₀|𝑢|^𝑝+𝛽^𝑝∫^𝑇₀|𝑣|^𝑝+𝛾^𝑝∫^𝑇₀|𝑤|^𝑝=1}. See Figure 5. A straight forward computation shows that 𝐼≡𝜆₃₁ on 𝑃.

Figure 5 

𝑃.

Consider a point 𝑧=𝛼𝑢+𝛽𝑣∈𝜕𝑃, that is, a point on the boundary of 𝑃 such that 𝛾=0. We construct the curve {𝑡^1/𝑝𝑧(𝑡𝑥)∶𝑏/𝑇≤𝑡≤1}. For convenience we denote 𝑧_𝑡(𝑥)=𝑡^1/𝑝𝑧(𝑡𝑥) and 𝑧_𝑡(𝑥)=𝛼𝑢_𝑡(𝑥)+𝛽𝑣_𝑡(𝑥). Elementary computations show that ∫^𝑇₀|𝑧_𝑡|^𝑝=1 and ∫^𝑇₀|(𝑧_𝑡)′|^𝑝=𝑡^𝑝∫^𝑇₀|(𝑧₁)′|^𝑝. Hence 𝐼(𝑧_𝑡) decreases as 𝑡 goes from 1 to 𝑏/𝑇. Observe that if 𝛼=𝛽 then 𝑧 is a positive multiple of 𝜙₃₁ on the interval [0,𝑏], and is thus a solution of−𝑦^′𝑝−2𝑦^′′−𝜆₃₁+𝑠𝑦^+𝑝−1+𝜆₃₁𝑦^−𝑝−1(),𝑦(0)(𝑏)=0in0,𝑏=𝑦=0. (7.3) It follows that 𝑧_𝑡 is a normalized two-node Fučík eigenfunction on [0,𝑏/𝑡] with positive first node. In particular we must have 𝑧_𝑏/𝑇=𝜙₂₁. Moreover, if we now consider {𝛼𝑢_𝑏/𝑇+𝛽𝑣_𝑏/𝑇∶𝛼,𝛽≥0 and 𝛼^𝑝∫^𝑇₀|𝑢_𝑏/𝑇|^𝑝+𝛽^𝑝∫^𝑇₀|𝑣_𝑏/𝑇|^𝑝=1} we see that this must be identical to {𝛼𝜙⁺₂₁−𝛽𝜙⁻₂₁∶𝛼,𝛽≥0 and 𝛼^𝑝∫^𝑇₀|𝜙⁺₂₁|^𝑝+𝛽^𝑝∫^𝑇₀|𝜙⁻₂₁|^𝑝=1}, which is precisely the segment that we used to construct 𝜕𝑄 in a previous section. Note that, we have a new figure 𝑃^′=𝑃∪{𝛼𝑢_𝑡+𝛽𝑣_𝑡∶𝛼𝑣+𝛽𝑢∈𝑃,𝑏/𝑇≤𝑡≤1}, which is still homeomorphic to 𝐷, but now includes 𝜙₂₁ on its boundary. See Figure 6. A similar construction is possible on the edge of 𝑃 where 𝛼=0, so that we arrive at 𝑃^′′ which is homeomorphic to 𝐷 and contains both 𝜙₂₁ and 𝜙₂₂. See Figure 7.

Figure 6 

𝑃′.

Figure 7 

𝑃′′.

Observe that the component of the boundary of 𝑃^′′ between 𝜙⁺₂₁/‖𝜙⁺₂₁‖_𝐿^𝑝 and 𝜙⁺₂₂/‖𝜙⁺₂₂‖_𝐿^𝑝 contains only nontrival nonnegative elements. For convenience call this part of the boundary (𝜕𝑃^′′)⁺. For each 𝑧∈(𝜕𝑃^′′)⁺ we form {(𝑡𝑧^𝑝+(1−𝑡)𝜙^𝑝₁₁)^1/𝑝∶0≤𝑡≤1}. It is easy to see that this is a curve on 𝒮 connecting 𝑧 to 𝜙₁₁ such that 𝐼 decreases from 𝑧 to 𝜙₁₁. A similar construction is possible along the boundary of 𝑃^′′ between −𝜙⁻₂₁/‖𝜙⁻₂₁‖_𝐿^𝑝 and −𝜙⁻₂₂/‖𝜙⁻₂₂‖_𝐿^𝑝, which we call (𝜕𝑃^′′)⁻.

Hence we can form 𝑃^′′′=𝑃^′′∪{(𝑡𝑧^𝑝+(1−𝑡)𝜙^𝑝₁₁)^1/𝑝∶𝑧∈(𝜕𝑃^′′)⁺,0≤𝑡≤1}∪{(−𝑡|𝑧|^𝑝−(1−𝑡)|𝜙₁₂|^𝑝)^1/𝑝∶𝑧∈(𝜕𝑃^′′)⁻, 0≤𝑡≤1}. See Figure 8. Once again this set is homeomorphic to 𝐷. Moreover, its boundary contains the points 𝜙₁₁, 𝜙₂₁, 𝜙₁₂, and 𝜙₂₂ in that order. Hence 𝑃^′′′∈𝒬. Since with each addition to 𝑃, we have decreased energy we see that max_{𝑃′′′}𝐼(𝑢)=max_𝑃𝐼(𝑢)=𝐼(𝜙₃₁)=𝜆₃₁. Henceinf_𝑄∈𝒬max_𝑢∈𝑄𝐼(𝑢)=𝜆₃₁, (7.4) and, we have the desired variational characterization.

Figure 8 

𝑃′′′.

Acknowledgment

This research was sponsored partly by Grant ME09109, Program KONTAKT, Ministry of Education, Youth and Sports, Czech Republic