Fifth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 10, 2003, pp 251–256. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

On the average value for nonconstant eigenfunctions of the p-Laplacian assuming

Neumann boundary data ^{∗}

Stephen B. Robinson

Abstract

We show that nonconstant eigenfunctions of the p-Laplacian do not necessarily have an average value of 0, as they must when p = 2. This fact has implications for deriving a sharp variational characterization of the second eigenvalue for a general class of nonlinear eigenvalue problems.

1 Introduction

In this paper we show that the nonconstant solutions of

−Δ_{p}u − λ|u|^{p−2}u = 0 a.e. in Ω,

\frac{∂u}{∂ν} = 0 on ∂Ω,

(1.1)

do not necessarily satisfy ∫_{Ω} u = 0. This fact has implications for deriving a sharp variational characterization of the second eigenvalue for a broad class of nonlinear eigenvalue problems including (1.1). We assume that Ω ⊂ ℝ^{N} is a smooth bounded domain, λ is a real number, and Δ_{p} is the p-Laplacian, i.e. Δ_{p}u := ∇ · (|∇u|^{p−2}∇u), for some p ∈ (1, ∞).

In some respects (1.1) is already well understood. Since Neumann boundary conditions are assumed, it is straightforward to see that the principle eigenvalue is λ_{1} = 0 with simple eigenspace W := span{1}. Recent work in [2], [3], [4], and [6] has provided a detailed description of the second eigenvalue, λ_{2}, which is defined as the smallest real number greater than λ_{1} such that (1.1) has a nontrivial solution. In particular, it is known that λ_{2} > 0, and that eigenfunctions associated with λ_{2} are sign-changing with exactly two nodal domains and are in the set V_{p} := {v ∈ W^{1,p}(Ω) : ∫_{Ω} |v|^{p−2}v = 0}. Also, λ_{2} satisfies variational characterizations that generalize from the linear case in a natural way. We should point out that the references above impose Dirichlet boundary conditions, but provide a framework that works just as well for (1.1). In section 2 we will provide a sketch of how some of these facts can be proved.

∗Mathematics Subject Classifications: 35P30, 35J20, 35J65.

Key words: Nonlinear eigenvalue problem, p-Laplacian, variational methods.

\begin{matrix}c \\ ◯ \end{matrix}2003 Southwest Texas State University.

Published February 28, 2003.

There are several situations where it is straightforward to see that second eigenfunctions have an average value of 0. Of course, if p = 2, then (1.1) reduces to the standard eigenvalue problem for the Laplace operator with Neumann boundary data, and it is clear that every nonconstant eigenfunction lies in V_{2} = W^{⊥} = {u ∈ W^{1,2}(Ω) : ∫_{Ω} u = 0}. For arbitrary p, if we examine the ODE case, then it is possible to exploit the symmetry of Ω = (a, b) to prove that nonconstant eigenfunctions once again satisfy ∫_{a}^{b} u = 0. This ODE argument can be extended to eigenfunctions on “boxes” in ℝ^{N} with N > 1, i.e. Ω = (a_{1}, b_{1})×⋯×(a_{N}, b_{N}). But what of the average value for second eigenfunctions over more general domains?

This question arose while studying eigenvalue problems for a class of quasilinear operators that generalize the p-Laplacian, i.e.

Q(u) := ∑_{1≤|α|≤m}(−1)^{|α|}D^{α}A_{α}(x, ξ′_{m}(u)),

where Q is a 2m-th order quasilinear operator satisfying general growth, ellipticity and monotonicity conditions. For boundary value problems associated with such operators some interesting existence theorems have been proved by Shapiro, et.al., where a second eigenvalue is defined and used as an upper bound in certain key growth estimates. This second eigenvalue is obtained via the minimization of an appropriate functional, essentially a Raleigh quotient, over the space V_{2} = {u ∈ W^{1,p}(Ω) : ∫_{Ω} u = 0}. (More details are provided in section 2 and in the references [7] and [9].) This allows something like an orthogonal splitting of the Banach Space W^{1,p}(Ω) so that saddle point theorems can be applied in a standard way. An open question that arose as a result of these papers was whether or not this orthogonal splitting leads to a sharp characterization of the second eigenvalue. Our main result in this paper shows that it does not. It follows that an improved characterization should lead to an improvement of the existence results in the papers listed above. These improved existence theorems are described in subsequent work.

2 Preliminaries

We begin with a standard variational formulation of the problem, and briefly present some straightforward properties and definitions. Details can be checked in the references. Let W^{1,p}(Ω) be defined in the usual way, as in [1]. Let

E(u) := ∫_{Ω} |∇u|^{p}, for u ∈ W^{1,p}(Ω).

It is well known that E is a C^{1} functional with

E′(u) · v = p ∫_{Ω} |∇u|^{p−2}∇u · ∇v.

Moreover, if we consider E constrained to the surface S := {u ∈ W^{1,p}(Ω) : ∫_{Ω} |u|^{p} = 1}, then any critical point, ϕ, satisfies

∫_{Ω} |∇ϕ|^{p−2}∇ϕ · ∇v = λ ∫_{Ω} |ϕ|^{p−2}ϕv  (2.1)

for some λ ∈ ℝ and all v ∈ W^{1,p}(Ω). Hence, the critical points of the constrained functional correspond to eigenfunctions, and the associated Lagrange multipliers correspond to eigenvalues. (Substitute v = ϕ into (2.1) to see that λ = E(ϕ).) Notice that by constraining the functional to the L^{p} unit sphere we are simply recognizing that all nontrivial eigenfunctions can be rescaled so that they are elements of S.

E clearly attains a global minimum of 0 at ±ϕ_{1} = ±(\frac{1}{|Ω|})^{\frac{1}{p}}. Also, it is clear that E(u) > 0 for any nonconstant u. Thus λ_{1} = 0 is a simple eigenvalue with eigenspace W := span{1}.

If λ > 0 is an eigenvalue with associated eigenfunction ϕ, then we can substitute v = 1 into (2.1) to see that ϕ ∈ V_{p}. Hence our search for critical points can be restricted to the set V_{p} ⋂ S. Members of this set are clearly sign-changing. Using the fact that V_{p} ⋂ S is weakly closed, and that E is bounded below and weakly lower semicontinuous, we see that E attains its positive infimum on S. Hence, one variational characterization of λ_{2} is

λ_{2} := inf_{S∩V_{p}} E.  (2.2)

Let ϕ_{2} represent an associated eigenfunction and consider the curve

h : ℝ → S : h(t) = \frac{ϕ_{2} + t}{||ϕ_{2} + t||_{L^{p}}} .

Then

E(h(t)) = \frac{∫_{Ω} |∇ϕ_{2}|^{p}}{∫_{Ω} |ϕ_{2} + t|^{p}} , and \frac{d}{dt}E(h(t)) = \frac{−p ∫_{Ω} |∇ϕ_{2}|^{p} ∫_{Ω} |ϕ_{2} + t|^{p−2}(ϕ_{2} + t)}{(∫_{Ω} |ϕ_{2} + t|^{p})^{2}}.

Thus E(h(t)) reaches a global maximum of λ_{2} only at t = 0. Moreover, lim_{t→±∞} h(t) = ±ϕ_{1} and lim_{t→±∞} E(h(t)) = 0. Thus h(t) can be modified to create a continuous curve γ : [−1, 1] → S such that γ(±1) = ±ϕ_{1}, γ(0) = ϕ_{2}, and such that E(γ(t)) achieves a maximum value of λ_{2} precisely when t = 0. Conversely, any continuous curve on S connecting ±ϕ_{1} must cross V_{p} and hence must contain a point, γ(t), where E(γ(t)) ≥ λ_{2}. Thus we deduce a second, equivalent, variational characterization of λ_{2} which is

λ_{2} := inf_{γ∈Γ} sup_{−1≤t≤1} E(γ(t)),  (2.3)

where Γ := {γ : [−1, 1] → S : γ is continuous, γ(±1) = ±ϕ_{1}}. The proof that ϕ_{2} has exactly 2 nodal domains relies on the fact that if ϕ_{2} has more than 2 nodal domains then a curve can be constructed that contradicts (2.3) . Details can be found in [4] or [6].

Let μ_{2} represent the parameter characterized in [9] and [7]. For homogeneous problems, such as (1.1), this reduces to

μ_{2} := inf_{S∩V_{2}} E.  (2.4)

If we compare the characterization (2.3) with (2.4), we observe that every curve in Γ must cross at least one point in V_{2}, and thus the maximum value of E over any such curve is at least as large as μ_{2}. It follows that μ_{2} ≤ λ_{2}. Now suppose that we can show that ϕ_{2} ∉ V_{2}. If we examine the special curve γ, constructed above, we see that γ crosses V_{2} at a point γ(t) ≠ ϕ_{2}, so E(γ(t)) < λ_{2}, and thus μ_{2} < λ_{2}. This would show that (2.4) is not a sharp characterization of λ_{2}. In section 3 we will prove that ϕ_{2} ∉ V_{2} for certain asymmetric domains. An interesting open question might be to classify the domains where μ_{2} = λ_{2}, and it is reasonable to conjecture that this depends upon a symmetry condition.

It is important to note that the quasilinear operators in [7] and [9] are not assumed to be homogeneous, so the associated eigenvalue problems could not be restricted to S. Hence, the more general characterization had to consider the infimum of \frac{E(u)}{∫_{Ω} |u|^{p}} over V_{2} ⋂ rS and then compute a lim inf as r → ∞.

3 Comparing λ_{2} and μ_{2}

Theorem 3.1 There is at least one domain Ω ⊂ ℝ^{N} such that the associated second eigenvalue, λ_{2}, has an associated eigenfunction, ϕ_{2}, that does not lie in V_{2}.

Proof Consider the problem

\begin{matrix}−Δ_{p}u − λ_{2}^{ϵ}|u|^{p−2}u = 0 in Ω_{ϵ}, \\ \frac{∂u}{∂ν} = 0 on ∂Ω_{ϵ}, \end{matrix}  (3.1)

where Ω_{ϵ} := ((0, 2) × (0, 2)) ⋃((2, 3) × (0, ϵ)) ⋃ ((3, 4) × (0, 1)) for 0 ≤ ϵ ≤ 1, and where λ_{2}^{ϵ} is characterized by (2.2) and (2.3). Ω_{0} will refer to the limiting case which is simply the union of the two disjoint rectangles. Let ϕ_{2,ϵ} ∈ V_{p} ⋂ S represent an associated second eigenfunction. When ϵ = 0 this will simply indicate a function that is a positive constant over one rectangle and a negative constant over the other, where the constants are balanced to fit the constraints.

First, we find an upper bound for λ_{2}^{ϵ}. Let

u_{2} := \begin{cases}1 & for (x, y) ∈ [0, 2] × [0, 2], \\ −2x + 5 & for (x, y) ∈ [2, 3] × [0, ϵ], \\ −1 & for (x, y) ∈ [3, 4] × [0, 1] \end{cases}

Also, let γ(α, β) = αu_{2}^{+} − βu_{2}^{−}, where u_{2}^{+} := max{u_{2}, 0}, u_{2}^{−} := max{−u_{2}, 0}, and α and β are nonnegative scalars such that α^{p}||u_{2}^{+}||_{L^{p}}^{p} + β^{p}||u_{2}^{−}||_{L^{p}}^{p} = 1.

Notice that γ is a curve on S connecting the points \frac{u_{2}^{+}}{||u_{2}^{+}||_{L^{p}}} and −\frac{u_{2}^{−}}{||u_{2}^{−}||_{L^{p}}}. By the Intermediate Value Theorem γ crosses the surface V_{p}. Hence the maximum of E(γ(α, β)) must be greater than λ_{2}^{ϵ}. However,

∇γ(α, β) = \begin{cases}(0, 0) & for (x, y) ∈ [0, 2] × [0, 2], \\ (−2α, 0) & for (x, y) ∈ [2, \frac{5}{2}] × [0, ϵ], \\ (−2β, 0) & for (x, y) ∈ (\frac{5}{2}, 3] × [0, ϵ], \\ (0, 0) & for (x, y) ∈ (3, 4] × [0, 1] \end{cases}

Thus ∫_{Ω_{ϵ}} |∇γ(α, β)|^{p} ≤ 2^{p} max{α^{p}, β^{p}}ϵ. But ||u_{2}^{+}||_{L^{p}}^{p} ≥ 4 and ||u_{2}^{−}||_{L^{p}}^{p} ≥ 1, so α^{p} ≤ \frac{1}{4} and β^{p} ≤ 1. Therefore ∫_{Ω_{ϵ}} |∇γ(α, β)|^{p} ≤ 2^{p}ϵ. It follows that λ_{2}^{ϵ} ≤ max_{(α,β)} E(γ(α, β)) ≤ 2^{p}ϵ, so lim_{ϵ→0} λ_{2}^{ϵ} = 0.

We will now show that ∫_{Ω_{ϵ}} ϕ_{2,ϵ} ≠ 0 for some ϵ. Since λ_{2}^{ϵ} → 0, straightforward estimates now show that ϕ_{2,ϵ} → ϕ_{2,0} in W^{1,p}(Ω_{0}), where ∇ϕ_{2,0} ≡ 0 and ∫_{Ω_{0}} |ϕ_{2,0}|^{p} = 1. Moreover, ∫_{Ω_{0}} |ϕ_{2,0}|^{p−2}ϕ_{2,0} = lim_{ϵ→0} ∫_{Ω_{ϵ}} |ϕ_{2,ϵ}|^{p−2}ϕ_{2,ϵ} = 0. It must be that there are constants a, b ∈ ℝ such that ϕ_{2,0} ≡ a in [0, 2] × [0, 2] and ϕ_{2,0} ≡ b in [3, 4] × [0, 1]. Moreover, it follows that 4|a|^{p} + |b|^{p} = 1, a and b have opposite signs, and 4|a|^{p−1} − |b|^{p−1} = 0. Thus |b| = 4^{\frac{1}{p−1}} |a|. It can now be checked that ∫_{Ω_{0}} ϕ_{2,0} = ±(4 − 4^{\frac{1}{p−1}} )|a| ≠ 0 for p ≠ 2. Hence ∫_{Ω_{ϵ}} ϕ_{2,ϵ} ≠ 0 for some ϵ > 0. □

As an immediate consequence we have the following statement.

Corollary 3.2 If Ω is the domain given in Theorem 3.1, then μ_{2} < λ_{2}.

References

[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

[2] A. Anane and N. Tsouli, On the second eigenvalue of the p-Laplacian, Nonlinear partial differential equations (Fés, 1994), 1-9, Pitman Res. Notes Math. Ser., 343, Longman, Harlow, 1996.

[3] M. Cuesta, D.G. de Figueiredo and J.-P. Gossez, The beginning of the Fučik spectrum for the p-Laplacian, J. Differential Equations 159 (1999), no.1, pp. 212-238.

[4] M. Cuesta, D. G. de Figueiredo, J.-P. Gossez, A nodal domain property for the p-Laplacian, C. R. Acad. Sci. Paris Sr. I Math. 330 (2000), no. 8, 669–673.

[5] D.G. de Figueiredo and J.-P. Gossez, On the first curve of the Fučik spectrum of an elliptic operator, Differential and Integral Equations 7 (1994), no.5-6, pp. 1285-1302.

[6] P. Drábek and S. Robinson, On the generalization of the Courant Nodal Domain Theorem, Journal of Differential Equations, vol. 181 (2002), pp. 58-71

[7] A.J. Rumbos and V.L. Shapiro, One-sided resonance for a quasilinear variational problem, Cont. Math., Vol. 208 (1997), pp. 277-299.

[8] V.L. Shapiro, Quasilinear ellipticity and the first eigenvalue, Comm in PDEs, 16 (1991), 1819-1855.

[9] V.L. Shapiro, Superlinear quasilinearity and the second eigenvalue, Nonlinear Anal., 44 (2001), no. 1, Ser. A: Theory Methods, 81-96.

[10] M. Struwe, Variational Methods; Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, New York, 1990.

Stephen B. Robinson Department of Mathematics, Wake Forest University, Winston-Salem, NC 27109, USA e-mail: sbr@wfu.edu