Precise asymptotic formulas for semilinear eigenvalue problems

Tetsutaro Shibata

The Division of Mathematical and Information Sciences, Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima, 739-8521, Japan shibata@mis.hiroshima-u.ac.jp

We consider the following nonlinear two-parameter problem

−u′′(x) + λu(x)^{q} = μu(x)^{p}, x ∈ I = (0, 1),
u(x) > 0, x ∈ I,  (0.1)
u(0) = u(1) = 0,

where 1 < q < p and λ, μ > 0 are parameters.

The purpose of this talk is to establish the asymptotic formulas for the eigencurve μ = μ(λ) with the exact second term as λ → ∞ by using a variational method. We also establish the critical relationship between p and q from a point of view of the decaying rate of the second term of μ(λ).

The study of two-parameter eigenvalue problems began with the oscillation theory and has been investigated by many authors. We refer to Atkinson [1], Binding and Browne [2], Cantrell [3], Faierman [4], Faierman and Rauch [5], Shibata [6, 7, 8], Sleeman [9], Turyn [10], Volkmer [11] and the references therein. One of the main problems in this area is to analyze the structure of the solution set {(λ, μ, u)} of (1.1) and the effective approach to this problem is to study the structure of the set S_{λ,μ} := {(λ, μ, ∥u∥_{p+1})} ⊂ R^{3} for large λ. In Shibata [8], by using a standard variational framework, the variational eigencurve μ = μ(λ) was defined to analyze S_{λ,μ} and a simple asymptotic formula for μ(λ) as λ → ∞ was established to understand the first term of μ(λ) as λ → ∞. However, the remainder estimate of μ(λ) has not been obtained. The purpose here is to obtain the exact second term of μ(λ) as λ → ∞. We emphasize that the second term depends deeply on the relationship between p and q. Finally, it should be mentioned that the asymptotic behavior of such eigencurve is also effected by the variational framework (cf. [6, 7]).

Notations and Definitions

Let H_{0}^{1}(I) be the usual real Sobolev space. ∥u∥_{r} denotes the usual L^{r}-norm. For u ∈ H_{0}^{1}(I)

E_{λ}(u) := \frac{1}{2}∥u′∥_{2}^{2} + \frac{1}{q + 1}λ∥u∥_{q+1}^{q+1},

M_{γ} := {u ∈ H_{0}^{1}(I) : ∥u∥_{p+1} = γ},

where γ > 0 is a fixed constant. For a given λ > 0, we call μ(λ) the variational eigenvalue when the following conditions are satisfied:

(λ, μ(λ), u_{λ}) ∈ R_{+} × R_{+} × M_{γ} satisfies (0.1).

E_{λ}(u_{λ}) = inf_{u∈M_{γ}} E_{λ}(u).

Then μ(λ) is obtained as a Lagrange multiplier and is represented explicitly as follows:

μ(λ) = \frac{∥u′_{λ}∥_{2}^{2} + λ∥u_{λ}∥_{q+1}^{q+1}}{γ^{p+1}} .

We discuss the asymptotic behavior of μ(λ) as λ → ∞ precisely.

References

[1] F. V. Atkinson, Multiparameter spectral theory, Bull. Amer. Math. Soc. 74 (1968), 1–27.

[2] P. Binding and P. J. Browne, Asymptotics of eigencurves for second order differential equations, I, J. Differential Equations 88 (1990), 30–45

[3] R. S. Cantrell, Multiparameter bifurcation problems for second order ordinary differential equations, Rocky Mountain J. Math. 12 (1982), 795–806 [4] M. Faierman, Two-parameter eigenvalue problems in ordinary differential equations (Pitman Research Notes in Math. Series 205), Longman, Essex (1991).

[5] M. Faierman and G. F. Roach, Eigenfunction expansions associated with a multiparameter system of differential equations, Differential and Integral Equations 2 (1989), 45–56.

[6] T. Shibata, Asymptotic behavior of eigenvalues of two-parameter nonlinear Sturm-Liouville problems, J. Anal. Math. 66 (1995), 277–294.

[7] T. Shibata, The effect of the variational framework on the spectral asymptotics for two-parameter nonlinear eigenvalue problems, to appear.

[8] T. Shibata, Two-parameter eigenvalue problems in nonlinear second order differential equations, Result. Math. 31 (1997), 136–147.

[9] B. D. Sleeman, The two parameter Sturm-Liouville problem for order differential equations, Proc. Roy. Soc. Edinburgh 69A (1971), 139–148.

[10] L. Turyn, Sturm-Liouville problems with several parameters, J. Differential equations 38 (1980), 239–259.

[11] H. Volkmer, On multiparameter theory, J. Math. Anal. Appl. 86 (1982), 44–53.