Research on the nonlinear elliptic eigenvalue problems by variational methods

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1. (1) We studied the nonlinear two-parameter problem -u"(x) + λu(x)^q = μu(x)^p,u(x) > 0,x ∈(0,1),u(0) = u(1) = 0. Here 1 < q < p are constants and λ,μ > 0 are parameters. We established precise asymptotic formulas with exact second term for variational eigencurve μ(λ) as λ→∞. We emphasize that the critical case concerning the decaying rate of the second term is p = (3q - 1)/2 and this kind of criticality is new.(2) We considered the nonlinear two-parameter problem u"(x) + μu(x)^p = λu(x)^q, u(x) > 0, x ∈ I = (0,1), u(0) = u(1) = 0, where 1 < q < p < 2q + 3 and λ,μ > 0 are parameters. We established the three-term spectral asymptotics for the eigencurve λ = λ(μ) as μ→∞ by using a variational method on the general level set due to Zeidler. The first and second terms of'λ(μ) do not depend on the relationship between p and q. However, the third term depends on the relationship between p and q, and the critical case is p = (3q-1)/2.2. We considered the nonlinear eigenvalue problem -Δu = λf(u), u > 0 in Ω,u = 0 on ?∂Ω, where Ω=B_R={x ∈ R^N : |x| <R} or A_<a,R> = {x ∈ R^N : a< |x| <R} (N【greater than or equal】2) and λ>0 is a parameter. It is known that under some conditions on f and g, the corresponding solution u_λ develops boundary layers when λ>> 1. We established the asymptotic formulas for the width of the boundary layers with exact second term and the estimate of the third term.3. We considered several elliptic partial differential equations and parabolic systems related to nonlinear eigenvalue problems and obtained some existence results and qualitative properties of the solutions.