Research on the nonlinear elliptic eigenvalue problems by variational methods

1. (1)研究了非线性双参数问题-u”（x）+ λu（x）^q = μu（x）^p，u（x）> 0，x ∈（0，1），u（0）= u（1）= 0.这里1 < q < p是常数，λ，μ > 0是参数。建立了变分特征曲线μ（λ）在λ→∞时具有精确第二项的精确渐近公式.我们强调，第二项衰减率的临界情形是p =（3q - 1）/2，这种临界性是新的。(2)考虑非线性双参数问题u”（x）+ μu（x）^p = λu（x）^q，u（x）> 0，x ∈ I =（0，1），u（0）= u（1）= 0，其中1 < q < p <2 q + 3，λ，μ > 0为参数.在Zeidler的一般水平集上，利用变分方法建立了特征曲线λ = λ（μ）在μ→∞时的三项谱渐近性.λ（μ）的第一项和第二项不依赖于p和q之间的关系。然而，第三项取决于p和q之间的关系，临界情况是p =（3q-1）/2.2。考虑非线性特征值问题-Δu = λf（u），u > 0在Ω上，u = 0在？其中Ω=B_R={x ∈ R^N：|X| <R}或A_<a，R> = {x ∈ R^N：a<|X| <R}（N[大于或等于]2）和λ>0是一个参数。在f和g满足一定条件时，当λ>> 1时，相应的解u_λ产生边界层。建立了边界层宽度的渐近公式，并给出了第二项的精确估计和第三项的估计.考虑了几类椭圆型偏微分方程和抛物型方程组的非线性特征值问题，得到了解的存在性结果和定性性质。

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.