Asymptotic behavior of eigenvalues of two-parameter nonlinear Sturm-Liouville problems

The nonlinear two-parameter Sturm-Liouville problemu ^"+μg(u)=λf(u) is studied for μ, λ>0. By using Ljusternik-Schnirelman theory on the general level set developed by Zeidler, we shall show the existence of ann-th variational eigenvalue λ=λ_n(μ). Furthermore, for specialf andg, the asymptotic formula of λ₁(μ)) as μ→∞ is established.     

Asymptotic behavior of solutions to multiplicative control problems for elliptic equations

The problems of optimal multiplicative control for the Helmholtz equation and the diffusion equation are studied. The control function is included multiplicatively in a mixed-type boundary condition specified on the entire domain boundary or its part. For each of the models under study, an iterative method for determining an approximate solution is constructed and theoretically substantiated for sufficiently large values of the regularization parameter.     

Asymptotic behavior of the eigenvalues of a boundary-value problem for a second-order elliptic operator-differential equation

We study the asymptotic behavior of the eigenvalues of a boundary-value problem with spectral parameter in the boundary conditions for a second-order elliptic operator-differential equation. The asymptotic formulas for the eigenvalues are obtained. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 8, pp. 1146–1152, August, 2006.     

Nonlinear eigenvalues for anisotropic quasilinear degenerate elliptic equations

We consider the Dirichlet problem for a class of anisotropic degenerate elliptic equations.     

一类奇摄动椭圆方程在摄动区域上的渐近解

研究了一类具有摄动边界的非线性椭圆方程摄动问题.经过极坐标变换,在适当的条件下,通过构建近似解以及校正项,利用上下解方法和微分不等式理论得到了解的渐近性态,并通过实际例子进行了验证.     

Concentration on Hopf-fibres for singularly perturbed elliptic equations

Singularly perturbed elliptic equations with superlinear nonlinearities of polynomial type are considered on an annulus in Rⁿ${\mathbb{R}}^n$, n≥4$n\ge 4$. It is shown that for small parameters there exist solutions which concentrate on manifolds of dimensions one, three and seven, which are given as Hopf-fibres.     

Asymptotic behavior and uniqueness of blow-up solutions of quasilinear elliptic equations

In this paper, we study the asymptotic behavior of solutions of the problem Δ _p u = f (u) in Ω, u = ∞ on ∂Ω, under general conditions on the function f, where Ω _p is the p-Laplace operator. We show that the technique used by the author for the special case p = 2 works in this more general setting, and that the behavior described by various authors for the case p = 2 is easily derived from this technique for the general case.     

Asymptotic behavior of eigenvalues of the Laplace operator in infinite cylinders perturbed by transverse extensions

We present sufficient conditions for the existence of an eigenvalue of the Laplace operator with zero Dirichlet conditions in a weakly perturbed infinite cylinder in the case of localized perturbations which are extensions along the transverse coordinates with coefficients depending on the longitudinal coordinate. If such an eigenvalue exists, then, for this eigenvalue, we obtain an asymptotic formula with respect to a small parameter characterizing the values of extensions.     

Asymptotic behavior at infinity for solutions of Emden-Fowler type equations

The semilinear equation Δu = |u|^σ?1 u is considered in the exterior of a ball in ? ⁿ , _n ≥ 3. It is shown that if the exponent σ is greater than a “critical” value (= n/n?2), then for x → ∞ the leading term of the asymptotics of any solution is a linear combination of derivatives of the fundamental solution. It is shown that there exist solutions with the indicated leading term of an asymptotics of such a type.     

Resonance at two consecutive eigenvalues for semilinear elliptic equations

The solvability of the Dirichlet problem for a semilinear elliptic equation is studied in some situations where the classical resonance conditions of Landesman and Lazer may fail.     

Conditions for two-peaked solutions of singularly perturbed elliptic equations

We obtain necessary conditions for the existence of two-peaked solutions of singularly perturbed elliptic equations. These conditions are related to the geometry of the domain. In particular, we prove there are no two-peaked solutions in a strictly convex domain. Received: 20 January 1997 / Revised version: 2 December 1997     

Asymptotic stability of two-parameter systems of delay differential equations

We establish exact efficient asymptotic stability criteria for two-parameter systems of two autonomous delay differential equations.     

Asymptotic paths for subsolutions of quasilinear elliptic equations

Letu be an entire lower semicontinuous subsolution to the quasilinear elliptic equation divA(x,u)=0 in ⁿ . It is shown that ifu is not bounded above, then there exists a path going to infinity along whichu tends to infinity. The result extends works of Talpur, Fuglede, and others. Growth aspects of subsolutions are also studied.     

Infinitely many solutions to perturbed elliptic equations

A new version of perturbation theory is developed which produces infinitely many sign-changing critical points for uneven functionals. The abstract result is applied to the following elliptic equations with a Hardy potential and a perturbation from symmetry: