In this paper we get the existence results of the nontrivial weak solution (λ,u) of the following eigenvalue problem of quasilinear elliptic systems-Dα (aαβ(x,u) Dβui) + 1/2 Dui aαβ(x,u)Dαuj Dβuj + h(x) ui = λ|u|p-2ui,   for x ∈ Rn, 1 ≤ i ≤ N and u = (u1, u2, ..., uN) ∈ E = {v = (v1, v2, ..., vN) | vi ∈ H1(Rn), 1 ≤ i ≤ N},where aαβ(x,u) satisfy the natural growth conditions. It seems that this kind of problem has never been dealt with before.

We consider the linear eigenvalue problem -Δu = λV(x)u, $u\in D_0^{1,2}\left(\Omega \right)$, and its nonlinear generalization $-{\Delta }_{p}u=\lambda V\left(x\right){\left|u\right|}^{p-2}u$, $u\in D_0^{1,p}\left(\Omega \right)$. The set Ω need not be bounded, in particular, $\Omega ={ℝ}^N$ is admitted. The weight function V may change sign and may have singular points. We show that there exists a sequence of eigenvalues ${\lambda }_n\to \infty$.

A discretized boundary value problem for the Laplace equation with the Dirichlet and Neumann boundary conditions on an equilateral triangle with a triangular mesh is transformed into a problem of the same type on a rectangle. Explicit formulae for all eigenvalues and all eigenfunctions are given.

A boundary value problem for the Laplace equation with Dirichlet and Neumann boundary conditions on an equilateral triangle is transformed to a problem of the same type on a rectangle. This enables us to use, e.g., the cyclic reduction method for computing the numerical solution of the problem. By the same transformation, explicit formulae for all eigenvalues and all eigenfunctions of the corresponding operator are obtained.

For a class of non-selfadjoint $h$–pseudodifferential operators with double characteristics, we give a precise description of the spectrum and establish accurate semiclassical resolvent estimates in a neighborhood of the origin. Specifically, assuming that the quadratic approximations of the principal symbol of the operator along the double characteristics enjoy a partial ellipticity property along a suitable subspace of the phase space, namely their singular space, we give a precise description of...

We consider a class of eigenvalue problems for polyharmonic operators, including Dirichlet and buckling-type eigenvalue problems. We prove an analyticity result for the dependence of the symmetric functions of the eigenvalues upon domain perturbations and compute Hadamard-type formulas for the Frechét differentials. We also consider isovolumetric domain perturbations and characterize the corresponding critical domains for the symmetric functions of the eigenvalues. Finally, we prove that balls are...

We consider the nonlinear eigenvalue problem $$-{div\left(\right|\nabla u|}^{p-2}{\nabla u)=\lambda g\left(x\right)|u|}^{p-2}u$$ in ${𝐑}^N$ with $p>1$. A condition on indefinite weight function $g$ is given so that the problem has a sequence of eigenvalues tending to infinity with decaying eigenfunctions in $W^{1,p}\left({𝐑}^N\right)$. A nonexistence result is also given for the case $p\ge N$.

We describe a simple linear algebra idea which has been used in different branches of mathematics such as bifurcation theory, partial differential equations and numerical analysis. Under the name of the Schur complement method it is one of the standard tools of applied linear algebra. In PDE and spectral analysis it is sometimes called the Grushin problem method, and here we concentrate on its uses in the study of infinite dimensional problems, coming from partial differential operators of mathematical...

One may produce the qth harmonic of a string of length π by applying the 'correct touch' at the node $\pi /q$ during a simultaneous pluck or bow. This notion was made precise by a model of Bamberger, Rauch and Taylor. Their 'touch' is a damper of magnitude b concentrated at $\pi /q$. The 'correct touch' is that b for which the modes, that do not vanish at $\pi /q$, are maximally damped. We here examine the associated spectral problem. We find the spectrum to be periodic and determined by a polynomial of degree $q-1$....