Variational Methods for Nonlinear Elliptic Eigenvalue Problems
نویسنده
چکیده
In the present note, we give a simple general proof for the existence of solutions of the following two types of variational problems: PROBLEM A. To minimize fa F(x> u, • • • , Du)dx over a subspace VofW>*(tt). PROBLEM B. TO minimize ƒ« F(x, w, • • • , Du)dx for u in V with / a G(x, u, • • • , D^u)dx^c. The solution of the first problem yields a weak solution of a corresponding elliptic boundary-value problem for the Euler-Lagrange equation
منابع مشابه
Existence of ground state solutions for a class of nonlinear elliptic equations with fast increasing weight
This paper is devoted to get a ground state solution for a class of nonlinear elliptic equations with fast increasing weight. We apply the variational methods to prove the existence of ground state solution.
متن کاملLp Embedding and Nonlinear Eigenvalue Problems for Unbounded Domains
Let R denote real iV-dimensional Euclidean space. Then it is a well-known fact that the imbedding of the Sobolev space Wi,2(R) in LP(R ) is bounded for 2g>pS2N/(N-2), but is definitely not compact. Consequently the theory of critical points for general isoperimetric variational problems defined over arbitrary unbounded domains in R has been little investigated despite its importance. Indeed the...
متن کاملNumerical Analysis of Nonlinear Eigenvalue Problems
We provide a priori error estimates for variational approximations of the ground state eigenvalue and eigenvector of nonlinear elliptic eigenvalue problems of the form −div(A∇u) + V u + f(u)u = λu, ‖u‖L2 = 1. We focus in particular on the Fourier spectral approximation (for periodic problems) and on the P1 and P2 finite-element discretizations. Denoting by (uδ, λδ) a variational approximation o...
متن کاملFree and constrained equilibrium states in a variational problem on a surface
We study the equilibrium states for an energy functional with a parametric force field on a region of a surface. Consideration of free equilibrium states is based on Lyusternik - Schnirelman's and Skrypnik's variational methods. Consideration of equilibrium states under a constraint of geometrical character is based on an analog of Skrypnik's method, described in [P. Vyridis, {it Bifurcation in...
متن کاملStability of patterns with arbitrary period for a Ginzburg-Landau equation with a mean field
We consider the following system of equations { At = Axx +A−A3 −AB, x ∈ R, t > 0, Bt = σBxx + μ(A)xx, x ∈ R, t > 0, where μ > σ > 0. It plays an important role as a Ginzburg-Landau equation with a mean field in several fields of the applied sciences. We study the existence and stability of periodic patterns with an arbitrary minimal period L. Our approach is by combining methods of nonlinear fu...
متن کامل