Variational methods for nonlinear elliptic eigenvalue problems

Abstract

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,The solution of the first problem yields a weak solution of a corresponding elliptic boundary-value problem for the Euler-Lagrange equation\a\zm From the solution of the second problem, we obtain a solution under corresponding boundary conditions of the nonlinear eigenvalue problem.(In §1, we give a complete self-contained treatment of the existence of minima of functionals on reflexive Banach spaces, a treatment which extends and strengthens earlier studies by Lusternik, E. Rothe, Vainberg, and others (see [ó]

Soliton solutions for quasilinear Schrödinger equations, I

“…Since we may assume u n ≥ 0 we have u ≥ 0. By the Lagrange multiplier theorem ( [7]) we conclude that u is a weak solution of (2)…”

On the existence of equilibrium states of an elastic beam on a nonlinear foundation

“…We formulate the problem as a fourthorder nonlinear boundary value problem and prove two theorems on the existence of solution of this boundary value problem. The proofs are based upon a paper by Browder [1].…”

“…Both Upon using the change of variables z=, A= E--"/' &= -ET' PROOF. Lemma 3.1 enables us to apply Theorem of [1] which implies the existence of a solution y of (3.1). The smoothness of y follows from the regularity theory of [4].…”

Existence of equilibrium states of hollow elastic cylinders submerged in a fluid

“…A related note on this subject was given by S. Antmann [3]. Several later works by Antmann [4][5][6] As in [1] we followed the mathematical tools provided by Browder in [2]. To facilitate our proofs we presented a generalization of Browder's results in Lemma 3.2 for semi-convex functionals in Hilbert spaces subject to several constraints.…”