TMNA - Volume 15 Number 2

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TOPOLOGICAL METHODS

IN

NONLINEAR ANALYSIS

Vol. 15, No. 2 June 2000

TABLE OF CONTENTS

Title and Author(s)

Page

A Topological Approach to Superlinear Indefinite Boundary Value Problems
D. Papini and F. Zanolin

ABSTRACT. We obtain the existence of infinitely many solutions with prescribed nodal properties for some boundary value problems associated to the second order scalar equation x + q(t)g(x) = 0, where g(x) has superlinear growth at infinity and q(t) changes sign.

203

Positive Solutions of a~Hammerstein Integral Equation with a Singular Nonlinear Term
M. M. Coclite

ABSTRACT. In this paper the existence of a positive measurable solution of the Hammerstein equation of the first kind with a singular nonlinear term at the origin is presented.

235

Blow Up Points of Solution Curves for a Semilinear Problem
J. Shi

ABSTRACT. We study a semilinear elliptic equation with an asymptotic linear nonlinearity. Exact multiplicity of solutions are obtained under various conditions on the nonlinearity and the spectrum set. Our method combines a bifurcation approach and Leray-Schauder degree theory.

251

Attractor and Dimension for Discretization of a Damped Wave Equation with Periodic Nonlinearity
S. Zhou

ABSTRACT. The existence and Hausdorff dimension of the global attractor for discretization of a damped wave equation with the periodic nonlinearity under the periodic boundary conditions are studied for any space dimension. The obtained Hausdorff dimension is independent of the mesh sizes and the space dimension and remains small for large damping, which conforms to the physics.

267

Multiple Solutions of Degenerate Perturbed Elliptic Problems Involving a Subcritical Sobolev Exponent
F. St. Cirstea and V. D. Radulescu

ABSTRACT. We study the degenerate elliptic equation $-\text{\rm div}(a(x)\nabla u)+b(x)u= K(x)\vert u\vert ^{p-2}u+g(x)\quad \text{\rm in } \RR^{N},$ where $N \geq 2$ and 2<p<2^*

. We assume that $a\not\equiv 0$ is a continuous, bounded and nonnegative function, while b and K are positive and essentially bounded in R^N. Under some assumptions on a, b and K, which control the location of zeros of a and the behaviour of a, b and K at infinity we prove that if the perturbation g is sufficiently small then the above problem has at least two distinct solutions in an appropriate weighted Sobolev space. The proof relies essentially on the Ekeland Variational Principle [8] and on the Mountain Pass Theorem without the Palais-Smale condition, established in Brezis-Nirenberg [6], combined with a weighted variant of the Brezis-Lieb Lemma [5], in order to overcome the lack of compactness.

285

A Global Bifurcation Result for Quasilinear Elliptic Equations in Orlicz-Sobolev Spaces
V. K. Le

ABSTRACT. The paper is concerned with a global bifurcation result for the equation $-\text{div} (A(|\nabla u|) \nabla u) = g(x,u,\lambda)$ in a general domain $\Omega$ with non necessarily radial solutions. Using a variational inequality formulation together with calculations of the Leray-Schauder degrees for mappings in Orlicz-Sobolev spaces, we show a global behavior (the Rabinowitz alternative) of the bifurcating branches.

301

Multiple Interior Layers of Solutions to Perturbed Elliptic Sine-Gordon Equation on an Interval
T. Shibata

ABSTRACT. We consider the perturbed elliptic Sine-Gordon ODE with two positive parameters $\mu$ and $\lambda$ , and show the existence of solutions which have 2n multiple interior layers for $\lambda \gg 1$ . We also determine the location of multiple interior layers as $\lambda\to\infty$ .

329

Cauchy Problems and Applications
C.-Y. Lin

ABSTRACT. Of concern is the Cauchy problem $\frac{du}{dt} \in Au,\quad u(0) = u_{0},\quad t > 0,$ where $u:[0,\infty)\to X$ X is a real Banach space, and $A:D(A)\subset X\to X$ is nonlinear and multi-valued. It is showed by the method of lines, combined with the Crandall-Liggett theorem that this problem has a limit solution, and that the limit solution is a unique strong one if A is what is called embeddedly quasi-demi-closed. In the case of linear, single-valued A, further results are given. An application to nonlinear partial differential equations in non-reflexive X is given.

359

On Some Properties of Dissipative Functional Differential Inclusions in a Banach Space
V. Obukhovskii and P. Zecca

ABSTRACT. For a semilinear functional differential inclusion of the form $y'(t) \in Ay(t)+f(t, x_t)$ satisfying a dissipativity condition in a separable Banach space we prove the existence of a periodic solution and a global compact attractor.

369

On Selection Theorems with Decomposable Values
S. M. Ageev and D. Repovs

ABSTRACT. The main result of the paper asserts that for every separable measurable space $(T,\goth F,\mu)$ , where $\goth F$ is the $\sigma$ -algebra of measurable subsets of T and $\mu$ is a nonatomic probability measure on $\goth F$ , every Banach space E and every paracompact space X, each dispersible closed-valued mapping $F: x \rightsquigarrow L_1(T,E)$ of X into the Banach space

of all Bochner integrable functions $u: T\to E$ admits a continuous selection. Our work generalizes some results of Goncarov and Tol'stonogov.

385

Corrections to ``Stable Maps of Genus Zero to Flag Spaces''
(Topol. Methods Nonlinear Anal. 11 (1998), 207--217)
Yu. I. Manin

401