TMNA - Volume 21 Number 2

TOPOLOGICAL METHODS

IN

NONLINEAR ANALYSIS

Vol. 21, No. 2 June 2003

TABLE OF CONTENTS

Title and Author(s)

Page

A Direct Topological Definition of the Fuller Index for Local Semiflows
Ch. C. Fenske

ABSTRACT. We define an index of Fuller type counting the periodic orbits of a local topological semiflow on ANR spaces avoiding smoothness assumptions and approximation processes.

195

Symmetry Results for Perturbed Problems and Related Questions
M. Grosi, F. Pacella and S. L. Yadava

ABSTRACT. In this paper we prove a symmetry result for positive solutions of the Dirichlet problem

(0.1)

$\cases -\Delta u=f(u) & \hbox{in }D,\\ u=0 & \hbox{on }\partial D,\endcases$

when f satisfies suitable assumptions and D is a small symmetric perturbation of a domain $\Omega$ for which the Gidas-Ni-Nirenberg symmetry theorem applies. We consider both the case when f has subcritical growth and $f(s)=s^{(N+2)/(N-2)}+\lambda s$, $N\ge3$, $\l$

211

Multiple Solutions for Asymptotically Linear Resonant Elliptic Problems
F. O. V. de Paiva

ABSTRACT. In this paper we establish the existence of multiple solutions for the semilinear elliptic problem

(1.1)

$\align -\Delta u =g(x,u) &\text{in }\Omega,\\u=0 &\text{on }\partial\Omega,\endalign$

where $\Omega \subset {\Bbb R}^N$ is a bounded domain with smooth boundary $\partial \Omega$ , a function $g\colon\Omega\times{\Bbb R}\to {\Bbb R}$ is of class C¹ such that g(x,0) = 0 and which is asymptotically linear at infinity. We considered both cases, resonant and nonresonant. We use critical groups to distinguish the critical points.

227

The Jumping Nonlinearity Problem Revisited: An Abstract Approach
D. G. Costa and H. Tehrani

ABSTRACT. We consider a class of nonlinear problems of the form Lu + g(x,u) = f, where L is an unbounded self-adjoint operator on a Hilbert space H of L²( $\Omega$ )-functions, $\Omega \subset {\Bbb R}^N$ an arbitrary domain, and $g\colon \Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a "jumping nonlinearity" in the sense that the limits $\lim_{s\rightarrow-\infty} \frac{g(x,s)}{s}=a \quad\text{\rm and}\quad \lim_{s\rightarrow\infty}\frac{g(x,s)}{s}=b$ exist and "jump" over an eigenvalue of the operator -L. Under rather general conditions on the operator L and for suitable a < b, we show that a solution to our problem exists for any $f\in H$ . Applications are given to the beam equation, the wave equation, and elliptic equations in the whole space R^N.

249

Morse Theory for Normal Geodesics in sub-Riemannian Manifolds with Codimension One Distributions
R. Giambo, F. Giannoni, P. Piccione and D. Tausk

ABSTRACT. We consider a Riemannian manifold (M,g) and a codimension one distribution $\Delta\subset T\M$ on M which is the orthogonal of a unit vector field Y on M. We do not make any nonintegrability assumption on $\Delta$ . The aim of the paper is to develop a Morse Theory for the sub-Riemannian action functional E on the space of horizontal curves, i.e. everywhere tangent to the distribution $\Delta$ . We consider the case of horizontal curves joining a smooth submanifold P of M and a fixed point $q\in\M$ . Under the assumption that P is transversal to $\Delta$ , it is known (see [19]) that the set of such curves has the structure of an infinite dimensional Hilbert manifold and that the critical points of E are the so called normal extremals (see [10]). We compute the second variation of E at its critical points, we define the notions of P-Jacobi field, of P-focal point and of exponential map and we prove a Morse Index Theorem. Finally, we prove the Morse relations for the critical points of E under the assumption of completeness for (M,g).

273

On a Radial Positive Solution to a Nonlocal Elliptic Equation
P. Fijalkowski and B. Przeradzki

ABSTRACT. The existence of a solution to the Dirichlet boundary value problem for nonlinear Poisson equations with the nonlocal nonlinear term $-\Delta u=f\bigg(u,\int (g\circ u)\bigg),\quad u\vert\partial U=0$ is proved by means of fixed point theorems for increasing compact operators.

293

Nontrivial Critical Groups in p-Laplacian Problems via the Yang Index
K. Perera

ABSTRACT. We construct and variationally characterize by a min-max procedure involving the Yang index a new sequence of eigenvalues of the p-Laplacian, and use the structure provided by this sequence to show that the associated variational functional always has a nontrivial critical group. As an application we obtain nontrivial solutions for a class of p-superlinear problems.

301

Parabolic Equations with Critical Nonlinearities
J. W. Cholewa and T. Dlotko

ABSTRACT. As well known the problem of global continuation of solutions to semilinear parabolic equations is completely solved when the nonlinear term is subordinated to an a-power of the main linear operator with $\alpha\in[0,1)$ . In this paper we study three examples of critical problems in which the mentioned subordination takes place with a = 1, i.e. the nonlinearity has the same order of magnitude as the linear main part. We use specific techniques of proving global solvability that fit well the considered examples for which general abstract methods fail.

311

On the Existence of Two Solutions for a General Class of Jumping Problems
A. Groli and M. Squassina

ABSTRACT. Via nonsmooth critical point theory we prove the existence of at least two solutions in $W^{1,p}_0(\Omega)$ . for a jumping problem involving the Euler equation of multiple integrals of calculus of variations under natural growth conditions. Some new difficulties arise in comparison with the study of the semilinear and also the quasilinear case.

325

Obstructions to the Extension Problem of Sobolev Mappings
T. Isobe

ABSTRACT. Let M and N be compact manifolds with $\partial M\ne\emptyset$ . We show that when 1 < p < dim M, there are two different obstructions to extending a map in $W^{1-1/p,p}(\partial M,N)$ to a map in $W^{1,p}(M,N)$ . We characterize one of these obstructions which is topological in nature. We also give properties of the other obstruction. For some cases, we give a characterization of $f\in W^{1-1/p,p}(\partial M,N)$ which has an extension $F\in W^{1,p}(M,N)$ .

345

On Sets of Constant Distance from a Planar Set
P. Pikuta

ABSTRACT. In this paper we prove that $d$-boundaries D_d = {x : dist(x,Z) = d} of a compact Z $\subset$ R² are closed absolutely continuous curves for d greater than some constant depending on Z. It is also shown that D_d is a trajectory of solution to the Cauchy Problem of a differential equation with a discontinuous right-hand side.

369

An Essential Map Theory for U_c^k and PK Maps
R. P. Agarwal and D. O'Regan

ABSTRACT. This paper presents a continuation theory for U_c^k maps. The analysis is elementary and relies on properties of retractions and fixed point theory for self maps. Also we present a separate theory for a certain subclass of U_c^k maps, namely the PK maps.

375

On Exact Topological Flows
A. Siemaszko and J. Szymanski

ABSTRACT. It is shown that group endomorphisms are exact flows if and only if they are exact in the measure-theoretic sense and that all flows which are exact with respect to an invariant measure with full support are exact. It is also proved that all locally eventually dense (led) flows have uniformly positive entropy (u.p.e.).

387