Raúl Manásevich

Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition

We show existence for a nonlinear fourth-order boundary value problem under a nonresonance condition involving a two-parameter linear eigenvalue problem. We also state extensions of this result to certain higher-order P.D.E. cases

Existence and multiplicity of solutions with prescribed period for a second order quasilinear O.D.E.

MANUEL A. DEL PINO School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A. RAIL F. MANASEVICH Departamento de Matematicas, F.C.F.M., Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile and ALEJANDRO E. MUR~A Division of Applied Mathematics, Brown University, Providence, RI 02912, U.S.A. (Received 20 September 1990; received for publication 13 March 1991)

MULTIPLE SOLUTIONS FOR THE p-LAPLACIAN UNDER GLOBAL NONRESONANCE

Via the study of a simple Dirichlet boundary value problem asso- ciated with the one-dimensional p-Laplacian, p > 1 , we show that in globally nonresonant problems for this differential operator the number of solutions may be arbitrarily large when p € (1, co)\{2} . From this point of view p = 2 turns out to be a very special case.

A non variational version of a max-min principle

@‘f(u) w, 4 2 milwl12 (1) then there exists a unique uo E H such that Vf (ug) = 0. Furthermoref (uO) =min f (u). In 1975 UEH Lazer, Landesman & Meyer (L.L.M.) [3] extended these results to the following situation. Let X and Y be two closed subspaces of H, such that H = X @ Y, X is finite dimensional and X and Y are not necessarily orthogonal. Let T = Vf, then T: H-+ H and is a C’ mapping. Its Frechet derivative at u E His given by T’(rc) = D’f (u). Let ml and ml be two positive constants such that Vu E H, Vx E X, and Vy E Y we have

Global existence of solutions for a chemotaxis-type system arising in crime modelling

We consider a nonlinear, strongly coupled, parabolic system arising in the modeling of burglary in residential areas. The system is of chemotaxis-type and involves a logarithmic sensivity function and specific interaction and relaxation terms. Under suitable assumptions on the data of the problem, we give a rigorous proof of the existence of a global and bounded, classical solution, thereby solving a problem left open in previous work on this model. Our proofs are based on the construction of approximate entropies and on the use of various functional inequalities. We also provide explicit numerical conditions for global existence when the domain in a square, including concrete cases involving values of the parameters which are expected to be physically relevant

Global Bifurcation of Solutions for Crime Modeling Equations

We study pattern formation in a quasi-linear system of two elliptic equations that was developed by Short et al. [Math. Models Methods Appl. Sci., 18 (2008), pp. 1249–1267] as a model for residential burglary. That model is based on the observation that the rate of burglaries of houses that have been burglarized recently and their close neighbors is typically higher than the average rate in the larger community, which creates hotspots for burglary. The patterns generated by the model describe the location of those hotspots. We prove that the system supports global bifurcation of spatially varying solutions from the spatially constant equilibrium, leading to the formation of spatial patterns. The analysis is based on recent results on global bifurcation in quasi-linear elliptic systems derived by Shi and Wang [J. Differential Equations, 7 (2009), pp. 2788–2812]. We show in some cases that near the bifurcation point the bifurcating spatial patterns are stable.

Solvability for a nonlinear three-point boundary value problemwith p-Laplacian-like operator at resonance

Let φ be an odd increasing homeomorphism from R onto R which satisfies φ(0)= 0 and let f : [a,b]×R×R → R be a function satisfying Caratheodory conditions. Separated two-point and periodic boundary value problems containing the nonlinear operator (φ(u′))′, or its more particular form, the so-called p-Laplace operator, have received a lot of attention lately (cf. [6, 7, 8, 14, 15] and the references therein). On the other hand, three-point (or m-point) boundary value problems for the case when (φ(u′))′ = u′′, that is, the linear operator, have been considered by many authors (cf. [3, 9, 10, 12, 13]). The purpose of this paper is to study the following three-point boundary value problem which contains the nonlinear operator (φ(u′))′, ( φ(u′) )′ = f (t,u,u′), u′(a)= 0, u(η)= u(b), (1.1)

On a pseudo Fuccik spectrum for strongly nonlinear second-order ODEs and an existence result

Abstract We study here the existence of solutions to the nonlinear Dirichlet problem (P): ( φ ( u ′))′ + f ( t , u ) = q ( t ), a.e. for t ∈ [ a, b ], u ( a ) = u ( b ) = 0, where φ is an increasing odd homeomorphism of R , f :[a,b] × R → R satisfies the Caratheodory assumptions and q ∈ L 1 ([a, b], R ) . With a view towards defining a pseudo Fucik spectrum (PFS) we study, using a time-mapping approach, the eigenvalue-like problem {( φ ( u ′))′ + Aφ ( u + ) − Bφ ( u − ) = 0, u ( a ) = 0 = u ( b ), where A > 0, B > 0. We show here that this PFS, which we denote by l , consists of a set of curves in ( R + ) 2 resembling the classical Fucik spectrum, i.e., when φ ( s ) = s . Our main existence result, which deals with nonresonance with respect to the PFS, can be roughly stated as follows: if for s > 0 sufficiently large and for almost every t ∈ [ a , b ], the pair ( f ( t , − s )/ φ (− s ), f ( t , s )/ φ ( s )) lies in a compact rectangle contained in an open component of ( R + ) 2 \ L which intersects the diagonal, then problem ( P ) has at least a solution .

MOUNTAIN PASS TYPE SOLUTIONS FOR QUASILINEAR ELLIPTIC INCLUSIONS

We establish the existence of weak solutions to the inclusion problem where Ω is a bounded domain in ℝN, , and ψ ∊ ℝ × ℝ is a maximal monotone odd graph. Under suitable conditions on ψ, g (which reduce to subcritical and superlinear conditions in the case of powers) we obtain the existence of non-trivial solutions which are of mountain pass type in an appropriate not necessarily reflexive Orlicz Sobolev space. The proof is based on a version of the Mountain Pass Theorem for a non-smooth case.

Contributions to Nonlinear Analysis

This volume represents a broad survey of current research in the fields of nonlinear analysis and nonlinear differential equations. It is dedicated to Djairo G. de Figueiredo on the occasion of his 70th birthday. The collection of 34 research and survey articles reflects the wide range of interests of Djairo de Figueiredo, including: - various types of nonlinear partial differential equations and systems, in particular equations of elliptic, parabolic, hyperbolic and mixed type; - equations of Schrodinger, Maxwell, Navier-Stokes, Bernoulli-Euler, Seiberg-Witten, Caffarelli-Kohn-Nirenberg; - existence, uniqueness and multiplicity of solutions; - critical Sobolev growth and connected phenomena; - qualitative properties, regularity and shape of solutions; - inequalities, a-priori estimates and asymptotic behavior; - various applications to models such as asymptotic membranes, nonlinear plates and inhomogeneous fluids. The contributions of so many distinguished mathematicians to this volume document the importance and lasting influence of the mathematical research of Djairo de Figueiredo. The book thus is a source of new ideas and results and should appeal to graduate students and mathematicians interested in nonlinear problems.