Patrick Habets

Multiple positive solutions of a one-dimensional prescribed mean curvature problem

We discuss existence, non-existence and multiplicity of positive solutions of the Dirichlet problem for the one-dimensional prescribed curvature equation [GRAPHICS] in connection with the changes of concavity of the function f. The proofs are based on an upper and lower solution method, we specifically develop for this problem, combined with a careful analysis of the time-map associated with some related autonomous equations.

CHAPTER 2 - The Lower and Upper Solutions Method for Boundary Value Problems

This chapter discusses the lower and upper solutions method for boundary value problems. The premises of the lower and upper solutions method can be traced back to Picard. In 1890 for partial differential equations and in 1893 for ordinary differential equations, he introduced monotone iterations from a lower solution. This is the starting point of the use of lower and upper solutions in connection with monotone methods. Recent results that extend the old idea of Picard to use lower and upper solutions with monotone methods are discussed. Two basic problems—namely, periodic and Dirichlet problems are considered.

Positive solutions of an indefinite prescribed mean curvature problem on a general domain

Abstract The existence of positive solutions is proved for the prescribed mean curvature problem where Ω ⊂ℝN is a bounded smooth domain, not necessarily radially symmetric. We assume that ∫0u f(x, s) ds is locally subquadratic at 0, ∫0u g(x, s) ds is superquadratic at 0 and λ > 0 is sufficiently small. A multiplicity result is also obtained, when ∫0u f(x, s) ds has an oscillatory behaviour near 0. We allow f and g to change sign in any neighbourhood of 0.

Existence and localization of solutions of second order elliptic problems using lower and upper solutions in the reversed order

where Ω is a bounded domain in R ,L is a linear second order elliptic operator for which the maximum principle holds, B is a linear first order boundary operator and f is a nonlinear Caratheodory function. We are concerned with the solvability of (1.1) in presence of lower and upper solutions. A classical basic result in this context says that if α is a lower solution and β is an upper solution satisfying

A seven-positive-solutions theorem for a superlinear problem

Abstract We consider the superlinear boundary value problem uʺ + aμ(t)uγ+1 = 0, u(0) = 0, u(1) = 0, where γ > 0 and aμ(t) is a sign indefinite weight of the form a+(t)−μa−(t). We prove, for μ positive and large, the existence of 2k − 1 positive solutions where k is the number of positive humps of aμ(t) which are separated by k − 1 negative humps. For sake of simplicity, the proof is carried on for the case k = 3 yielding to 7 positive solutions. Our main argument combines a modified shooting method in the phase plane with some properties of the blow up solutions in the intervals where the weight function is negative.

An Overview of the Method of Lower and Upper Solutions for ODEs

The method of lower and upper solutions deals mainly with existence results for boundary value problems. In this presentation, we will restrict attention to second order ODE problems with separated boundary conditions. Although some of the ideas can be traced back to E. Picard [16], the method of lower and upper solutions was firmly established by G. Scorza Dragoni [20]. This 1931 paper considered upper and lower solutions which are C2; in 1938, the same author extended his method to the L1;-Caratheodory case [21]. Upper and lower solutions with corners were considered by M. Nagumo in 1954 [13]. Since then a multitude of variants have been introduced. The Definitions 2.1 and 3.1 we present here tend to be general enough for applications and simple enough to model the geometric intuition built into the concept.

A two-point problem with nonlinearity depending only on the derivative

We study the existence and multiplicity of solutions for a two-point boundary value problem at resonance in the first eigenvalue; the nonlinearity depends only on the first derivative and has finite limits at

Examples of the nonexistence of a solution in the presence of upper and lower solutions

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Classification of stability-like concepts and their study using vector Lyapunov functions

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On pairs of positive solutions for a singular boundary value

Standard results for boundary value problems involving second-order ordinary differential equations ensure that the existence of a well-ordered pair of lower and upper solutions together with a Nagumo condition imply existence of a solution. In this note we introduce some examples which show that existence is not guaranteed if no Nagumo condition is satisfied.