Philip Korman

Exact Multiplicity Results for Boundary Value Problems with Nonlinearities Generalizing Cubic

Using techniques of bifurcation theory we present two exact multiplicity results for boundary value problems of the type The first result concerns the case when the nonlinearity is independent of x and behaves like a cubic in u. The second one deals with a class of nonlinearities with explicit x dependence.

On the existence and uniqueness of positive steady states in the volterra-lotka ecological models with diffusion

Using the theory of quasimonotone increasing systems developed by P. J. McKenna and W. Walter, we give a rather detailed analysis of the steady state solutions for the Volterra-Lotka model of two cooperating species, and prove some new nonexistence results for the competing species case. We indicate generalizations to the case of n > 2 species

An exact multiplicity result for a class of semilinear equations

For a class of Dirichlet problems in two dimensions, generalizing the model case we show existence of a critical so that there are exactly 0, 1 or 2 nontrivial solutions (in fact, positive), depending on whether We show that all solutions lie on a single smooth solution curve, and study some properties of this curve. We use bifurcation approach. The Curial thing is to show that any nontrivial solution of the corresponding linearized problem is of one sign.

On the exactness of an S-shaped bifurcation curve

For a class of two-point boundary value problems we prove exactness of S-shaped bifurcation curve. Our result applies to a problem from combustion theory, which involves nonlinearities like e for a > 0.

A general monotone scheme for elliptic systems with applications to ecological models

We consider weakly-coupled elliptic systems of the type with each f i being either an increasing or a decreasing function of each u j . Assuming the existence of coupled super- and subsolutions, we prove the existence of solutions, and provide a constructive iteration scheme to approximate the solutions. We then apply our results to study the steady-states of two-species interaction in the Volterra–Lotka model with diffusion.

Solution curves for semilinear equations on a ball

We show that the set of positive solutions of semilinear Dirichlet problem on a ball of radius R in RI Au+ Af(u) = O for lxl < R, u = O on lxl = R consists of smooth curves. Our results can be applied to compute the direction of bifurcation. We also give an easy proof of a uniqueness theorem due to Smoller and Wasserman (1984).

Uniqueness and exact multiplicity of solutions for a class of fourth-order semilinear problems

Using techniques of bifurcation theory, we give exact multiplicity and uniqueness results for the fourth-order Dirichlet problem, which describes deflection of an elastic beam, subjected to a nonlinear force, and clamped at the end points. The crucial part of this approach was to show positivity of non-trivial solutions of the corresponding linearized problem.

Chapter 6 Global Solution Branches and Exact Multiplicity of Solutions for Two Point Boundary Value Problems

Publisher Summary This chapter discusses the solutions of the two point boundary value problems. The chapter describes the equation curves of solutions, u = u(x,λ) . The advantage of this approach is that some parts of the solution curve are easy to understand, and it also becomes clear what are the tougher parts of the solution curve that one needs to study—the turning points. The most detailed results are obtained when one considers positive solutions of autonomous problems, i.e. when f = f (u) . Since in that case both the length and the position of the interval (a, b) are irrelevant, and since positive solutions are symmetric with respect to the midpoint of the interval, it is convenient to pose the problem on the interval (- 1, 1).

Global solution curves for semilinear elliptic equations

Continuation of Solutions in General Domain Curves of Positive Solutions on Balls Symmetry Breaking Curves with Infinitely Many Turns Numerical Computation of Solutions Solutions of Annular Domains Curves of Solutions to Hamiltonian Systems S-Shapes Bifurcation Curves for Two Point Problems Infinitely Many Solution Curves with Pitchfork Bifurcation Elastic Beam Equations Prescribed Mean Curvature Equation.

A maximum principle for fourth order ordinary differential equations

We present a maximum principle for fourth order ordinary differential equations, based on a new approach involving counting of inflection points. We use our results to compute solutions of nonlinear equations describing static displacements of a uniform beam