Kenichiro Umezu

On eigenvalue problems with Robin type boundary conditions having indefinite coefficients

A Robin type boundary condition with a sign-changing coefficient is treated. First, the associated linear elliptic eigenvalue problem is studied, where the existence of a principal eigenvalue is discussed by the use of a variational approach. Second, the associated semilinear elliptic boundary value problem of logistic type is studied and the one parameter-dependent structure of positive solutions is investigated, where results obtained are due to the construction of suitable super- and subsolutions by using the principal positive eigenfunctions of the linear eigenvalue problem.

MULTIPLICITY OF POSITIVE SOLUTIONS UNDER NONLINEAR BOUNDARY CONDITIONS FOR DIFFUSIVE LOGISTIC EQUATIONS

In this paper we consider the existence and multiplicity of positive solutions of a nonlinear elliptic boundary-value problem with nonlinear boundary conditions which arises in population dynamics. While bifurcation problems from lines of trivial solutions are studied, the existence of bifurcation positive solutions from infinity is discussed. The former will be caught by the reduction to a bifurcation equation following the Lyapunov and Schmidt procedure. The latter will be based on a variational argument depending on the corresponding constrained minimization problem.

An indefinite concave-convex equation under a Neumann boundary condition I

We investigate the problem (Pλ) −Δu = λb(x)|u|q−2u + a(x)|u|p−2u in Ω, ∂u/∂n = 0 on ∂Ω, where Ω is a bounded smooth domain in RN (N ≥ 2), 1 < q < 2 < p, λ ∈ R, and a, b ∈

A loop type component in the non-negative solutions set of an indefinite elliptic problem

{C^alpha }left( {overline Omega } right)

Non-existence of Positive Solutions for Diffusive Logistic Equations with Nonlinear Boundary Conditions

Cα(Ω¯) with 0 < α < 1. Under certain indefinite type conditions on a and b, we prove the existence of two nontrivial nonnegative solutions for small |λ|. We then characterize the asymptotic profiles of these solutions as λ → 0, which in some cases implies the positivity and ordering of these solutions. In addition, this asymptotic analysis suggests the existence of a loop type component in the non-negative solutions set. We prove the existence of such a component in certain cases, via a bifurcation and a topological analysis of a regularized version of (Pλ).

Loop Type Subcontinua of Positive Solutions for Indefinite Concave-Convex Problems

We investigate positive steady states of an indefinite superlinear reaction-diffusion equation arising from population dynamics, coupled with a nonlinear boundary condition. Both the equation and the boundary condition depend upon a positive parameter

Positive Solutions of Semilinear Elliptic Boundary Value Problems in Chemical Reactor Theory

lambda