Positive solutions of nonlinear elliptic boundary value problems

Nonlinear Elliptic Boundary Value Problems

It is the object of the present note to present a new nonlinear version of the orthogonal projection method for proving the existence of solutions of nonlinear elliptic boundary value problems. The key point in this method is the application of a new general theorem concerning the solvability of nonlinear functional equations in a reflexive Banach space involving operators which may not be cont...

Multiple Positive Solutions for Elliptic Boundary Value Problems

We extend ODE results of Henderson and Thompson, see [10], to a large class of boundary value problems for both ODEs and PDEs. Our method of proof combines upper and lower solutions with degree theory.

Nonexistence of Positive Solutions of Nonlinear Boundary Value Problems

We discuss the nonexistence of positive solutions for nonlinear boundary value problems. In particular, we discuss necessary restrictions on parameters in nonlocal problems in order that (strictly) positive solutions exist. We consider cases that can be written in an equivalent integral equation form which covers a wide range of problems. In contrast to previous work, we do not use concavity ar...

Existence of Three Positive Solutions to Nonlinear Boundary Value Problems

Multiple Positive Solutions of Nonlinear Boundary Value Problems

where w, p are positive and continuous on (0, 1), which allows singularities at the endpoints, and f(y) ≥ 0 and continuous for y ∈ R. For a given positive integer N , we provide conditions on the nonlinear function f which guarantee existence of N positive solutions. Our motivation comes from previous work in the case w(x) ≡ p(x) ≡ 1 of Henderson and Thompson [6], which uses a fixed point theor...