Symmetric positive solutions of nonlinear boundary value problems

Positive Symmetric Solutions of Singular Semipositone Boundary Value Problems

Using the method of upper and lower solutions, we prove that the singular boundary value problem, −u = f(u) u in (0, 1), u(0) = 0 = u(1) , has a positive solution when 0 < α < 1 and f : R → R is an appropriate nonlinearity that is bounded below; in particular, we allow f to satisfy the semipositone condition f(0) < 0. The main difficulty of this approach is obtaining a positive subsolution, whi...

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...

Positive Solutions of Second Order Nonlinear Difference Boundary Value Problems

We study a class of second order nonlinear difference boundary value problems with separated boundary conditions. A series of criteria are obtained for the existence of one, two, arbitrary number, and even an infinite number of positive solutions. A theorem for the nonexistence of positive solutions is also derived. Several examples are given to demonstrate the applications. Our results improve...