Constrained topological degree and positive solutions of fully nonlinear boundary value problems

Abstract

In the first part of the paper we provide a construction of an abstract homotopy invariant detecting zeros of maps of the formon an open subset U of a neighborhood retract M being invariant with respect to the resolvents of A. The construction is performed under the assumption that resolvents of A are completely continuous. In the second part we derive index formulae for isolated zeros and apply them to show the existence of nontrivial positive steady state solutions for a class of nonlinear reaction-diffusion equations and equations with one-dimensional p-Laplacian with possibly non-positive perturbations as well as some controlled Neumann-like problems.

Stationary Solutions and Connecting Orbits for p-Laplace Equation

“…nf (x, r n u, µ n ), x ∈ [0, l], u ∈ R, n ≥ 1. By (4) and (5), |f (x, u)| ≤ M|u| p−1 for some M > 0. Therefore, r 1−p |f (x, ru)| ≤ M|u| p−1 and consequently…”

Positive stationary solutions for p-Laplacian problems with nonpositive perturbation

“…is an L-retract (see [2] and [5]), i.e. there exist a retraction r : B(M * , η) → M * with some η > 0 and a constant L > 0 such that…”

“…We end this section with a general result, which allows us to compute the degree is specific situations (comp. [5,Prop. 4.2]).…”

“…Next, we apply this general framework to a class of partial differential equations with p-Laplacian under Dirichlet boundary conditions. In the paper we employ general ideas from [5], where a setting suitable for the one dimensional p-Laplacian was introduced.…”

Nonlocal problems in Hilbert spaces

“…Advanced topological methods were recently used for the study of (1), based on suitable topological degrees (see e.g. [1], [8] and [10]). Recent results in this context can be found in [3], [4], [6], [7], [8] and [12].…”