High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems

Abstract

Abstract. This paper analyzes the existence and structure of the positive solutions of a very simple superlinear indefinite semilinear elliptic prototype model under non-homogeneous boundary conditions, measured by M ≤ ∞. Rather strikingly, there are ranges of values of the parameters involved in its setting for which the model admits an arbitrarily large number of positive solutions, as a result of their fast oscillatory behavior, for sufficiently large M . Further, using the amplitude of the superlinear term as the main bifurcation parameter, we can ascertain the global bifurcation diagram of the positive solutions. This seems to be the first work where these multiplicity results have been documented.

Generating an arbitrarily large number of isolas in a superlinear indefinite problem

“…Our main result establishes that the unique (rather intricate) component constructed in [11], and computed in [15] for c 0 = c 1 , splits into an arbitrary large number of compact components (isolas) as λ approximates −∞, plus an additional unbounded component establishing a homotopy between the unique positive solution of (1.1) for b = 0, denoted by u 0 , whose existence and uniqueness were proven, e.g., in [13], and the metasolution 1] where, for each c > 0, ℓ c (t) stands for the unique (large) solution of  −u ′′ = λu − cu p in (0, α) u(0) = M, u(α) = +∞.…”

“…We start by collecting some well known results from [11,15] which are pivotal for our analysis. They can be shortly summarized as follows:…”

“…A rather natural question is to study how the bifurcation diagrams of [11], which consist of a unique global (connected) component with an increasing number of secondary loops for λ ↓ −∞, perturb into the bifurcation diagrams constructed here as c 1 perturbs from c 0 . As this singular perturbation analysis from the case where the equilibrium Ω coincides with m 0 = m 1 requires a substantial amount of additional work, we will stop our discussion here.…”

“…Moreover, it is also known that the spatial heterogeneities of the weight function a b (t), measured by the number of simple zeros of a b (t) in the context of Problem (1.1), might provoke multiplicity of positive solutions, not only for λ < 0, but also for λ ≥ 0 (see [6][7][8]). Rather astonishingly, even in the simplest case when a b (t) exhibits a single hump, it has been recently found in [11] that the complexity of the solution set of (1.1) in the symmetric case c 0 = c 1 , using b as the main bifurcation parameter, increases arbitrarily as λ approximates −∞, through an increasing series of twists of the branch bifurcating from the unique solution of Problem (1.1) for b = 0 and associated secondary bifurcations from it.…”

Study of a logistic equation with local and non-local reaction terms

“…This class of local problems has been considered with other boundary conditions, for example, non-homogeneous Dirichlet boundary conditions, see [9] and [18], where multiplicity results are shown. We do not consider the non-local counterpart in this paper.…”