Superlinear Indefinite Problems Research Articles

Global bifurcation diagrams of positive solutions for a class of 1D superlinear indefinite problems* *Partially supported by the Research Grant PGC2018-097104-B-IOO of the Spanish Ministry of Science, Innovation and Universities, and the Institute of Inter-disciplinar Mathematics (IMI) of Complutense University. M Fencl has been supported by the project SGS-2019-010 of the University of West Bohemia, the project 18-03253S of the

This paper analyzes the structure of the set of positive solutions of a class of one-dimensional superlinear indefinite bvp’s. It is a paradigm of how mathematical analysis aids the numerical study of a problem, whereas simultaneously its numerical study confirms and illuminates the analysis. On the analytical side, we establish the fast decay of the positive solutions as λ ↓ −∞ in the region where a(x) < 0 (see (1.1)), as well as the decay of the solutions of the parabolic counterpart of the model (see (1.2)) as λ ↓ −∞ on any subinterval of [0, 1] where u 0 = 0, provided u 0 is a subsolution of (1.1). This result provides us with a proof of a conjecture of [26] under an additional condition of a dynamical nature. On the numerical side, this paper ascertains the global structure of the set of positive solutions on some paradigmatic prototypes whose intricate behavior is far from predictable from existing analytical results.

Characterizing the formation of singularities in a superlinear indefinite problem related to the mean curvature operator

The aim of this paper is characterizing the development of singularities by the positive solutions of the quasilinear indefinite Neumann problem−(u′/1+(u′)2)′=λa(x)f(u)in (0,1),u′(0)=0,u′(1)=0, where λ∈R is a parameter, a∈L∞(0,1) changes sign once in (0,1) at the point z∈(0,1), and f∈C(R)∩C1[0,+∞) is positive and increasing in (0,+∞) with a potential, ∫0sf(t)dt, superlinear at +∞. In this paper, by providing a precise description of the asymptotic profile of the derivatives of the solutions of the problem as λ→0+, we can characterize the existence of singular bounded variation solutions of the problem in terms of the integrability of this limiting profile, which is in turn equivalent to the condition(∫xza(t)dt)−12∈L1(0,z)and(∫xza(t)dt)−12∈L1(z,1). No previous result of this nature is known in the context of the theory of superlinear indefinite problems.

On one-dimensional superlinear indefinite problems involving the $$\phi $$ ϕ -Laplacian

Let $$\Omega := ( a,b ) \subset \mathbb {R}$$ , $$m\in L^{1} ( \Omega ) $$ and $$\phi :\mathbb {R\rightarrow R}$$ be an odd increasing homeomorphism. We consider the existence of positive solutions for problems of the form $$\begin{aligned} \left\{ \begin{array} [c]{ll} -\phi ( u^{\prime } ) ^{\prime }=m ( x ) f ( u) &{}\quad \text {in } \Omega ,\\ u=0 &{}\quad \text {on } \partial \Omega , \end{array} \right. \end{aligned}$$ where $$f: [ 0,\infty ) \rightarrow [ 0,\infty ) $$ is a continuous function which is, roughly speaking, superlinear with respect to $$\phi $$ . Our approach combines the Guo-Krasnoselskiĭ fixed-point theorem with some estimates on related nonlinear problems. We mention that our results are new even in the case $$m\ge 0$$ .

High multiplicity of positive solutions for superlinear indefinite problems with homogeneous Neumann boundary conditions

We prove that a class of superlinear indefinite problems with homogeneous Neumann boundary conditions admits an arbitrarily high number of positive solutions, provided that the parameters of the problem are adequately chosen. The sign-changing weight in front of the nonlinearity is taken to be piecewise constant, which allows us to perform a sharp phase-plane analysis, firstly to study the sets of points reached at the end of the regions where the weight is negative, and then to connect such sets through the flow in the positive part. Moreover, we study how the number of solutions depends on the amplitude of the region in which the weight is positive, using the latter as the main bifurcation parameter and constructing the corresponding global bifurcation diagrams.

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

This paper analyzes the effect of an asymmetric weight on the bifurcation diagrams relative to a class of superlinear indefinite problems which admit an arbitrarily high number of positive solutions for certain values of the parameters involved in their formulation. The main result is that the secondary bifurcations which occur in the symmetric case (see López-Gómez et al. (2014)) give rise, in the asymmetric case, to bounded components of solutions, whose number grows arbitrarily as the number of the solutions of the problem grows.

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

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 $M\leq \infty$. 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.

The uniqueness of the linearly stable positive solution for a class of superlinear indefinite problems with nonhomogeneous boundary conditions

This paper proves the uniqueness of the positive linearly stable steady-state for a paradigmatic class of superlinear indefinite parabolic problems arising in population dynamics, under nonhomogeneous Dirichlet conditions on the boundary of the domain. The result is absolutely non-trivial, since examples are known for which the model admits an arbitrarily large number of steady-states. Our proof is based on some local and global continuation techniques. Optimal existence and multiplicity results are also obtained through some additional monotonicity and topological techniques.

The Effect of Varying Coefficients on the Dynamics of a Class of Superlinear Indefinite Reaction–Diffusion Equations

In this paper we analyze how the dynamics of a class of superlinear indefinite reaction–diffusion equations varies as the nodal behavior of a coefficient changes. To perform this analysis we use both theoretical and numerical tools. The analysis aids the numerical study, and the numerical study confirms and completes the analysis. The numerics in addition provides us with some further results for which-at-first glance analytical tools are not available yet. Our main analytical result shows that the problem possesses a unique positive solution which is linearly asymptotically stable if the trivial state is linearly unstable and the model admits some positive solution. This result is a relevant feature for superlinear indefinite problems, since our numerical computations show how these models can have an arbitrarily large number of positive solutions if the trivial state is unstable.