Imperfect bifurcations via topological methods in superlinear indefinite problems

In [5] the structure of the bifurcation diagrams of a class of superlinear indefinite problems with a symmetric weight was ascertained, showing that they consist of a primary branch and secondary loops bifurcating from it. In [4] it has been proved that, when the weight is asymmetric, the bifurcation diagrams are no longer connected since parts of the primary branch and loops of the symmetric case form an arbitrarily high number of isolas. In this work we give a deeper insight on this phenomenon, studying how the secondary bifurcations break as the weight is perturbed from the symmetric situation. Our proofs rely on the approach of [5,4], i.e. on the construction of certain Poincar\'e maps and the study of how they vary as some of the parameters of the problems change, constructing in this way the bifurcation diagrams.Comment: 13 pages, 7 figure

Multiple positive solutions to elliptic boundary blow-up problems

We prove the existence of multiple positive radial solutions to the sign-indefinite elliptic boundary blow-up problem $$\left\{\begin{array}{ll} \Delta u + \bigl(a^+(\vert x \vert) - \mu a^-(\vert x \vert)\bigr) g(u) = 0, & \; \vert x \vert < 1, \\ u(x) \to \infty, & \; \vert x \vert \to 1, \end{array} \right.$$ where $g$ is a function superlinear at zero and at infinity, $a^+$ and $a^-$ are the positive/negative part, respectively, of a sign-changing function $a$ and $\mu > 0$ is a large parameter. In particular, we show how the number of solutions is affected by the nodal behavior of the weight function $a$. The proof is based on a careful shooting-type argument for the equivalent singular ODE problem. As a further application of this technique, the existence of multiple positive radial homoclinic solutions to $$\Delta u + \bigl(a^+(\vert x \vert) - \mu a^-(\vert x \vert)\bigr) g(u) = 0, \qquad x \in \mathbb{R}^N,$$ is also considered

On the existence and multiplicity of positive solutions for some indefinite nonlinear eigenvalue problem

This paper concerns with the existence, uniqueness and/or multiplicity, and stability of positive solutions of an indefinite weight elliptic problem with concave or convex nonlinearity. We use mainly bifurcation method to obtain our results.Ministerio de Ciencia y Tecnologí

Multiple positive solutions of a Sturm-Liouville boundary value problem with conflicting nonlinearities

We study the second order nonlinear differential equation \begin{equation*} u"+ \sum_{i=1}^{m} \alpha_{i} a_{i}(x)g_{i}(u) - \sum_{j=0}^{m+1} \beta_{j} b_{j}(x)k_{j}(u) = 0, \end{equation*} where $\alpha_{i},\beta_{j}>0$, $a_{i}(x), b_{j}(x)$ are non-negative Lebesgue integrable functions defined in $\mathopen{[}0,L\mathclose{]}$, and the nonlinearities $g_{i}(s), k_{j}(s)$ are continuous, positive and satisfy suitable growth conditions, as to cover the classical superlinear equation $u"+a(x)u^{p}=0$, with $p>1$. When the positive parameters $\beta_{j}$ are sufficiently large, we prove the existence of at least $2^{m}-1$ positive solutions for the Sturm-Liouville boundary value problems associated with the equation. The proof is based on the Leray-Schauder topological degree for locally compact operators on open and possibly unbounded sets. Finally, we deal with radially symmetric positive solutions for the Dirichlet problems associated with elliptic PDEs.Comment: 23 pages, 6 PNG figure

Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential

We prove the existence and multiplicity of periodic solutions as well as solutions presenting a complex behavior for the one-dimensional nonlinear Schrödinger equation -ε2u′′+V(x)u=f(u), where the potential V(x) approximates a two-step function. The term f(u) generalizes the typical p-power nonlinearity considered by several authors in this context. Our approach is based on some recent developments of the theory of topological horseshoes, in connection with a linked twist maps geometry, which are applied to the discrete dynamics of the Poincaré map. We discuss the periodic and the Neumann boundary conditions. The value of the term ε>0, although small, can be explicitly estimated