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

Multiple positive solutions for a superlinear problem: a topological approach

We study the multiplicity of positive solutions for a two-point boundary value problem associated to the nonlinear second order equation $u''+f(x,u)=0$. We allow $x \mapsto f(x,s)$ to change its sign in order to cover the case of scalar equations with indefinite weight. Roughly speaking, our main assumptions require that $f(x,s)/s$ is below $\lambda_{1}$ as $s\to 0^{+}$ and above $\lambda_{1}$ as $s\to +\infty$. In particular, we can deal with the situation in which $f(x,s)$ has a superlinear growth at zero and at infinity. We propose a new approach based on the topological degree which provides the multiplicity of solutions. Applications are given for $u'' + a(x) g(u) = 0$, where we prove the existence of $2^{n}-1$ positive solutions when $a(x)$ has $n$ positive humps and $a^{-}(x)$ is sufficiently large.Comment: 36 pages, 3 PNG figure

Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems

We prove the existence of positive periodic solutions for the second order nonlinear equation $u" + a(x) g(u) = 0$, where $g(u)$ has superlinear growth at zero and at infinity. The weight function $a(x)$ is allowed to change its sign. Necessary and sufficient conditions for the existence of nontrivial solutions are obtained. The proof is based on Mawhin's coincidence degree and applies also to Neumann boundary conditions. Applications are given to the search of positive solutions for a nonlinear PDE in annular domains and for a periodic problem associated to a non-Hamiltonian equation.Comment: 41 page

Positive solutions to indefinite problems: a topological approach

The present Ph.D. thesis is devoted to the study of positive solutions to indefinite problems. In particular, we deal with the second order nonlinear differential equation u'' + a(t) g(u) = 0, where g : [0,+ 1e[\u2192[0,+ 1e[ is a continuous nonlinearity and a : [0,T]\u2192R is a Lebesgue integrable sign-changing weight. We analyze the Dirichlet, Neumann and periodic boundary value problems on [0,T] associated with the equation and we provide existence, nonexistence and multiplicity results for positive solutions. In the first part of the manuscript, we investigate nonlinearities g(u) with a superlinear growth at zero and at infinity (including the classical superlinear case g(u)=u^p, with p>1). In particular, we prove that there exist 2^m-1 positive solutions when a(t) has m positive humps separated by negative ones and the negative part of a(t) is sufficiently large. Then, for the Dirichlet problem, we solve a conjecture by G\uf3mez\u2010Re\uf1asco and L\uf3pez\u2010G\uf3mez (JDE, 2000) and, for the periodic problem, we give a complete answer to a question raised by Butler (JDE, 1976). In the second part, we study the super-sublinear case (i.e. g(u) is superlinear at zero and sublinear at infinity). If a(t) has m positive humps separated by negative ones, we obtain the existence of 3^m-1 positive solutions of the boundary value problems associated with the parameter-dependent equation u'' + \u3bb a(t) g(u) = 0, when both \u3bb>0 and the negative part of a(t) are sufficiently large. We propose a new approach based on topological degree theory for locally compact operators on open possibly unbounded sets, which applies for Dirichlet, Neumann and periodic boundary conditions. As a byproduct of our method, we obtain infinitely many subharmonic solutions and globally defined positive solutions with complex behavior, and we deal with chaotic dynamics. Moreover, we study positive radially symmetric solutions to the Dirichlet and Neumann problems associated with elliptic PDEs on annular domains. Furthermore, this innovative technique has the potential and the generality needed to deal with indefinite problems with more general differential operators. Indeed, our approach apply also for the non-Hamiltonian equation u'' + cu' + a(t) g(u) = 0. Meanwhile, more general operators in the one-dimensional case and problems involving PDEs will be subjects of future investigations

Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case

We study the periodic and the Neumann boundary value problems associated with the second order nonlinear differential equation \begin{equation*} u'' + c u' + \lambda a(t) g(u) = 0, \end{equation*} where $g \colon \mathopen{[}0,+\infty\mathclose{[}\to \mathopen{[}0,+\infty\mathclose{[}$ is a sublinear function at infinity having superlinear growth at zero. We prove the existence of two positive solutions when $\int_{0}^{T} a(t) \!dt < 0$ and $\lambda > 0$ is sufficiently large. Our approach is based on Mawhin's coincidence degree theory and index computations.Comment: 26 page