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

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

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

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