@article {1534-0392_2017_3_1083, title = {Multiple positive solutions of a sturm-liouville boundary value problem with conflicting nonlinearities}, journal = {Communications on Pure \& Applied Analysis}, volume = {16}, number = {1534-0392_2017_3_108}, year = {2017}, pages = {1083}, abstract = {

We study the second order nonlinear differential equation

\begindocument $ u{\textquoteright}{\textquoteright} + \sum\limits_i = 1^m α_ia_i(x)g_i(u) - \sum\limits_j = 1^m + 1 β_jb_j(x)k_j(u) = 0,\rm $ \enddocument

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.

}, keywords = {Leray-Schauder topological degree;, positive solutions, Sturm-Liouville boundary conditions, Superlinear indefinite problems}, issn = {1534-0392}, doi = {10.3934/cpaa.2017052}, url = {http://aimsciences.org//article/id/1163b042-0c64-4597-b25c-3494b268e5a1}, author = {Guglielmo Feltrin} } @article {FELTRIN20174255, title = {Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree}, journal = {Journal of Differential Equations}, volume = {262}, number = {8}, year = {2017}, pages = {4255 - 4291}, abstract = {

We study the periodic boundary value problem associated with the second order nonlinear differential equationu"+cu'+(a+(t)-μa-(t))g(u)=0, where g(u) has superlinear growth at zero and at infinity, a(t) is a periodic sign-changing weight, c∈R and μ\>0 is a real parameter. Our model includes (for c=0) the so-called nonlinear Hill{\textquoteright}s equation. We prove the existence of 2m-1 positive solutions when a(t) has m positive humps separated by m negative ones (in a periodicity interval) and μ is sufficiently large, thus giving a complete solution to a problem raised by G.J. Butler in 1976. The proof is based on Mawhin{\textquoteright}s coincidence degree defined in open (possibly unbounded) sets and applies also to Neumann boundary conditions. Our method also provides a topological approach to detect subharmonic solutions.

}, keywords = {Coincidence degree, Multiplicity results, Neumann boundary value problems, Positive periodic solutions, subharmonic solutions, Superlinear indefinite problems}, issn = {0022-0396}, doi = {https://doi.org/10.1016/j.jde.2017.01.009}, url = {http://www.sciencedirect.com/science/article/pii/S0022039617300219}, author = {Guglielmo Feltrin and Fabio Zanolin} }