@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} } @mastersthesis {2016, title = {Positive solutions to indefinite problems: a topological approach}, year = {2016}, note = {The research work described in this Ph.D. thesis has produced 10 papers.}, school = {SISSA}, abstract = {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{\textquoteright}{\textquoteright} + a(t) g(u) = 0, where g : [0,+$\infty$[{\textrightarrow}[0,+$\infty$[ is a continuous nonlinearity and a : [0,T]{\textrightarrow}R 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{\'o}mez-Re{\~n}asco and L{\'o}pez-G{\'o}mez (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{\textquoteright}{\textquoteright} + λ a(t) g(u) = 0, when both λ>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{\textquoteright}{\textquoteright} + cu{\textquoteright} + a(t) g(u) = 0. Meanwhile, more general operators in the one-dimensional case and problems involving PDEs will be subjects of future investigations.}, keywords = {positive solutions}, author = {Guglielmo Feltrin} } @article {BOSCAGGIN2011259, title = {A note on a superlinear indefinite Neumann problem with multiple positive solutions}, journal = {Journal of Mathematical Analysis and Applications}, volume = {377}, number = {1}, year = {2011}, pages = {259 - 268}, abstract = {

We prove the existence of three positive solutions for the Neumann problem associated to u"+a(t)uγ+1=0, assuming that a(t) has two positive humps and ∫0Ta-(t)dt is large enough. Actually, the result holds true for a more general class of superlinear nonlinearities.

}, keywords = {Indefinite weight, Nonlinear boundary value problems, positive solutions, Shooting method}, issn = {0022-247X}, doi = {https://doi.org/10.1016/j.jmaa.2010.10.042}, url = {http://www.sciencedirect.com/science/article/pii/S0022247X10008796}, author = {Alberto Boscaggin} }