01736nas a2200181 4500008004100000022001400041245010800055210006900163300000900232490000700241520106900248653003901317653002301356653004001379653003601419100002301455856007601478 2017 eng d a1534-039200aMultiple positive solutions of a sturm-liouville boundary value problem with conflicting nonlinearities0 aMultiple positive solutions of a sturmliouville boundary value p a10830 v163 a

We study the second order nonlinear differential equation

\begindocument $ u'' + \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.

10aLeray-Schauder topological degree;10apositive solutions10aSturm-Liouville boundary conditions10aSuperlinear indefinite problems1 aFeltrin, Guglielmo uhttp://aimsciences.org//article/id/1163b042-0c64-4597-b25c-3494b268e5a102865nas a2200121 4500008004100000245007000041210006900111260001000180520240500190653002302595100002302618856010202641 2016 en d00aPositive solutions to indefinite problems: a topological approach0 aPositive solutions to indefinite problems a topological approach bSISSA3 aThe 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,+∞[→[0,+∞[ is a continuous nonlinearity and a : [0,T]→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 Gómez‐Reñasco and López‐Gómez (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'' + 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'' + 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.10apositive solutions1 aFeltrin, Guglielmo uhttps://www.math.sissa.it/publication/positive-solutions-indefinite-problems-topological-approach00897nas a2200181 4500008004100000022001400041245008800055210006900143300001400212490000800226520028400234653002200518653003800540653002300578653002000601100002300621856007100644 2011 eng d a0022-247X00aA note on a superlinear indefinite Neumann problem with multiple positive solutions0 anote on a superlinear indefinite Neumann problem with multiple p a259 - 2680 v3773 a

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.

10aIndefinite weight10aNonlinear boundary value problems10apositive solutions10aShooting method1 aBoscaggin, Alberto uhttp://www.sciencedirect.com/science/article/pii/S0022247X10008796