Positive solutions for super-sublinear indefinite problems: High multiplicity results via coincidence degree

Boscaggin, Alberto; Feltrin, Guglielmo; Zanolin, Fabio

doi:10.1090/tran/6992

Positive solutions for super-sublinear indefinite problems: High multiplicity results via coincidence degree
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by Alberto Boscaggin, Guglielmo Feltrin and Fabio Zanolin PDF

Trans. Amer. Math. Soc. 370 (2018), 791-845 Request permission

Abstract:

We study the periodic boundary value problem associated with the second order non-linear equation \begin{equation*} u'' + \bigr {(} \lambda a^{+}(t) - \mu a^{-}(t) \bigr {)} g(u) = 0, \end{equation*} where $g(u)$ has superlinear growth at zero and sublinear growth at infinity. For $\lambda , \mu$ positive and large, we prove the existence of $3^{m}-1$ positive $T$-periodic solutions when the weight function $a(t)$ has $m$ positive humps separated by $m$ negative ones (in a $T$-periodicity interval). As a byproduct of our approach we also provide an abundance of positive subharmonic solutions and symbolic dynamics. The proof is based on coincidence degree theory for locally compact operators on open unbounded sets and also applies to Neumann and Dirichlet boundary conditions. Finally, we deal with radially symmetric positive solutions for the Neumann and the Dirichlet problems associated with elliptic PDEs.

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