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- W4300920218 abstract "We investigate the existence, non-existence, multiplicity of positive periodic solutions, both harmonic (i.e., $T$-periodic) and subharmonic (i.e., $kT$-periodic for some integer $k geq 2$) to the equation begin{equation*} Biggl{(} dfrac{u'}{sqrt{1-(u')^{2}}} Biggr{)}' + lambda a(t) g(u) = 0, end{equation*} where $lambda > 0$ is a parameter, $a(t)$ is a $T$-periodic sign-changing weight function and $g colon mathopen{[}0,+inftymathclose{[} to mathopen{[}0,+inftymathclose{[}$ is a continuous function having superlinear growth at zero. In particular, we prove that for both $g(u)=u^{p}$, with $p>1$, and $g(u)= u^{p}/(1+u^{p-q})$, with $0 leq q leq 1 < p$, the equation has no positive $T$-periodic solutions for $lambda$ close to zero and two positive $T$-periodic solutions (a 'small' one and a 'large' one) for $lambda$ large enough. Moreover, in both cases the 'small' $T$-periodic solution is surrounded by a family of positive subharmonic solutions with arbitrarily large minimal period. The proof of the existence of $T$-periodic solutions relies on a recent extension of Mawhin's coincidence degree theory for locally compact operators in product of Banach spaces, while subharmonic solutions are found by an application of the Poincar'e--Birkhoff fixed point theorem, after a careful asymptotic analysis of the $T$-periodic solutions for $lambda to +infty$." @default.
- W4300920218 created "2022-10-04" @default.
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- W4300920218 date "2018-05-17" @default.
- W4300920218 modified "2023-10-01" @default.
- W4300920218 title "Positive periodic solutions to an indefinite Minkowski-curvature equation" @default.
- W4300920218 doi "https://doi.org/10.48550/arxiv.1805.06659" @default.
- W4300920218 hasPublicationYear "2018" @default.
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