A variational approach to superlinear semipositone elliptic problems

Costa, David; Quoirin, Humberto; Tehrani, Hossein

doi:10.1090/proc/13426

A variational approach to superlinear semipositone elliptic problems
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by David G. Costa, Humberto Ramos Quoirin and Hossein Tehrani PDF

Proc. Amer. Math. Soc. 145 (2017), 2661-2675 Request permission

Abstract:

In this paper we present a variational approach to a class of elliptic problems with superlinear semipositone nonlinearities. We consider the parametrized family of problems \[ \left \{ \begin {array}{lll} -\Delta u =\lambda a(x)(f(u)-l)& \textrm {in } & \Omega ,\\ u = 0 & \textrm {on } & \partial \Omega , \end {array}\right . \] with $l>0$, $a$ continuous, and $f$ subcritical and superlinear at infinity. We obtain positive solutions of such problems for $0< \lambda < \lambda _0$ by combining a suitable rescaling with a continuity argument. In doing so, we require $f$ to be of regular variation at infinity, so that $f$ does not need to be asymptotic to a power. Furthermore, $a$ may vanish in open parts of $\Omega$ or change sign.

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