Existence and nonexistence of positive solutions to some fully nonlinear equation in one dimension

Abstract

In this paper, we consider the existence (and nonexistence) of solutions to \[ -\mathcal{M}_{λ,Λ}^\pm (u'') + V(x) u = f(u) \quad {\rm in} \ \mathbf{R} \] where ${\mathcal{M}}_{\lambda ,\Lambda }^{+}$ and ${\mathcal{M}}_{\lambda ,\Lambda }^{-}$ denote the Pucci operators with $0<\lambda \le \Lambda <\infty$, $V(x)$ is a bounded function, $f(s)$ is a continuous function and its typical example is a power-type nonlinearity $f(s)=|s{|}^{p-1}s$ $(p>1)$. In particular, we are interested in positive solutions which decay at infinity, and the existence (and nonexistence) of such solutions is proved.

29 pages. Journal of Functional Analysis, 2018

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• Compactness via monotonicity in nonsmooth critical point theory, with application to Born-Infeld type equations
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