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# Positive Solutions in Exterior Domains of Nonlinear Field Equations

Ponente: Liliane Maia
$-\Delta u+V(x)u=f(u),\qquad x \in \Omega \subset \mathbb{R}^{N}, \qquad u(x) \to 0\;\;\text{as}\;\; |x| \to \infty,$
for $$N\geq3$$, $$\Omega$$ a regular unbounded exterior domain, when either the nonlinearity $$f$$ is subcritical and superquadratic or asymptotically linear at infinity, in case $$V$$ approaches a positive constant limit at infinity, or $$f$$ is subcritical at infinity and
supercritical near the origin if the potential $$V$$ vanishes at infinity.
Under a suitable decay assumption on the potential $$V$$, we show that the