| Contents |
| We consider the question of nonexistence of positive supersolutions of the equation
$$
-\Delta u = f(u) \qquad \hbox{in }\mathbb{R}^N\setminus B_{R}
\eqno (1)
$$
where $N\ge 3$ and $f$ is a continuous, positive function defined in $(0,\infty)$.
For the model case $f(t)=t^p$, $p>0$, it is well-known that positive
supersolutions do not exist provided $p\le \frac{N}{N-2}$, while they can be constructed if
$p>\frac{N}{N-2}$ (see [2] for a simple proof). It has also been proved that this holds
true if $f(t)$ is comparable to a power near $t=0$, even with some more general
fully nonlinear second order operators (cf. [1]).
We extend this result to arbitrary nonlinearities, and show that if the condition
$$
\int_0^\delta \frac{f(t)}{t^\frac{2(N-1)}{N-2}} dt=\infty
$$
holds for some small positive $\delta$, then no positive supersolutions of (1) exist.
This condition is shown to be optimal.
We also consider some generalizations of this result to problems with gradient terms
and/or weights, and to more general operators like the $p-$Laplacian or the Pucci's
maximal operators.
The case $N=2$ is also briefly considered.
\bigskip
\bigskip
\noindent {\bf References}
\bigskip
\noindent [1] S. N. Armstrong, B. Sirakov, {\it Nonexistence of positive supersolutions of elliptic
equations via the maximum principle}. Comm. Partial Differential Equations 36 (2011), no. 11, 2011--2047.
\medskip
\noindent [2] P. Quittner, P. Souplet, {\it Superlinear parabolic problems. Blow-up, global existence and
steady states}. Birkh\"auser Advanced Texts: Basel Textbooks, Birkh\"auser Verlag, Basel, 2007. |
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