| Abstract: |
| In this seminar we present an existence and uniqueness result, in the spirit of the celebrated
paper by Brezis and Oswald (Nonlinear Anal., 1986), for the following
sublinear the Dirichlet problem
\begin{equation*}
\mathrm{(P)}\qquad\quad \begin{cases}
\LL_{p,s}u = f(x,u) & \text{in $\Omega$}, \
u \gneqq 0 & \text{in $\Omega$}, \
u \equiv 0 & \text{in $\R^n\setminus \Omega$},
\end{cases}
\end{equation*}
where $\LL_{p,s}$ is the sum of a quasilinear local and a nonlocal operator, i.e.,
$$\LL_{p,s} = -\Delta_p + (-\Delta)^s_p.$$
Under standard assumptions on the nonlinearity
$f$, we show that if $u$ solves (P), then $u>0$ in $\Omega$; moreover,
we give precise conditions under which such a solution
exists and is unique. |
|