$W^{1,N}$ versus $C^1$ local minimizer for a singular functional with Neumann boundary condition

Résumé

Let $\Omega\subset\R^N,$ be a bounded domain with smooth boundary. Let $g:\R^+\to\R^+$ be a continuous on $(0,+\infty)$
non-increasing and satisfying $$c_1=\liminf_{t\to 0^+}g(t)t^{\delta}\leq\underset{t\to 0^+}{\limsup} g(t)t^{\delta}=c_2,$$
for some $c_1,c_2>0$ and $0<\delta<1.$ Let $f(x,s) = h(x,s)e^{bs^{\frac{N}{N-1}}},$ $b>0$ is a constant.
Consider the singular functional
$I: W^{1,N}(\Omega)\to \R$ defined as
\begin{eqnarray*}
&&I(u)
\eqdef\frac{1}{N}\|u\|^N_{W^{1,N}(\Omega)}-\int_{\Omega}G(u^+)\,{\rm d} x
-\int_{\Omega}F(x,u^+) \,{\rm d} x\nonumber\\
&& -\frac{1}{q+1}||u||^{q+1}_{L^{q+1}(\partial\Omega)}
\nonumber
\end{eqnarray*}
where $F(x,u)=
\int_0^sf(x,s)\,{\rm d}s$, $G(u)=\int_0^s g(s)\,{\rm d}s$. We show that if
$u_0\in C^1(\overline{\Omega})$ satisfying $u_0\geq \eta \mbox{dist}(x,\partial\Omega)$,
for some $0<\eta$, is a local minimum of $I$ in the $C^1(\overline{\Omega})\cap C_0(\overline{\Omega})$
topology, then it is also a local minimum in $W^{1,N}(\Omega)$ topology. This result is useful
%for proving multiple solutions to the associated Euler-lagrange equation ${\rm (P)}$ defined below.
to prove the multiplicity of positive solutions to critical growth problems with co-normal
boundary conditions.