Let Ω be a smooth bounded domain in RN, N≥2; let a, f, h be smooth functions on
$\overline{\varOmega }$
, f being positive on
$\overline{\varOmega }$
and a satisfying the following condition:
∫Ω|∇u|+∫Ωa|u|≥C∫Ω|u|, ∀u∈W1,10(Ω), where C is some positive constant.
We look for some u∈BV(Ω), u not identically 0, which satisfies:
$\[\left\{\begin{array}{l}-\mathop{\mathrm{div}}\sigma+a(x)\mathop{\mathrm{sign}}(u)=f(x)|u|^{1^{*}-2}u+h(x)|u|^{q-2}u\quad \mbox{in}\ \varOmega ,\\\sigma\in L^{\infty}(\varOmega ,\mathbf{R}^{N}),\quad \sigma\cdot\nabla u=|\nabla u|\quad \mbox{in}\ \varOmega ,\\-\sigma\cdot\vec{n}u=|u|\quad\mbox{on}\ \curpartial \varOmega ,\end{array}\right.\]$
where 1*=N/(N−1) denotes the critical Sobolev exponent for the embedding of W1,1(Ω) and BV(Ω) into Lk(Ω), q is a real in ]1,1*[ and sign(u) is some L∞ function such that sign(u) u=|u|.
