On some nonlinear equations involving the 1‐Laplacian with critical Sobolev growth and perturbation terms

Let Ω be a smooth bounded domain in R^N, 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∈W^1,1₀(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 W^1,1(Ω) and BV(Ω) into L^k(Ω), q is a real in ]1,1^*[ and sign(u) is some L^∞ function such that sign(u) u=|u|.