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

Article type: Research Article

Authors: Wigniolle, Jérôme

Affiliations: University of Cergy‐Pontoise, 2, avenue Adolphe Chauvin, 95302 Cergy‐Pontoise, France E‐mail: jerwig@club‐internet.fr

Abstract: 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|.

Journal: Asymptotic Analysis, vol. 35, no. 3-4, pp. 207-234, 2003