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We deal with existence and uniqueness of nonnegative solutions to: −Δu=f(x),inΩ,∂u∂ν+λ(x)u=g(x)uη,on∂Ω,\left\{\begin{array}{ll}-\Delta u=f\left(x),\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ \frac{\partial u}{\partial \nu }+\lambda \left(x)u=\frac{g\left(x)}{{u}^{\eta }},\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where η≥0\eta \ge 0 and f,λf,\lambda , and gg are the nonnegative integrable functions. The set Ω⊂RN(N>2)\Omega \subset {{\mathbb{R}}}^{N}\left(N\gt 2) is open and bounded with smooth boundary, and ν\nu denotes its unit outward normal vector. More generally, we handle equations driven by monotone operators of pp-Laplacian type jointly with nonlinear boundary conditions. We prove the existence of an entropy solution and check that, under natural assumptions, this solution is unique. Among other features, we study the regularizing effect given to the solution by both the absorption and the nonlinear boundary term.

In this paper, we study the Γ-limit, as p → 1 , of the functional J p ⁢ (u) = ∫ Ω | ∇ ⁡ u | p + β ⁢ ∫ ∂ ⁡ Ω | u | p ∫ Ω | u | p , where Ω is a smooth bounded open set in ℝ N , p > 1 and β is a real number. Among our results, for β > - 1 , we derive an isoperimetric inequality for Λ ⁢ (Ω , β) = inf u ∈ BV ⁡ (Ω) , u ≢ 0 ⁡ | D ⁢ u | ⁢ (Ω) + min ⁡ (β , 1) ⁢ ∫ ∂ ⁡ Ω | u | ∫ Ω | u | which is the limit as p → 1 + of λ ⁢ (Ω , p , β) = min u ∈ W 1 , p ⁢ (Ω) ⁡ J p ⁢ (u) . We show that among all bounded and smooth open sets with given volume, the ball maximizes Λ ⁢ (Ω , β) when β ∈ (- 1 , 0) and minimizes Λ ⁢ (Ω , β) when β ∈ [ 0 , ∞) . [ABSTRACT FROM AUTHOR]