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In this paper we prove existence of nonnegative bounded solutions for the non-autonomous prescribed mean curvature problem in non-parametric form on an open bounded domain Ω of R N . The mean curvature, that depends on the location of the solution u itself, is asked to be of the form f(x)h(u), where f is a nonnegative function in L N , ∞ (Ω) and h : R + ↦ R + is merely continuous and possibly unbounded near zero. As a preparatory tool for our analysis we propose a purely PDE approach to the prescribed mean curvature problem not depending on the solution, i.e. h ≡ 1 . This part, which has its own independent interest, aims to represent a modern and up-to-date account on the subject. Uniqueness is also handled in presence of a decreasing nonlinearity. The sharpness of the results is highlighted by mean of explicit examples. [ABSTRACT FROM AUTHOR]

In this paper we study existence and uniqueness of solutions to Dirichlet problems as g (u) - div Du 1 + | D u | 2 = f in Ω , u = 0 on ∂ Ω , where Ω is an open bounded subset of R N ( N ≥ 2 ) with Lipschitz boundary, g : R → R is a continuous function and f belongs to some Lebesgue space. In particular, under suitable saturation and sign assumptions, we explore the regularizing effect given by the absorption term g(u) in order to get solutions for data f merely belonging to L 1 (Ω) and with no smallness assumptions on the norm. We also prove a sharp boundedness result for data in L N (Ω) as well as uniqueness if g is increasing. [ABSTRACT FROM AUTHOR]

We study the asymptotic behaviour, as $p\to 1^+$ , of the solutions of the following inhomogeneous Robin boundary value problem: P \begin{equation*} \begin{cases} \displaystyle -\Delta_p u_p = f & \text{ in }\Omega,\\ \displaystyle |\nabla u_p|^{p-2}\nabla u_p\cdot \nu +\lambda |u_p|^{p-2}u_p = g & \text{ on } \partial\Omega, \end{cases} \end{equation*} where $\Omega$ is a bounded domain in $\mathbb {R}^{N}$ with sufficiently smooth boundary, $\nu$ is its unit outward normal vector and $\Delta _p v$ is the $p$ -Laplacian operator with $p>1$. The data $f\in L^{N,\infty }(\Omega)$ (which denotes the Marcinkiewicz space) and $\lambda,\,g$ are bounded functions defined on $\partial \Omega$ with $\lambda \ge 0$. We find the threshold below which the family of $p$ –solutions goes to 0 and above which this family blows up. As a second interest we deal with the $1$ -Laplacian problem formally arising by taking $p\to 1^+$ in (P). [ABSTRACT FROM AUTHOR]

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]

We deal with existence and uniqueness of positive solutions of an elliptic boundary value problem modeled by − Δ p u = f u γ + g u q in Ω , u = 0 on ∂ Ω , where Ω is an open bounded subset of ℝ N where Ω is an open bounded subset of ℝ N , Δpu := ÷(|∇u|p− 2∇u) is the usual p-Laplacian operator, γ ≥ 0 and 0 ≤ q ≤ p − 1; f and g are nonnegative functions belonging to suitable Lebesgue spaces. [ABSTRACT FROM AUTHOR]

In this paper we prove existence of nonnegative solutions to parabolic Cauchy–Dirichlet problems with (eventually) singular superlinear gradient terms. The model equation is u t - Δ p u = g (u) | ∇ u | q + h (u) f (t , x) in (0 , T) × Ω , where Ω is an open bounded subset of R N with N > 2 , 0 < T < + ∞ , 1 < p < N , and q < p is superlinear. The functions g , h are continuous and possibly satisfying g (0) = + ∞ and/or h (0) = + ∞ , with different rates. Finally, f is nonnegative and it belongs to a suitable Lebesgue space. We investigate the relation among the superlinear threshold of q, the regularity of the initial datum and the forcing term, and the decay rates of g , h at infinity. [ABSTRACT FROM AUTHOR]

We study the Dirichlet problem for an elliptic equation involving the 1-Laplace operator and a reaction term, namely: where is an open bounded set having Lipschitz boundary, f ∈ L1(Ω) is nonnegative, and h is a continuous real function that may possibly blow up at zero. We investigate optimal ranges for the data in order to obtain existence, nonexistence and (whenever expected) uniqueness of nonnegative solutions. [ABSTRACT FROM AUTHOR]

We study existence and uniqueness of nonnegative solutions to a problem which is modeled by - Δ p u = u - θ | ∇ u | p + f u - γ in Ω , u = 0 on ∂ Ω , where Ω is an open bounded subset of R N ( N ≥ 2 ), Δ p is the p-Laplacian operator ( 1 < p < N ), f ∈ L 1 (Ω) is nonnegative and θ , γ ≥ 0 . Examples and extensions are discussed at the end of the paper. [ABSTRACT FROM AUTHOR]

We prove existence of solutions for a class of singular elliptic problems with a general measure as source term whose model is where Ω is an open bounded subset of RN. Here γ > 0, f is a nonnegative function on Ω, and μ is a nonnegative bounded Radon measure on Ω. [ABSTRACT FROM AUTHOR]

In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form - Δ 1 u = h (u) f in Ω , u ≥ 0 in Ω , u = 0 on ∂ Ω. Here Δ 1 is the 1-Laplace operator, Ω is a bounded open subset of R N with Lipschitz boundary, h(s) is a continuous function which may become singular at s = 0 + , and f is a nonnegative datum in L N , ∞ (Ω) with suitable small norm. Uniqueness of solutions is also shown provided h is decreasing and f > 0 . As a preparatory tool for our method a general theory for the same problem involving the p-Laplacian (with p > 1 ) as principal part is also established. The main assumptions are further discussed in order to show their optimality. [ABSTRACT FROM AUTHOR]