Singular quasilinear and Hessian equations and inequalities

Annals of Mathematics, 2008

The existence problem is solved, and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of Lane-Emden type, including the following two model problems:

Communications on Pure and Applied Mathematics, 2013

Contemporary Mathematics, 2013

Let Ω ⊂ R N be a smooth bounded domain, H a Caratheodory function defined in Ω × R × R N , and µ a bounded Radon measure in Ω. We study the problem −∆ p u + H(x, u, ∇u) = µ in Ω, u = 0 on ∂Ω,

Communications in Contemporary Mathematics, 2020

In this paper, we study the global regularity estimates in Lorentz spaces for gradients of solutions to quasilinear elliptic equations with measure data of the form [Formula: see text] where [Formula: see text] is a finite signed Radon measure in [Formula: see text], [Formula: see text] is a bounded domain such that its complement [Formula: see text] is uniformly [Formula: see text]-thick and [Formula: see text] is a Carathéodory vector-valued function satisfying growth and monotonicity conditions for the strongly singular case [Formula: see text]. Our result extends the earlier results [19,22] to the strongly singular case [Formula: see text] and a recent result [18] by considering rough conditions on the domain [Formula: see text] and the nonlinearity [Formula: see text].

Advances in Calculus of Variations, 2021

We study quasilinear elliptic equations of the type -Δp⁢u=σ⁢uq+μ{-\Delta_{p}u=\sigma u^{q}+\mu} in ℝn{\mathbb{R}^{n}} in the case 0<q<p-1{0<q<p-1}, where μ and σ are nonnegative measurable functions, or locally finite measures, and Δp⁢u=div⁡(|∇⁡u|p-2⁢∇⁡u){\Delta_{p}u=\operatorname{div}(\lvert\nabla u\rvert^{p-2}\nabla u)} is the p-Laplacian. Similar equations with more general local and nonlocal operators in place of Δp{\Delta_{p}} are treated as well. We obtain existence criteria and global bilateral pointwise estimates for all positive solutions u: u⁢(x)≈(𝐖p⁢σ⁢(x))p-qp-q-1+𝐊p,q⁢σ⁢(x)+𝐖p⁢μ⁢(x),x∈ℝn,u(x)\approx({\mathbf{W}}_{p}\sigma(x))^{\frac{p-q}{p-q-1}}+{\mathbf{K}}_{p,q}% \sigma(x)+{\mathbf{W}}_{p}\mu(x),\quad x\in\mathbb{R}^{n}, where 𝐖p{{\mathbf{W}}_{p}} and 𝐊p,q{{\mathbf{K}}_{p,q}} are, respectively, the Wolff potential and the intrinsic Wolff potential, with the constants of equivalence depending only on p, q, and n. The contributions of μ and σ in these pointwi...

Calculus of Variations and Partial Differential Equations, 2006

Advances in Nonlinear Analysis, 2016

In this paper, we prove the gradient estimate for renormalized solutions to quasilinear elliptic equations with measure data on variable exponent Lebesgue spaces with BMO coefficients in a Reifenberg flat domain.

Eprint Arxiv 1011 3169, 2010

The existence of positive solutions is considered for the Dirichlet problem \[ \left\{ \begin{array} [c]{rcll}% -\Delta_{p}u & = & \lambda\omega_{1}(x)\left\vert u\right\vert ^{q-2}% u+\beta\omega_{2}(x)\left\vert u\right\vert ^{a-1}u|\nabla u|^{b} & \text{in }\Omega\\ u & = & 0 & \text{on }\partial\Omega, \end{array} \right. \] where $\lambda$ and $\beta$ are positive parameters, $a$ and $b$ are positive constants satisfying $a+b\leq p-1$, $\omega_{1}(x)$ and $\omega_{2}(x)$ are nonnegative weights and $1<q\leq p$. The homogeneous case $q=p$ is handled by making $q\rightarrow p^{-}$ in the sublinear case $1<q<p,$ which is based on the sub- and super-solution method. The core of the proof of this problem is then generalized to the Dirichlet problem $-\Delta_{p}u=f(x,u,\nabla u)$ in $\Omega$, where $f$ is a nonnegative, continuous function satisfying simple, geometrical hypotheses. This approach might be considered as a unification of arguments dispersed in various papers, with the advantage of handling also nonlinearities that depend on the gradient, even in the $p$-growth case. It is then applied to the problem $-\Delta_{p}u=\lambda\omega(x)u^{q-1}\left( 1+|\nabla u|^{p}\right) $ with Dirichlet boundary conditions in the domain $\Omega$.

ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, 2009

We consider elliptic partial differential equations and provide a method constructing solutions with critical integrability properties. We illustrate the technique by studying isotropic equations and equations in non-divergence form in the plane. 2 KARI ASTALA, DANIEL FARACO, AND LÁSZLÓ SZÉKELYHIDI JR Here R 2×2 sym represents the space of symmetric matrices with real entries. Theorem 1.1 was obtained by Leonetti and Nesi in [20] where they discovered that under the assumptions (1), (2) the solution u = f , where f is a K-quasiregular mapping, with the same K. Since the work of the first author [1] says that a K quasiregular mapping in W 1,q is immediately in W 1,p whenever 2K K+1 < q < p < 2K K−1 , Theorem 1.1 follows. In addition, in the theorem we have included the end point q = 2K K+1. This case follows from the recent result of Petermichl and Volberg (see [4],[28],[10]). The classical examples built on the radial stretching u(x) = (x|x| 1 K −1) show that for general σ the range of exponents p, q can not be improved without extra assumptions. On the other hand, there was the hope that if the range of σ was restricted then the gradients would enjoy higher integrability. A basic example pointing towards this direction is the work of Piccinini and Spagnolo [29]. There it is shown that if σ(x) = ρ(x)I, where ρ is a real valued function with 1/K ≤ ρ(x) ≤ K, then u has a better Hölder regularity than in the case of a general σ. Our first theorems show, however, that for Sobolev regularity one can not improve any of the critical exponents 2K K+1 , 2K K−1 even if the essential range of σ consists of only two matrices. Theorem 1.2. Let Ω be a bounded domain in R 2 and let K > 1. Then there exists a measurable function ρ 1 : Ω → { 1 K , K} such that the solution u 1 ∈ W 1,2 (Ω) to the equation (3) div ρ 1 (x)∇u 1 (x) = 0 in Ω u 1 (x) = x 1 on ∂Ω satisfies for every B(x, r) ⊂ Ω the condition (4)

2000

This article is concerned with the regularity of the entrop y solution of � −div(|x| p|∇u|p 2∇u) = f(x) in , u = 0 on @, where is a smooth bounded domain of RN such that 0 ∈ , 1 &lt; p &lt; N , and &lt; (N − p)/p. Assuming f ∈ Lq(, |x|�(q 1)dx) for some q ≥