Related papers: The obstacle problem for nonlinear degenerate equa…
We consider a strongly nonlinear elliptic problem with the homogeneous Dirichlet boundary condition. The growth and the coercivity of the elliptic operator is assumed to be indicated by an inhomogeneous anisotropic $\mathcal{N}$-function.…
In this paper we establish well posedness of the Neumann problem with boundary data in $L^2$ or the Sobolev space $\dot W^2_{-1}$, in the half space, for linear elliptic differential operators with coefficients that are constant in the…
We study a degenerate elliptic system with variable exponents. Using the variational approach and some recent theory on weighted Lebesgue and Sobolev spaces with variable exponents, we prove the existence of at least two distinct nontrivial…
In this paper, we study the regularity of the solution for the obstacle problem associated with the linearized Monge-Amp\`ere operator: \begin{align*} \begin{cases} &u\geq\varphi \text{\quad in } \Omega &L_{ w}u=\tr( W D^{2}u)\leq 0…
In this paper, we investigate the existence of weak solution for a fractional type problems driven by a nonlocal operator of elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions. We first extend…
In this paper we consider nonlinear elliptic PDEs of the type $$-\Delta_p u+a(x)|u|^{p-2}u=|u|^{p^*-2}u \qquad \mbox{ in }\Omega,$$ where $1<p<N$ and $p^*=Np/(N-p)$ is the critical Sobolev exponent, and allowing the asymptotic behavior of…
In this paper we give a comprehensive treatment of a two-penalty boundary obstacle problem for a divergence form elliptic operator, motivated by applications to fluid dynamics and thermics. Specifically, we prove existence, uniqueness and…
In this paper, we consider the thin obstacle problem in $\mathbb{R}^2$ with data at infinity. We first prove the existence and uniqueness of it. Then we show that its symmetric solutions are actually half-space solutions. Our results are…
In this article, we show the existence of a unique entropy solution to the following problem: \begin{equation} \begin{split} (-\Delta)_{p,\alpha}^su&= f(x)h(u)+g(x) ~\text{in}~\Omega,\\ u&>0~\text{in}~\Omega,\\ u&=…
We prove an existence and uniqueness result for quasilinear Stochastic PDEs with obstacle (OSPDE in short). Our method is based on analytical technics coming from the parabolic potential theory. The solution is expressed as a pair $(u,\nu)$…
We prove the existence and uniqueness of solutions to a Dirichlet problem \[ \begin{cases} Lu = f + v^{-1}\text{Div}(v{\bf e} h), & x \in \Omega; u = 0, & x \in \partial \Omega, \end{cases}\] where $L$ is a degenerate, linear, second order…
We study regularity properties for solutions to elliptic equations that are degenerate or singular along orthogonal hyperplanes. The degenerate ellipticity is carried out by a weight term which is the monomial product of different powers of…
We prove the existence and $C^{1,\alpha}$ regularity of solutions to nonlocal fully nonlinear elliptic equations with gradient constraints. We do not assume any regularity about the constraints; so the constraints need not be $C^1$ or…
The aim of this work is to prove existence and uniqueness results for a doubly nonlinear elliptic problem that is essential for solving the associated parabolic problem using Rothe's method (discretizing time). We work under very weak…
In the present paper we prove existence results for solutions to nonlinear elliptic Neumann problems whose prototype is \begin{equation*} \begin{cases} -\Delta_{p} u -\text{div} (c(x)|u|^{p-2}u)) =f & \text{in}\ \Omega, \\ \left( |\nabla…
We study second-order hyperbolic equations with degenerate elliptic operators and non-homogeneous Dirichlet boundary inputs. We establish existence and regularity of weak solutions in weighted Sobolev spaces under mild assumptions on the…
In this paper we study the existence of solutions of thedegererate elliptic system.
Given $\Omega(\subseteq\;R^{1+m})$, a smooth bounded domain and a nonnegative measurable function $f$ defined on $\Omega$ with suitable summability. In this paper, we will study the existence and regularity of solutions to the quasilinear…
In this paper we deal with an inhomogeneous parabolic Dirichlet problem involving the 1-Laplacian operator. We show the existence of a unique solution when data belong to $L^1(0,T;L^2(\Omega))$ for every $T>0$. As a consequence, global…
In this paper, we establish the existence, uniqueness and stability results for the obstacle problem associated with a degenerate nonlinear diffusion equation perturbed by conservative gradient noise. Our approach revolves round introducing…