Related papers: Semilinear elliptic equations on rough domains
We establish some existence and regularity results to the Dirichlet problem, for a class of quasilinear elliptic equations involving a partial differential operator, depending on the gradient of the solution. Our results are formulated in…
This paper is concerned with the homogenization of Dirichlet problem of elliptic systems in a bounded, smooth domain of finite type. Both the coefficients of the elliptic operator and the Dirichlet boundary data are assumed to be periodic…
This paper deals with an existence and uniqueness result of the weak solution for a quasilinear elliptic PDE with nonlinear Robin boundary conditions.This problem is defined on a domain whose boundary is the union of two disjoint…
The necessity of a Maximum Principle arises naturally when one is interested in the study of qualitative properties of solutions to partial differential equations. In general, to ensure the validity of these kind of principles one has to…
We present a notion of weak solution for the Dirichlet problem driven by the fractional Laplacian, following the Stampacchia theory. Then, we study semilinear problems on bounded domains $\Omega$ with two different boundary conditions at…
In this paper, we study the nonhomogeneous Dirichlet problem concerning general semilinear elliptic equations in divergence form. We establish that the boundary Lipschitz regularity of solutions under some more weaker conditions on the…
The class of problems treated here are elliptic partial differential equations with a homogeneous boundary condition and a non-linear perturbation obtained by composition with a fixed smooth function. The existence of solutions is obtained…
We introduce a new boundary Harnack principle in Lipschitz domains for equations with right hand side. Our approach, which uses comparisons and blow-ups, will adapt to more general domains as well as other types of operators. We prove the…
Take an open domain $\Omega \subset \mathbb R^n$ whose boundary may be composed of pieces of different dimensions. For instance, $\Omega$ can be a ball on $\mathbb R^3$, minus one of its diameters $D$, or $\Omega \subset \mathbb R^3$ could…
We investigate quantitative properties of nonnegative solutions $u(x)\ge 0$ to the semilinear diffusion equation $\mathcal{L} u= f(u)$, posed in a bounded domain $\Omega\subset {\mathbb R}^N$ with appropriate homogeneous Dirichlet or outer…
We are interested in regularity properties of semi-stable solutions for a class of singular semilinear elliptic problems with advection term defined on a smooth bounded domain of a complete Riemannian manifold with zero Dirichlet boundary…
We study the critical points of the solution of second elliptic equations in divergence and diagonal form with a bounded and positive definite coefficient, under the assumption that the statement of the Hopf lemma holds (sign assumptions on…
In this paper, we demonstrate the existence of positive solutions for certain weakly coupled elliptic systems of sublinear growth under homogeneous Dirichlet boundary conditions. Our findings generalize existing results related to sublinear…
In this paper we study the asymptotic behavior of the solutions of a class of nonlinear elliptic problems posed in a 2-dimensional domain that degenerates into a line segment (a thin domain) when a positive parameter $\varepsilon$ goes to…
In this work we consider higher dimensional thin domains with the property that both boundaries, bottom and top, present oscillations of weak type. We consider the Laplace operator with Neumann boundary conditions and analyze the behavior…
In this paper we introduce new characterizations of spectral fractional Laplacian to incorporate nonhomogeneous Dirichlet and Neumann boundary conditions. The classical cases with homogeneous boundary conditions arise as a special case. We…
In this paper, we analyze an eigenvalue problem for nonlinear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We prove bifurcation results from trivial solutions and from infinity for…
We establish a positivity property for a class of semilinear elliptic problems involving indefinite sublinear nonlinearities. Namely, we show that any nontrivial nonnegative solution is positive for a class of problems the strong maximum…
We consider the negative Laplacian subject to mixed boundary conditions on a bounded domain. We prove under very general geometric assumptions that slightly above the critical exponent $\frac{1}{2}$ its fractional power domains still…
In this paper we prove existence of (viscosity) solutions of Dirichlet problems concerning fully nonlinear elliptic operator, which are either degenerate or singular when the gradient of the solution is zero. For this class of operators it…