Related papers: An elliptic problem with two singularities
We study the existence and the properties of solutions to the Dirichlet problem for uniformly elliptic second-order Hamilton-Jacobi-Bellman operators, depending on the principal eigenvalues of the operator.
In this paper, we study the existence of a solution for a class of Dirichlet problems with a singularity and a convection term. Precisely, we consider the existence of a positive solution to the Dirichlet problem $$-\Delta_p u =…
The classical Dirichlet problem for a second-order strongly elliptic system with constant coefficients in a Jordan domain is considered. We show that the solution of the problem can be represented as a functional series in powers of the…
We study parabolic operators H = $\partial$t -- div $\lambda$,x A(x, t)$\nabla$ $\lambda$,x in the parabolic upper half space R n+2 + = {($\lambda$, x, t) : $\lambda$ > 0}. We assume that the coefficients are real, bounded, measurable,…
We study the existence of positive solutions for a class of double phase Dirichlet equations which have the combined effects of a singular term and of a parametric superlinear term. The differential operator of the equation is the sum of a…
We consider a nonlinear elliptic equation driven by the Dirichlet $p$-Laplacian with a singular term and a $(p-1)$-linear perturbation which is resonant at $+\infty$ with respect to the principal eigenvalue. Using variational tools,…
We consider the Dirichlet problem for second-order linear elliptic equations in divergence form \begin{equation*} -\mathrm{div }(A\nabla u)+\mathbf{b} \cdot \nabla u+\lambda u=f+\mathrm{div } \mathbf{F}\quad \text{in }…
In this paper we use a natural iteration technique to prove existence of solutions to nonlinear Dirichlet problems. Among the examples included is the prescribed mean curvature equation. The nature of the technique allows applications to…
We study the existence and uniqueness for weak solutions to some classes of anisotropic elliptic Dirichlet problems with data belonging to the natural dual space.
The present study investigates a linear-quadratic Dirichlet control problem governed by a non-coercive elliptic equation posed on a possibly non-convex polygonal domain. Tikhonov regularization is carried out in an energy seminorm. The…
In this note we set up the elliptic and the parabolic Dirichlet problem for linear nonlocal operators. As opposed to the classical case of second order differential operators, here the "boundary data" are prescribed on the complement of a…
A second-order regularity theory is developed for solutions to a class of quasilinear elliptic equations in divergence form, including the $p$-Laplace equation, with merely square-integrable right-hand side. Our results amount to the…
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…
In this paper, we study the solvability of the nonlinear Dirichlet problem with sum of the operators of independent non standard growths in a bounded domain $\Omega \subset \mathbb{R}^{n}$. We obtain sufficient conditions and show the…
In this paper we are mainly concerned with nontrivial positive solutions to the Dirichlet problem for the degenerate elliptic equation \begin{gather} -\frac{\partial^2 u}{\partial x^2} -\left|x\right|^{2k}\frac{\partial^2 u}{\partial…
We study the discrete logarithm problem for the multiplicative group and for elliptic curves over a finite field by using a lifting of the corresponding object to an algebraic number field and global duality. We introduce the…
This paper studies the existence of positive normalized solutions to the singular elliptic equation \[ -\Delta u + \lambda u = u^{-r} + u^{p-1} \quad \text{in } \Omega, \] with the Dirichlet boundary condition $u=0$ on $\partial\Omega$ and…
In this paper we study the local behavior of a solution to second order elliptic operators with sharp singular coefficients in lower order terms. One of the main results is the bound on the vanishing order of the solution, which is a…
The present paper establishes the first result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a BMO anti-symmetric part. In…
We derive a priori second order estimates for solutions of a class of fully nonlinear elliptic equations on Riemannian manifolds under some very general structure conditions. We treat both equations on closed manifolds, and the Dirichlet…