Search results for "ELLIPTIC PROBLEMS"

Bounded and unbounded solutions for a class of quasi-linear elliptic problems with a quadratic gradient term

2001

Abstract Our aim in this article is to study the following nonlinear elliptic Dirichlet problem: − div [a(x,u)·∇u]+b(x,u,∇u)=f, in Ω; u=0, on ∂Ω; where Ω is a bounded open subset of RN, with N>2, f∈L m (Ω) . Under wide conditions on functions a and b, we prove that there exists a type of solution for this problem; this is a bounded weak solution for m>N/2, and an unbounded entropy solution for N/2>m⩾2N/(N+2). Moreover, we show when this entropy solution is a weak one and when can be taken as test function in the weak formulation. We also study the summability of the solutions.

Existence and comparison results for a singular semilinear elliptic equation with a lower order term

2014

This paper deals with the homogeneous Dirichlet problem for a singular semilinear elliptic equation with a first order term. When the datum is bounded we prove an existence result and we show that any solution can be compared with the solution to a suitable symmetrized problem.

Rank structured approximation method for quasi--periodic elliptic problems

2016

We consider an iteration method for solving an elliptic type boundary value problem $\mathcal{A} u=f$, where a positive definite operator $\mathcal{A}$ is generated by a quasi--periodic structure with rapidly changing coefficients (typical period is characterized by a small parameter $\epsilon$) . The method is based on using a simpler operator $\mathcal{A}_0$ (inversion of $\mathcal{A}_0$ is much simpler than inversion of $\mathcal{A}$), which can be viewed as a preconditioner for $\mathcal{A}$. We prove contraction of the iteration method and establish explicit estimates of the contraction factor $q$. Certainly the value of $q$ depends on the difference between $\mathcal{A}$ and $\mathcal…

Elliptic problems involving the 1–Laplacian and a singular lower order term

2018

Steady‐state solutions of the aerotaxis problem

2022

We study the steady-state system of aerotaxis equations in higher dimensions.It is shown that the existence and multiplicity of solutions depend on the totalmass of the colony of bacteria, the energy function, and the boundary conditions.

The Radó–Kneser–Choquet theorem for $p$-harmonic mappings between Riemannian surfaces

2020

In the planar setting the Rad\'o-Kneser-Choquet theorem states that a harmonic map from the unit disk onto a Jordan domain bounded by a convex curve is a diffeomorphism provided that the boundary mapping is a homeomorphism. We prove the injectivity criterion of Rad\'o-Kneser-Choquet for $p$-harmonic mappings between Riemannian surfaces. In our proof of the injecticity criterion we approximate the $p$-harmonic map with auxiliary mappings that solve uniformly elliptic systems. We prove that each auxiliary mapping has a positive Jacobian by a homotopy argument. We keep the maps injective all the way through the homotopy with the help of the minimum principle for a certain subharmonic expressio…