Analía Silva

Multiple solutions for the p-Laplace operator with critical growth

Abstract In this paper we show the existence of at least three nontrivial solutions to the following quasilinear elliptic equation − Δ p u = | u | p ∗ − 2 u + λ f ( x , u ) in a smooth bounded domain Ω of R N with homogeneous Dirichlet boundary conditions on ∂ Ω , where p ∗ = N p / ( N − p ) is the critical Sobolev exponent and Δ p u = div ( | ∇ u | p − 2 ∇ u ) is the p -Laplacian. The proof is based on variational arguments and the classical concentration compactness method.

Multiple Solutions for the p(x)− Laplace Operator with Critical Growth

Abstract The aim of this paper is to extend previous results regarding the multiplicity of solutions for quasilinear elliptic problems with critical growth to the variable exponent case. We prove, in the spirit of [4], the existence of at least three nontrivial solutions to the quasilinear elliptic equation −Δp(x)u = |u|q(x)−2u + λ f (x, u) in a smooth bounded domain Ω of RN with homogeneous Dirichlet boundary conditions on ∂Ω. We assume that {q(x) = p∗(x)} ≠ ø, where p∗(x) = Np(x)/(N − p(x)) is the critical Sobolev exponent for variable exponents and Δp(x)u = div(|∇u|p(x)−2∇u) is the p(x)−laplacian. The proof is based on variational arguments and the extension of concentration compactness method for variable exponent spaces.

Existence of solution to a critical trace equation with variable exponent

In this paper we study sufficient local conditions for the existence of non-trivial solution to a critical equation for the

Opinion Formation Models with Heterogeneous Persuasion and Zealotry

p(x)-

The concentration-compactness principle for fractional order Sobolev spaces in unbounded domains and applications to the generalized fractional Brezis–Nirenberg problem

Laplacian where the critical term is placed as a source through the boundary of the domain. The proof relies on a suitable generalization of the concentration--compactness principle for the trace embedding for variable exponent Sobolev spaces and the classical mountain pass theorem.

Concentration-compactness principle for variable exponent spaces and applications.

In this work an opinion formation model with heterogeneous agents is proposed. Each agent is supposed to have different power of persuasion, and besides its own level of zealotry, that is, an individual willingness to being convinced by other agent. In addition, our model includes zealots or stubborn agents, agents that never change opinions. We derive a Bolzmann-like equation for the distribution of agents on the space of opinions, which is approximated by a transport equation with a nonlocal drift term. We study the long-time asymptotic behavior of solutions, characterizing the limit distribution of agents, which consists of the distribution of stubborn agents, plus a delta function at the mean of their opinions, weighted by they power of persuasion. Moreover, explicit bounds on the rate of convergence are given, and the time to convergence is shown to decrease when the number of stubborn agents increases. This is a remarkable fact observed in agent based simulations in different works.

On the Sobolev embedding theorem for variable exponent spaces in the critical range

In this paper we extend the well-known concentration-compactness principle for the Fractional Laplacian operator in unbounded domains. As an application we show sufficient conditions for the existence of solutions to some critical equations involving the fractional p-Laplacian in the whole

Existence of solution to a critical equation with variable exponent

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