Search results for "Nonlinear system"

Positive solutions for the Neumann p-Laplacian

2017

We examine parametric nonlinear Neumann problems driven by the p-Laplacian with asymptotically ( $$p-1$$ )-linear reaction term f(z, x) (as $$x\rightarrow +\infty $$ ). We determine the existence, nonexistence and minimality of positive solutions as the parameter $$\lambda >0$$ varies.

Equivalence of viscosity and weak solutions for the $p(x)$-Laplacian

2010

We consider different notions of solutions to the $p(x)$-Laplace equation $-\div(\abs{Du(x)}^{p(x)-2}Du(x))=0$ with $ 1<p(x)<\infty$. We show by proving a comparison principle that viscosity supersolutions and $p(x)$-superharmonic functions of nonlinear potential theory coincide. This implies that weak and viscosity solutions are the same class of functions, and that viscosity solutions to Dirichlet problems are unique. As an application, we prove a Rad\'o type removability theorem.

The annular decay property and capacity estimates for thin annuli

2016

We obtain upper and lower bounds for the nonlinear variational capacity of thin annuli in weighted $\mathbf{R}^n$ and in metric spaces, primarily under the assumptions of an annular decay property and a Poincar\'e inequality. In particular, if the measure has the $1$-annular decay property at $x_0$ and the metric space supports a pointwise $1$-Poincar\'e inequality at $x_0$, then the upper and lower bounds are comparable and we get a two-sided estimate for thin annuli centred at $x_0$, which generalizes the known estimate for the usual variational capacity in unweighted $\mathbf{R}^n$. Most of our estimates are sharp, which we show by supplying several key counterexamples. We also character…

Perturbed eigenvalue problems for the Robin p-Laplacian plus an indefinite potential

2020

AbstractWe consider a parametric nonlinear Robin problem driven by the negativep-Laplacian plus an indefinite potential. The equation can be thought as a perturbation of the usual eigenvalue problem. We consider the case where the perturbation$$f(z,\cdot )$$f(z,·)is$$(p-1)$$(p-1)-sublinear and then the case where it is$$(p-1)$$(p-1)-superlinear but without satisfying the Ambrosetti–Rabinowitz condition. We establish existence and uniqueness or multiplicity of positive solutions for certain admissible range for the parameter$$\lambda \in {\mathbb {R}}$$λ∈Rwhich we specify exactly in terms of principal eigenvalue of the differential operator.

Solutions and positive solutions for superlinear Robin problems

2019

We consider nonlinear, nonhomogeneous Robin problems with a (p − 1)-superlinear reaction term, which need not satisfy the Ambrosetti-Rabinowitz condition. We look for positive solutions and prove existence and multiplicity theorems. For the particular case of the p-Laplacian, we prove existence results under a different geometry near the origin.We consider nonlinear, nonhomogeneous Robin problems with a (p − 1)-superlinear reaction term, which need not satisfy the Ambrosetti-Rabinowitz condition. We look for positive solutions and prove existence and multiplicity theorems. For the particular case of the p-Laplacian, we prove existence results under a different geometry near the origin.

$C^{1,��}$ regularity for the normalized $p$-Poisson problem

2017

We consider the normalized $p$-Poisson problem $$-��^N_p u=f \qquad \text{in}\quad ��.$$ The normalized $p$-Laplacian $��_p^{N}u:=|D u|^{2-p}��_p u$ is in non-divergence form and arises for example from stochastic games. We prove $C^{1,��}_{loc}$ regularity with nearly optimal $��$ for viscosity solutions of this problem. In the case $f\in L^{\infty}\cap C$ and $p&gt;1$ we use methods both from viscosity and weak theory, whereas in the case $f\in L^q\cap C$, $q&gt;\max(n,\frac p2,2)$, and $p&gt;2$ we rely on the tools of nonlinear potential theory.

On the accurate determination of nonisolated solutions of nonlinear equations

1981

A simple but efficient method to obtain accurate solutions of a system of nonlinear equations with a singular Jacobian at the solution is presented. This is achieved by enlarging the system to a higher dimensional one whose solution in question is isolated. Thus it can be computed e. g. by Newton's method, which is locally at least quadratically convergent and selfcorrecting, so that high accuracy is attainable.

Reply to 'The super-quadratic growth of high-harmonic signal as a function of pressure'

2010

1982

The molecular weight distribution (MWD) of a high polymer is calculated from a weakly perturbed Zimm-plot of the classical light scattering on dilute solutions of Gaussian polymer coils (theta state). A typical Zimm-plot is simulated corresponding to the measurements of high accuracy as would be obtained by using the laser photometer described by Hack and Meyerhoff. The accuracy as published by these authors for small dissymmetries is used. Two numerical methods for calculating the MWD are briefly described and tested, both using an empirical formula for the Laplace image of the calculated MWD.

Experimental study of bifurcations in modified FitzHugh-Nagumo cell

2003

A nonlinear electrical circuit is proposed as a basic cell for modelling the FitzHugh-Nagumo equation with a modified excitability. Depending on initial conditions and parameters, experiments show various dynamics including stability with excitation threshold, bistability and oscillations.