Results 71 to 80 of about 406 (185)

On Bounds for Norms of Reparameterized ReLU Artificial Neural Network Parameters: Sums of Fractional Powers of the Lipschitz Norm Control the Network Parameter Vector

Mathematical Methods in the Applied Sciences, Volume 49, Issue 4, Page 2135-2160, 15 March 2026.
ABSTRACT It is an elementary fact in the scientific literature that the Lipschitz norm of the realization function of a feedforward fully connected rectified linear unit (ReLU) artificial neural network (ANN) can, up to a multiplicative constant, be bounded from above by sums of powers of the norm of the ANN parameter vector.
Arnulf Jentzen, Timo Kröger
wiley   +1 more source

On elliptic systems with Sobolev critical exponent

Discrete and Continuous Dynamical Systems, 2016
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openaire   +3 more sources

Breaking Barriers in High‐Order Spectral Methods: The Intrinsic Matrix Approach

International Journal for Numerical Methods in Engineering, Volume 127, Issue 5, 15 March 2026.
ABSTRACT This paper introduces a unified framework in Hilbert spaces for applying high‐order differential operators in bounded domains using Chebyshev, Legendre, and Fourier spectral methods. By exploiting the banded structure of differentiation matrices and embedding boundary conditions directly into the operator through a scaling law relating ...
Osvaldo Guimarães, José R. C. Piqueira
wiley   +1 more source

Existence results for elliptic systems involving critical Sobolev exponents

Electronic Journal of Differential Equations, 2004
n this paper, we study the existence and nonexistence of positive solutions of an elliptic system involving critical Sobolev exponent perturbed by a weakly coupled term.
Mohammed Bouchekif, Yasmina Nasri
doaj  

Infinitely many positive solutions for p-Laplacian equations with singular and critical growth terms

Boundary Value Problems
In this paper, we study the existence of multiple solutions for the following nonlinear elliptic problem of p-Laplacian type involving a singularity and a critical Sobolev exponent { − Δ p u = u p ∗ − 1 + λ | u | γ − 1 u , in Ω , u = 0 , on ∂ Ω ...
Chen-Xi Wang, Hong-Min Suo
doaj   +1 more source

A note on problems involving critical Sobolev exponents

Differential and Integral Equations, 1995
Existence of a nontrivial solution of the semilinear elliptic equation \(- \Delta u= | u|^{p-2} u+ f(x,u)\) in \(G\), \(u=0\) on \(\partial G\) is shown in this paper. Here \(G\) denotes a smooth bounded domain in \(\mathbb{R}^ N\) \((N\geq 4)\), \(p-1= (N+2)/ (N-2)\) (the so-called critical Sobolev exponent) and \(f(x,u)= \lambda u+ g(x,u)\) is ...
Costa, D. G., Silva, E. A.
openaire   +3 more sources

The Numbers of Positive Solutions by the Lusternik-Schnirelmann Category for a Quasilinear Elliptic System Critical with Hardy Terms

Abstract and Applied Analysis, 2019
In this paper, we study the quasilinear elliptic system with Sobolev critical exponent involving both concave-convex and Hardy terms in bounded domains.
Mustapha Khiddi
doaj   +1 more source

On Dirac equation with a potential and critical Sobolev exponent

Communications on Pure and Applied Analysis, 2015
The authors consider a Dirac equation with a nonlinear zero order term and additional potential on a closed spin manifold. The nonlinearity is critical in the sense of Sobolev embedding -- analogously to the nonlinearity of the Yamabe equation. They prove existence of solutions under some mild assumption on the spectrum of the Dirac operator with ...
Gong, Wenmin, Lu, Guangcun
openaire   +2 more sources

Non-homogeneous problem for fractional Laplacian involving critical Sobolev exponent

Electronic Journal of Differential Equations, 2017
In this article, we study the existence of positive solutions for the nonhomogeneous fractional equation involving critical Sobolev exponent $$\displaylines{ (-\Delta)^{s} u +\lambda u=u^p+\mu f(x), \quad u>0\quad \text{in } \Omega,\cr u =0, \quad \
Kun Cheng, Li Wang
doaj