Weak solution of a Neumann boundary value problem with 𝑝(𝑥)-Laplacian-like operator

Abstract

In this paper, we study the existence of a weak solution for a class of Neumann boundary value problems for equations involving the p ⁢ ( x ) p(x) -Laplacian-like operator. Using a topological degree theory for a class of demicontinuous operators of generalized ( S + ) (S_{+}) -type and the theory of the variable exponent Sobolev spaces, we establish the existence of a weak solution of this problem. Our results extend and generalize several corresponding results from the existing literature.

Nonlocal τ(m)‐Laplacian‐like problem with logarithmic nonlinearity and without Ambrosetti–Rabinowitz condition on compact Riemannian manifolds

“…This growing interest is motivated not only by the attraction of natural phenomena such as the movement of drops, waves, and bubbles but also by their importance in various practical fields, including industrial, biomedical, pharmaceutical, and microfluidic systems. In the context of the study of capillarity phenomena, many results have been obtained; see, for example, earlier studies [1][2][3][4][5][6][7][8]. The fundamental and common assumption of these works is the Ambrosetti-Rabinowitz condition [9].…”

On a new p(x)-Kirchhoff type problems with p(x)-Laplacian-like operators and Neumann boundary conditions

“…The operator −div ∇u 1 + |∇u| 2 is most often denoted by the specified mean curvature operator and ∇u √ 1+|∇u| 2 is the Kirchhoff stress term. Elliptic boundary value problems involving the mean curvature operator play apivotal role in the mathematical analysis of several physical or geometrical issues,such as capillarity phenomena for incompressible or compressible fluids, mathematicalmodels in physiology or in electrostatics, flux-limited diffusion phenomena, prescribedmean curvature problems for Cartesian surfaces in the Euclidean space: relevantreferences on these topics include [3,7,8,13].…”