Abstract
We are concerned with the existence of nontrivial weak solutions for a class of generalized minimal surface equations with subcritical growth and Dirichlet boundary condition. In relationship with the values of several variable exponents, we establish two sufficient conditions for the existence of solutions. In the first part of this paper, we prove the existence of a non-negative solution. Next, we are concerned with the existence of infinitely many solutions in a symmetric abstract setting.
Highlights
Introduction and abstract settingThe Plateau problem is associated with the study of minimal surfaces
This problem consists in finding a surface with least area in R2 that spans a given closed curve with smooth boundary. Such a surface is described by solutions of the following minimal surface equation
Meusnier in 1776 in his works on minimal surfaces related with the Plateau problem
Summary
The main results in this paper are concerned with the existence of nontrivial solutions for a nonhomogeneous perturbation of the differential operator div (1 + |∇u|2)(p(x)−2)/2∇u and corresponding to a power-type reaction term with variable exponent. Relevant applications of nonlinear problems with variable exponent are developed in the monographs [7], [8] and [22]. Radulescu [21] for recent related results devoted to the mathematical analysis of some problems driven by differential operators with variable exponent. We recall some basic properties of the Lebesgue and Sobolev spaces with variable exponent. We recall some basic facts that concern the Lebesgue and Sobolev function spaces with variable exponent.
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