Weak and entropy solutions to nonlinear Neumann boundary-value problems with variable exponents

作者：Ouaro Stanislas^*; Soma Safimba

来源：Complex Variables and Elliptic Equations, 2011, 56(7-9): 829-851.

DOI：10.1080/17476933.2010.504840

• generalized Lebesgue and Sobolev spaces
• weak solution
• entropy solution
• p(x)-Laplace operator

In this article, we study the following nonlinear Neumann boundary-value problem - div a(x, del u) + vertical bar u vertical bar(p(x)-2) u = f in Omega, partial derivative u/partial derivative nu = 0 on partial derivative Omega, where Omega is a smooth bounded open domain in R-N, N >= 3, partial derivative u/partial derivative nu is the outer unit normal derivative on partial derivative Omega, div a(x, del u) a p(x)-Laplace type operator. We prove the existence and uniqueness of a weak solution for f is an element of L(p-)' (Omega), the existence and uniqueness of an entropy solution for L-1-data f independent of u and the existence of weak solutions for f dependent on u. The functional setting involves Lebesgue and Sobolev spaces with variable exponents.

• 出版日期2011