On the sub-supersolution method for p(x)-Laplacian equations

By Xianling Fan. 2007

199 citations recorded by the source corpus.

Abstract This paper deals with the sub-supersolution method for the p ( x ) -Laplacian equations. A sub-supersolution principle for the Dirichlet problems involving the p ( x ) -Laplacian is established. It is proved that the local minimizers in the C 1 topology are also local minimizers in the W 1 , p ( x ) topology for given energy functionals. A strong comparison theorem for the p ( x ) -Laplacian equations is presented. Some applications of the abstract theorems obtained in this paper to the eigenvalue problems for the p ( x ) -Laplacian equations are given.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/469f915e6c0f887b0bab0222fede7a3176760b79

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Anisotropic nonlinear Neumann problems (2011, result): “For constant exponent Dirichlet Sobolev spaces, the result was obtained by Brezis–Nirenberg [7] (for p = 2), Garcia Azorero–Manfredi–Peral Alonso [21] (for p > 1) and Guo–Zhang [25] (for p 2). For variable exponent Dirichlet Sobolev spaces, the result is due to Fan [ 15 ], while for the constant exp”
Local "superlinearity"and "sublinearity" for the p-Laplacian (2009, background): “[18], [23], [22], [16].”
Existence of Positive Solutions for p(x)-Laplacian Equations with a Singular Nonlinear Term (2014, background): “For example, the regularity of weak solutions for differential equations with variable exponent was studied in [1, 7], and existence of solutions for variable exponent problems was studied in a series of papers [3, 9, 12, 20, 24, 30, 33, 36, 39, 44, 45].”
Remarks on eigenvalue problems involving the p(x)-Laplacian (2009, methodology): “In [14] it was proved by using the sub–supersolution method that, roughly speaking (see Theorem 3.”