Study on the Existence and Uniqueness of Solution of Generalized Capillarity Problem

Li Wei,Liling Duan,Haiyun Zhou

doi:10.1155/2012/154307

Abstract

By using the perturbation theories on sums of ranges of nonlinear accretive mappings of Calvert and Gupta (1978), the abstract result on the existence and uniqueness of the solution inLp(Ω)of the generalized Capillarity equation with nonlinear Neumann boundary value conditions, where2N/(N+1)<p<+∞andN≥1denotes the dimension ofRN, is studied. The equation discussed in this paper and the methods here are a continuation of and a complement to the previous corresponding results. To obtain the results, some new techniques are used in this paper.

Highlights

Introduction and PreliminarySince the p-Laplacian operator −Δp with p / 2 arises from a variety of physical phenomena, such as nonNewtonian fluids, reaction-diffusion problems, and petroleum extraction, it becomes a very popular topic in mathematical fields.We began our study on this topic in 1995
We showed in Wei and Agarwal 7 that 1.2 has solutions in Ls Ω under some conditions, where 2N/ N 1 < p ≤ s < ∞, 1 ≤ q < ∞ if p ≥ N, and 1 ≤ q ≤ Np/ N − p if p < N, for N ≥ 1
Let Ω be a bounded domain in RN and g : Ω × R → R be a function satisfying Caratheodory’s conditions such that i g x, · is monotonically increasing on R; ii the mapping u ∈ Lp Ω → g x, u x ∈ Lp Ω is well defined, where 2N/ N 1 < p < ∞ and N ≥ 1

Summary

Introduction

Introduction and PreliminarySince the p-Laplacian operator −Δp with p / 2 arises from a variety of physical phenomena, such as nonNewtonian fluids, reaction-diffusion problems, and petroleum extraction, it becomes a very popular topic in mathematical fields.We began our study on this topic in 1995. Let J denote the duality mapping from X into 2X∗ defined by The duality mapping J : X → 2X∗ is said to be satisfying Condition I if there exists a function η : X → 0, ∞ such that

Results

Conclusion