Open Access Research

The estimates on the energy functional of an elliptic system with Neumann boundary conditions

Jing Zeng

Author Affiliations

School of Mathematics and Computer Sciences, Fujian Normal University, Fuzhou, 350007, P.R. China

Boundary Value Problems 2013, 2013:194  doi:10.1186/1687-2770-2013-194

Published: 28 August 2013

Abstract

We consider an elliptic system of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M2">View MathML</a> in Ω with Neumann boundary conditions, where Ω is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M3">View MathML</a> domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M4">View MathML</a>, f and g are nonlinearities having superlinear and subcritical growth at infinity. We prove the existence of nonconstant positive solutions of the system, and estimate the energy functional on a configuration space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M5">View MathML</a> by a different technique, which is an important step in the proof of the solution’s concentrative property. We conclude that the least energy solutions of the system concentrate at the point of boundary, which maximizes the mean curvature of Ω.

Keywords:
elliptic system; estimates; energy functional