International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 62601, 12 pages

A nonexistence result for a nonlinear PDE with Robin condition

Brahim Khodja

Department of Mathematics, Faculty of Sciences, University of Annaba, P.O. Box 12, Annaba, Algeria

Received 16 May 2005; Revised 24 April 2006; Accepted 4 May 2006

Copyright © 2006 Brahim Khodja. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Under the assumption λ>0 and f verifying f(x,y,0)=0 in D, 2F(x,y,u)uf(x,y,u)0, u0, and if Ω=R×D, we show the convexity of function E(t)=D|u(t,x,y)|2dxdy, where u:Ω is a solution of problem λ(2u/t2)(/x)(p(x,y)(u/x))(/y)(q(x,y)(u/y))+f(x,y,u)=0 in Ω, u+ε(u/n)=0 on Ω, considered in H2(Ω)L(Ω), p,q:D¯ are two nonnull functions on D, ε is a positive real number, and D=]a1,b1[×]a2,b2[, (F(x,y,s)=0sf(x,y,t)dt).