Topological degree methods for a Neumann problem governed by nonlinear elliptic equation
Adil Abbassi, Chakir Allalou, Abderrazak Kassidi
Abstract
Abstract In this paper, we will use the topological degree, introduced by Berkovits, to prove existence of weak solutions to a Neumann boundary value problems for the following nonlinear elliptic equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>-</m:mo> <m:mi>d</m:mi> <m:mi>i</m:mi> <m:mi>v</m:mi> <m:mi> </m:mi> <m:mi> </m:mi> <m:mi>a</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mo>∇</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>b</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mrow> <m:mrow> <m:mo>|</m:mo> <m:mi>u</m:mi> <m:mo>|</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>λ</m:mi> <m:mi>H</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mo>∇</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> - div\,\,a\left( {x,u,\nabla u} \right) = b\left( x \right){\left| u \right|^{p - 2}}u + \lambda H\left( {x,u,\nabla u} \right), where Ω is a bounded smooth domain of N .