Some existence results for quasilinear elliptic equations in a finite cylinder

Abstract

We study the existence of solutionsu in the finite cylinderS a=(−a, a)×ω of the quasilinear elliptic equation $$\sum\limits_{i,j = 1}^n {a_{i,j} (x)\partial _{i,j} u + f(x,u,\nabla u) = 0} $$ with Dirichlet boundary condition on flat parts of σS a and Neumann condition on the curved parts. In this paper, we focus on the technicality caused by the “corners” ofS a. We prove the existence of such solutions provided that suitable sub and super solutions are known and under the condition that the coefficientsa 1,i,i#1 vanish on the corners. We also prove a more general result in ℝ2.