Multiple solutions for semilinear elliptic equations with nonlinear boundary conditions

Abstract

We consider the elliptic problem with nonlinear boundary conditions: $$displaylines{ -Delta u +bu=f(x,u)quadhbox{in }Omega,cr -partial_{u}u=|u|^{q-1}u-g(u)quadhbox{on }partialOmega, }$$ where $Omega$ is a bounded domain in $mathbb{R}^n$. Proving the existence of solutions of this problem relies essentially on a variational argument. However, since $L^{q+1}(partialOmega)subset H^1(Omega)$ does not hold for large q, the standard variational method can not be applied directly. To overcome this difficulty, we use approximation methods and uniform a priori estimates for solutions of approximate equations.

Keywords

• Nonlinear boundary conditions