Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 33, pp. 1-9.

Title: Multiple solutions for semilinear elliptic equations
       with nonlinear boundary conditions

Authors: Junichi Harada  (Waseda Univ., Tokyo, Japan)
         Mitsuharu Otani (Waseda Univ., Tokyo, Japan)

Abstract:
 We consider the elliptic problem with nonlinear boundary conditions:
 $$\displaylines{
 -\Delta u +bu=f(x,u)\quad\hbox{in }\Omega,\cr
 -\partial_{\nu}u=|u|^{q-1}u-g(u)\quad\hbox{on }\partial\Omega,
 }$$
 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 $L^{q+1}(\partial\Omega)\subset H^1(\Omega)$
 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.

Submitted November 17, 2011. Published February 23, 2012.
Math Subject Classifications: 35J20.
Key Words: Nonlinear boundary conditions