Multiple solutions for semilinear elliptic equations with nonlinear boundary conditions

We consider the elliptic problem with nonlinear boundary conditions: Δu + bu = f (x, u) in Ω, - ∂ _νu = {Pipe}u{Pipe} ^q-1u - g(u) on dΩ, where Ω is a bounded domain in ℝ ⁿ. Proving the existence of solutions of this problem relies essentially on a variational argument. However, since L ^q+1(∂Ω) ⊂ H ¹(Ω) 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.