Semilinear elliptic equations with nonlinear boundary conditions

We consider the following elliptic problem with a nonlinear boundary condition: - Δ u + b u = | u |^{p - 1} u in Ω, - frac(∂ u, ∂ n) = | u |^{q - 1} u - g (u) on ∂ Ω, where 1 < q < p and p < (N + 2) / (N - 2), if N ≥ 3. The existence of solutions to this problem is discussed under suitable conditions on g (u). Our proof relies on the variational argument. However, since L^{q + 1} (∂ Ω) ⊂ H¹ (Ω) does not hold for large q, we cannot apply the variational method in a direct way. To overcome this difficulty, some approximation problems are introduced and uniform a priori estimates for solutions of approximate equations are established.