Existence results for superlinear elliptic equations with nonlinear boundary value conditions

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In this paper, we study the existence of solutions for the following superlinear elliptic equation with nonlinear boundary value condition { arrayll -Δu+u=|u|ʳ⁻²u &in Ω, ∂ u/∂ ν=|u|q⁻²u &on∂Ω, array. where Ω⊂ Rⁿ, N≥ 3 is a bounded domain with smooth boundary. We will prove the existence results for the above equation under four different cases: (i) Both q and r are subcritical; (ii) r is critical and q is subcritical; (iii) r is subcritical and q is critical; (iv) Both q and r are critical.