On Multiplicity of Positive Solutions for Quasilinear Equation with Co-normal Boundary Condition

Let Ω ⊂ I N , N ≥ 3, be a bounded domain with C 2 boundary, p∗ = N −p , the R critical exponent for the Sobolev imbedding. In this work, we are interested in the following problem: ⎧ ∗ ⎨ −Δp u + up−1 = up −1 in Ω, (Pλ ) u > 0 ⎩ p−2 ∂u q |∇u| = λu on ∂Ω, ∂ν where λ > 0, 0 ≤ q < p − 1. We show that there exists 0 < Λ < ∞ such that for suitable ranges of p and q, (Pλ ) admits at least two solutions in W 1,p (Ω) if λ ∈ (0, Λ) and no solution if λ > Λ. The proof of these assertions is done by first finding the local minimum for the variational functional associated to (Pλ ) and then applying mountain pass arguments to obtain a saddle point type solution. In the critical case we are considering, there are technical reasons which make the mountain pass argument work for only certain ranges of p and q.