A bifurcation result involving Sobolev trace embedding and the duality mapping of W1,p

We consider the perturbed nonlinear boundary condition problem

{-Δpu=|u|p-2u+f(λ,x,u) in Ω|∇u|p-2∇u.ν=λρ(x)|u|p-2u on Γ.$$\left\{ {\matrix{ { - \Delta _p u} \hfill & = \hfill & {\left| u \right|^{p - 2} u + f\left( {\lambda ,x,u} \right)\,{\rm{in}}\,\Omega } \hfill \cr {\left| {\nabla u} \right|^{p - 2} \nabla u.\nu } \hfill & = \hfill & {\lambda \rho \left( x \right)\left| u \right|^{p - 2} u\,{\rm{on}}\,\Gamma .} \hfill \cr } } \right.$$

Using the Sobolev trace embedding and the duality mapping defined on W¹,^p(Ω), we prove that this problem bifurcates from the principal eigenvalue λ₁ of the eigenvalue problem

{-Δpu=|u|p-2u in Ω|∇u|p-2∇u.ν=λρ(x)|u|p-2u on Γ.$$\left\{ {\matrix{ { - \Delta _p u} \hfill & = \hfill & {\left| u \right|^{p - 2} u\,{\rm{in}}\,\Omega } \hfill \cr {\left| {\nabla u} \right|^{p - 2} \nabla u.\nu } \hfill & = \hfill & {\lambda \rho \left( x \right)\left| u \right|^{p - 2} u\,{\rm{on}}\,\Gamma .} \hfill \cr } } \right.$$