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We consider the perturbed nonlinear boundary condition problem { - Δ p u = | u | p - 2 u + f ( λ , x , u )   in   Ω | ∇ u | p - 2 ∇ u . ν = λ ρ ( x ) | u | p - 2 u   on   Γ . $$\left\{ {\matrix{  { - \Delta _p u} \hfill &amp; = \hfill &amp; {\left| u \right|^{p - 2} u + f\left( {\lambda ,x,u} \right)\,{\rm{in}}\,\Omega } \hfill \cr  {\left| {\nabla u} \right|^{p - 2} \nabla u.\nu } \hfill &amp; = \hfill &amp; {\lambda \rho \left( x \right)\left| u \right|^{p - 2} u\,{\rm{on}}\,\Gamma .} \hfill \cr } } \right.$$ Using the Sobolev trace embedding and the duality mapping defined on  W 1 , p (Ω), we prove that this problem bifurcates from the principal eigenvalue  λ 1  of the eigenvalue problem { - Δ p u = | u | p - 2 u   in   Ω | ∇ u | p - 2 ∇ u . ν = λ ρ ( x ) | u | p - 2 u   on   Γ . $$\left\{ {\matrix{  { - \Delta _p u} \hfill &amp; = \hfill &amp; {\left| u \right|^{p - 2} u\,{\rm{in}}\,\Omega } \hfill \cr  {\left| {\nabla u} \right|^{p - 2} \nabla u.\nu } \hfill &amp; = \hfill &amp; {\lambda \rho \left( x \right)\left| u \right|^{p - 2} u\,{\rm{on}}\,\Gamma .} \hfill \cr } } \right.$$

A bifurcation result involving Sobolev trace embedding and the duality mapping of W 1 ,p

A bifurcation result involving Sobolev trace embedding and the duality mapping of W ¹ ^,p