Steklov problems for the p −Laplace operator involving L q -norm

In this paper, we are concerned with the study of the spectrum for the nonlinear Steklov problem of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mrow> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mtable columnalign="left"> <m:mtr columnalign="left"> <m:mtd columnalign="left"> <m:mrow> <m:msub> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> <m:mi>p</m:mi> </m:msub> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mrow> <m:mrow> <m:mo>|</m:mo> <m:mi>u</m:mi> <m:mo>|</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> </m:mrow> </m:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mtext>in</m:mtext> <m:mi> </m:mi> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> <m:mtr columnalign="left"> <m:mtd columnalign="left"> <m:mrow> <m:msup> <m:mrow> <m:mrow> <m:mrow> <m:mo>|</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>|</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mfrac> <m:mrow> <m:mo>∂</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo>∂</m:mo> <m:mi>v</m:mi> </m:mrow> </m:mfrac> <m:mo>=</m:mo> <m:mi>λ</m:mi> <m:msubsup> <m:mrow> <m:mrow> <m:mrow> <m:mo>‖</m:mo> <m:mi>u</m:mi> <m:mo>‖</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mrow> <m:mi>q</m:mi> <m:mo>,</m:mo> <m:mo>∂</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mi>q</m:mi> </m:mrow> </m:msubsup> <m:msup> <m:mrow> <m:mrow> <m:mrow> <m:mo>|</m:mo> <m:mi>u</m:mi> <m:mo>|</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mrow> <m:mi>q</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> </m:mrow> </m:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mtext>on</m:mtext> <m:mi> </m:mi> <m:mo>∂</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mrow> </m:mrow> </m:math> \{ {{{{Δ _p}u = {{| u |}p - 2}u} & {{in}\,Ω ,} {{{| {∇ u} |}p - 2}{{∂ u} {∂ v}} = λ \| u \|q,∂ Ωp - q{{| u |}q - 2}u} & {{on}\,∂ Ω ,} } } . where Ω is a smooth bounded domain in ℝ N ( N ≥ 1), λ is a real number which plays the role of eigenvalue and the unknowns u ∈ W 1, p (Ω). Using the Ljusterneck-Shnirelmann theory on C 1 manifold and Sobolev trace embedding we prove the existence of an increasing sequence positive of eigenvalues (λ k ) k ≥1 , for the above problem. We then establish that the first eigenvalue is simple and isolated.