Type: Article
Publication Date: 2022-04-13
Citations: 0
DOI: https://doi.org/10.2478/mjpaa-2022-0016
Abstract 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> \left\{ {\matrix{{{\Delta _p}u = {{\left| u \right|}^{p - 2}}u} \hfill & {{\rm{in}}\,\Omega ,} \hfill \cr {{{\left| {\nabla u} \right|}^{p - 2}}{{\partial u} \over {\partial v}} = \lambda \left\| u \right\|_{q,\partial \Omega }^{p - q}{{\left| u \right|}^{q - 2}}u} \hfill & {{\rm{on}}\,\partial \Omega ,} \hfill \cr } } \right. 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.
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