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- W2887352661 startingPage "186" @default.
- W2887352661 abstract "Abstract We consider the perturbed nonlinear boundary condition problem <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mtable> <m:mtr> <m:mtd> <m:mrow> <m:mo>-</m:mo> <m:msub> <m:mi>Δ</m:mi> <m:mi>p</m:mi> </m:msub> <m:mi>u</m:mi> </m:mrow> </m:mtd> <m:mtd> <m:mo>=</m:mo> </m:mtd> <m:mtd> <m:mrow> <m:msup> <m:mrow> <m:mrow> <m:mo>|</m:mo> <m:mi>u</m:mi> <m:mo>|</m:mo> </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:mo>+</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>λ</m:mi> <m:mo>,</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi> </m:mi> <m:mtext>in</m:mtext> <m:mi> </m:mi> <m:mi>Ω</m:mi> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mrow> <m:msup> <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:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo>∇</m:mo> <m:mi>u</m:mi> <m:mo>.</m:mo> <m:mi>ν</m:mi> </m:mrow> </m:mtd> <m:mtd> <m:mo>=</m:mo> </m:mtd> <m:mtd> <m:mrow> <m:mi>λ</m:mi> <m:mi>ρ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mrow> <m:mo>|</m:mo> <m:mi>u</m:mi> <m:mo>|</m:mo> </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:mi> </m:mi> <m:mtext>on</m:mtext> <m:mi> </m:mi> <m:mi>Γ</m:mi> <m:mo>.</m:mo> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mrow> </m:mrow> </m:math> $$left{ {matrix{ { - Delta _p u} hfill & = hfill & {left| u right|^{p - 2} u + fleft( {lambda ,x,u} right),{rm{in}},Omega } hfill cr {left| {nabla u} right|^{p - 2} nabla u.nu } hfill & = hfill & {lambda rho left( x right)left| u right|^{p - 2} u,{rm{on}},Gamma .} hfill cr } } right.$$ Using the Sobolev trace embedding and the duality mapping defined on W 1 , p (Ω), we prove that this problem bifurcates from the principal eigenvalue λ 1 of the eigenvalue problem <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mtable> <m:mtr> <m:mtd> <m:mrow> <m:mo>-</m:mo> <m:msub> <m:mi>Δ</m:mi> <m:mi>p</m:mi> </m:msub> <m:mi>u</m:mi> </m:mrow> </m:mtd> <m:mtd> <m:mo>=</m:mo> </m:mtd> <m:mtd> <m:mrow> <m:msup> <m:mrow> <m:mrow> <m:mo>|</m:mo> <m:mi>u</m:mi> <m:mo>|</m:mo> </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:mi> </m:mi> <m:mtext>in</m:mtext> <m:mi> </m:mi> <m:mi>Ω</m:mi> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mrow> <m:msup> <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:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo>∇</m:mo> <m:mi>u</m:mi> <m:mo>.</m:mo> <m:mi>ν</m:mi> </m:mrow> </m:mtd> <m:mtd> <m:mo>=</m:mo> </m:mtd> <m:mtd> <m:mrow> <m:mi>λ</m:mi> <m:mi>ρ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mrow> <m:mo>|</m:mo> <m:mi>u</m:mi> <m:mo>|</m:mo> </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:mi> </m:mi> <m:mtext>on</m:mtext> <m:mi> </m:mi> <m:mi>Γ</m:mi> <m:mo>.</m:mo> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mrow> </m:mrow> </m:math> $$left{ {matrix{ { - Delta _p u} hfill & = hfill & {left| u right|^{p - 2} u,{rm{in}},Omega } hfill cr {left| {nabla u} right|^{p - 2} nabla u.nu } hfill & = hfill & {lambda rho left( x right)left| u right|^{p - 2} u,{rm{on}},Gamma .} hfill cr } } right.$$" @default.
- W2887352661 created "2018-08-22" @default.
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- W2887352661 date "2017-12-01" @default.
- W2887352661 modified "2023-10-04" @default.
- W2887352661 title "A bifurcation result involving Sobolev trace embedding and the duality mapping of W 1 ,p" @default.
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- W2887352661 doi "https://doi.org/10.1515/mjpaa-2017-0015" @default.
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