Nonlinear eigenvalue problems and Fredholm alternative
Pavel Drábek · 2024
Let us start this series of lectures by giving the following simple motivation which arises in such a fundamental subject as the Sobolev imbedding theorems. It is well known that for https://www.w3.org/1998/Math/MathML" display="inline"> Ω ⊂ ℝ N https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332555/ddb532f6-ff62-48a6-bdc1-effe7fe2ef8d/content/ieq0001.tif "/> , a domain, the continuous imbedding 1.1 https://www.w3.org/1998/Math/MathML" display="block"> W 0 1 , p ( Ω ) → L q ( Ω ) https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332555/ddb532f6-ff62-48a6-bdc1-effe7fe2ef8d/content/eqn0001.tif "/> holds provided p ≥ 1, q ≥ 1 and N ≥ 1 satisfy certain relations and that under some additional restrictions this imbedding is compact 1.2 https://www.w3.org/1998/Math/MathML" display="block"> W 0 1 , p ( Ω ) → → L q ( Ω ) https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332555/ddb532f6-ff62-48a6-bdc1-effe7fe2ef8d/content/eqn0002.tif "/> (see e.g. Adams [ A ] or Kufner, John and Fučík [ KJF ]). Denoting by ||·|| q and ||·|| 1, p the norm in L q (Ω) and in https://www.w3.org/1998/Math/MathML" display="inline"> W 0 1 , p ( Ω ) https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332555/ddb532f6-ff62-48a6-bdc1-effe7fe2ef8d/content/ieq0002.tif "/> , respectively, the imbedding (1.1) expressed in terms of norms reads as follows: there exists C > 0 independent of https://www.w3.org/1998/Math/MathML" display="inline"> u ∈ W 0 1 , p ( Ω ) https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332555/ddb532f6-ff62-48a6-bdc1-effe7fe2ef8d/content/ieq0003.tif "/> such that 1.3 https://www.w3.org/1998/Math/MathML" display="block"> ‖ u ‖ q ≤ C ‖ u ‖ 1 , p https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332555/ddb532f6-ff62-48a6-bdc1-effe7fe2ef8d/content/eqn0003.tif "/> holds for any https://www.w3.org/1998/Math/MathML" display="inline"> u ∈ W 0 1 , p ( Ω ) https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332555/ddb532f6-ff62-48a6-bdc1-effe7fe2ef8d/content/ieq0004.tif "/> . Due to the Friedrichs inequality (see [ A ], [ KJF ]) the last assertion can be restated also as 1.4 https://www.w3.org/1998/Math/MathML" display="block"> ‖ u ‖ q ≤ C ‖ ∇ u ‖ p https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332555/ddb532f6-ff62-48a6-bdc1-effe7fe2ef8d/content/eqn0004.tif "/> for any https://www.w3.org/1998/Math/MathML" display="inline"> u ∈ W 0 1 , p ( Ω ) https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332555/ddb532f6-ff62-48a6-bdc1-effe7fe2ef8d/content/ieq0005.tif "/> , where C > 0 does not depend on u . To make the notation clear we note that https://www.w3.org/1998/Math/MathML" display="block"> ‖ ∇ u ‖ p = ∫ Ω | ∇ u ( x ) | p d x 1 / p . https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332555/ddb532f6-ff62-48a6-bdc1-effe7fe2ef8d/content/eqn0005.tif "/> The following natural question arises when studying more carefully (1.4).