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Communications in Analysis and Mechanics

2024, Volume 16, Issue 4: 872-895. doi: 10.3934/cam.2024038
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Brezis Nirenberg type results for local non-local problems under mixed boundary conditions

  • Department of Mathematics, Indian Institute of Technology Jodhpur, Rajasthan 342030, India
  • Received: 13 February 2024 Revised: 30 July 2024 Accepted: 08 November 2024 Published: 26 November 2024

  • 35A15, 35J25, 35J20

  • In this paper, we are concerned with an elliptic problem with mixed Dirichlet and Neumann boundary conditions that involve a mixed operator (i.e., the combination of classical Laplace operator and fractional Laplace operator) and critical nonlinearity. Also, we focus on identifying the optimal constant in the mixed Sobolev inequality, which we show is never achieved. Furthermore, by using variational methods, we provide an existence and nonexistence theory for both linear and superlinear perturbation cases.

    Citation: Lovelesh Sharma. Brezis Nirenberg type results for local non-local problems under mixed boundary conditions[J]. Communications in Analysis and Mechanics, 2024, 16(4): 872-895. doi: 10.3934/cam.2024038

    Related Papers:

  • In this paper, we are concerned with an elliptic problem with mixed Dirichlet and Neumann boundary conditions that involve a mixed operator (i.e., the combination of classical Laplace operator and fractional Laplace operator) and critical nonlinearity. Also, we focus on identifying the optimal constant in the mixed Sobolev inequality, which we show is never achieved. Furthermore, by using variational methods, we provide an existence and nonexistence theory for both linear and superlinear perturbation cases.



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