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Interior Bubbling Solutions for an Elliptic Equation with Slightly Subcritical Nonlinearity

Author

Listed:
  • Khalil El Mehdi

    (Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraydah 51452, Saudi Arabia
    Faculté des Sciences et Techniques, Université de Nouakchott, Nouakchott 2373, Mauritania
    These authors contributed equally to this work.)

  • Fatimetou Mohamed Salem

    (Faculté des Sciences et Techniques, Université de Nouakchott, Nouakchott 2373, Mauritania
    These authors contributed equally to this work.)

Abstract

In this paper, we considered the Neumann elliptic equation ( P ε ) : − Δ u + K ( x ) u = u ( n + 2 ) / ( n − 2 ) − ε , u > 0 in Ω , ∂ u / ∂ ν = 0 on ∂ Ω , where Ω is a smooth bounded domain in R n , n ≥ 6 , ε is a small positive real and K is a smooth positive function on Ω ¯ . Using refined asymptotic estimates of the gradient of the associated Euler–Lagrange functional, we constructed simple and non-simple interior bubbling solutions of ( P ε ) which allowed us to prove multiplicity results for ( P ε ) provided that ε is small. The existence of non-simple interior bubbling solutions is a new phenomenon for the positive solutions of subcritical problems.

Suggested Citation

  • Khalil El Mehdi & Fatimetou Mohamed Salem, 2023. "Interior Bubbling Solutions for an Elliptic Equation with Slightly Subcritical Nonlinearity," Mathematics, MDPI, vol. 11(6), pages 1-28, March.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:6:p:1471-:d:1100358
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