Interior Bubbling Solutions with Residual Mass for a Neumann Problem with Slightly Subcritical Exponent

Khulud Alfaleh and Khalil El Mehdi ()
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Khulud Alfaleh: Department of Mathematics, College of Science, Qassim University, Buraydah 51542, Saudi Arabia
Khalil El Mehdi: Department of Mathematics, College of Science, Qassim University, Buraydah 51542, Saudi Arabia

Mathematics, 2025, vol. 13, issue 8, 1-30

Abstract: In this paper, we consider the Neumann elliptic problem ( I ε ) : − Δ u + V ( x ) u = u p − ε , u > 0 in Ω , ∂ u / ∂ ν = 0 on ∂ Ω , where Ω is a bounded smooth domain in R n , n ≥ 3 , p + 1 = 2 n / ( n − 2 ) is the critical Sobolev exponent, ε is a small positive parameter and V is a smooth positive function on Ω ¯ . First, in contrast to the case where solutions converge weakly to zero, we rule out the existence of interior bubbling solutions with a non-zero weak limit in small-dimensional domains. Second, for large-dimensional domains, we construct both simple and non-simple interior bubbling solutions with residual mass. This construction allows us to establish the multiplicity results. The proofs of these results are based on refined asymptotic expansions of the gradient of the associated Euler–Lagrange functional.

Keywords: partial differential equations; blow-up phenomena; critical Sobolev exponent (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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