Revisiting an idea of Brézis and Nirenberg

By Chun-Hsiung Hsia, Changshou Lin, H. Wadade. 2010

58 citations recorded by the source corpus.

Abstract Let n ⩾ 3 and Ω be a C 1 bounded domain in R n with 0 ∈ ∂ Ω . Suppose ∂Ω is C 2 at 0 and the mean curvature of ∂Ω at 0 is negative, we prove the existence of positive solutions for the equation: (0.1) { Δ u + λ u n + 2 n − 2 + u 2 ∗ ( s ) − 1 | x | s = 0 in Ω , u = 0 on ∂ Ω , where λ > 0 , 0 s 2 , 2 ∗ ( s ) = 2 ( n − s ) n − 2 and n ⩾ 4 . For n = 3 , the existence result holds for 0 s 1 . Under the same assumption of the domain Ω, for p ⩽ 2 ∗ ( s ) − 1 , we also prove the existence of a positive solution for the following equation: (0.2) { Δ u − λ u p + u 2 ∗ ( s ) − 1 | x | s = 0 in Ω , u = 0 on ∂ Ω , where λ > 0 and 1 ⩽ p n n − 2 .

Subfields: Applied Mathematics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/11fc84fb246bc162a7c6da3933042ac800cd5951

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In Applied Mathematics

In Computational Theory and Mathematics

In Control and Systems Engineering

Citations in the corpus

A Nonlinear Elliptic PDE with Two Sobolev–Hardy Critical Exponents (2011, methodology): “[17] when s2 = 0, 0 < s1 < 2 and λ > 0.”
On the sharp constant for the weighted Trudinger–Moser type inequality of the scaling invariant form (2014, background): “Moreover, the inequality (1.9) in Corollary 1.2 was obtained as a particular case of the result in Nagayasu and Wadade [18, Corollary 1.3] which does not include the consideration for its sharp constant with respect to α.”