Bubbles clustered inside for almost-critical problems
Mohamed Ben Ayed, Khalil El Mehdi · Open Mathematics · 2025
Abstract We investigate the existence of blowing-up solutions of the following almost-critical problem: − Δ u + V ( x ) u = u p − ε , u > 0 in Ω , u = 0 on ∂ Ω , -\Delta u+V\left(x)u={u}^{p-\varepsilon },\hspace{1.0em}u\gt 0\hspace{0.25em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,u=0\hspace{0.25em}\hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.25em}\partial \Omega , where Ω \Omega is a bounded regular domain in R n {{\mathbb{R}}}^{n} , n ≥ 4 n\ge 4 , ε \varepsilon is a small positive parameter, p + 1 = ( 2 n ) ∕ ( n − 2 ) p+1=\left(2n)/\left(n-2) is the critical Soblolev exponent, and the potential V V is a smooth positive function. We find solutions that exhibit bubbles clustered inside as ε \varepsilon goes to zero. To the best of our knowledge, this is the first existence result for interior non-simple blowing-up positive solutions to Dirichlet problems in general domains. Our results are proven through delicate asymptotic estimates of the gradient of the associated Euler-Lagrange functional.