Nonexistence of interior bubbling solutions for slightly supercritical elliptic problems

Abstract

Abstract In this paper, we consider the Neumann elliptic problem ( P ε ) $(\mathcal{P}_{\varepsilon})$ : − Δ u + μ u = u ( ( n + 2 ) / ( n − 2 ) ) + ε $-\Delta u +\mu u = u^{(({n+2})/({n-2}))+\varepsilon}$ , u > 0 $u>0$ in Ω, ∂ u / ∂ ν = 0 ${\partial u}/{\partial \nu}=0$ on ∂Ω, where Ω is a smooth bounded domain in R n $\mathbb{R}^{n}$ , n ≥ 4 $n\geq 4$ , ε is a small positive real, and μ is a fixed positive number. We show that, in contrast with the three dimensional case, ( P ε ) $(\mathcal{P}_{\varepsilon})$ has no solution blowing up at only interior points as ε goes to zero. The proof strategy consists in testing the equation by appropriate vector fields and then using refined asymptotic estimates in the neighborhood of bubbles, we obtain equilibrium conditions satisfied by the concentration parameters. The careful analysis of these balancing conditions allows us to obtain our results.

Keywords

• Partial differential equations
• Neumann elliptic problems
• Bubbling solutions
• Blow-up
• Critical Sobolev exponent