Positive Solutions for some almost Critical Brezis-Nirenberg Type Problems in Bounded and Exterior Domains

We study the equation

$$-\Delta{u}=\vert{x}\vert^{\alpha}u^{p_{\alpha}^{\ast}+\varepsilon}+\lambda_{\varepsilon}\vert{x}\vert^{\beta}{u}\quad\text{in}\;\Omega,$$

under the condition u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝ^N, N ≥ 5, which is symmetric respect to x₁, x₂, 2026;, x_N and contains the origin, α > −2, −2 < β < N − 4, \(p_{\alpha}^{\ast}={N+2\alpha+2\over{N-2}}\), ε > 0 is a small parameter and λ_ε > 0 depends on ε, with λ_ε → 0 as ε → 0. Our main focus lies in finding positive solutions that take the form of a tower of bubbles of order α, exhibiting concentration at the origin as ε tends to zero. Furthermore, we extend our study to the equation

$$-\Delta{u}=\vert{x}\vert^{\alpha}u^{p_{\alpha}^{\ast}-\varepsilon}-\lambda_{\varepsilon}\vert{x}\vert^{\beta}{u}\quad\text{in}\;\mathbb{R}^{N}\;\backslash\;B_{1},$$

where B₁ is the unit ball centered at the origin, under Dirichlet zero boundary condition and an additional vanishing condition at infinity. In this context, we discover positive solutions that take the form of a tower of bubbles of order α, progressively flattening as ε tends to zero.