Boundary Blowing Up Solutions for an Elliptic Neumann Problem with Nearly Critical Exponent

In this paper, we investigate the nonlinear problem $(P_\varepsilon): -\Delta u + V(x)u = f u^{\frac{n+2}{n-2} - \varepsilon}$, $u > 0$ in $\Omega$ and $\partial u/\partial \nu = 0$ on $\partial \Omega$, where $\Omega$ is a bounded regular domain in $\mathbb{R}^n$, with $n \geq 4$, $\varepsilon$ is a small positive parameter, $V$ and $f$ are smooth positive functions on $\overline{\Omega}$. Under certain conditions involving the function $f$ and the mean curvature of the boundary, we construct boundary blowing up solutions, leading to a multiplicity result for $(P_\varepsilon)$. The proof of these results involves expanding the gradient of the associated functional and testing the equation with suitable vector fields. This process imposes constraints on the concentration parameters, and a careful analysis of these constraints leads to the conclusions presented.