Some sufficient conditions for the existence of positive solutions to the equation \(-\Delta u + a(x)u = u^{2^* -1}\) in bounded domains (Q1912387)

The author studies the equation in the title for bounded domains \(\Omega\subset \mathbb{R}^n\) with \(n\geq 3\). As usual \(2^*= 2n/(n- 2)\). The nonnegative coefficient \(a(x)\) is supposed to be of types like \(a(x)= \overline\alpha(x)+ \sum \lambda^2_i \alpha_i[\lambda_i(x- x_i)]\), where \(i\) runs through a finite set, \(\lambda_i\) are real numbers and \(\alpha_i\) are nontrivial functions in \(L^{n/2}(\mathbb{R}^n)\), \(\overline \alpha\in L^{n/2}(\Omega)\). The author shows existence of multiple solutions for large \(\lambda_i\). The proofs are refined variational arguments.