Existence of positive solutions for Brezis-Nirenberg type problems involving an inverse operator

This article concerns the existence of positive solutions for thesecond order equation involving a nonlocal term $$ -\Delta u=\gamma (-\Delta)^{-1} u+|u|^{p-1}u, under Dirichlet boundary conditions. We prove depending on real parameter \(\gamma>0\), and up to critical value exponent \(p\), i.e. when \(1<p\leq 2^*-1\), where \(2^*=\frac{2N}{N-2}\) is Sobolev exponent. For \(p=2^*-1\), this leads us Brezis-Nirenberg type problem, cf. \cite{BN}, but, in our particular case, linear term. The effect that has changes dimensions which classical technique based minimizers constant, ensures solution, going from \(N\geq 4\) \(N\geq7\) problem. more information see https://ejde.math.txstate.edu/Volumes/2021/52/abstr.html