\(H^1 \cap L^p\) versus \(C^1\) local minimizers (Q1724661)

Summary: We show that a local minimizer of \(\Phi\) in the \(C^1\) topology must be a local minimizer in the \(H^1 \cap L^p\) topology, under suitable assumptions for the functional \(\Phi=(1/2)\int_\Omega |\nabla u|^2+(1/p)\int_ \Omega |u|^p-\int_\Omega F(x,u)\) with supercritical exponent \(p>2^*=2n/(n-2)\). This result can be used to establish a solution to the corresponding equation admitting sub- and supersolution. Hence, we extend the conclusion proved by \textit{H. Brézis} and \textit{L. Nirenberg} [C. R. Acad. Sci., Paris, Sér. I 317, No. 5, 465--472 (1993; Zbl 0803.35029)], the subcritical and critical case.