Sign changing bump solutions for Schrödinger equations involving critical growth and indefinite potential wells
In this paper, the authors consider the following equation of the form \[ -\Delta u+(\lambda a(x)-\delta)u=\left| u \right| ^{2^\ast -2}u\; \text{ in } \mathbb{R}^N, \] where \(N\geq 4\), \(\lambda>0\) is a parameter, \(a(x)\geq 0\) is a given potential, and \(2^\ast\) is the critical Sobolev exponent. Under the assumption that the zero sets \(\mathrm{int} a^{-1}(0)\) admit more than one isolated connected components, the paper studies the question of whether this equation has a family of solutions \(u_\lambda\) which converges, as \(\lambda\to \infty,\) to the least energy solution in some selected isolated zero sets of \(a(x)\) and to \(0\) elsewhere. This paper considers the case that \(\delta>0\) be arbitrarily large such that the operator \((-\Delta+\lambda a(x)-\delta)\) might be indefinite for \(\lambda>0,\) which in turn implies that the least energy solution of the limit problem corresponding to the equation is of linking type solution. To describe their main result, suppose that the zero sets \(\mathrm{int} a^{-1}(0)\) have isolated connected components \(\Omega_1, \dots , \Omega_k, k\geq 2.\) The authors prove that for \(\lambda\) sufficiently large, the equation admits, for any \(i\in \{1,2, \dots ,k\},\) a solution which is trapped in a neighborhood of \(\Omega_i\) provided that the interior of \(\Omega_i\;(i=1, \dots ,k)\) is not empty and \(\partial \Omega_i\;(i=1, \dots, k)\) is smooth. The key ingredient in the proof is a flow argument together with a combination of global linking and local linking, originally this idea is due to \textit{T. Bartsch} and the second author [Discrete Contin. Dyn. Syst. 33, No. 1, 7--26 (2013; Zbl 1284.35390)].
scientific article; zbMATH DE number 6484233