Naimen, Daisuke; Shibata, Masataka Existence and multiplicity of positive solutions of a critical Kirchhoff type elliptic problem in dimension four. (English) Zbl 1463.35223 Differ. Integral Equ. 33, No. 5-6, 223-246 (2020). The authors consider the nonlocal critical Kirchhoff problem \[\left\{\begin{array}{ll}\displaystyle{-\left(1+\alpha \int_\Omega |\nabla u|^2dx\right)\Delta u=\lambda u^{q}+u^3},\ \ \ \ &\text{in}\ \ \Omega,\\ u>0,\ \ \ \ &\text{in}\ \ \Omega,\\ u=0 &\text{on}\ \ \partial\Omega,\end{array}\right.\] where \(\Omega\subset\mathbb{R}^4\) is a smooth bounded domain, \(\alpha,\lambda\in(0,+\infty)\) and \(q\in[1,3)\), and establish existence, non-existence and multiplicity results on varying of the parameters \(\alpha,\lambda\) and \(q\). One of the main feature in these results is that, for \(q=1\), the above problem, in contrast with the local case \(\alpha=0\), admits at least one solution, and even multiple solutions, for \(\lambda>\lambda_1\) sufficiently near to \(\lambda_1\) (here \(\lambda_1\) is the first eigenvalue of \(-\Delta\) on \(\Omega\)) provided that \(\alpha\) belongs to a suitable subinterval of \((0,+\infty)\).For \(q\in(1,3)\), a previous result of one of the authors is improved by removing a lower bound on \(\alpha\), and the same result is also extended to the case \(\alpha\) small with no condition on \(\lambda\).The proofs are based on variational methods involving the Nehari manifold and the Concentration Compactness Principle. Reviewer: Giovanni Anello (Messina) Cited in 2 Documents MSC: 35J20 Variational methods for second-order elliptic equations 35J25 Boundary value problems for second-order elliptic equations 35J62 Quasilinear elliptic equations Keywords:nonlocal Kirchhoff equation; multiplicity of solutions; positive solution; Nehari manifold; concentration compactness PDFBibTeX XMLCite \textit{D. Naimen} and \textit{M. Shibata}, Differ. Integral Equ. 33, No. 5--6, 223--246 (2020; Zbl 1463.35223)