Ricceri, Biagio On an elliptic Kirchhoff-type problem depending on two parameters. (English) Zbl 1192.49007 J. Glob. Optim. 46, No. 4, 543-549 (2010). Summary: On a bounded domain \(\Omega\subset \mathbb R^n\), we consider a non-local problem of the type \[ \begin{cases} -K\left(\int_\Omega|\nabla u(x)|^2\,dx\right)\Delta u =\lambda f(x,u)+\mu g(x,u) &\text{in }\Omega, \\ u=0 &\text{on }\partial\Omega.\end{cases} \]Under rather general assumptions on \(K\) and \(f\), we prove, in particular, that there exists \(\lambda^*> 0\) such that, for each \(\lambda > \lambda ^*\) and each Carathéodory function \(g\) with a sub-critical growth, the above problem has at least three weak solutions for every \(\mu \geq 0\) small enough. Cited in 92 Documents MSC: 49J20 Existence theories for optimal control problems involving partial differential equations 35J20 Variational methods for second-order elliptic equations Keywords:Kirchhoff-type problem; variational methods; three critical points theorem PDFBibTeX XMLCite \textit{B. Ricceri}, J. Glob. Optim. 46, No. 4, 543--549 (2010; Zbl 1192.49007) Full Text: DOI arXiv References: [1] Alves C.O., Corrêa F.S.J.A., Ma T.F.: Positive solutions for a quasilinear elliptic equations of Kirchhoff type. Comput. Math. Appl. 49, 85–93 (2005) · Zbl 1130.35045 · doi:10.1016/j.camwa.2005.01.008 [2] Chipot M., Lovat B.: Some remarks on non local elliptic and parabolic problems. Nonlinear Anal. 30, 4619–4627 (1997) · Zbl 0894.35119 · doi:10.1016/S0362-546X(97)00169-7 [3] He X., Zou W.: Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Anal. 70, 1407–1414 (2009) · Zbl 1157.35382 · doi:10.1016/j.na.2008.02.021 [4] Ma T.F.: Remarks on an elliptic equation of Kirchhoff type. Nonlinear Anal. 63, e1967–e1977 (2005) · Zbl 1224.35140 · doi:10.1016/j.na.2005.03.021 [5] Mao A., Zhang Z.: Sign-changing and multiple solutions of Kirchhoff type problems without the P. S. condition. Nonlinear Anal. 70, 1275–1287 (2009) · Zbl 1160.35421 · doi:10.1016/j.na.2008.02.011 [6] Perera K., Zhang Z.T.: Nontrivial solutions of Kirchhoff-type problems via the Yang index. J. Differ. Equ. 221, 246–255 (2006) · Zbl 1357.35131 · doi:10.1016/j.jde.2005.03.006 [7] Ricceri, B.: A further three critical points theorem. Nonlinear Anal. (2009) (to appear) · Zbl 1187.47057 [8] Zhang Z.T., Perera K.: Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow. J. Math. Anal. Appl. 317, 456–463 (2006) · Zbl 1100.35008 · doi:10.1016/j.jmaa.2005.06.102 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.