Tarantello, Gabriella A note on a semilinear elliptic problem. (English) Zbl 0786.35060 Differ. Integral Equ. 5, No. 3, 561-565 (1992). The author studies the fourth order problem \[ \Delta^ 2u+c \Delta u=b [(u+1)_ +-1] \text{ in } \Omega,\quad u=\Delta u=0 \text{ on }\partial \Omega, \] for bounded domains \(\Omega \subset \mathbb{R}^ n\) and for \(c<\lambda_ 1\), the first Dirichlet eigenvalue of the Laplace operator. It is shown that \(b \geq \lambda_ 1\) \((\lambda_ 1-c)\) is necessary and sufficient for the existence of nontrivial solutions, and that \(b \geq \lambda_ 1\) \((\lambda_ 1-c)>0\) is sufficient for the existence of negative solutions.(Reviewer’s remark: In fact \(b \geq \lambda_ 1\) \((\lambda_ 1-c)>0\) is also necessary for the existence of negative solutions, as can be seen from the inequality preceding (1.3).) There are some connections to second order problems with similar nonlinearities, see [P. J. Mc Kenna, Arch. Ration. Mech. Anal. 98, 167-190 (1987; Zbl 0676.35003)]. Reviewer: B.Kawohl (Heidelberg) Cited in 58 Documents MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations 35J40 Boundary value problems for higher-order elliptic equations Keywords:biharmonic operator; semilinear elliptic problem Citations:Zbl 0676.35003 PDFBibTeX XMLCite \textit{G. Tarantello}, Differ. Integral Equ. 5, No. 3, 561--565 (1992; Zbl 0786.35060)