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A note on a semilinear elliptic problem. (English) Zbl 0786.35060
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)].

35J65 Nonlinear boundary value problems for linear elliptic equations
35J40 Boundary value problems for higher-order elliptic equations