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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)].

##### 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