Mathematica Bohemica, first online, pp. 1-15


Nonlinear fourth order problems with asymptotically linear nonlinearities

Abir Amor Ben Ali, Makkia Dammak

Received January 11, 2022.   Published online April 3, 2023.

Abstract:  We investigate some nonlinear elliptic problems of the form $\Delta^2v + \sigma(x) v= h(x,v)\quadin \Omega,\quad v=\Delta v=0 \quadon \partial\Omega, \eqno({\rm P})$ where $\Omega$ is a regular bounded domain in $\mathbb{R}^N$, $N\geq2$, $\sigma(x)$ a positive function in $L^{\infty}(\Omega)$, and the nonlinearity $h(x,t)$ is indefinite. We prove the existence of solutions to the problem (P) when the function $h(x,t)$ is asymptotically linear at infinity by using variational method but without the Ambrosetti-Rabinowitz condition. Also, we consider the case when the nonlinearities are superlinear and subcritical.
Keywords:  asymptotically linear; mountain pass theorem; biharmonic equation; Cerami sequence
Classification MSC:  35A15, 35J35, 35J60, 35J91

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Affiliations:   Abir Amor Ben Ali (corresponding author), Mathematics Department, Faculty of Mathematical Sciences, Physics and Natural Sciences of Tunis, University of Tunis El Manar, Tunis, Tunisia, e-mail: e-mail: abir.amorbenali@gmail.com; Makkia Dammak, Mathematics Department, Faculty of Sciences of Sfax, University of Sfax, Soukra Rd. km 4, BP 1171-3000 Sfax, Tunisia, e-mail: makkia.dammak@gmail.com


 
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