Nonlinear fourth order problems with asymptotically linear nonlinearities

Amor Ben Ali, Abir ; Dammak, Makkia

Mathematica Bohemica, Tome 149 (2024) no. 2, pp. 209-223

Abstract (VO)

We investigate some nonlinear elliptic problems of the form $$ \Delta ^{2}v + \sigma (x) v= h(x,v)\quad \mbox {in}\ \Omega ,\quad v=\Delta v=0 \quad \mbox {on}\ \partial \Omega , \eqno ({\rm P}) $$ where $\Omega $ is a regular bounded domain in $\mathbb {R}^{N}$, $N\geq 2$, $\sigma (x)$ 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.

Détail

DOI : 10.21136/MB.2023.0008-22

Classification : 35A15, 35J35, 35J60, 35J91
Keywords: asymptotically linear; mountain pass theorem; biharmonic equation; Cerami sequence

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