Nonlinear fourth order problems with asymptotically linear nonlinearities

Abir Amor Ben Ali; Makkia Dammak

Mathematica Bohemica (2024)

  • Volume: 149, Issue: 2, page 209-223
  • ISSN: 0862-7959

Abstract

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We investigate some nonlinear elliptic problems of the form Δ 2 v + σ ( x ) v = h ( x , v ) in Ω , v = Δ v = 0 on ∂ Ω , ( P ) where Ω is a regular bounded domain in ℝ N , N ≥ 2 , σ ( x ) a positive function in L ∞ ( Ω ) , 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.

How to cite

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Amor Ben Ali, Abir, and Dammak, Makkia. "Nonlinear fourth order problems with asymptotically linear nonlinearities." Mathematica Bohemica 149.2 (2024): 209-223. <http://eudml.org/doc/299420>.

@article{AmorBenAli2024,
abstract = {We investigate some nonlinear elliptic problems of the form \[ \Delta ^\{2\}v + \sigma (x) v= h(x,v)\quad \mbox\{in\}\ \Omega ,\quad v=\Delta v=0 \quad \mbox\{on\}\ \partial \Omega , \qquad \mathrm \{(\{\rm P\})\}\] where $\Omega $ is a regular bounded domain in $\mathbb \{R\}^\{N\}$, $N\ge 2$, $\sigma (x)$ 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.},
author = {Amor Ben Ali, Abir, Dammak, Makkia},
journal = {Mathematica Bohemica},
keywords = {asymptotically linear; mountain pass theorem; biharmonic equation; Cerami sequence},
language = {eng},
number = {2},
pages = {209-223},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Nonlinear fourth order problems with asymptotically linear nonlinearities},
url = {http://eudml.org/doc/299420},
volume = {149},
year = {2024},
}

TY - JOUR
AU - Amor Ben Ali, Abir
AU - Dammak, Makkia
TI - Nonlinear fourth order problems with asymptotically linear nonlinearities
JO - Mathematica Bohemica
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 149
IS - 2
SP - 209
EP - 223
AB - We investigate some nonlinear elliptic problems of the form \[ \Delta ^{2}v + \sigma (x) v= h(x,v)\quad \mbox{in}\ \Omega ,\quad v=\Delta v=0 \quad \mbox{on}\ \partial \Omega , \qquad \mathrm {({\rm P})}\] where $\Omega $ is a regular bounded domain in $\mathbb {R}^{N}$, $N\ge 2$, $\sigma (x)$ 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.
LA - eng
KW - asymptotically linear; mountain pass theorem; biharmonic equation; Cerami sequence
UR - http://eudml.org/doc/299420
ER -

References

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