Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
asymptotically linear; mountain pass theorem; biharmonic equation; Cerami sequence
asymptotically linear; mountain pass theorem; biharmonic equation; Cerami sequence
Summary:
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.
References:
[1] Abid, I., Dammak, M., Douchich, I.: Stable solutions and bifurcation problem for asymptotically linear Helmholtz equations. Nonlinear Funct. Anal. Appl. 21 (2016), 15-31. Zbl 1353.35130
[2] Abid, I., Jleli, M., Trabelsi, N.: Weak solutions of quasilinear biharmonic problems with positive, increasing and convex nonlinearities. Anal. Appl., Singap. 6 (2008), 213-227. DOI 10.1142/s0219530508001134 | MR 2429357 | Zbl 1160.35470
[3] Alnaser, L. A., Dammak, M.: Biharmonic problem with indefinite asymptotically linear nonlinearity. Int. J. Math. Comput. Sci. 16 (2021), 1355-1370. MR 4294409 | Zbl 1473.35176
[4] Ambrosetti, A., Rabinowitz, P. H.: Dual variational methods in critical points theory and applications. J. Funct. Anal. 14 (1973), 349-381. DOI 10.1016/0022-1236(73)90051-7 | MR 0370183 | Zbl 0273.49063
[5] Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011). DOI 10.1007/978-0-387-70914-7 | MR 2759829 | Zbl 1220.46002
[6] Cerami, G.: An existence criterion for the critical points on unbounded manifolds. Ist. Lombardo Accad. Sci. Lett., Rend., Sez. A 112 (1978), 332-336 Italian. MR 0581298 | Zbl 0436.58006
[7] Chen, S., Santos, C. A., Yang, M., Zhou, J.: Bifurcation analysis for a modified quasilinear equation with negative exponent. Adv. Nonlinear Anal. 11 (2022), 684-701. DOI 10.1515/anona-2021-0215 | MR 4344369 | Zbl 1486.35034
[8] Costa, D. G., Magalhães, C. A.: Variational elliptic problems which are nonquadratic at infinity. Nonlinear Anal., Theory Methods Appl. 23 (1994), 1401-1412. DOI 10.1016/0362-546x(94)90135-x | MR 1306679 | Zbl 0820.35059
[9] Costa, D. G., Miyagaki, O. H.: Nontrivial solutions for perturbations of the $p$-Laplacian on unbounded domains. J. Math. Anal. Appl. 193 (1995), 737-755. DOI 10.1006/jmaa.1995.1264 | MR 1341038 | Zbl 0856.35040
[10] D'Ambrosio, L., Mitidieri, E.: Entire solutions of certain fourth order elliptic problems and related inequalities. Adv. Nonlinear Anal. 11 (2022), 785-829. DOI 10.1515/anona-2021-0217 | MR 4379602 | Zbl 1485.35016
[11] Dammak, M., Jaidane, R., Jerbi, C.: Positive solutions for an asymptotically linear biharmonic problems. Nonlinear Funct. Anal. Appl. 22 (2017), 59-78. Zbl 1368.35053
[12] El-Abed, A., Ali, A. A. B., Dammak, M.: Schrödinger equation with asymptotically linear nonlinearities. Filomat 36 (2022), 629-639. DOI 10.2298/FIL2202629E | MR 4394295
[13] Jeanjean, L.: On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on $\Bbb{R}^N$. Proc. R. Soc. Edinb., Sect. A, Math. 129 (1999), 787-809. DOI 10.1017/s0308210500013147 | MR 1718530 | Zbl 0935.35044
[14] Lian, W., Rădulescu, V. D., Xu, R., Yang, Y., Zhao, N.: Global well-posedness for a class of fourth-order nonlinear strongly damped wave equations. Adv. Calc. Var. 14 (2021), 589-611. DOI 10.1515/acv-2019-0039 | MR 4319045 | Zbl 1476.35048
[15] Liu, Y., Wang, Z.: Biharmonic equations with asymptotically linear nonlinearities. Acta Math. Sci., Ser. B, Engl. Ed. 27 (2007), 549-560. DOI 10.1016/s0252-9602(07)60055-1 | MR 2339395 | Zbl 1150.35037
[16] Martel, Y.: Uniqueness of weak extremal solutions of nonlinear elliptic problems. Houston J. Math. 23 (1997), 161-168. MR 1688823 | Zbl 0884.35037
[17] Mironescu, P., Rădulescu, V. D.: A bifurcation problem associated to a convex, asymtotically linear function. C. R. Acad. Sci., Paris, Sér. I 316 (1993), 667-672. MR 1214413 | Zbl 0799.35025
[18] Mironescu, P., Rădulescu, V. D.: The study of a bifurcation problem associated to an asymptotically linear function. Nonlinear Anal., Theory Methods Appl. 26 (1996), 857-875. DOI 10.1016/0362-546x(94)00327-e | MR 1362758 | Zbl 0842.35008
[19] Papageorgiou, N. S., Rădulescu, V. D., Repovš, D. D.: Existence and multiplicity of solutions for double-phase Robin problems. Bull. Lond. Math. Soc. 52 (2020), 546-560. DOI 10.1112/blms.12347 | MR 4171387 | Zbl 1447.35131
[20] Rabinowitz, P. H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. Regional Conference Series in Mathematics 65. AMS, Providence (1986). DOI 10.1090/cbms/065 | MR 0845785 | Zbl 0609.58002
[21] Sâanouni, S., Trabelsi, N.: A bifurcation problem associated to an asymptotically linear function. Acta Math. Sci., Ser. B, Engl. Ed. 36 (2016), 1731-7146. DOI 10.1016/s0252-9602(16)30102-3 | MR 3548320 | Zbl 1374.35051
[22] Sâanouni, S., Trabelsi, N.: Bifurcation for elliptic forth-order problems with quasilinear source term. Electronic J. Differ. Equ. 92 (2016), Article ID 92, 16 pages. MR 3489976 | Zbl 1342.35036
[23] Schechter, M.: Superlinear elliptic boundary value problems. Manuscr. Math. 86 (1995), 253-265. DOI 10.1007/bf02567993 | MR 1323791 | Zbl 0839.35048
[24] Stuart, C. A., Zhou, H. S.: A variational problem related to self-trapping of an electromagnetic field. Math. Methods Appl. Sci. 19 (1996), 1397-1407. DOI 10.1002/(sici)1099-1476(19961125)19:17<1397::aid-mma833>3.0.co;2-b | MR 1414401 | Zbl 0862.35123
[25] Stuart, C. A., Zhou, H. S.: Applying the mountain pass theorem to asymptotically linear elliptic equation on $\Bbb{R}^N$. Commun. Partial Differ. Equations 24 (1999), 1731-1758. DOI 10.1080/03605309908821481 | MR 1708107 | Zbl 0935.35043
[26] Zahed, H.: Existence investigation of a fourth order semi-linear weighted problem. Int. J. Math. Comput. Sci. 16 (2021), 687-704. MR 4195463 | Zbl 1455.35106
[27] Zahed, H., Alnaser, L. A.: Elliptic weighted problem with indefinite asymptotically linear nonlinearity. J. Math. Stat. 17 (2021), 13-21. DOI 10.3844/jmssp.2021.13.21
[28] Zhou, H. S.: An application of a mountain pass theorem. Acta Math. Sin., Engl. Ser. 18 (2002), 27-36. DOI 10.1007/s101140100147 | MR 1894835 | Zbl 1018.35020
Search
Partner of
