×

Polyharmonic Kirchhoff-type problem without Ambrosetti-Rabinowitz condition. (English) Zbl 1532.35155

Summary: In this paper, we will study the polyharmonic Kirchhoff-type problem with singular exponential nonlinearity without the Ambrosetti-Rabinowitz condition: \[ \begin{cases} -M \left( \int_\Omega |\nabla^m u|^{\frac{n}{m}} \right) \Delta_{\frac{n}{m}}^mu=\frac{f(x,u)}{|x|^\sigma} \quad &\text{in } \Omega, \\ u=\nabla u =\cdots=\nabla^{m-1}u=0 & \text{on } \partial\Omega, \end{cases} \] where \(\Omega \subset \mathbb{R}^n\) is a bounded domain with smooth boundary, \( 0<\sigma <n\), \(n\geq 2m\geq 2\), \(M\) is a Kirchhoff function and \(f(x,u)\) has critical exponential growth. We use a suitable version of the Mountain Pass Theorem to prove the existence of a positive ground state solution for this problem.

MSC:

35J30 Higher-order elliptic equations
35J62 Quasilinear elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A15 Variational methods applied to PDEs
Full Text: DOI

References:

[1] Kirchoff, G.Mechanik. Leipzig: Teubner; 1883.
[2] Figueiredo, GM, Severo, UB.Ground state solution for a Kirchhoff problem with exponential critical growth. Milan J Math. 2016;84:23-39. · Zbl 1348.35064
[3] Autuori, G, Colasuonno, F, Pucci, P.On the existence of stationary solutions for higher-order-Kirchhoff problems. Commun Contemp Math. 2014;16(5):1450002. (43 pages). · Zbl 1325.35129
[4] Goyal, S, Sreenadh, K.The Nehari manifold for a quasilinear polyharmonic equation with exponential nonlinearities and a sign-changing weight function. Adv Nonlinear Anal. 2015;4(3):177-200. · Zbl 1321.35029
[5] Mishra, PK, Goyal, S, Sreenadh, K.Polyharmonic Kirchhoff type equations with singular exponential nonlinearities. Commun Pur Appl Anal. 2016;15(5):1689-1717. · Zbl 1351.35043
[6] Arora, R, Giacomoni, J, Mukherjee, T, et al. Polyharmonic Kirchhoff problems involving exponential non-linearity of Choquard type with singular weights. Nonlinear Analysis. 2020;196:111779. · Zbl 1437.35687
[7] Giacomoni, J, Kumar Mishra, P, Sreenadh, K.Fractional Kirchhoff equation with critical exponential nonlinearity. Complex Var Elliptic Equ. 2016;61(9):1241-1266. · Zbl 1344.35164
[8] Ambrosetti, A, Rabinowitz, PH.Dual variational methods in critical point theory and applications. J Funct Anal. 1973;14:349-381. · Zbl 0273.49063
[9] Jeanjean, L.On the existence of bounded Palais-Smale sequence and applications to a Landesman-Lazer problem set on \(R^N \). Proc Roy Soc Edinburgh Sect A. 1999;129 A:787-809. · Zbl 0935.35044
[10] Li, GB, Zhou, HS.Asymptotically linear Dirichlet problem for the p-Laplacian. Nonlinear Anal. 2001;43:1043-1055. · Zbl 0983.35046
[11] Lam, N, Lu, G.Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti-Rabinowitz condition. J Geom Anal. 2014;24:118-143. · Zbl 1305.35069
[12] Alves, CO, Figueiredo, GM.On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in \(hbox{R}^n \). J Differ Equ. 2009;246(3):1288-1311. · Zbl 1160.35024
[13] Lam, N, Lu, GZ.Existence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth in \(R^N \). J Funct Anal. 2012;262(3):1132-1165. · Zbl 1236.35050
[14] Lam, N, Lu, GZ.Existence of nontrivial solutions to polyharmonic equations with subcritical and critical exponential growth. Discrete Contin Dyn Syst. 2012;32(6):2187-2205. · Zbl 1245.35064
[15] Moser, J.A sharp form of an inequality by N. Trudinger. Indiana Univ Math J. 1970/71;20:1077-1092. · Zbl 0203.43701
[16] Lam, N, Lu, GZ.Sharp singular Adams inequalities in high order Sobolev spaces. Meth Appl Anal. 2011;19(3):243-266. · Zbl 1319.46027
[17] Costa, D, Miyagaki, O.Nontrivial solutions for perturbations of the p-Laplacian on unbounded domains. J Math Anal Appl. 1995;193:737-755. · Zbl 0856.35040
[18] Cerami, G.An existence criterion for the critical points on unbounded manifolds. Istit Lombardo Accad Sci Lett Rend A. 1978;112(2):332-336. (Italian). · Zbl 0436.58006
[19] Cerami, G.On the existence of eigenvalues for a nonlinear boundary value problem. Ann Mat Pura Appl. 1980;124:161-179. (Italian). · Zbl 0441.35054
[20] Chang, KC.Critical point theory and its applications. Shanghai: Shanghai Kexue Jishu Chubanshe; 1986. p. 316. (). · Zbl 0698.58002
[21] de Figueiredo, DG, Miyagaki, OH, Ruf, B.Elliptic equations in \(R^2\) with nonlinearities in theoretical growth range. Calc Var Partial Differ Equ. 1995;3(2):139-153. · Zbl 0820.35060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.