A fourth order elliptic problem of Kirchhoff type and finding infinitely many weak solution for it

Subject Areas : Statistics

Karimeh Ardeshiri ¹ , Somayeh Khademloo ² , ghasem alizadeh afrouzi ³

1 - Department of Mathematics, Mazandaran University, Mazandaran, Iran
2 - Department of Mathematics, Noshirvani University of Technology, Babol, Iran
3 - Department of Mathematics, Mazandaran University, Mazandaran, Iran

Received: 2019-07-17 Accepted : 2020-09-15 Published : 2021-07-23

Keywords: بی‌نهایت جواب ضعیف, نظریه‌ی نقطه‌ی بحرانی, پتانسیل چند تکینگی معکوس مربعی, کیرشهف مرتبه‌ی چهار, معادله‌ی بیضوی,

Abstract :

This paper is devoted to the existence of infinitely many solutions for a forth order elliptic Kirchhoff problem involving Multi-singular inverse square potentials in a bounded domain using the methods in nonlinear analysis, precisely the variational methods. Nonlinear analysis is a powerful tool for solving many physical models and a technique for proving the solvability of them.Among the methods can be considered in nonlinear analysis, variational methods can prove the existence and multiplicity of solutions without finding the exact value of them.Accordingly, we shall be told, perhaps, that we can find in this field of analysis one of the most applications of analysis in solving the real models of natural problems.The most important feature of the problem proposed in this article, is the existence of singular points in the domain. Using the critical point theory, we prove that there exists an interval in which, the problem has a sequence of distinct weak solutions.In other word, the existence of infinitely many solutions for this problem can be proved.This problem is of the type time independent Poison-Schrodinger equation appearing in an interesting physical context.

References:

[1] D. Cao, P. Han, Solutions to critical elliptic equations with multisingular inverse square potentials, Differential Equations. 224 (2006) 332-372.

 

[2] V. Benci and D. Fortunato, Discreteness conditions of the spectrum of Schrödinger operators, J. Math. Anal. Appl, 64(3) (1978) 695-700.

 

[3] N. Ghoussoub, C. Yuan, Multiple solutions for quasilinear PDEs involving critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc. 352 (2000) 5703-5743.

 

[4] P. N. Srikanth, Uniqueness of solutions of ninlinear Dirichlet problems, Differential Integral Equations .6 (1993) 663-670.

 

[5] M. Willem, Minimax theorems, Progress in Nonlinear Differential

 

[6] Adimurthi, M. Grossi, S. Santra, maximum principle and related eigenvalue problem, J. Funct. Anal. 240 (2006) 36-83.

 

[7] Y. An, R. Liu, Existence of nontrivial solutions of an asymptotically linear fourth-order elliptic equations, Nonlinear Anal. 68 (2008) 3325-3331.

 

[8] V. Benci, D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal. 11 (1998) 283-293.

 

[9] E. Berchio, A. Farina, A. Ferrero, F. Gazzola, Existence and stability of entire solutions to a semi linear fourth order elliptic problem, Differential Equations. 252(3) (2012)2596-2616.

 

[10] F. Catrina, Z.Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence) and symmetry of extremal functions, Comm. Pure Appl. Math. 56 (2001) 229-258.

 

[11] D. Cao, P. Han, Solutions to critical elliptic equations with multisingular inverse square potentials, Differential Equations. 224 (2006) 332-372.

 

[12] K.S. Chou, C.W. Chu, On the best constant for a weighted Sobolev-Hardy inequality, J. Lond. Math. Soc. 48 (1993) 137-151.

 

[13] D. R. Dunninger, Maximum principles for solutions of some fourthorder elliptic equations, Math. Anal. Appl. 37(1972) 655-658.

 

[14] M. Ferrara, S. Khademloo, S. Heidarkhani, Multiplicity results for perturbed fourth-order Kirchhoff type elliptic problems, Appl. Math. Comput. 234 (2014), 316-325.

 

[15] M. Ferrara, G. Molica Bisci, Existence results for elliptic problems with Hardy potential, Bull. Sci. math. 138 (2014) 846-859.

 

[16] F. Gazzola, H.C. Grunau, G. Sweers, Polyharmonic boundary value problems, LNM 1991 Springer (2010).

 

[17] Giovani M. Figueiredo, Rúbia G. Nascimento, Multiplicity of solutions for equations involving a nonlocal term and the biharmonic operator, Electron. J. Diff. Equ. 2016 (2016) 217, 1–15.

 

[18] A. Hadjian, M. Ramezani, Existence of Infinitely many solutions for forth-order equations depending on two parameters, Electron. J. Diff. Equ. 2017 (2017) 117, 1–10.

 

[19] T. S. Hsu, Multiple positive solutions for semi linear elliptic equations involving multi-singular inverse square potentials and concave–convex nonlinearities, Nonlinear Analysis. 74 (2011) 3703-3715.

 

[20] Y. Huang, X. Liu, Sign-changing solutions for p-biharmonic equations with Hardy potential, J. Math. Anal. Appl. 412 (2014), 142-154.

 

[21] G. Kirchhoff, Vorlesungen uber Mathematische Physik Mechanik, Teubner, leipzig, Germany, 1883.

 

[22] Y. Li, F. Li, J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differ. Equ. 253 (2012), 2285-2294.

 

[23] M. Massar, Existence and multiplicity solutions for nonlocal elliptic problems, Electron. J. Diff. Equ. 2013 (2013), No. 75, pp. 1-14.

 

[24] F. Wang, Y. An, Existence and multiplicity of solutions for a forthorder elliptic equation, Boundary Value Problems 2012 (2012) 6, 1-9.

 

[25] L. Wei, X. Cheng, Z. Feng, Exact behavior of positive solutions to elliptic equations with multi-singular inverse square potentials, Discrete and continuous dynamical systems. 36 (2016) 12, 7169-7189.

 

[26] M. XU, C. BAI, Existence of Infinitely many solutions for perturbed Kirchhoff type elliptic problems with Hardy potential, Electronic Journal of Differential Equations. 2015 (2015), 268, 1-9.

 

[27] E. Zeidler, Nonlinear Functional Analysis and Its Applications, voll. II, Springer, Berlin-HeidelberNew York, 1985.