Existence and multiplicity of multi-bump solutions for the double phase Kirchhoff problems with convolution term in R N

Abstract

In this paper, we study a class of the $(p, q)$ Kirchhoff type problems with convolution term in $R^{N}$ . With the appropriate assumptions on potential function V and convolution term f, together with the penalization techniques, Morse iterative method and variational method, the existence and multiplicity of multi-bump solutions are obtained for this problem. In some sense, our results also generalize some known results.

Keywords

Double phase Kirchhoff problems potential well multi-bump solutions convolution term variational methods

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References

C.A.

Alves and

Yang , Investigating the multiplicity and concentration behaviour of solutions for a quasilinear Choquard equation via penalization method, Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), 23–58. doi:10.1017/S0308210515000311.

C.O.

Alves and

G.M.

Figueiredo , Multi-bump solutions for a Kirchhoff-type problem, Adv. Nonlinear Anal. 5 (2016), 1–26. doi:10.1515/anona-2015-0101.

C.O.

Alves ,

A.B.

Nóbrega and

M.B.

Yang , Multi-bump solutions for Choquard equation with deepening potential well, Cal Var Partial Differential Equations 55(3) (2016), 48. doi:10.1007/s00526-016-0984-9.

C.O.

Alves and

Yang , Existence of positive multi-bump solutions for a Schrödinger–Poisson system in

R^{3}

, Discrete Contin. Dyn. Syst. 36 (2013), 5881–5910. doi:10.3934/dcds.2016058.

Ambrosio and

V.D.

Rădulescu , Fractional double-phase patterns: Concentration and multiplicity of solutions, J. Math. Pures Appl. 142 (2020), 101–145. doi:10.1016/j.matpur.2020.08.011.

Ambrosio and

Repovš , Multiplicity and concentration results for a

(p, q)

-Laplacian problem in

R^{N}

, Z. Angew. Math. Phys. 72 (2021), 1–33. doi:10.1007/s00033-020-01428-z.

Bartolo ,

A.M.

Candela and

Salvatore , On a class of superlinear

(p, q)

-Laplacian type equations on

R^{N}

, J. Math. Anal. Appl. 438(1) (2016), 29–41. doi:10.1016/j.jmaa.2016.01.049.

Chen and

Li , Multiple solutions for the nonhomogeneous Kirchhoff equation on

R^{N}

, Nonlinear Anal. Real World Appl. 14 (2013), 1477–1486. doi:10.1016/j.nonrwa.2012.10.010.

del Pino and

P.L.

Felmer , Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations 4 (1996), 121–137. doi:10.1007/BF01189950.

10.

Y.H.

Ding and

Tanaka , Multiplicity of positive solutions of a nonlinear Schrödinger equation, Manuscripta Math. 112 (2003), 109–135.

11.

G.M.

Figueiredo , Existence of positive solutions for a class of

p & q

elliptic problems with critical growth on

R^{N}

, J. Math. Anal. Appl. 378 (2011), 507–518. doi:10.1016/j.jmaa.2011.02.017.

12.

G.M.

Figueiredo , Existence of positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl. 401 (2013), 706–713. doi:10.1016/j.jmaa.2012.12.053.

13.

Guo and

Hu , Multi-bump solutions for nonlinear Choquard equation with ootential wells and a general nonlinearity, Acta Math. Sci. 40 (2020), 316–340. doi:10.1007/s10473-020-0202-x.

14.

He and

Zou , Existence and concentration of positive solutions for a Kirchhoff equation in

R^{3}

, J. Differential Equations 252 (2012), 1813–1834. doi:10.1016/j.jde.2011.08.035.

15.

Iannizzotto ,

Mosconi and

Squassina , Global Hölder regularity for the fractional p-Laplacian, Rev. Mat. Iberoam. 32 (2016), 1353–1392. doi:10.4171/RMI/921.

16.

Ji and

V.D.

Rădulescu , Multi-bump solutions for the nonlinear magnetic Schrödinger equation with exponential critical growth in

R^{2}

, Manuscripta Math. 164 (2021), 509–542. doi:10.1007/s00229-020-01195-1.

17.

Ji and

V.D.

Rădulescu , Multi-bump solutions for the nonlinear magnetic Choquard equation with deepening potential well, J. Differential Equations 306 (2022), 251–279. doi:10.1016/j.jde.2021.10.030.

18.

Kirchhoff , Mechanik, Teubner, Leipzig, 1883.

19.

Li , Some properties of weak solutions of nonlinear scalar field equations, Ann. Acad. Sci. Fenn. Math. 14 (1989), 27–36. doi:10.5186/aasfm.1989.1425.

20.

Li ,

Li and

Shi , Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations 253 (2012), 2285–2294. doi:10.1016/j.jde.2012.05.017.

21.

Liang ,

Chung and

Zhang , Multi-bump solutions for fractional Schrödinger equation with electromagnetic fields and critical nonlinearity, Adv. Differential Equations 25 (2020), 423–456. doi:10.57262/ade/1594692077.

22.

Liang ,

Pucci and

Zhang , Multiple solutions for critical Choquard–Kirchhoff type equations, Adv. Nonlinear Anal. 10 (2021), 400–419. doi:10.1515/anona-2020-0119.

23.

Liang and

Shi , Existence of multi-bump solutions for a class of Kirchhoff type problems in

R^{3}

, J. Math. Phys. 54 (2013), 121510. doi:10.1063/1.4850835.

24.

Liang and

Shi , On multi-bump solutions of nonlinear Schrödinger equation with electromagnetic fields and critical nonlinearity in

R^{N}

, Calc. Var. Partial Differential Equations 56(25) (2017), 1–29.

25.

Liang and

Zhang , Existence of multi-bump solutions for a class of quasilinear Schrödinger equations in

R^{N}

involving critical growth, Milan J. Math. 83 (2015), 55–90. doi:10.1007/s00032-015-0236-z.

26.

Liang and

Zhang , Multi-bump solutions for a class of Kirchhoff type problems with critical growth in

R^{N}

, Topol. Methods Nonlinear Anal. 48 (2016), 71–101.

27.

E.H.

Lieb , Existence and uniqueness of the minimizing solution of Choquard nonlinear equation, Stud. Appl. Math. 57 (1977), 93–105. doi:10.1002/sapm197757293.

28.

E.H.

Lieb and

Loss , Analysis, 2nd edn, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001.

29.

P.L.

Lions , The Choquard equation and related questions, Nonlinear Anal. 4 (1980), 1063–1072. doi:10.1016/0362-546X(80)90016-4.

30.

Ma and

Zhao , Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal. 195 (2010), 455–467. doi:10.1007/s00205-008-0208-3.

31.

Miranda , Un’ osservazione Su un teorema di Brouwer, Bol. Un. Mat. Ital. 3 (1940), 5–7.

32.

Moroz and

Van Schaftingen , Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal. 265 (2013), 153–184. doi:10.1016/j.jfa.2013.04.007.

33.

N.S.

Papageorgiou ,

Pudelko and

V.D.

Rădulescu , Non-autonomous

(p, q)

-equations with unbalanced growth, Math. Ann. (2022). doi:10.1007/s00208-022-02381-0.

34.

N.S.

Papageorgiou ,

Qin and

V.D.

Rădulescu , Nonlinear eigenvalue problems for the

(p, q)

-Laplacian, Bull. Sci. Math. 172 (2021), 103038. doi:10.1016/j.bulsci.2021.103039.

35.

N.S.

Papageorgiou ,

V.D.

Rădulescu and

D.D.

Repovs˘ , Double-phase problems with reaction of arbitrary growth, Z. Angew. Math. Phys. 69(4) (2018), 108. doi:10.1007/s00033-018-1001-2.

36.

Pekar , Untersuchung Uber die Elektronentheorie der Kristalle, Akademie Verlag, 1954.

37.

Wang ,

Tian ,

Xu and

Zhang , Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations 253 (2012), 2314–2351. doi:10.1016/j.jde.2012.05.023.

38.

Willem , Minimax Theorems, Birkhäuser, Boston, MA, 1996.

39.

Xie ,

Wang and

Zhang , Existence of solutions for the

(p, q)

-Laplacian equation with nonlocal Choquard reaction, Appl. Math. Lett. 135 (2023), 108418. doi:10.1016/j.aml.2022.108418.

40.

Zhang ,

Zhang and

V.D.

Rădulescu , Double phase problems with competing potentials: Concentration and multiplication of ground states, Math. Z. 301 (2022), 4037–4078. doi:10.1007/s00209-022-03052-1.

41.

Zhang and

Zhang , Multiplicity and concentration of positive solutions for fractional unbalanced double phase problems, J. Geom. Anal. 32 (2022), 235. doi:10.1007/s12220-022-00983-3.

42.

Zhang ,

Zhang and

V.D.

Rădulescu , Concentrating solutions for singularly perturbed double phase problems with nonlocal reaction, J. Differential Equations 347 (2023), 56–103. doi:10.1016/j.jde.2022.11.033.