ON KIRCHHOFF-HARDY TYPE PROBLEMS INVOLVING DOUBLE PHASE OPERATORS

doi:10.1007/s10473-025-0504-0

Abstract

Abstract: This paper is devoted to demonstrating several multiplicity results of nontrivial weak solutions to double phase problems of Kirchhoff type with Hardy potentials. The main features of the paper are the appearance of non-local Kirchhoff coefficients and the Hardy potential, the absence of the compactness condition of Palais-Smale, and the $L^{\infty}$-bound for any possible weak solution. To establish multiplicity results, we utilize the fountain theorem and the dual fountain theorem as main tools. Also, we give the $L^{\infty}$-bound for any possible weak solution by exploiting the De Giorgi iteration method and a truncated energy technique. As an application, we give the existence of a sequence of infinitely many weak solutions converging to zero in $L^{\infty}$-norm. To derive this result, we employ the modified functional method and the dual fountain theorem.

Key words: double phase problems, Musielak-Orlicz-Sobolev spaces, variational methods, multiple solutions, De Giorgi iteration method

CLC Number:

35B38

Yun-Ho KIM, Taek-Jun JEONG, Jun-Yeob, SHIM. ON KIRCHHOFF-HARDY TYPE PROBLEMS INVOLVING DOUBLE PHASE OPERATORS[J].Acta mathematica scientia,Series B, 2025, 45(5): 1814-1854.

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References

[1] Ahn J H, Kim I H, Kim Y H, Zeng S. Existence results and $L^{\infty}$-bound of solutions to Kirchhoff-Schrödinger-Hardy type equations involving double phase operators. Results Math, 2024, 79: 1-48
[2] Alves C O, Liu S B. On superlinear $p(x)$-Laplacian equations in $\mathbb{R}^N$. Nonlinear Anal, 2010, 73: 2566-2579
[3] Ambrosetti A, Rabinowitz P. Dual variational methods in critical point theory and applications. J Funct Anal, 1973, 14: 349-381
[4] Bahrouni A, Rădulescu V D, Repovš D D. Double phase transonic flow problems with variable growth: nonlinear patterns and stationary waves. Nonlinearity, 2019, 32: 2481-2495
[5] Baroni P, Colombo M, Mingione G. Regularity for general functionals with double phase. Calc Var Partial Differential Equations, 2018, 57: 1-43
[6] Bonanno G, Marano S. On the structure of the critical set of non-differentiable functions with a weak compactness condition. Appl Anal, 2010, 89: 1-10
[7] Brezis H. Functional Analysis, Sobolev Spaces and Partial Differential Equation. New York: Springer, 2011
[8] Brezis H, Lieb E. A relation between pointwise convergence of functions and convergence of functionals. Proc Amer Math Soc, 1983, 88: 486-490
[9] Cen J, Kim S J, Kim Y H, Zeng S. Multiplicity results of solutions to the double phase anisotropic variational problems involving variable exponent. Adv Differential Equations, 2023, 28: 467-504
[10] Cen J, Vetro C, Zeng S. A multiplicity theorem for double phase degenerate Kirchhoff problems. Appl Math Lett, 2023, 146: 1-6
[11] Choudhuri D. Existence and Hölder regularity of infinitely many solutions to a $p$ Kirchhoff type problem involving a singular nonlinearity without the Ambrosetti-Rabinowitz (AR) condition. Z Angew Math Phys, 2021, 72: 1-36
[12] Colasuonno F, Squassina M.Eigenvalues for double phase variational integrals. Ann Mat Pura Appl, 2016, 195: 1917-1959
[13] Colombo M, Mingione G. Regularity for double phase variational problems. Arch Ration Mech Anal, 2015, 215: 443-496
[14] Crespo-Blanco Á, Gasiński L, Harjulehto P, Winkert P. A new class of double phase variable exponent problems: Existence and uniqueness. J Differential Equations, 2022, 323: 182-228
[15] Díaz J I. Nonlinear Partial Differential Equations and Free Boundaries. Vol 1: Elliptic Equations. Boston-London-Melbourne: Pitman Advanced Publishing Program, 1985
[16] Drábek P, Kufner A, Nicolosi F. Quasilinear Elliptic Equations with Degenerations and Singularities. Berlin: De Gruyter, 1997
[17] Drábek P. Nonlinear eigenvalue problem for $p$-Laplacian in $\mathbb{R}^N$. Math Nachr, 1995, 173: 131-139
[18] Fabian M, Habala P, Hajék P, et al. Banach Space Theory: The Basis for Linear and Nonlinear Analysis. New York: Springer, 2011
[19] Ferrara M, Bisci G M. Existence results for elliptic problems with Hardy potential. Bull Sci Math, 2014, 138: 846-859
[20] Fiscella A. A double phase problem involving Hardy potentials. Appl Math Optim, 2022, 85: 1-16
[21] Fiscella A, Pinamonti A. Existence and multiplicity results for Kirchhoff-type problems on a double-phase setting. Mediterr J Math, 2023, 20: 1-19
[22] Fiscella A, Pucci P. Kirchhoff-Hardy fractional problems with lack of compactness. Adv Nonlinear Stud, 2017, 17: 429-456
[23] Garcia Azozero J P, Peral Alonso I. Hardy inequalities and some critical elliptic and parabolic problems. J Differential Equations, 1998, 144: 441-476
[24] Harjulehto P, Hästö P. Orlicz Spaces and Generalized Orlicz Spaces. Cham: Springer, 2019
[25] Heinz H P. Free Ljusternik-Schnirelman theory and the bifurcation diagrams of certain singular nonlinear problems. J Differential Equations, 1987, 66: 263-300
[26] Ho K, Winkert P. Infinitely many solutions to Kirchhoff double phase problems with variable exponents. Appl Math Lett, 2023, 145: 1-8
[27] Hurtado E J, Miyagaki O H, Rodrigues R S. Existence and multiplicity of solutions for a class of elliptic equations without Ambrosetti-Rabinowitz type conditions. J Dynam Differential Equations, 2018, 30: 405-432
[28] Joe W J, Kim S J, Kim Y H, Oh M W. Multiplicity of solutions for double phase equations with concave-convex nonlinearities. J Appl Anal Comput, 2021, 11: 2921-2946
[29] Khodabakhshi M, Aminpour A M, Afrouzi G A, Hadjian A. Existence of two weak solutions for some singular elliptic problems. Rev R Acad Cienc Exactas Fís Nat Ser A Math RACSAM, 2016, 110: 385-393 % https://doi.org/10.1007/s13398-015-0239-1
[30] Khodabakhshi M, Hadjian A. Existence of three weak solutions for some singular elliptic problems. Complex Var Elliptic Equ, 2018, 63: 68-75
[31] Kim Y H, Ahn J H, Lee J, Zeng S. Multiplicity and a-priori bounds of solutions to Kirchhoff-Schrödinger-Hardy type equations involving the $p$-Laplacian. submitted. Kim Y H, Jeong T J. Multiplicity results of solutions to the double phase problems of Schrödinger-Kirchhoff type with concave-convex nonlinearities. Mathematics, 2024, 12: 1-35
[32] Kirchhoff G R.Vorlesungen über Mathematische Physik. Leipzig: Teubner, 1876
[33] Lee J I, Kim Y H. Multiplicity of radially symmetric small energy solutions for quasilinear elliptic equations involving nonhomogeneous operators. Mathematics, 2020, 8: 1-15
[34] Lin X, Tang X H. Existence of infinitely many solutions for $p$-Laplacian equations in $\Bbb R^{N}$. Nonlinear Anal, 2013, 92: 72-81
[35] Liu W, Dai G. Existence and multiplicity results for double phase problem. J Differential Equations, 2018, 265: 4311-4334
[36] Nachman A, Callegari A. A nonlinear singular boundary value problem in the theory of pseudoplastic fluids. SIAM J Appl Math, 1980, 38: 275-281
[37] Piersanti P, Pucci P. Entire solutions for critical $p$-fractional Hardy Schrödinger Kirchhoff equations. Publ Mat, 2018, 62: 3-36
[38] Pucci P, Xiang M, Zhang B. Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional $p$-Laplacian in $\mathbb{R}^N$. Calc Var Partial Differential Equations, 2015, 54: 2785-2806
[39] Ricceri B. A further three critical points theorem. Nonlinear Anal, 2009, 71: 4151-4157
[40] Ricceri B. Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions. Adv Nonlinear Anal, 2024, 13: 1-7
[41] Simon J.Régularité de la solution dune équation non linéaire dans $\mathbb{R}^N$//Bénilan P, Robert J. Journees d'Analyse Non Lineaire. Berlin: Springer, 1978
[42] Tan Z, Fang F. On superlinear $p(x)$-Laplacian problems without Ambrosetti and Rabinowitz condition. Nonlinear Anal, 2012, 75: 3902-3915
[43] Vergara V, Zacher R. A priori bounds for degenerate and singular evolutionary partial integro-differential equations. Nonlinear Anal, 2010, 73: 3572-3585
[44] Wang Z Q. Nonlinear boundary value problems with concave nonlinearities near the origin. NoDEA Nonlinear Differential Equations Appl, 2001, 8: 15-33
[45] Willem M. Minimax Theorems. Basel: Birkhauser, 1996
[46] Zhao M, Song Y, Repovš, D D. On the $p$-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity. Demonstr Math, 2024, 57: 1-18
[47] Zhikov V V. Averaging of functionals of the calculus of variations and elasticity theory. Mathematics of the USSR-Izvestiya, 1986, 50: 675-710
[48] Zhikov V V. On Lavrentiev's phenomenon. Russ J Math Phys, 1995, 3: 249-269