Multiple solutions to a transmission problem with a critical Hardy-Sobolev exponential source term. (English) Zbl 1540.35211
Summary: In the paper there are established many results for a transmission problem with critical Hardy-Sobolev exponential source term \(\frac{u^3}{|x|}\) in \(\mathbb{R}^3\). We obtain that there are at least three weakly nontrivial solutions when a positive coefficient of nonhomogeneous term is enough small using the variational method. Next infinitely many classical solutions are obtained when the coefficient equals to zero. Moreover, a new compactness condition is derived with the help of Brezis-Lieb’s lemma and Mazur’s lemma.
MSC:
| 35J62 | Quasilinear elliptic equations |
| 35B33 | Critical exponents in context of PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A15 | Variational methods applied to PDEs |
References:
| [1] | Ambrosetti, A.; Rabinowitz, PH, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 4, 349-381, 1973 · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7 |
| [2] | Badiale, M.; Serra, E., Semilinear Elliptic Equations for Beginners, 2011, London: Existence Results via the Variational Approach. Springer-Verlag, London · Zbl 1214.35025 · doi:10.1007/978-0-85729-227-8 |
| [3] | Bernstein, SN, Sur une classe d’équations fonctionnelles aux dérivées partielles, Izv. Ros. Akad. Nauk Ser. Mat., 4, 1, 1-26, 1940 · JFM 66.0471.01 |
| [4] | Brézis, H.; Lieb, EH, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88, 3, 486-490, 1983 · Zbl 0526.46037 · doi:10.1090/S0002-9939-1983-0699419-3 |
| [5] | Chu, CM; Liu, JQ, Multiple solutions for a nonlocal problem, Appl. Math. Lett., 145, 2023 · Zbl 1520.35070 · doi:10.1016/j.aml.2023.108773 |
| [6] | Chu, CM; Xiao, YX, The multiplicity of nontrivial solutions for a new \(p(x)\)-Kirchhoff-type elliptic problem, J. Funct. Spaces, 2021, 1569376, 2021 · Zbl 1473.35259 |
| [7] | Daoues, A.; Hammami, A.; Saoudi, K., Multiplicity results of nonlocal singular PDEs with critical Sobolev-Hardy exponent, Electron. J. Differential Equ., 2023, 10, 2023 · Zbl 1509.35343 · doi:10.58997/ejde.2023.10 |
| [8] | Daoues, A.; Hammami, A.; Saoudi, K., Existence and multiplicity of solutions for a nonlocal problem with critical Sobolev-Hardy nonlinearities, Mediterr. J. Math., 17, 167, 2020 · Zbl 1448.35190 · doi:10.1007/s00009-020-01601-8 |
| [9] | Diestel, J., Sequences and Series in Banach Spaces, 1984, New York: Springer-Verlag, New York · Zbl 0542.46007 · doi:10.1007/978-1-4612-5200-9 |
| [10] | Ekeland, I., On the variational principle, J. Math. Anal. Appl., 47, 2, 324-353, 1974 · Zbl 0286.49015 · doi:10.1016/0022-247X(74)90025-0 |
| [11] | Evans, L.C.: Partial Differential Equations, 2nd edn. AMS, Providence, Rhode Island (2010) · Zbl 1194.35001 |
| [12] | Ghoussoub, N.; Yuan, CG, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352, 12, 5703-5743, 2000 · Zbl 0956.35056 · doi:10.1090/S0002-9947-00-02560-5 |
| [13] | Huang, HH; Sun, CT, Anomalous wave propagation in a one-dimensional acoustic metamaterial having simultaneously negative mass density and Young’s modulus, J. Acoust. Soc. Amer., 132, 4, 2887-2895, 2012 · doi:10.1121/1.4744977 |
| [14] | Kirchhoff, GR, Vorlesungen über Matematische Physik: Mechanik, 1876, Leipzig: Druck und von B.G. Teubner, Leipzig · JFM 08.0542.01 |
| [15] | Lei, CY; Chu, CM; Suo, HM, Positive solutions for a nonlocal problem with singularity, Electron. J. Differential Equ., 2017, 85, 2017 · Zbl 1370.35086 |
| [16] | Lei, CY; Liao, JF; Suo, HM, Multiple positive solutions for a class of nonlocal problems involving a sign-changing potential, Electron. J. Differential Equ., 2017, 9, 2017 · Zbl 1357.35107 |
| [17] | Lions, JL, Quelques Méthodes de Résolution des Problèmes aux Limites Non-linéaires, 1969, Paris: Dunod, Paris · Zbl 0189.40603 |
| [18] | Liu, ZY; Zhang, DL, A new Kirchhoff-Schrödinger-Poisson type system on the Heisenberg group, Differ. Integral Equ., 34, 11, 621-639, 2021 · Zbl 1513.35179 |
| [19] | Ono, K., Global existence, decay, and blow-up of solutions for some mildly degenerate, J. Differential Equ., 137, 2, 273-301, 1997 · Zbl 0879.35110 · doi:10.1006/jdeq.1997.3263 |
| [20] | Qian, XT, Multiplicity of positive solutions for a class of nonlocal problem involving critical exponent, Electron. J. Qual. Theory Differ. Equ., 2021, 57, 2021 · Zbl 1488.35060 |
| [21] | Renardy M., Rogers, R.C.: An Introduction to Partial Differential Equations. Second Edition, Texts in Applied Mathematics 13, Springer-Verlag, New York (2004) · Zbl 1072.35001 |
| [22] | Routh, EJ, The advanced part of A treatise on the dynamics of a system of rigid bodies: Being part II of a treatise on the whole subject - with numerous examples, 1905, Ltd, London: Macmillan and Co., Ltd, London · JFM 36.0749.03 |
| [23] | Rsheed Mohamed Alotaibi, S.; Saoudi, K., Regularity and multiplicity of solutions for a nonlocal problem with critical Sobolev-Hardy nonlinearities, J. Korean Math. Soc., 57, 3, 747-775, 2020 · Zbl 1437.35280 |
| [24] | Strauss, WA, Nonlinear Wave Equations, 1989, Providence, Rhode Island: AMS, Providence, Rhode Island |
| [25] | Shi, ZG; Qian, XT, Multiple nontrivial solutions for a nonlocal problem with sublinear nonlinearity, Adv. Math. Phys., 2021, 6671882, 2021 · Zbl 1479.35432 · doi:10.1155/2021/6671882 |
| [26] | Talenti, G., Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110, 1, 353-372, 1976 · Zbl 0353.46018 · doi:10.1007/BF02418013 |
| [27] | Taniguchi, T., Existence and asymptotic behaviour of solutions to weakly damped wave equations of Kirchhoff type with nonlinear damping and source terms, J. Math. Anal. Appl., 361, 2, 566-578, 2010 · Zbl 1183.35044 · doi:10.1016/j.jmaa.2009.07.010 |
| [28] | Tao, MF; Zhang, BL, Solutions for nonhomogeneous fractional (p, q)-Laplacian systems with critical nonlinearities, Adv. Nonlinear Anal., 11, 1, 1332-1351, 2022 · Zbl 1489.35118 · doi:10.1515/anona-2022-0248 |
| [29] | Wang, Y., The third solution for a Kirchhoff-type problem with a critical exponent, J. Math. Anal. Appl., 526, 1, 2023 · Zbl 1518.35369 · doi:10.1016/j.jmaa.2023.127174 |
| [30] | Wang, Y.; Suo, HM; Lei, CY, Multiple positive solutions for a nonlocal problem involving critical exponent, Electron. J. Differential Equ., 2017, 275, 2017 · Zbl 1386.35007 |
| [31] | Wang, Y.; Suo, HM; Wei, W., Classical solutions for a kind of new Kirchhoff-type problems without boundary constraint, Acta Math. Sci. Ser. A, 40, 4, 857-868, 2020 · Zbl 1474.35016 |
| [32] | Wang, Y.; Wei, QP; Suo, HM, Three solutions for a new Kirchhoff-type problem, Differ. Equ. Appl., 14, 1, 1-16, 2022 · Zbl 1491.35224 |
| [33] | Wang, Y.; Wei, W.; Xiong, ZH; Yang, J., Positive solution for a nonlocal problem with strong singular nonlinearity, Open Math., 21, 1, 20230103, 2023 · Zbl 1526.35019 · doi:10.1515/math-2023-0103 |
| [34] | Wang, Y.; Yang, X., Infinitely many solutions for a new Kirchhoff type equation with subcritical exponent, Appl. Anal., 101, 3, 1038-1051, 2022 · Zbl 1486.35215 · doi:10.1080/00036811.2020.1767288 |
| [35] | Yagdjian, K., The Lax-Mizohata theorem for Kirchhoff-type equations, J. Differential Equ., 171, 2, 346-369, 2001 · Zbl 0988.35121 · doi:10.1006/jdeq.2000.3842 |
| [36] | Zhu, GL; Duan, CP; Zhang, JJ; Zhang, HX, Ground states of coupled critical Choquard equations with weighted potentials, Opuscula Math., 42, 2, 337-354, 2022 · Zbl 1492.35030 · doi:10.7494/OpMath.2022.42.2.337 |
| [37] | Zhou, S.; Liu, ZS; Zhang, JJ, Groundstates for Choquard type equations with weighted potentials and Hardy-Littlewood-Sobolev lower critical exponent, Adv. Nonlinear Anal., 11, 1, 141-158, 2022 · Zbl 1473.35249 · doi:10.1515/anona-2020-0186 |
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.
