This paper considers the following fractional $ (p, q) $-Laplacian equation:
$ (-\Delta)_{p}^{s} u+(-\Delta)_{q}^{s} u+V(x)\left(|u|^{p-2} u+|u|^{q-2} u\right) = \lambda f(u)+|u|^{q^*_s-2}u \quad \text { in } \mathbb{R}^{N}, $
where $ s \in(0, 1), \lambda > 0, 2 < p < q < \frac{N}{s} $, $ (-\Delta)_{t}^{s} $ with $ t \in\{p, q\} $ is the fractional $ t $-Laplacian operator, and potential $ V $ is a continuous function. Using constrained variational methods, a quantitative Deformation Lemma and Brouwer degree theory, we prove that the above problem has a least energy sign-changing solution $ u_{\lambda} $ under suitable conditions on $ f $, $ V $ and $ \lambda $. Moreover, we show that the energy of $ u_{\lambda} $ is strictly larger than two times the ground state energy.
Citation: Kun Cheng, Shenghao Feng, Li Wang, Yuangen Zhan. Least energy sign-changing solutions for a class of fractional $ (p, q) $-Laplacian problems with critical growth in $ \mathbb{R}^N $[J]. AIMS Mathematics, 2023, 8(6): 13325-13350. doi: 10.3934/math.2023675
This paper considers the following fractional $ (p, q) $-Laplacian equation:
$ (-\Delta)_{p}^{s} u+(-\Delta)_{q}^{s} u+V(x)\left(|u|^{p-2} u+|u|^{q-2} u\right) = \lambda f(u)+|u|^{q^*_s-2}u \quad \text { in } \mathbb{R}^{N}, $
where $ s \in(0, 1), \lambda > 0, 2 < p < q < \frac{N}{s} $, $ (-\Delta)_{t}^{s} $ with $ t \in\{p, q\} $ is the fractional $ t $-Laplacian operator, and potential $ V $ is a continuous function. Using constrained variational methods, a quantitative Deformation Lemma and Brouwer degree theory, we prove that the above problem has a least energy sign-changing solution $ u_{\lambda} $ under suitable conditions on $ f $, $ V $ and $ \lambda $. Moreover, we show that the energy of $ u_{\lambda} $ is strictly larger than two times the ground state energy.
| [1] |
C. O. Alves, V. Ambrosio, T. Isernia, Existence, multiplicity and concentration for a class of fractional $p \& q$ Laplacian problems in $\mathbb{R}^{N}$, Commun. Pure Appl. Anal., 18 (2019), 2009–2045. https://doi.org/10.3934/cpaa.2019091 doi: 10.3934/cpaa.2019091
|
| [2] | V. Ambrosio, Multiple solutions for a fractional $p$-Laplacian equation with sign-changing potential, preprint paper, arXiv: 1603.05282, 2016. https://doi.org/10.48550/arXiv.1603.05282 |
| [3] |
V. Ambrosio, T. Isernia, Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional $p$-Laplacian, Discrete Contin. Dyn. Syst., 38 (2018), 5835–5881. https://doi.org/10.3934/dcds.2018254 doi: 10.3934/dcds.2018254
|
| [4] |
V. Ambrosio, V. D. R$\check{a}$dulescu, Fractional double-phase patterns: concentration and multiplicity of solutions, J. Math. Pures Appl., 142 (2020), 101–145. https://doi.org/10.1016/j.matpur.2020.08.011 doi: 10.1016/j.matpur.2020.08.011
|
| [5] |
S. Barile, G. M. Figueiredo, Existence of least energy positive, negative and nodal solutions for a class of $p \& q$-problems with potentials vanishing at infinity, J. Math. Anal. Appl., 427 (2015), 1205–1233. https://doi.org/10.1016/j.jmaa.2015.02.086 doi: 10.1016/j.jmaa.2015.02.086
|
| [6] |
T. Bartsch, Z. Liu, T. Weth, Sign changing solutions of superlinear Schrödinger equations, Commun. Part. Diff. Eq., 29 (2004), 25–42. https://doi.org/10.1081/PDE-120028842 doi: 10.1081/PDE-120028842
|
| [7] |
T. Bartsch, T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. Henri Poincaré, Anal. NonLinéaire, 22 (2005), 259–281. https://doi.org/10.1016/j.anihpc.2004.07.005 doi: 10.1016/j.anihpc.2004.07.005
|
| [8] |
T. Bartsch, T. Weth, M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math., 96 (2005), 1–18. https://doi.org/10.1007/BF02787822 doi: 10.1007/BF02787822
|
| [9] |
G. Bonanno, G. Molica Bisci, V. R$\breve{a}$dulescu, Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz-Sobolev spaces, Nonlinear Anal., 75 (2012), 4441–4456. https://doi.org/10.1016/j.na.2011.12.016 doi: 10.1016/j.na.2011.12.016
|
| [10] |
X. Cabré, Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 31 (2014), 23–53. https://doi.org/10.1016/j.anihpc.2013.02.001 doi: 10.1016/j.anihpc.2013.02.001
|
| [11] |
L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Part. Diff. Eq., 32 (2007), 1245–1260. https://doi.org/10.1080/03605300600987306 doi: 10.1080/03605300600987306
|
| [12] |
A. D. Castro, J. Cossio, J. M. Neuberger, A sign-changing solution for a superlinear Dirichlet problem, Rocky Mt. J. Math., 27 (1997), 1041–1053. https://doi.org/10.1216/rmjm/1181071858 doi: 10.1216/rmjm/1181071858
|
| [13] |
A. D. Castro, T. Kuusi, G. Palatucci, Local behavior of fractional $p$-minimizers, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 33 (2016), 1279–1299. https://doi.org/10.1016/j.anihpc.2015.04.003 doi: 10.1016/j.anihpc.2015.04.003
|
| [14] |
A. D. Castro, T. Kuusi, G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal., 267 (2014), 1807–1836. https://doi.org/10.1016/j.jfa.2014.05.023 doi: 10.1016/j.jfa.2014.05.023
|
| [15] |
X. Chang, Z. Nie, Z. Q. Wang, Sign-changing solutions of fractional $p$-laplacian problems, Adv. Nonlinear Stud., 19 (2019), 29–53. https://doi.org/10.1515/ans-2018-2032 doi: 10.1515/ans-2018-2032
|
| [16] |
X. Chang, Z. Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differ. Equ., 256 (2014), 2965–2992. https://doi.org/10.1016/j.jde.2014.01.027 doi: 10.1016/j.jde.2014.01.027
|
| [17] |
C. Chen, J. Bao, Existence, nonexistence, and multiplicity of solutions for the fractional $p \& q$-Laplacian equation in $ \mathbb{R}^N$, Bound Value Probl., 2016 (2016), 153. https://doi.org/10.1186/s13661-016-0661-0 doi: 10.1186/s13661-016-0661-0
|
| [18] |
W. Chen, S. Deng, Existence, nonexistence, and multiplicity of solutions for the fractional $p \& q$-Laplacian equation in $ \mathbb{R}^N$, Nonlinear Anal. Real World Appl., 27 (2016), 80–92. https://doi.org/10.1016/j.nonrwa.2015.07.009 doi: 10.1016/j.nonrwa.2015.07.009
|
| [19] |
W. Chen, C. Li, Maximum principles for the fractional $p$-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735–758. https://doi.org/10.1016/j.aim.2018.07.016 doi: 10.1016/j.aim.2018.07.016
|
| [20] | C. D. Filippis, G. Palatucci, Hölder regularity for nonlocal double phase equations, J. Differ. Equ., 267 (2020), 547–586. https://doi.org/0.1016/j.jde.2019.01.017 |
| [21] |
P. Felmer, A. Quaas, J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237–1262. https://doi.org/10.1017/S0308210511000746 doi: 10.1017/S0308210511000746
|
| [22] |
G. M. Figueiredo, Existence of positive solutions for a class of $p \& q$ elliptic problems with critical growth on $ \mathbb{R}^N$, J. Math. Anal. Appl., 378 (2011), 507–518. https://doi.org/10.1016/j.jmaa.2011.02.017 doi: 10.1016/j.jmaa.2011.02.017
|
| [23] |
R. F. Gabert, R. S. Rodrigues, Existence of sign-changing solution for a problem involving the fractional Laplacian with critical growth nonlinearities, Complex Var. Elliptic Equ., 65 (2020), 272–292. https://doi.org/10.1080/17476933.2019.1579208 doi: 10.1080/17476933.2019.1579208
|
| [24] | C. He, G. Li, The regularity of weak solutions to nonlinear scalar field elliptic equations containing $p \& q$-Laplacians, Ann. Acad. Sci. Fenn., Math., 33 (2008), 337–371. |
| [25] |
A. Iannizzotto, S. Mosconi, M. Squassina, Global Hölder regularity for the fractional $p$-Laplacian, Rev. Mat. Iberoam., 32 (2016), 1353–1392. https://doi.org/10.4171/RMI/921 doi: 10.4171/RMI/921
|
| [26] |
T. Isernia, Fractional $p \& q$-Laplacian problems with potentials vanishing at infinity, Opusc. Math., 40 (2020), 93–110. https://doi.org/10.7494/OpMath.2020.40.1.93 doi: 10.7494/OpMath.2020.40.1.93
|
| [27] | I. Kuzin, S. Pohozaev, Entire solutions of semilinear Elliptic equations, Basel: Birkhäuser, 1995. |
| [28] |
N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298–305. https://doi.org/10.1016/S0375-9601(00)00201-2 doi: 10.1016/S0375-9601(00)00201-2
|
| [29] |
N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108. https://doi.org/10.1103/PhysRevE.66.056108 doi: 10.1103/PhysRevE.66.056108
|
| [30] |
G. Li, X. Liang, The existence of nontrivial solutions to nonlinear elliptic equation of $p \& q$-Laplacian type on $\mathbb{R}^{N}$, Nonlinear Anal., 71 (2009), 2316–2334. https://doi.org/10.1016/j.na.2009.01.066 doi: 10.1016/j.na.2009.01.066
|
| [31] | C. Miranda, Un'osservazione su un teorema di Brouwer, Boll Un Mat. Ital., 3 (1940), 5–7. |
| [32] |
D. Mugnai, N. S. Papageorgiou, Wang's multiplicity result for superlinear $(p, q)$-equations without the Ambrosetti-Rabinowitz condition, Trans. Amer. Math. Soc., 366 (2014), 4919–4937. https://doi.org/10.1090/S0002-9947-2013-06124-7 doi: 10.1090/S0002-9947-2013-06124-7
|
| [33] |
E. D. Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521–573. https://doi.org/10.1016/j.bulsci.2011.12.004 doi: 10.1016/j.bulsci.2011.12.004
|
| [34] |
G. Palatucci, The Dirichlet problem for the $p$-fractional Laplace equation, Nonlinear Anal., 177 (2018), 699–732. https://doi.org/10.1016/j.na.2018.05.004 doi: 10.1016/j.na.2018.05.004
|
| [35] |
P. Pucci, M. Xiang, B. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional $p$-Laplacian in $\mathbb{R}^{N}$, Calc. Var. Partial Differ. Equ., 54 (2015), 2785–2806. https://doi.org/10.1007/s00526-015-0883-5 doi: 10.1007/s00526-015-0883-5
|
| [36] |
S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbb{R}^{N}$, J. Math. Phys., 54 (2013), 031501. https://doi.org/10.1063/1.4793990 doi: 10.1063/1.4793990
|
| [37] |
Z. Wang, H. Zhou, Radial sign-changing solution for fractional Schrödinger equation, Discrete Contin. Dyn. Syst., 36 (2016), 499–508. https://doi.org/10.3934/dcds.2016.36.499 doi: 10.3934/dcds.2016.36.499
|
| [38] | M. Willem, Progress in nonlinear differential equations and their applications, In: Minimax theorems, Berlin: Springer, 1997. |
| [39] |
M. Wu, Z. Yang, A class of $p \& q$-Laplacian type equation with potentials eigenvalue problem in $\mathbb{R}^{N}$, Bound Value Probl., 2009 (2009), 185319. https://doi.org/10.1155/2009/185319 doi: 10.1155/2009/185319
|
| [40] |
J. Zhang, W. Zhang, V. D. R$\check{a}$dulescu, Double phase problems with competing potentials: concentration and multiplication of ground states, Math. Z., 301 (2022), 4037–4078. https://doi.org/10.1007/s00209-022-03052-1 doi: 10.1007/s00209-022-03052-1
|
| [41] |
W. Zhang, J. Zhang, Multiplicity and concentration of positive solutions for fractional unbalanced double-phase problems, J. Geom. Anal., 32 (2022), 235. https://doi.org/10.1007/s12220-022-00983-3 doi: 10.1007/s12220-022-00983-3
|
| [42] |
W. Zhang, J. Zhang, V. D. R$\check{a}$dulescu, Concentrating solutions for singularly perturbed double phase problems with nonlocal reaction, J. Differ. Equ., 347 (2023), 56–103. https://doi.org/10.1016/j.jde.2022.11.033 doi: 10.1016/j.jde.2022.11.033
|
| [43] |
W. Zhang, S. Yuan, L. Wen, Existence and concentration of ground-states for fractional Choquard equation with indefinite potential, Adv. Nonlinear Anal., 11 (2022), 1552–1578. https://doi.org/10.1515/anona-2022-0255 doi: 10.1515/anona-2022-0255
|
| [44] |
Y. Zhang, X. Tang, V. D. R$\check{a}$dulescu, Concentration of solutions for fractional double-phase problems: critical and supercritical cases, J. Differ. Equ., 302 (2021), 139–184. https://doi.org/10.1016/j.jde.2021.08.038 doi: 10.1016/j.jde.2021.08.038
|
Kun Cheng, Shenghao Feng, Li Wang, Yuangen Zhan. Least energy sign-changing solutions for a class of fractional $ (p, q) $-Laplacian problems with critical growth in $ \mathbb{R}^N $[J]. AIMS Mathematics, 2023, 8(6): 13325-13350. doi: 10.3934/math.2023675
DownLoad: