October  2019, 39(10): 5825-5846. doi: 10.3934/dcds.2019256

Sign-changing solutions for some nonhomogeneous nonlocal critical elliptic problems

Departamento de Matemática, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile

Received  September 2018 Revised  May 2019 Published  July 2019

We construct multiple sign-changing solutions for the nonhomogeneous nonlocal equation
$ (-\Delta_{\Omega})^{s} u = \left| u\right| ^{\frac{4}{N-2s}}u +\varepsilon f(x)\quad \mbox{in }\Omega, $
under zero Dirichlet boundary conditions in a bounded domain
$ \Omega $
in
$ \mathbb{R}^{N} $
,
$ N>4s $
,
$ s\in (0,1] $
, with
$ f\in L^{\infty}(\Omega) $
,
$ f\geq 0 $
and
$ f\neq0 $
. Here,
$ \varepsilon>0 $
is a small parameter, and
$ (-\Delta_{\Omega})^{s} $
represents a type of nonlocal operator sometimes called the spectral fractional Laplacian. We show that the number of sign-changing solutions goes to infinity as
$ \varepsilon\rightarrow 0 $
when it is assumed that
$ \Omega $
and
$ f $
have certain smoothness and possess certain symmetries, and we are also able to establish accurately the contribution of the nonhomogeneous term in the found solutions. Our proof relies on the Lyapunov-Schmidt reduction method.
Citation: Salomón Alarcón, Jinggang Tan. Sign-changing solutions for some nonhomogeneous nonlocal critical elliptic problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5825-5846. doi: 10.3934/dcds.2019256
References:
[1]

S. Alarcón, Double-spike solutions for a critical inhomogeneous elliptic problem in domains with small holes, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 671-692. doi: 10.1017/S0308210506000928. Google Scholar

[2]

D. Applebaum, Lévy processes–from probability to finance and quantum groups, Not. Am. Math. Soc., 51 (2004), 1336-1347. Google Scholar

[3]

B. BarriosE. ColoradoA. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162. doi: 10.1016/j.jde.2012.02.023. Google Scholar

[4]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405. Google Scholar

[5]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025. Google Scholar

[6]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. Google Scholar

[7]

D. Cao and H. Zhou, On the existence of multiple solutions of nonhomogeneous elliptic equations involving critical Sobolev exponents, Z. Angew. Math. Phys., 47 (1996), 89-96. doi: 10.1007/BF00917575. Google Scholar

[8]

A. Capella, Solutions of a pure critical exponent problem involving the half-Laplacian in annular-shaped domains, Commun. Pure Appl. Anal., 10 (2011), 1645-1662. doi: 10.3934/cpaa.2011.10.1645. Google Scholar

[9]

A. CapellaJ. DávilaL. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384. doi: 10.1080/03605302.2011.562954. Google Scholar

[10]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. Google Scholar

[11]

W. ChoiS. Kim and K. Lee, Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian, J. Funct. Anal., 266 (2014), 6531-6598. doi: 10.1016/j.jfa.2014.02.029. Google Scholar

[12]

M. ClappM. del Pino and M. Musso, Multiple solutions for a non-homogeneous elliptic equation at the critical exponent, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 69-87. doi: 10.1017/S0308210500003085. Google Scholar

[13]

E. ColoradoA. de Pablo and U. Sánchez, Perturbations of a critical fractional equation, Pacific J. Math., 271 (2014), 65-85. doi: 10.2140/pjm.2014.271.65. Google Scholar

[14]

J. DávilaM. del Pino and Y. Sire, Nondegeneracy of the bubble in the critical case for nonlocal equations, Proc. Amer. Math. Soc., 141 (2013), 3865-3870. doi: 10.1090/S0002-9939-2013-12177-5. Google Scholar

[15]

J. DávilaL. López and Y. Sire, Bubbling solutions for nonlocal elliptic problems, Rev. Mat. Iberoam., 33 (2017), 509-546. doi: 10.4171/RMI/947. Google Scholar

[16]

M. del PinoP. Felmer and M. Musso, Two-bubble solutions in the super-critical Bahri-Coron's problem, Calc. Var. Partial Differential Equations, 16 (2003), 113-145. doi: 10.1007/s005260100142. Google Scholar

[17]

G. Grubb, Regularity of spectral fractional Dirichlet and Neumann problems, Math. Nachr., 289 (2016), 831-844. doi: 10.1002/mana.201500041. Google Scholar

[18]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Fractional differential equations: a emergent field in applied and mathematical sciences, Factorization, Singular Operators and Related Problems (eds. S. Samko, A. Lebre and A.F. dos Santos), Proceedings of the Conference in Honour of Professor Georgii Litvinchuk, Springer-Science+Business Media, 2003,151–173. doi: 10.1007/978-94-017-0227-0_11. Google Scholar

[19] N. Krall and A. W. Trivelpiece, Principles of Plasma Physics, Academic Press, New York, London, 1973. Google Scholar
[20]

M. Musso, Sign-changing blowing-up solutions for a non-homogeneous elliptic equation at the critical exponent, J. Fixed Point Theory Appl., 19 (2017), 345-361. doi: 10.1007/s11784-016-0356-2. Google Scholar

[21]

O. Rey, Concentration of solutions to elliptic equations with critical nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 201-218. doi: 10.1016/S0294-1449(16)30245-1. Google Scholar

[22]

X. ShangY. Yang and J. Zhang, Positive solutions of nonhomogeneous fractional laplacian problem with critical exponent, Commun. Pure Appl. Anal., 13 (2014), 567-584. doi: 10.3934/cpaa.2014.13.567. Google Scholar

[23]

J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 42 (2011), 21-41. doi: 10.1007/s00526-010-0378-3. Google Scholar

[24]

J. Tan, Positive solutions for non local elliptic problems, Discrete Contin. Dyn. Syst., 33 (2013), 837-859. doi: 10.3934/dcds.2013.33.837. Google Scholar

[25]

G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 281-304. doi: 10.1016/S0294-1449(16)30238-4. Google Scholar

[26]

A. UpadhyayaJ.-P. RieuJ. A. Glazier and Y. Sawada, Anomalous diffusion and non-Gaussian velocity distribution of Hydra cells in cellular aggregates, Physica A, 293 (2001), 549-558. doi: 10.1016/S0378-4371(01)00009-7. Google Scholar

show all references

References:
[1]

S. Alarcón, Double-spike solutions for a critical inhomogeneous elliptic problem in domains with small holes, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 671-692. doi: 10.1017/S0308210506000928. Google Scholar

[2]

D. Applebaum, Lévy processes–from probability to finance and quantum groups, Not. Am. Math. Soc., 51 (2004), 1336-1347. Google Scholar

[3]

B. BarriosE. ColoradoA. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162. doi: 10.1016/j.jde.2012.02.023. Google Scholar

[4]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405. Google Scholar

[5]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025. Google Scholar

[6]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. Google Scholar

[7]

D. Cao and H. Zhou, On the existence of multiple solutions of nonhomogeneous elliptic equations involving critical Sobolev exponents, Z. Angew. Math. Phys., 47 (1996), 89-96. doi: 10.1007/BF00917575. Google Scholar

[8]

A. Capella, Solutions of a pure critical exponent problem involving the half-Laplacian in annular-shaped domains, Commun. Pure Appl. Anal., 10 (2011), 1645-1662. doi: 10.3934/cpaa.2011.10.1645. Google Scholar

[9]

A. CapellaJ. DávilaL. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384. doi: 10.1080/03605302.2011.562954. Google Scholar

[10]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. Google Scholar

[11]

W. ChoiS. Kim and K. Lee, Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian, J. Funct. Anal., 266 (2014), 6531-6598. doi: 10.1016/j.jfa.2014.02.029. Google Scholar

[12]

M. ClappM. del Pino and M. Musso, Multiple solutions for a non-homogeneous elliptic equation at the critical exponent, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 69-87. doi: 10.1017/S0308210500003085. Google Scholar

[13]

E. ColoradoA. de Pablo and U. Sánchez, Perturbations of a critical fractional equation, Pacific J. Math., 271 (2014), 65-85. doi: 10.2140/pjm.2014.271.65. Google Scholar

[14]

J. DávilaM. del Pino and Y. Sire, Nondegeneracy of the bubble in the critical case for nonlocal equations, Proc. Amer. Math. Soc., 141 (2013), 3865-3870. doi: 10.1090/S0002-9939-2013-12177-5. Google Scholar

[15]

J. DávilaL. López and Y. Sire, Bubbling solutions for nonlocal elliptic problems, Rev. Mat. Iberoam., 33 (2017), 509-546. doi: 10.4171/RMI/947. Google Scholar

[16]

M. del PinoP. Felmer and M. Musso, Two-bubble solutions in the super-critical Bahri-Coron's problem, Calc. Var. Partial Differential Equations, 16 (2003), 113-145. doi: 10.1007/s005260100142. Google Scholar

[17]

G. Grubb, Regularity of spectral fractional Dirichlet and Neumann problems, Math. Nachr., 289 (2016), 831-844. doi: 10.1002/mana.201500041. Google Scholar

[18]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Fractional differential equations: a emergent field in applied and mathematical sciences, Factorization, Singular Operators and Related Problems (eds. S. Samko, A. Lebre and A.F. dos Santos), Proceedings of the Conference in Honour of Professor Georgii Litvinchuk, Springer-Science+Business Media, 2003,151–173. doi: 10.1007/978-94-017-0227-0_11. Google Scholar

[19] N. Krall and A. W. Trivelpiece, Principles of Plasma Physics, Academic Press, New York, London, 1973. Google Scholar
[20]

M. Musso, Sign-changing blowing-up solutions for a non-homogeneous elliptic equation at the critical exponent, J. Fixed Point Theory Appl., 19 (2017), 345-361. doi: 10.1007/s11784-016-0356-2. Google Scholar

[21]

O. Rey, Concentration of solutions to elliptic equations with critical nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 201-218. doi: 10.1016/S0294-1449(16)30245-1. Google Scholar

[22]

X. ShangY. Yang and J. Zhang, Positive solutions of nonhomogeneous fractional laplacian problem with critical exponent, Commun. Pure Appl. Anal., 13 (2014), 567-584. doi: 10.3934/cpaa.2014.13.567. Google Scholar

[23]

J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 42 (2011), 21-41. doi: 10.1007/s00526-010-0378-3. Google Scholar

[24]

J. Tan, Positive solutions for non local elliptic problems, Discrete Contin. Dyn. Syst., 33 (2013), 837-859. doi: 10.3934/dcds.2013.33.837. Google Scholar

[25]

G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 281-304. doi: 10.1016/S0294-1449(16)30238-4. Google Scholar

[26]

A. UpadhyayaJ.-P. RieuJ. A. Glazier and Y. Sawada, Anomalous diffusion and non-Gaussian velocity distribution of Hydra cells in cellular aggregates, Physica A, 293 (2001), 549-558. doi: 10.1016/S0378-4371(01)00009-7. Google Scholar

[1]

Zhengping Wang, Huan-Song Zhou. Radial sign-changing solution for fractional Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 499-508. doi: 10.3934/dcds.2016.36.499

[2]

Mateus Balbino Guimarães, Rodrigo da Silva Rodrigues. Elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2697-2713. doi: 10.3934/cpaa.2013.12.2697

[3]

Yohei Sato, Zhi-Qiang Wang. On the least energy sign-changing solutions for a nonlinear elliptic system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2151-2164. doi: 10.3934/dcds.2015.35.2151

[4]

Yanfang Peng, Jing Yang. Sign-changing solutions to elliptic problems with two critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2015, 14 (2) : 439-455. doi: 10.3934/cpaa.2015.14.439

[5]

Tsung-Fang Wu. On semilinear elliptic equations involving critical Sobolev exponents and sign-changing weight function. Communications on Pure & Applied Analysis, 2008, 7 (2) : 383-405. doi: 10.3934/cpaa.2008.7.383

[6]

Gabriele Cora, Alessandro Iacopetti. Sign-changing bubble-tower solutions to fractional semilinear elliptic problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6149-6173. doi: 10.3934/dcds.2019268

[7]

Yohei Sato. Sign-changing multi-peak solutions for nonlinear Schrödinger equations with critical frequency. Communications on Pure & Applied Analysis, 2008, 7 (4) : 883-903. doi: 10.3934/cpaa.2008.7.883

[8]

Wei Long, Shuangjie Peng, Jing Yang. Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 917-939. doi: 10.3934/dcds.2016.36.917

[9]

Jun Yang, Yaotian Shen. Weighted Sobolev-Hardy spaces and sign-changing solutions of degenerate elliptic equation. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2565-2575. doi: 10.3934/cpaa.2013.12.2565

[10]

A. El Hamidi. Multiple solutions with changing sign energy to a nonlinear elliptic equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 253-265. doi: 10.3934/cpaa.2004.3.253

[11]

Aixia Qian, Shujie Li. Multiple sign-changing solutions of an elliptic eigenvalue problem. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 737-746. doi: 10.3934/dcds.2005.12.737

[12]

M. Ben Ayed, Kamal Ould Bouh. Nonexistence results of sign-changing solutions to a supercritical nonlinear problem. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1057-1075. doi: 10.3934/cpaa.2008.7.1057

[13]

Guirong Liu, Yuanwei Qi. Sign-changing solutions of a quasilinear heat equation with a source term. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1389-1414. doi: 10.3934/dcdsb.2013.18.1389

[14]

Yuxin Ge, Monica Musso, A. Pistoia, Daniel Pollack. A refined result on sign changing solutions for a critical elliptic problem. Communications on Pure & Applied Analysis, 2013, 12 (1) : 125-155. doi: 10.3934/cpaa.2013.12.125

[15]

Yuanxiao Li, Ming Mei, Kaijun Zhang. Existence of multiple nontrivial solutions for a $p$-Kirchhoff type elliptic problem involving sign-changing weight functions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 883-908. doi: 10.3934/dcdsb.2016.21.883

[16]

Wen Zhang, Xianhua Tang, Bitao Cheng, Jian Zhang. Sign-changing solutions for fourth order elliptic equations with Kirchhoff-type. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2161-2177. doi: 10.3934/cpaa.2016032

[17]

Huxiao Luo, Xianhua Tang, Zu Gao. Sign-changing solutions for non-local elliptic equations with asymptotically linear term. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1147-1159. doi: 10.3934/cpaa.2018055

[18]

Bartosz Bieganowski, Jaros law Mederski. Nonlinear SchrÖdinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (1) : 143-161. doi: 10.3934/cpaa.2018009

[19]

Teodora-Liliana Dinu. Entire solutions of the nonlinear eigenvalue logistic problem with sign-changing potential and absorption. Communications on Pure & Applied Analysis, 2003, 2 (3) : 311-321. doi: 10.3934/cpaa.2003.2.311

[20]

Angela Pistoia, Tonia Ricciardi. Sign-changing tower of bubbles for a sinh-Poisson equation with asymmetric exponents. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5651-5692. doi: 10.3934/dcds.2017245

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (83)
  • HTML views (105)
  • Cited by (0)

Other articles
by authors

[Back to Top]