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May  2018, 17(3): 1147-1159. doi: 10.3934/cpaa.2018055

Sign-changing solutions for non-local elliptic equations with asymptotically linear term

School of Mathematics and Statistics, Central South University, Changsha, Hunan, 410083, P. R. China

* Corresponding author: XHT

Received  January 2017 Revised  November 2017 Published  January 2018

Fund Project: XHT is supported by NNSF grant No.11571370.

In this article, we study the existence of sign-changing solutions for a problem driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary condition
$\left\{ \begin{array}{ll}-\mathcal{L}_Ku = f(x,u) &\text{in}~Ω, \\u = 0 &\text{in}~\mathbb{R}^n\setminusΩ, \end{array} \right.\ \ \ \ \ \ \ \ \ \left( 1 \right)$
where $Ω\subset\mathbb{R}^n(n≥2)$ is a bounded, smooth domain and $f(x, u)$ is asymptotically linear at infinity with respect to $u$. By introducing some new ideas and combining constraint variational method with the quantitative deformation lemma, we prove that there exists a sign-changing solution of problem (1).
Citation: Huxiao Luo, Xianhua Tang, Zu Gao. Sign-changing solutions for non-local elliptic equations with asymptotically linear term. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1147-1159. doi: 10.3934/cpaa.2018055
References:
[1]

C. O. Alves and M. A. S. Souto, Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains, Z. Angew. Math. Phys., 65 (2014), 1153-1166. 

[2]

V. Ambrosio and T. Isernia, Sign-changing solutions for a class of fractional Schrödinger equations with vanishing potentials, preprint, arXiv: 1609.09003.

[3]

S. Barile and 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. 

[4]

B. BarriosE. ColoradoA. de Pablo and U. Sánchez, On some critical problems for the fractional Laplace operator, J. Diff. Eqns., 252 (2012), 6133-6162. 

[5]

T. BartschZ. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equations, Commmun. Part. Diff. Eq., 29 (2004), 25-42. 

[6]

T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 259-281. 

[7]

T. BartschT. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math., 96 (2005), 1-18. 

[8]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations, Ⅱ, Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375. 

[9]

C. BrändleE. ColoradoA. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. R. Soc. Edinb., 143(A) (2013), 39-71. 

[10]

L. CaffarelliJ. M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179. 

[11]

L. CaffarelliS. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461. 

[12]

A. CapellaJ. DacilaL. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some nonlocal semilinear equations, Commmun. Part. Diff. Eq., 36 (2011), 1353-1384. 

[13]

A. CastroJ. Cossio and J. Neuberger, A sign-changing solution for a superlinear Dirichlet problem, Rocky Mountain J. Math., 27 (1997), 1041-1053. 

[14]

S. Y. A. Chang and M. González, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432. 

[15]

S. T. ChenY. B. Li and X. H. Tang, Sign-changing solutions for asymptotically linear Schrodinger equation in bounded domains, Electron. J. Differ. Eq., 317 (2016), 1-9. 

[16]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. 

[17]

Z. GaoX. H. Tang and W. Zhang, Least energy sign-changing solutions for nonlinear problems involving fractional laplacian, Electron. J. Differ. Eq., 238 (2016), 1-6. 

[18]

Z. L. Liu and J. X. Sun, Invariant Sets of Descending Flow in Critical Point Theory with Applications to Nonlinear Differential Equations, J. Diff. Eqns., 172 (2001), 257-299. 

[19]

X. Y. Lin and X. H. Tang, An asymptotically periodic and asymptotically linear Schrödinger equation with indefinite linear part, Comput. Math. Appl., 70 (2015), 726-736. 

[20]

C. Miranda, Un'osservazione sul teorema di Brouwer, Boll. Unione Mat. Ital., 3 (1940), 5-7. 

[21]

E. S. Noussair and J. Wei, On the effect of the domain geometry on the existence and profile of nodal solution of some singularly perturbed semilinear Dirichlet problem, Indiana Univ. Math. J., 46 (1997), 1255-1271. 

[22]

M. SchechterZ. Q. Wang and W. Zou, New Linking Theorem and Sign-Changing Solutions, Commmun. Part. Diff. Eq., 29 (2005), 471-488. 

[23]

M. Schechter and W. Zou, Sign-changing critical points from linking type theorems, Trans. Amer. Math. Soc., 358 (2006), 5293-5318. 

[24]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. 

[25]

R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. 

[26]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2006), 67-112. 

[27]

X. H. Tang, Non-nehari-manifold method for asymptotically linear Schrödinger equation, J. Aust. Math. Soc., 98 (2015), 104-116. 

[28]

X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials, Disc. Contin. Dyn. Syst.-Series A., 37 (2017), 4973-5002. 

[29]

Z. Wang and H. Zhou, Radial sign-changing solution for fractional Schrödinger equation, Discrete Contin. Dyn. Syst., 36 (2016), 499-508. 

[30] M. Willem, Minimax Theorems, Birkhäuser, Basel, 1996. 
[31]

W. ZhangX. H. Tang and J. Zhang, Infinitely many radial and non-radial solutions for a fractional Schrödinger equation, Comput. Math. Appl., 71 (2016), 737-747. 

show all references

References:
[1]

C. O. Alves and M. A. S. Souto, Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains, Z. Angew. Math. Phys., 65 (2014), 1153-1166. 

[2]

V. Ambrosio and T. Isernia, Sign-changing solutions for a class of fractional Schrödinger equations with vanishing potentials, preprint, arXiv: 1609.09003.

[3]

S. Barile and 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. 

[4]

B. BarriosE. ColoradoA. de Pablo and U. Sánchez, On some critical problems for the fractional Laplace operator, J. Diff. Eqns., 252 (2012), 6133-6162. 

[5]

T. BartschZ. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equations, Commmun. Part. Diff. Eq., 29 (2004), 25-42. 

[6]

T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 259-281. 

[7]

T. BartschT. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math., 96 (2005), 1-18. 

[8]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations, Ⅱ, Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375. 

[9]

C. BrändleE. ColoradoA. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. R. Soc. Edinb., 143(A) (2013), 39-71. 

[10]

L. CaffarelliJ. M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179. 

[11]

L. CaffarelliS. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461. 

[12]

A. CapellaJ. DacilaL. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some nonlocal semilinear equations, Commmun. Part. Diff. Eq., 36 (2011), 1353-1384. 

[13]

A. CastroJ. Cossio and J. Neuberger, A sign-changing solution for a superlinear Dirichlet problem, Rocky Mountain J. Math., 27 (1997), 1041-1053. 

[14]

S. Y. A. Chang and M. González, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432. 

[15]

S. T. ChenY. B. Li and X. H. Tang, Sign-changing solutions for asymptotically linear Schrodinger equation in bounded domains, Electron. J. Differ. Eq., 317 (2016), 1-9. 

[16]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. 

[17]

Z. GaoX. H. Tang and W. Zhang, Least energy sign-changing solutions for nonlinear problems involving fractional laplacian, Electron. J. Differ. Eq., 238 (2016), 1-6. 

[18]

Z. L. Liu and J. X. Sun, Invariant Sets of Descending Flow in Critical Point Theory with Applications to Nonlinear Differential Equations, J. Diff. Eqns., 172 (2001), 257-299. 

[19]

X. Y. Lin and X. H. Tang, An asymptotically periodic and asymptotically linear Schrödinger equation with indefinite linear part, Comput. Math. Appl., 70 (2015), 726-736. 

[20]

C. Miranda, Un'osservazione sul teorema di Brouwer, Boll. Unione Mat. Ital., 3 (1940), 5-7. 

[21]

E. S. Noussair and J. Wei, On the effect of the domain geometry on the existence and profile of nodal solution of some singularly perturbed semilinear Dirichlet problem, Indiana Univ. Math. J., 46 (1997), 1255-1271. 

[22]

M. SchechterZ. Q. Wang and W. Zou, New Linking Theorem and Sign-Changing Solutions, Commmun. Part. Diff. Eq., 29 (2005), 471-488. 

[23]

M. Schechter and W. Zou, Sign-changing critical points from linking type theorems, Trans. Amer. Math. Soc., 358 (2006), 5293-5318. 

[24]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. 

[25]

R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. 

[26]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2006), 67-112. 

[27]

X. H. Tang, Non-nehari-manifold method for asymptotically linear Schrödinger equation, J. Aust. Math. Soc., 98 (2015), 104-116. 

[28]

X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials, Disc. Contin. Dyn. Syst.-Series A., 37 (2017), 4973-5002. 

[29]

Z. Wang and H. Zhou, Radial sign-changing solution for fractional Schrödinger equation, Discrete Contin. Dyn. Syst., 36 (2016), 499-508. 

[30] M. Willem, Minimax Theorems, Birkhäuser, Basel, 1996. 
[31]

W. ZhangX. H. Tang and J. Zhang, Infinitely many radial and non-radial solutions for a fractional Schrödinger equation, Comput. Math. Appl., 71 (2016), 737-747. 

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