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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 & 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.   Google Scholar

[2]

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

[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.   Google Scholar

[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.   Google Scholar

[5]

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

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[13]

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

[14]

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

[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.   Google Scholar

[16]

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

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[20]

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

[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.   Google Scholar

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[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.   Google Scholar

[29]

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

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

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.   Google Scholar

[2]

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

[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.   Google Scholar

[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.   Google Scholar

[5]

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

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[13]

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

[14]

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

[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.   Google Scholar

[16]

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

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[20]

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

[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.   Google Scholar

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[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.   Google Scholar

[29]

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

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

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