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

Huxiao Luo; Xianhua Tang; Zu Gao

doi:10.3934/cpaa.2018055

Article Contents

2018, Volume 17, Issue 3: 1147-1159. Doi: 10.3934/cpaa.2018055

This issue Previous Article Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth Next Article Nonlocal heat equations: Regularizing effect, decay estimates and Nash inequalities

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
* Corresponding author: XHT

Received: December 31, 2016

Revised: October 31, 2017

Published: May 2018

XHT is supported by NNSF grant No.11571370

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Abstract

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).

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Mathematics Subject Classification: Primary: 35R11; Secondary: 58E30.

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	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.
	V. Ambrosio and T. Isernia, Sign-changing solutions for a class of fractional Schrödinger equations with vanishing potentials, preprint, arXiv: 1609.09003.
	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.
	B. Barrios , E. Colorado , A. de Pablo and U. Sánchez , On some critical problems for the fractional Laplace operator, J. Diff. Eqns., 252 (2012) , 6133-6162.
	T. Bartsch , Z. Liu and T. Weth , Sign changing solutions of superlinear Schrödinger equations, Commmun. Part. Diff. Eq., 29 (2004) , 25-42.
	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.
	T. Bartsch , T. Weth and M. Willem , Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math., 96 (2005) , 1-18.
	H. Berestycki and P. L. Lions , Nonlinear scalar field equations, Ⅱ, Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983) , 347-375.
	C. Brändle , E. Colorado , A. de Pablo and U. Sánchez , A concave-convex elliptic problem involving the fractional Laplacian, Proc. R. Soc. Edinb., 143(A) (2013) , 39-71.
	L. Caffarelli , J. M. Roquejoffre and Y. Sire , Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010) , 1151-1179.
	L. Caffarelli , S. 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.
	A. Capella , J. Dacila , L. Dupaigne and Y. Sire , Regularity of radial extremal solutions for some nonlocal semilinear equations, Commmun. Part. Diff. Eq., 36 (2011) , 1353-1384.
	A. Castro , J. Cossio and J. Neuberger , A sign-changing solution for a superlinear Dirichlet problem, Rocky Mountain J. Math., 27 (1997) , 1041-1053.
	S. Y. A. Chang and M. González , Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011) , 1410-1432.
	S. T. Chen , Y. 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.
	E. Di Nezza , G. Palatucci and E. Valdinoci , Hitchhiker's guide to the fractional sobolev spaces, Bull. Sci. Math., 136 (2012) , 521-573.
	Z. Gao , X. H. Tang and W. Zhang , Least energy sign-changing solutions for nonlinear problems involving fractional laplacian, Electron. J. Differ. Eq., 238 (2016) , 1-6.
	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.
	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.
	C. Miranda , Un'osservazione sul teorema di Brouwer, Boll. Unione Mat. Ital., 3 (1940) , 5-7.
	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.
	M. Schechter , Z. Q. Wang and W. Zou , New Linking Theorem and Sign-Changing Solutions, Commmun. Part. Diff. Eq., 29 (2005) , 471-488.
	M. Schechter and W. Zou , Sign-changing critical points from linking type theorems, Trans. Amer. Math. Soc., 358 (2006) , 5293-5318.
	R. Servadei and E. Valdinoci , Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013) , 2105-2137.
	R. Servadei and E. Valdinoci , Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012) , 887-898.
	L. Silvestre , Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2006) , 67-112.
	X. H. Tang , Non-nehari-manifold method for asymptotically linear Schrödinger equation, J. Aust. Math. Soc., 98 (2015) , 104-116.
	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.
	Z. Wang and H. Zhou , Radial sign-changing solution for fractional Schrödinger equation, Discrete Contin. Dyn. Syst., 36 (2016) , 499-508.
	M. Willem, Minimax Theorems, Birkhäuser, Basel, 1996.
	W. Zhang , X. H. Tang and J. Zhang , Infinitely many radial and non-radial solutions for a fractional Schrödinger equation, Comput. Math. Appl., 71 (2016) , 737-747.