Topological Methods in Nonlinear Analysis

Nonlocal Schrödinger equations for integro-differential operators with measurable kernels

Ronaldo C. Duarte and Marco A. S. Souto

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Abstract

In this paper we investigate the existence of positive solutions for the problem $$ -\mathcal{L}_{K}u+V(x)u=f(u) $$% in $\mathbb R^N$, where $-\mathcal{L}_{K}$ is an integro-differential operator with measurable kernel $K$. Under apropriate hypotheses, we prove by variational methods that this equation has a nonnegative solution.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 54, Number 1 (2019), 383-406.

Dates
First available in Project Euclid: 22 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1563760821

Digital Object Identifier
doi:10.12775/TMNA.2019.056

Citation

Duarte, Ronaldo C.; Souto, Marco A. S. Nonlocal Schrödinger equations for integro-differential operators with measurable kernels. Topol. Methods Nonlinear Anal. 54 (2019), no. 1, 383--406. doi:10.12775/TMNA.2019.056. https://projecteuclid.org/euclid.tmna/1563760821


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References

  • G. Alberti and G. Bellettini, A nonlocal anisotropic model for phase transitions, Math. Ann. 310 (1998), 527–560.
  • C.O. Alves and O.H. Miyagaki, A critical nonlinear fractional elliptic equation with saddle-like potential in $\mathbb{R}^{N}$, J. Math. Phys. 57 (2016), 081501.
  • C.O. Alves and M.A.S. Souto, Existence of solutions for a class of elliptc equations in $\mathbb{R}^{n}$ with vanishing potentials, J. Differential Equations 252 (2012), 5555–5568.
  • V. Ambrosio, Ground state for superlinear fractional Schrödinger equations in $\mathbb{R}^{N}$, Ann. Acad. Sci. Fenn. Math. 41 (2016), 745–756.
  • V. Ambrosio, A fractional Landesman–Lazer type problem set on $\mathbb{R}^{N}$, Matematiche (Catania) 71 (2017), 99–116.
  • A.L. Bertozzi, J.B. Garnett and T. Laurent, Characterization of radially symmetric finite time blowup in multidimensional aggregation equations, SIAM J. Math. Anal. 44 (2012), 651–681.
  • G.M. Bisci and V.D. Radulescu, Ground state solutions of scalar field fractional Schrödinger equations, Calc. Var. Partial Differential Equations 54 (2015), 2985–3008.
  • C. Bucur, Some observations on the Green function for the ball in the fractional Laplace framework, Commun. Pure Appl. Anal. 15 (2016), 657–699.
  • C. Bucur and E. Valdinoci, Nonlocal diffusion and applications, Lecture Notes of the Unione Matematica Italiana, Springer International Publishing (ISBN 978-3-319-28738-6), 20 (2016), pp.,xii+155.
  • X. Cabre and X. Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math. 58 (2005), 1678–1732.
  • L. Caffarelli and L. Silvestre, An extension problem related to the fractional laplacian, Comm. Partial Differential Equations 32 (2007), 1245–1260.
  • X. Chang, Ground states of some fractional Schrödinger equations on $\mathbb{R}^{N}$, Proc. Edinb. Math. Soc. (2) 58 (2015), 305–321.
  • X. Chang, Ground state solutions of asymptotically linear fractional Schrödinger equations, J. Math. Phys. 54 (2013), 061504.
  • C. Chen, Infinitely many solutions for fractional Schrödinger equations in $\mathbb{R}^{N}$, Electron. J. Differential Equations 88 (2016), 1–15.
  • M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phys. 53 (2012), 043507.
  • P. d'Avenia, M. Squassina and M. Zenari, Fractional logarithmic Schrödinger equations, Math. Methods Appl. Sci. 38 (2015), 5207–5216.
  • A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional p-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2015), 1279–1299.
  • E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573.
  • S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Matematiche (Catania) 68 (2013), 201–216.
  • R.C. Duarte and M.A.S. Souto, Fractional Schrödinger–Poisson equations with general nonlinearities, Electron. J. Differential Equations 319 (2016), 1–19.
  • M.M. Fall and E. Valdinoci, Uniqueness and nondegeneracy of positive solutions of $(-\Delta)^{s}u + u = u^{p}$ in $\mathbb{R}^{N}$ when $s$ is close to $1$, Comm. Math. Phys. 329 (2014), 383–404.
  • P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional laplacian, Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), 1237–1262.
  • G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Math. Univ. Parma 5 (2014), 315–328.
  • G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul. 7 (2008), 1005–1028.
  • T. Gou and H. Sun, Solutions of nonlinear Schrödinger equation with fractional laplacian without the Ambrosetti–Rabinowitz condition, Appl. Math. Comput. 257 (2015), 409–416.
  • S. Khoutir and H. Chen, Existence of infinitely many high energy solutions for a fractional Schrödinger equation in $\mathbb{R}^{N}$, Appl. Math. Lett. 61 (2016), 156–162.
  • R. Lehrer, L.A. Maia and M. Squassina, Asymptotically linear fractional Schrodinger equations, Complex Var. Elliptic Equ. 60 (2015), 529–558.
  • E.C. Oliveira, F.S. Costa and J. Jr. Vaz, The fractional Schrödinger equation for delta potentials, J. Math. Phys. 51 (2010), 123517.
  • S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbb{R}^{N}$, J. Math. Phys. 54 (2013), 031501.
  • S. Secchi, On fractional Schrödinger equations in $\mathbb{R}^N$ without the Ambrosetti–Rabinowitz condition, Topol. Methods Nonlinear Anal. 47 (2016), 19–41.
  • R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl. 389 (2012), 887–898.
  • R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst. 33 (2013), 2105–2137.
  • D. Siegel and E. Talvila, Pointwise growth estimates of the Riesz potential, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 5 (1999), 185–194.
  • M. Souza and Y.L. Araújo, On nonlinear perturbations of a periodic fractional Schrödinger equation with critical exponential growth, Math. Nachr. 5 (2016), 610–625.
  • K. Teng, Multiple solutions for a class of fractional Schrödinger equations in $\mathbb{R}^{N}$, Nonlinear Anal. Real World Appl. 21 (2015), 76–86.
  • K. Teng and X. He, Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent, Commun. Pure Appl. Anal. 15 (2016), 991–1008.
  • Y. Wan and Z. Wang, Bound state for fractional Schrödinger equation with saturable nonlinearity, Appl. Math. Comput. 273 (2016), 735–740.
  • Q. Wang, D. Zhao and K. Wang, Existence of solutions to nonlinear fractional Schrödinger equations with singular potentials, Electron. J. Differential Equations 218 (2016), 1–19.
  • M. Willem, Minimax Theorems, Birkhäuser, 1996.
  • J. Xu, Z. Wei and W. Dong, Existence of weak solutions for a fractional Schrödinger equation, Commun. Nonlinear Sci. Numer. Simul. 22 (2015), 1215–1222.
  • L. Yang and Z. Liu, Multiplicity and concentration of solutions for fractional Schrödinger equation with sublinear perturbation and steep potential well, Comput. Math. Appl. 72 (2016), 1629–1640.
  • W. Zhang, X. Tang and J. Zhang, Infinitely many radial and non-radial solutions for a fractional Schrödinger equation, Comput. Math. Appl. 71 (2016), 737–747.
  • H. Zhang, J. Xu and F. Zhang, Existence and multiplicity of solutions for superlinear fractional Schrödinger equations in $\mathbb{R}^{N}$, J. Math. Phys. 56 (2015), 091502.
  • X. Zhang, B. Zhang and D. Repovs, Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials, Nonlinear Anal. 142 (2016), 48–68.