### Nonlocal Schrödinger equations for integro-differential operators with measurable kernels

#### 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.

#### Keywords

#### 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 RN , J. Math. Phys. 57 (2016), 081501.

C.O. Alves and M.A.S. Souto, Existence of solutions for a class of elliptc equations in Rn with vanishing potentials, J. Differential Equations 252 (2012), 5555–5568.

V. Ambrosio, Ground state for superlinear fractional Schrödinger equations in RN , Ann. Acad. Sci. Fenn. Math. 41 (2016), 745–756.

V. Ambrosio, A fractional Landesman–Lazer type problem set on RN , 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 RN , 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 RN , 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 (−∆)s u + u = up in RN 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 RN , 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 RN , J. Math. Phys. 54 (2013), 031501.

S. Secchi, On fractional Schrödinger equations in RN 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 RN , 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 RN , 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.

### Refbacks

- There are currently no refbacks.