EXISTENCE AND CONCENTRATION OF SOLUTION FOR A NON-LOCAL REGIONAL SCHRÖDINGER EQUATION WITH COMPETING POTENTIALS

ALVES, CLAUDIANOR O. ; LEDESMA, CÉSAR E. TORRES

Glasgow mathematical journal, Tome 61 (2019) no. 2, pp. 441-460

Résumé

In this paper, we study the existence and concentration phenomena of solutions for the following non-local regional Schrödinger equation$$\begin{equation*}\left\{\begin{array}{l}\epsilon^{2\alpha}(-\Delta)_\rho^{\alpha} u + Q(x)u = K(x)|u|^{p-1}u,\;\;\mbox{in}\;\; \mathbb{R}^n,\\u\in H^{\alpha}(\mathbb{R}^n)\end{array}\right.\end{equation*}$$where ε is a positive parameter, 0 < α < 1, $1<p<\frac{n+2\alpha}{n-2\alpha}$, n > 2α; (−Δ)ρα is a variational version of the regional fractional Laplacian, whose range of scope is a ball with radius ρ(x) > 0, ρ, Q, K are competing functions.

Détail

DOI : 10.1017/S0017089518000289

Bibliographie

[1] 1. Cheng, M., Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phys. 53 (2012), 043507. Google Scholar

[2] 2. Chen, G. and Zheng, Y., Concentration phenomenon for fractional nonlinear Schrödinger equations, Comm. Pure Appl. Anal. 13 (6) (2014), 2359–2376. Google Scholar

[3] 3. Dávila, J., Del Pino, M. and Wei, J., Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differ. Equ. 256 (2014), 858–892. Google Scholar

[4] 4. Dipierro, S., Palatucci, G. and Valdinoci, E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Matematiche 68 (2013), 201–216. Google Scholar

[5] 5. Felmer, P., Quaas, A. and Tan, J., Positive solutions of nonlinear Schrödinger equation with the fractional laplacian, Proc. Edinburgh: Sect. A Math. 142 (6) (2012), 1237–1262. Google Scholar

[6] 6. Felmer, P. and Torres, C., Non-linear Schrödinger equation with non-local regional diffusion, Calc. Var. Partial Diff. Equ. 54 (2015), 75–98. Google Scholar

[7] 7. Felmer, P. and Torres, C., Radial symmetry of ground states for a regional fractional nonlinear Schrödinger equation, Comm. Pure Appl. Anal. 13 (2014), 2395–2406. Google Scholar

[8] 8. Guan, Q.-Y., Integration by parts formula for regional fractional Laplacian, Commun. Math. Phys. 266 (2006), 289–329. Google Scholar

[9] 9. Guan, Q.-Y. and Ma, Z. M., The reflected α-symmetric stable processes and regional fractional Laplacian. Probab. Theory Relat. Fields 134 (2006), 649–694. Google Scholar

[10] 10. Ishii, H. and Nakamura, G., A class of integral equations and approximation of p-Laplace equations, Calc. Var. 37 (2010), 485–522. Google Scholar

[11] 11. Secchi, S., Ground state solutions for nonlinear fractional Schrödinger equations in ℝn, J. Math. Phys. 54 (2013), 031501. Google Scholar

[12] 12. Shang, X. and Zhang, J., Concentrating solutions of nonlinear fractional Schrödinger equation with potentials, J. Differ. Equ. 258 (2015), 1106–1128. Google Scholar

[13] 13. Shang, X. and Zhang, J., Existence and multiplicity solutions of fractional Schrödinger equation with competing potential functions, Complex Variables Elliptic Equ. 61 (2016), 1435–1463. Google Scholar

[14] 14. Torres, C., Symmetric ground state solution for a non-linear Schrödinger equation with non-local regional diffusion, Complex Variables Elliptic Equ., (2016) Google Scholar | DOI

[15] 15. Torres, C., Multiplicity and symmetry results for a nonlinear Schrödinger equation with non-local regional diffusion, Math. Meth. Appl. Sci. 39 (2016), 2808–2820. Google Scholar

[16] 16. Torres, C., Nonlinear Dirichlet problem with non local regional diffusion, Fract. Cal. Appl. Anal. 19 (2) (2016), 379–393. Google Scholar

[17] 17. Willem, M., Minimax theorems (Birkhäuser, Boston, Basel, Berlin, 1996). Google Scholar