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RETRACTED: Solvability of boundary value problems for the general Schrödinger equation via Schrödinger-type identity methods. (English) Zbl 1441.39009

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 196, Article ID 111828, 15 p. (2020); retraction notice ibid. 202, Article ID 112150, 1 p. (2021).
Summary: In this article we consider a free boundary value problem for the general Schrödinger equations with density-dependent viscosity coefficients by applying a powerful technique based on the Schrödinger-type identity method. Under certain assumptions imposed on the initial data, there exists a unique solution for this equation mentioned above. Finally, we also give an application, which reveals that the Schrödinger-type identity method is effective and simple.
Editorial remark: According to the retraction note, this paper has been retracted at the request of the Editors-in-Chief since it turned out that this article was mistakenly accepted by a previous handling editor based upon the positive advice of at least one inappropriate reviewer report. Furthermore, the original submission contained false affiliations.

MSC:

39A27 Boundary value problems for difference equations
39A12 Discrete version of topics in analysis
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
Full Text: DOI

References:

[1] Anderson, R. M.; May, R. M., Population Biology of Infectious Diseases (1982), Springer-Verlag: Springer-Verlag Berlin
[2] Anderson, R. M.; May, R. M.; Anderson, B., Infectious Diseases of Humans: Dynamics and Control (1992), Oxford University Press: Oxford University Press Oxford
[3] López-Gómez, J.; Marquez, V.; Wolanski, N., Blow up results and localization of blow up points for the heat equations with a nonlinear boundary conditions, J. Differential Equations, 92, 384-401 (1991) · Zbl 0735.35016
[4] Sattinger, D., (Topics in Stability and Bifurcation Theory. Topics in Stability and Bifurcation Theory, Lectures Notes in Mathematics, vol. 309 (1973), Springer: Springer Berlin/New York) · Zbl 0248.35003
[5] Walter, W., On existence and nonexistence in the large of solutions of parabolic differential equations with a nonlinear boundary condition, SIAM J. Math. Anal., 6, 1, 85-90 (1975) · Zbl 0268.35052
[6] Sun, D., Schrödinger-type identity to the existence and uniqueness of a solution to the stationary Schrödinger equation, Bound. Value Probl., 2019, 60 (2019) · Zbl 1524.34224
[7] He, H.; Pang, Z., A modified Schrödinger-type identity: uniqueness of solutions for singular boundary value problem for the Schrödinger equation, Bound. Value Probl., 2019, 147 (2019) · Zbl 1524.35257
[8] Amann, H., Nonlinear elliptic equations with nonlinear boundary conditions, (Eckhaus, W., New Developments in Differential Equations. New Developments in Differential Equations, Math Studies, vol. 21 (1976), North-Holland: North-Holland Amsterdam), 43-63 · Zbl 0345.35045
[9] Andreu, F.; Mazón, J. M.; Toledo, J.; Rossi, J. D., Porous medium equation with absorption and a nonlinear boundary condition, Nonlinear Anal., 49, 541-563 (1992) · Zbl 1011.35109
[10] Chipot, M.; Fila, M.; Quittner, P., Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions, Acta Math. Univ. Comen., 60, 35-103 (1991) · Zbl 0743.35038
[11] Diekmann, O., Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Differential Equations, 33, 58-73 (1979) · Zbl 0377.45007
[12] Kermack, W. O.; M’Kendrick, A. D., A contribution to the mathematical theory of epidemics, Proc. R. Soc. A, 115, 700-721 (1927) · JFM 53.0517.01
[13] Qiao, L.; Ren, Y., Integral representations for the solutions of infinite order of the stationary Schrödinger equation in a cone, Monatsh. Math., 173, 593-603 (2014) · Zbl 1288.31007
[14] Xiang, M.; Zhang, B.; Radulescu, V. D., Superlinear Schrödinger-Kirchhoff type problems involving the fractional \(p\)-Laplacian and critical exponent, Adv. Nonlinear Anal., 9, 1, 690-709 (2020) · Zbl 1427.35340
[15] Pucci, P.; Xiang, M.; Zhang, B., Existence results for Schrödinger-Choquard-Kirchhoff equations involving the fractional \(p\)-Laplacian, Adv. Calc. Var., 12, 3, 253-275 (2019) · Zbl 1431.35233
[16] Xiang, M.; Radulescu, V. D.; Zhang, B., Fractional Kirchhoff problems with critical Trudinger-Moser nonlinearity, Calc. Var. Partial Differential Equations, 58, 2, 27 (2019), Art. 57 · Zbl 1407.35216
[17] Lyaghfouri, A., The evolution dam problem for nonlinear Darcy’s law and Dirichlet boundary conditions, Port. Math., 56, 1, 1-38 (1999) · Zbl 0933.35196
[18] Moser, J., A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math., 13, 3, 457-468 (1960) · Zbl 0111.09301
[19] Assing, S.; Manthey, R., The behavior of solutions of stochastic differential inequalities, Probab. Theory Related Fields, 103, 493-514 (1995) · Zbl 0844.60031
[20] Carrillo, J.; Gilardi, G., La vitesse de propagation dans le problème de la digue, Ann. Fac. Sc. Toulouse, XI, 3, 7-28 (1990) · Zbl 0723.76094
[21] Chen, X. F.; Friedman, A., A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal., 32, 778-800 (1998) · Zbl 0972.35193
[22] Stein, E. M., Singular Integrals and Differentiability Properties of Functions (1970), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0207.13501
[23] Willem, M., Minimax Theorems (1996), Birkhäuser: Birkhäuser Boston · Zbl 0856.49001
[24] Zhikov, V. V., Averaging of functionals of the calculus of variations and elasticity theory, Izv. SSSR Ser. Matem., 50, 4, 675-711 (1986)
[25] Kassay, G.; Radulescu, V. D., (Equilibrium Problems and Applications. Equilibrium Problems and Applications, Mathematics in Science and Engineering (2018), Elsevier Academic Press: Elsevier Academic Press London) · Zbl 1448.47005
[26] Huang, J., Existence of weak solutions for the Schrödinger equation and its application, Ann. Acad. Sci. Fenn. Math., 44, 1101-1120 (2019) · Zbl 07551045
[27] Guliyev, V. S.; Guliyev, R. V.; Omarova, M. N.; Ragusa, M. A., Schrödinger type operators on local generalized Morrey spaces related to certain nonnegative potentials, Discrete Contin. Dyn. Syst. Ser. B, 25, 2, 671-690 (2020) · Zbl 1428.42020
[28] Polidoro, S.; Ragusa, M. A., Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term, Rev. Mat. Iberoam., 24, 3, 1011-1046 (2008) · Zbl 1175.35081
[29] Ragusa, M. A.; Tachikawa, A., Boundary regularity of minimizers of \(p ( x )\)-energy functionals, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33, 2, 451-476 (2016) · Zbl 1333.49052
[30] Papageorgiou, N. S.; Radulescu, V. D.; Repovs, D. D., (Nonlinear Analysis-Theory and Methods. Nonlinear Analysis-Theory and Methods, Springer Monographs in Mathematics (2019), Springer: Springer Cham) · Zbl 1414.46003
[31] Rawashdeh, Z., A free boundary problem for a predator-prey model, Nonlinearity, 20, 1883-1892 (2001) · Zbl 1126.35111
[32] Carrillo, J., On the uniqueness of the solution of the evolution Dam problem, Nonlinear Anal. Theory Methods Appl., 22, 573-607 (1994) · Zbl 0810.76086
[33] Rossi, J. D., The blow-up rate for a semilinear parabolic equation with a nonlinear boundary condition, Acta Math. Univ. Comen., 67, 343-350 (1998) · Zbl 0924.35017
[34] Murray, J. D., (Mathematical Biology. II. Spatial Models and Biomedical Applications. Mathematical Biology. II. Spatial Models and Biomedical Applications, Interdisciplinary Applied Mathematics, vol. 18 (2003), Springer-Verlag: Springer-Verlag New York) · Zbl 1006.92002
[35] Rass, L.; Radcliffe, J., Spatial Deterministic Epidemics (2003), AMS: AMS Providence, RI · Zbl 1018.92028
[36] Rabinowitz, P. H., Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7, 487-513 (1971) · Zbl 0212.16504
[37] Rossi, J. D., Elliptic problems with nonlinear boundary conditions and the Sobolev trace theorem, (Chipot, M.; Quittner, P., Handbook of Differential Equations: Stationary Partial Differential Equations, Vol. 2 (1995), Elsevier), 311-406, (Chapter 5) · Zbl 1207.35160
[38] Serrin, J., Local behavior of solutions of quasilinear elliptic equations, Acta Math., 111, 247-302 (1964) · Zbl 0128.09101
[39] Gilardi, G., A new approach to evolution free boundary problems, Comm. Partial Differential Equations, 4, 1099-1123 (1979) · Zbl 0426.35096
[40] Dibenedetto, E.; Friedman, A., Periodic behaviour for the evolutionary dam problem and related free boundary problems, Comm. Partial Differential Equations, 11, 1297-1377 (1986) · Zbl 0624.35084
[41] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1983), Springer: Springer New York · Zbl 0562.35001
[42] Acerbi, E.; Fusco, N., A transmission problem in the calculus of variations, Calc. Var. Partial Differential Equations, 2, 1-16 (1994) · Zbl 0791.49041
[43] Kokologiannaki, Ch. G.; Krasniqi, V., \(q\)- Completely monotonic and q-Bernstein functions, J. Appl. Math. Stat. Inform., 10, 2, 43-50 (2014) · Zbl 1334.26018
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