In this paper, we consider the following semilinear Schrödinger equation:
$ \begin{eqnarray*} \left\{ \begin{array}{ll} -\Delta u+V(x)u = a(x)g(u)&{\mbox{for}}\; x\in \mathbb{R}^{N} ,\\ u(x)\rightarrow0&{\mbox{as}}\; |x|\rightarrow \infty , \end{array} \right. \end{eqnarray*} $
where $ a(x) > 0 $ for all $ \mathbb{R}^{N} $. Under some different superlinear conditions on $ g(u) $, we obtain the existence of solutions for the above problem. In order to regain the compactness of the Sobolev embedding, a competing condition between $ a(x) $ and $ V(x) $ is introduced.
Citation: Xia Su, Chunhua Deng. Solutions for Schrödinger equations with variable separated type nonlinear terms[J]. AIMS Mathematics, 2023, 8(12): 30487-30500. doi: 10.3934/math.20231557
In this paper, we consider the following semilinear Schrödinger equation:
$ \begin{eqnarray*} \left\{ \begin{array}{ll} -\Delta u+V(x)u = a(x)g(u)&{\mbox{for}}\; x\in \mathbb{R}^{N} ,\\ u(x)\rightarrow0&{\mbox{as}}\; |x|\rightarrow \infty , \end{array} \right. \end{eqnarray*} $
where $ a(x) > 0 $ for all $ \mathbb{R}^{N} $. Under some different superlinear conditions on $ g(u) $, we obtain the existence of solutions for the above problem. In order to regain the compactness of the Sobolev embedding, a competing condition between $ a(x) $ and $ V(x) $ is introduced.
[1] |
N. Ackermann, A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Anal., 234 (2006), 277–320. https://doi.org/10.1016/j.jfa.2005.11.010 doi: 10.1016/j.jfa.2005.11.010
![]() |
[2] |
T. Bartsch, Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^{N}$, Commun. Partial Differ. Equ., 20 (1995), 1725–1741. https://doi.org/10.1080/03605309508821149 doi: 10.1080/03605309508821149
![]() |
[3] |
R. Castro López, G. H. Sun, O. Camacho-Nieto, C. Yáñez-Márquez, S. H. Dong, Analytical traveling-wave solutions to a generalized Gross-Pitaevskii equation with some new time and space varying nonlinearity coefficients and external fields, Phys. Lett. A, 381 (2017), 2978–2985. https://doi.org/10.1016/j.physleta.2017.07.012 doi: 10.1016/j.physleta.2017.07.012
![]() |
[4] |
Y. H. Ding, S. X. Luan, Multiple solutions for a class of nonlinear Schrödinger equations, J. Differ. Equ., 207 (2004), 423–457. https://doi.org/10.1016/j.jde.2004.07.030 doi: 10.1016/j.jde.2004.07.030
![]() |
[5] |
X. D. Fang, A. Szulkin, Multiple solutions for a quasilinear Schrödinger equation, J. Differ. Equ., 254 (2013), 2015–2032. https://doi.org/10.1016/j.jde.2012.11.017 doi: 10.1016/j.jde.2012.11.017
![]() |
[6] |
Y. S. Guo, W. Li, S. H. Dong, Gaussian solitary solution for a class of logarithmic nonlinear Schrödinger equation in (1+n) dimensions, Results Phys., 44 (2023), 106187. https://doi.org/10.1016/j.rinp.2022.106187 doi: 10.1016/j.rinp.2022.106187
![]() |
[7] |
S. B. Liu, On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differ. Equ., 45 (2012), 1–9. https://doi.org/10.1007/s00526-011-0447-2 doi: 10.1007/s00526-011-0447-2
![]() |
[8] |
Y. Q. Li, Z. Q. Wang, J. Zeng, Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. H. Poincaré Anal. NonLinéaire, 23 (2006), 829–837. https://doi.org/10.1016/j.anihpc.2006.01.003 doi: 10.1016/j.anihpc.2006.01.003
![]() |
[9] |
P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270–291. https://doi.org/10.1007/BF00946631 doi: 10.1007/BF00946631
![]() |
[10] | P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Providence, RI: American Mathematical Society, 1986. |
[11] |
B. Sirakov, Existence and multiplicity of solutions of semi-linear elliptic equations in $\mathbb{R}^{N}$, Calc. Var. Partial Differ. Equ., 11 (2000), 119–142. https://doi.org/10.1007/s005260000010 doi: 10.1007/s005260000010
![]() |
[12] |
X. H. Tang, New super-quadratic conditions on ground state solutions for superlinear Schrödinger equation, Adv. Nonlinear Stud., 14 (2014), 361–373. https://doi.org/10.1515/ans-2014-0208 doi: 10.1515/ans-2014-0208
![]() |
[13] |
E. Toon, P. Ubilla, Existence of positive solutions of Schrödinger equations with vanishing potentials, Discrete Contin. Dyn. Syst., 40 (2020), 5831–5843. https://doi.org/10.3934/dcds.2020248 doi: 10.3934/dcds.2020248
![]() |
[14] |
E. Toon, P. Ubilla, Hamiltonian systems of Schrödinger equations with vanishing potentials, Commun. Contemp. Math., 24 (2022), 2050074. https://doi.org/10.1142/S0219199720500741 doi: 10.1142/S0219199720500741
![]() |
[15] |
D. B. Wang, H. B. Zhang, W. Guan, Existence of least-energy sign-changing solutions for Schrödinger-Poisson system with critical growth, J. Math. Anal. Appl., 479 (2019), 2284–2301. https://doi.org/10.1016/j.jmaa.2019.07.052 doi: 10.1016/j.jmaa.2019.07.052
![]() |
[16] |
L. L. Wan, C. L. Tang, Existence of solutions for non-periodic superlinear Schrödinger equations without (AR) condition, Acta Math. Sci., 32 (2012), 1559–1570. https://doi.org/10.1016/s0252-9602(12)60123-4 doi: 10.1016/s0252-9602(12)60123-4
![]() |
[17] |
T. F. Wu, Multiple positive solutions for a class of concave-convex elliptic problems in $\mathbb{R}^{N}$ involving sign-changing weight, J. Funct. Anal., 258 (2010), 99–131. https://doi.org/10.1016/j.jfa.2009.08.005 doi: 10.1016/j.jfa.2009.08.005
![]() |
[18] |
D. L. Wu, F. Y. Li, H. X. Lin, Existence and nonuniqueness of solutions for a class of asymptotically linear nonperiodic Schrödinger equations, J. Fixed Point Theory Appl., 24 (2022), 72. https://doi.org/10.1007/s11784-022-00975-4 doi: 10.1007/s11784-022-00975-4
![]() |
[19] |
Q. Y. Zhang, Q. Wang, Multiple solutions for a class of sublinear Schrödinger equations, J. Math. Anal. Appl., 389 (2012), 511–518. https://doi.org/10.1016/j.jmaa.2011.12.003 doi: 10.1016/j.jmaa.2011.12.003
![]() |
[20] |
H. Zhang, J. X. Xu, F. B. Zhang, On a class of semilinear Schrödinger equations with indefinite linear part, J. Math. Anal. Appl., 414 (2014), 710–724. https://doi.org/10.1016/j.jmaa.2014.01.001 doi: 10.1016/j.jmaa.2014.01.001
![]() |
[21] |
Q. Zheng, D. L. Wu, Multiple solutions for Schrödinger equations involving concave-convex nonlinearities without $(AR)$-type condition, Bull. Malays. Math. Sci. Soc., 44 (2021), 2943–2956. https://doi.org/10.1007/s40840-021-01096-w doi: 10.1007/s40840-021-01096-w
![]() |
[22] |
X. Zhong, W. Zou, Ground state and multiple solutions via generalized Nehari manifold, Nonlinear Anal., 102 (2014), 251–263. https://doi.org/10.1016/j.na.2014.02.018 doi: 10.1016/j.na.2014.02.018
![]() |