# American Institute of Mathematical Sciences

June  2010, 28(2): 827-844. doi: 10.3934/dcds.2010.28.827

## On some Schrödinger equations with non regular potential at infinity

 1 Dipartimento di Matematica, Politecnico di Bari, Via E. Orabona, 4, 70125 Bari, Italy 2 Dipartimento di Matematica, Università di Roma "Tor Vergata”, Via della Ricerca Scientifica, 1, 00133 Roma, Italy

Received  February 2010 Published  April 2010

In this paper we study the existence of solutions $u\in H^1(\R^N)$ forthe problem $-\Delta u+a(x)u=|u|^{p-2}u$, where $N\ge 2$ and $p$ issuperlinear and subcritical.The potential $a(x)\in L^\infty(\R^N)$ is such that $a(x)\ge c>0$ butis not assumed to have a limit at infinity.Considering different kinds of assumptions on the geometry of $a(x)$we obtain two theorems stating the existence of positive solutions.Furthermore, we prove that there are no nontrivial solutions, when adirection exists along which the potential is increasing.
Citation: Giovanna Cerami, Riccardo Molle. On some Schrödinger equations with non regular potential at infinity. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 827-844. doi: 10.3934/dcds.2010.28.827
 [1] Lars Diening, Michael Růžička. An existence result for non-Newtonian fluids in non-regular domains. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 255-268. doi: 10.3934/dcdss.2010.3.255 [2] Yinbin Deng, Wei Shuai. Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2273-2287. doi: 10.3934/cpaa.2014.13.2273 [3] César E. Torres Ledesma. Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well. Communications on Pure & Applied Analysis, 2016, 15 (2) : 535-547. doi: 10.3934/cpaa.2016.15.535 [4] Xing Cheng, Ze Li, Lifeng Zhao. Scattering of solutions to the nonlinear Schrödinger equations with regular potentials. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 2999-3023. doi: 10.3934/dcds.2017129 [5] Shu-Yu Hsu. Non-existence and behaviour at infinity of solutions of some elliptic equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 769-786. doi: 10.3934/dcds.2004.10.769 [6] Liping Wang, Chunyi Zhao. Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1707-1731. doi: 10.3934/dcds.2017071 [7] Zuji Guo. Nodal solutions for nonlinear Schrödinger equations with decaying potential. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1125-1138. doi: 10.3934/cpaa.2016.15.1125 [8] Suna Ma, Huiyuan Li, Zhimin Zhang. Novel spectral methods for Schrödinger equations with an inverse square potential on the whole space. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1589-1615. doi: 10.3934/dcdsb.2018221 [9] Toshiyuki Suzuki. Scattering theory for semilinear Schrödinger equations with an inverse-square potential via energy methods. Evolution Equations & Control Theory, 2019, 8 (2) : 447-471. doi: 10.3934/eect.2019022 [10] Mingwen Fei, Huicheng Yin. Nodal solutions of 2-D critical nonlinear Schrödinger equations with potentials vanishing at infinity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2921-2948. doi: 10.3934/dcds.2015.35.2921 [11] Thomas Bartsch, Zhongwei Tang. Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 7-26. doi: 10.3934/dcds.2013.33.7 [12] Wu Chen, Zhongxue Lu. Existence and nonexistence of positive solutions to an integral system involving Wolff potential. Communications on Pure & Applied Analysis, 2016, 15 (2) : 385-398. doi: 10.3934/cpaa.2016.15.385 [13] Xiaoyan Lin, Yubo He, Xianhua Tang. Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1547-1565. doi: 10.3934/cpaa.2019074 [14] Ran Zhuo, Yan Li. Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1595-1611. doi: 10.3934/dcds.2019071 [15] Minbo Yang, Yanheng Ding. Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part. Communications on Pure & Applied Analysis, 2013, 12 (2) : 771-783. doi: 10.3934/cpaa.2013.12.771 [16] Tai-Chia Lin, Tsung-Fang Wu. Existence and multiplicity of positive solutions for two coupled nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2911-2938. doi: 10.3934/dcds.2013.33.2911 [17] Robert Magnus, Olivier Moschetta. The non-linear Schrödinger equation with non-periodic potential: infinite-bump solutions and non-degeneracy. Communications on Pure & Applied Analysis, 2012, 11 (2) : 587-626. doi: 10.3934/cpaa.2012.11.587 [18] Wojciech M. Zajączkowski. Long time existence of regular solutions to non-homogeneous Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1427-1455. doi: 10.3934/dcdss.2013.6.1427 [19] Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations & Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337 [20] Miaomiao Niu, Zhongwei Tang. Least energy solutions of nonlinear Schrödinger equations involving the half Laplacian and potential wells. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1215-1231. doi: 10.3934/cpaa.2016.15.1215

2018 Impact Factor: 1.143