Schrödinger-Poisson systems with radial potentials and discontinuous quasilinear nonlinearity

Anmin Mao, Hejie Chang


We consider the following Schrödinger-Poisson system: $$ \begin{cases} -\triangle u+V(|x|)u+\phi u= Q(|x|) f(u) &\hbox{in } \mathbb{R}^3,\\ -\triangle \phi=u^{2} & \hbox{in } \mathbb{R}^3, \end{cases} $$ \noindent with more general radial potentials $V,Q$ and discontinuous nonlinearity $f$. The Lagrange functional may be locally {\it Lipschitz}. Using nonsmooth critical point theorem, we obtain the multiplicity results of radial solutions, we also show concentration properties of the solutions. This is in contrast with some recent papers concerning similar problems by using the classical Sobolev embedding theorems.


Schrödinger-Poisson equation; radial potentials; weighted Sobolev embedding

Full Text:



A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc. 7 (2005), 117–144.

A. Azzollini, P. D’Avenia and A. Pomponio, On the Schrödinger–Maxwell equations under the effect of a general nonlinear term, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), 779–791.

A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger–Maxwell equations, J. Math. Anal. Appl. 345 (2008), 90–108.

T. Bartsch, Infinitely many solutions of a symmetric Dirichlet problem, Nonlinear Anal. 20 (1993), 1205–1216.

T. Bartsch and M. Willem, On an elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc. 123 (1995), 3555–3561.

V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger–Maxwell equations, Topol. Methods Nonlinear Anal. 11 (1998), 283–293.

D. Bonheure and C. Mercuri, Embedding theorems and existence results for nonlinear Schrödinger–Poisson systems with unbounded and vanishing potentials, J. Differential Equations 251 (2011), 1056–1085.

K.C. Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102–129.

S.J. Chen and C.L. Tang, High energy solutions for the superlinear Schrödinger–Maxwell equations, Nonlinear Anal. 71 (2009), 4927–4934.

F.H. Clarck, A new approach to Lagrange multipliers, Math. Oper. Res. 1 (1976), 165–174.

T. D’Aprile and J. Wei, On bound states concentrating on spheres for the Maxwell–Schrödinger equation, SIAM J. Math. Anal. 37 (2005), 321–342.

P. De Nápoli and M.C. Mariani, Mountain pass solutions to equations of p-Laplacian type, Nonlinear Anal. 54 (2003), 1205–1219.

I. Ianni, Sign-changing radial solutions for the Schrödinger–Poisson–Slater problem, Topol. Methods Nonlinear Anal. 41 (2013), 365–385.

S. Kim and J. Seok, On nodal solutions of the nonlinear Schrödinger–Poisson equations, Comm. Contemp. Math. 14 (2012), 12450041–12450057.

A. Li, H. Cai and J. Su, Qusilinear elliptic equations with sigular potentials and bounded discontinuous nonlinearities, Topol. Methods Nonlinear Anal. 43 (2014), 439–450.

C. Mercuri, Positive solutions of nonlinear Schrödinger–Poisson system with radial potentials vanishing at infinity, Rend. Lincei Mat. Appl. 19 (2008), 211–227.

D. Ruiz, Semiclassical states for coupled Schrödinger–Maxwell equations: concentration around a sphere, Math. Models Methods Appl. Sci. 15 (2005), 141–164.

J.B. Su, Quasilinear elliptic equations on R3 with singular potentials and bounded nonlinearity, Z. Angew. Math. Phys. 63 (2012), 51–62.

J.B. Su and R.S. Tian, Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations, Commun. Pure Appl. Anal. 9 (2010), no. 4, 885–904.

J.B. Su and R.S. Tian, Weighted Sobolev type embeddings and coercive quasilinear elliptic equations on RN , Proc. Amer. Math. Soc. 140 (2012), 891–903.

J.B. Su, Z.Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differential Equations 238 (2007), 201–219.

Z.Q. Wang, Nonlinear boundary value problems with concave nonlinearities near the origin, NoDEA Nonlinear Differential Equations Appl. 8 (2001), 15–33.

Z.P. Wang and H.S. Zhou, Positive solution for a nonlinear stationary Schrödinger–Poisson system in R3 , Discrete Contin. Dyn. Syst. 18 (2007), 809–816.

M. Willem, Minimax Theorems. Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser, Boston, 1996.

L. Zhao and F. Zhao, On the existence of solutions for the Schrödinger–Poisson equations, J. Math. Anal. Appl. 346 (2008), 155–169.

H. Zhu, Asymptotically linear Schrödinger–Poisson systems with potentials vanishing at infinity, J. Math. Anal. Appl. 380 (2011), 501–510.


  • There are currently no refbacks.

Partnerzy platformy czasopism