Topological Methods in Nonlinear Analysis

Sign changing solutions of nonlinear Schrödinger equations

Thomas Bartsch and Zhi-Qiang Wang

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Abstract

We are interested in solutions $u\in H^1({\mathbb R}^N)$ of the linear Schrödinger equation $-\delta u +b_{\lambda} (x) u =f(x,u)$. The nonlinearity $f$ grows superlinearly and subcritically as $\vert u\vert \to\infty$. The potential $b_{\lambda}$ is positive, bounded away from $0$, and has a potential well. The parameter $\lambda$ controls the steepness of the well. In an earlier paper we found a positive and a negative solution. In this paper we find third solution. We also prove that this third solution changes sign and that it is concentrated in the potential well if $\lambda \to \infty$. No symmetry conditions are assumed.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 13, Number 2 (1999), 191-198.

Dates
First available in Project Euclid: 29 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1475178878

Mathematical Reviews number (MathSciNet)
MR1742220

Zentralblatt MATH identifier
0961.35150

Citation

Bartsch, Thomas; Wang, Zhi-Qiang. Sign changing solutions of nonlinear Schrödinger equations. Topol. Methods Nonlinear Anal. 13 (1999), no. 2, 191--198. https://projecteuclid.org/euclid.tmna/1475178878


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References

  • T. Bartsch, K.-C. Chang, and Z.-Q. Wang, On the Morse indices of sign changing solution of nonlinear elliptic problems , Math. Z., to appear \ref
  • T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\R^n$ , Comm. Partial Differential Equations, 20 , 1725–1741 (1995) \ref ––––, On the existence of sign changing solutions for semilinear Dirichlet problems , Topol. Methods Nonlinear Anal., 7 , 115–131 (1996) \ref
  • T. Bartsch and M. Willem, Infinitely many nonradial solutions of a Euclidean scalar field equation , J. Funct. Anal., 117 , 447–460 (1993) \ref ––––, Infinitely many radial solutions of a semilinear elliptic problem on $\R^N$ , Arch. Rational Mech. Anal., 124 , 261–276 (1993) \ref
  • A. Castro, J. Cossio and J. Neuberger, Sign changing solutions for a superlinear Dirichlet problem , Rocky Mountain J. Math., 27 , 1041–1053 (1997) \ref
  • C. N. Chen, Some existence and bifurcation results for solutions of nonlinear Sturm-Liouville eigenvalue problems , Math. Z., 208 , 177–192 (1991) \ref \key 8
  • E. Dancer and Y. Du, Existence of sign changing solutions for some semilinear problems with jumping nonlinearities at zero , Proc. Roy. Soc. Edinburgh Sect. A, 124 , 1165–1176 (1994) \ref \key 9 ––––, Multiple solutions of some semilinear elliptic equations via the generalized Conley index , J. Math. Anal. Appl., 189 , 848–871 (1995) \ref
  • C. Jones, T. Küpper, and H. Plakties, A shooting argument with oscillation for semilinear elliptic radially symmetric equations , Proc. Roy. Soc. Edinburgh Sect. A, 108 A , 165–180 (1988) \ref
  • P. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part II. , Ann. Inst. H. Poincaré. Anal. Non Linéaire, 1 , 223–283 (1984) \ref
  • P. Rabinowitz, On a class of nonlinear Schrödinger equations , Z. Angew. Math. Phys., 43 , 270–291 (1992) \ref
  • Z.-Q. Wang, On a superlinear elliptic equation , Ann. Inst. H. Poincaré. Anal. Non Linéaire, 8 , 43–58 (1991)