Taiwanese Journal of Mathematics

NON-NEHARI MANIFOLD METHOD FOR SUPERLINEAR SCHRÖDINGER EQUATION

X. H. Tang

Full-text: Open access

Abstract

We consider the boundary value problem \begin{equation} \tag{0.1} \left\{ \begin{array}{ll} -\triangle u+V(x)u=f(x, u), \ \ \ \  & x\in \Omega,\\ u=0, \ \ \ \ & x\in \partial\Omega, \end{array} \right. \end{equation} where $ \Omega \subset \mathbb R^N$ be a bounded domain, $\inf_{\Omega}V(x)\gt -\infty$, $f$ is a superlinear, subcritical nonlinearity. Inspired by previous work of Szulkin and Weth (2009)) [21] and (2010) [22], we develop a more direct and simpler approach on the basis of one used in [21], to deduce weaker conditions under which problem (0.1) has a ground state solution of generalized Nehari type or infinity many nontrivial solutions. Unlike the Nehari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the generalized Nehari manifold by using the diagonal method.

Article information

Source
Taiwanese J. Math., Volume 18, Number 6 (2014), 1957-1979.

Dates
First available in Project Euclid: 21 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500667506

Digital Object Identifier
doi:10.11650/tjm.18.2014.3541

Mathematical Reviews number (MathSciNet)
MR3284041

Zentralblatt MATH identifier
1357.35163

Subjects
Primary: 35J20: Variational methods for second-order elliptic equations 35J60: Nonlinear elliptic equations

Keywords
Schrödinger equation strongly indefinite functional superlinear diagonal method boundary value problem ground state solutions of Nehari-Pankov type

Citation

Tang, X. H. NON-NEHARI MANIFOLD METHOD FOR SUPERLINEAR SCHRÖDINGER EQUATION. Taiwanese J. Math. 18 (2014), no. 6, 1957--1979. doi:10.11650/tjm.18.2014.3541. https://projecteuclid.org/euclid.twjm/1500667506


Export citation

References

  • A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
  • A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge Studies in Advanced Mathematics 104, Cambridge University Press, Cambridge, 2007.
  • A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
  • T. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Anal., 7 (1983), 241-273.
  • D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987.
  • L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on $\R^N$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809.
  • G. B. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.
  • G. B. Li and C. Wang, The existence of a nontrivial solution to a nonlinear elliptic problem of linking type without the Ambrosetti-Rabinowitz condition, Annales Academi${\ae}$ Scientiarum Fennic${\ae}$ Mathematica, 36 (2011), 461-480.
  • G. B. Li and H. S. Zhou, Asympotically linear Dirichlet problem for $p$-Laplacian, Nonlinear Anal., 43 (2001), 1043-1055.
  • G. B. Li and H. S. Zhou, Multiple solutions to $p$-Laplacian problems with asympotic nonlinearity as $u^{p-1}$ at infinity, J. London Math. Soc., 65(2) (2002), 123-138.
  • G. B. Li and H. S. Zhou, Dirichlet problem of $p$-Laplacian with nonlinear term $f(x; t)\sim t^{p-1}$ at infinity, In: Morse theory, Minimax theory and their applications to nonlinear partial differential equations, edited by H. Brezis, S. J. Li, J. Q. Liu and P. H. Rabinowitz, New Stud. Adv. Math., 1 (2003), 77-89.
  • S. Liu, On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), 1-9.
  • T.-C. Ouyang and J.-P. Shi, Exact multiplicity of positive solutions for a class of semilinear problem, J. Differential Equations, 146 (1998), 121-156.
  • T.-C. Ouyang and J.-P. Shi, Exact multiplicity of positive solutions for a class of semilinear problem: II, J. Differential Equations, 158 (1999), 94-151.
  • A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.
  • P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Series Math. 65, Amer. Math. Soc., Providence, R.I., 1986.
  • M. Schechter, Minimax Systems and Critical Point Theory, Birkhäuser, Boston, 2009.
  • M. Schechter and W. Zou, Superlinear problems, Pacific J. Math., 214 (2004), 145-160.
  • M. Struwe, Variational Methods, Springer-Verlag, Berlin, 1990.
  • J. Sun and Z. Wang, Spectral Analysis for Linear Operators, Science Press, Beijing, 2005 (Chinese).
  • A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257(12) (2009), 3802-3822.
  • A. Szulkin and T. Weth, The method of Nehari manifold, in: Handbook of Nonconvex Analysis and Applications, D. Y. Gao and D. Motreanu eds., International Press, Boston, 2010, pp. 597-632.
  • X. H. Tang, Infinitely many solutions for semilinear Schrödinger equations with sign-changing potential and nonlinearity, J. Math. Anal. Appl., 401 (2013), 407-415.
  • X. H. Tang, New super-quadratic conditions on ground state solutions for superlinear Schrödinger equation, Advance Nonlinear Studies, 14 (2014), 349-361.
  • M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.