## Topological Methods in Nonlinear Analysis

### Solutions to a nonlinear Schrödinger equation with periodic potential and zero on the boundary of the spectrum

Jarosław Mederski

#### Abstract

We study the following nonlinear Schrödinger equation \begin{equation*} \begin{cases} -\Delta u + V(x) u = g(x,u) & \hbox{for } x\in\mathbb R^N,\\ u(x)\to 0 & \hbox{as } |x|\to\infty, \end{cases} \end{equation*} where $V\colon \mathbb R^N\to\mathbb R$ and $g\colon \mathbb R^N\times\mathbb R\to\mathbb R$ are periodic in $x$. We assume that $0$ is a right boundary point of the essential spectrum of $-\Delta+V$. The superlinear and subcritical term $g$ satisfies a Nehari type monotonicity condition. We employ a Nehari manifold type technique in a strongly indefitnite setting and obtain the existence of a ground state solution. Moreover, we get infinitely many geometrically distinct solutions provided that $g$ is odd.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 46, Number 2 (2015), 755-771.

Dates
First available in Project Euclid: 21 March 2016

https://projecteuclid.org/euclid.tmna/1458588660

Digital Object Identifier
doi:10.12775/TMNA.2015.067

Mathematical Reviews number (MathSciNet)
MR3494969

Zentralblatt MATH identifier
1362.35289

#### Citation

Mederski, Jarosław. Solutions to a nonlinear Schrödinger equation with periodic potential and zero on the boundary of the spectrum. Topol. Methods Nonlinear Anal. 46 (2015), no. 2, 755--771. doi:10.12775/TMNA.2015.067. https://projecteuclid.org/euclid.tmna/1458588660

#### References

• S. Alama and Y.Y. Li, On ”multibump” bound states for certain semilinear elliptic equations, Indiana Univ. Math. J. 41 (1992), no. 4, 983–1026.
• C.O. Alves, M.A.S. Souto and M. Montenegro, Existence of solution for two classes of elliptic problems in $\R^N$ with zero mass, J. Differential Equations 252 (2012), 5735–5750.
• M. Badiale, L. Pisani and S. Rolando, Sum of wheighted Lebesgue spaces and nonlinear elliptic equations, Nonlinear Differential Equations Appl. 18 (2011), 369–405.
• T. Bartsch and Y. Ding, On a nonlinear Schrödinger equation with periodic potential, Math. Ann. 313 (1999), no. 1, 15–37.
• T. Bartsch and J. Mederski, Ground and bound state solutions of semilinear time-harmonic Maxwell equations in a bounded domain, Arch. rational Mech. Anal. 215 (1), (2015), 283–306.
• H. Berestycki and P.L. Lions, Nonlinear scalar field equations. \romI. Existence of a ground state, Arch. Rational Mech. Anal. 82, (1983), 313–345.
• B. Buffoni, L. Jeanjean and C.A. Stuart, Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc. 119 (1993), no. 1, 179–186.
• V. Coti-Zelati and P. Rabinowitz, Homoclinic type solutions for a semilinear elliptic \romPDE on $\R^n$, Comm. Pure Appl. Math. 45 (1992), no. 10, 1217–1269.
• Y. Ding and C. Lee, Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms, J. Differential Equations 222 (2006), no. 1, 137–163.
• G. Evequoz and T. Weth, Real solutions to the nonlinear Helmholtz equation with local nonlinearity, Arch. Rat. Mech. Anal. 211 (2014), 359–388.
• D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Springer–Verlag, Berlin, 2001.
• 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), no. 4, 787–809.
• W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. Differential Equations 3 (1998), 441–472.
• Y. Li, Z.-Q. Wang and J. Zeng, Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), no. 6, 829–837.
• S. Liu, On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations 45 (2012), no. 1–2, 1–9.
• Z. Liu and Z.-Q. Wang, On the Ambrosetti–Rabinowitz superlinear condition, Adv. Nonlinear Stud. 4 (2004), 561–572.
• O.H. Miyagaki and M.A.S. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations 245 (2008), no. 12, 3628–3638.
• A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math. 73 (2005), 259–287.
• ––––, Lecture Notes on Schrödinger Equations, Nova Science Pub Inc. 2008.
• P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), 270–291.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, Analysis of Operators, Vol. IV, Academic Press, New York, 1978.
• M. Struwe, Variational Methods, Springer 2008.
• M. Schechter and W. Zou, Weak linking theorems and Schrödinger equations with critical Sobolev exponent, ESAIM Control Optim. Calc. Var. 9 (2003), 601–619.
• A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal. 257 (2009), no. 12, 3802–3822.
• ––––, The method of Nehari manifold. Handbook of nonconvex analysis and applications, Handbook of Nonconvex Analysis and Applications, Int. Press, Somerville, 2010, 597–632.
• C. Troestler and M. Willem, Nontrivial solution of a semilinear Schrödinger equation, Comm. Partial Differential Equations 21 (1996), 1431–1449.
• M. Willem, Minimax Theorems, Birkhäuser Verlag, 1996.
• M. Willem and W. Zou, On a Schrödinger equation with periodic potential and spectrum point zero, Indiana Univ. Math. J. 52 (2003), no. 1, 109–132.
• M. Yang, W. Chen and Y. Ding, Solutions for periodic Schrödinger equation with spectrum zero and general superlinear nonlinearities, J. Math. Anal. Appl. 364 (2010), no. 2, 404–413.