Advances in Differential Equations

Generalized linking theorem with an application to a semilinear Schrödinger equation

Wojciech Kryszewski and Andrzej Szulkin

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Abstract

Consider the semilinear Schrödinger equation (*) $-\Delta u + V(x)u = f(x,u)$, $u\in H^1(\mathbf {R} ^N)$. It is shown that if $f$, $V$ are periodic in the $x$-variables, $f$ is superlinear at $u=0$ and $\pm\infty$ and 0 lies in a spectral gap of $-\Delta+V$, then (*) has at least one nontrivial solution. If in addition $f$ is odd in $u$, then (*) has infinitely many (geometrically distinct) solutions. The proofs rely on a degree-theory and a linking-type argument developed in this paper.

Article information

Source
Adv. Differential Equations, Volume 3, Number 3 (1998), 441-472.

Dates
First available in Project Euclid: 19 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1366399849

Mathematical Reviews number (MathSciNet)
MR1751952

Zentralblatt MATH identifier
0947.35061

Subjects
Primary: 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)
Secondary: 35J60: Nonlinear elliptic equations 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 47J30: Variational methods [See also 58Exx]

Citation

Kryszewski, Wojciech; Szulkin, Andrzej. Generalized linking theorem with an application to a semilinear Schrödinger equation. Adv. Differential Equations 3 (1998), no. 3, 441--472. https://projecteuclid.org/euclid.ade/1366399849


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