Differential and Integral Equations

Minimal nodal solutions of a Schrödinger equation with critical nonlinearity and symmetric potential

Mónica Clapp and Yanheng Ding

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We study the nonlinear Schr\"{o}dinger equation \[ -\Delta u+\lambda a(x)u=\mu u+u^{2^{\ast }-1},\hbox{ \ }u\in \mathbb{R}^{N}, \] with critical exponent $2^{\ast }=2N/(N-2),$ $N\geq 4,$ where $a\geq 0$ has a potential well and is invariant under an orthogonal involution of $\mathbb{R} ^{N}$. Using variational methods we establish existence and multiplicity of solutions which change sign exactly once. These solutions localize near the potential well for $\mu $ small and $\lambda $ large.

Article information

Differential Integral Equations, Volume 16, Number 8 (2003), 981-992.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations
Secondary: 35B33: Critical exponents 35B38: Critical points 35J20: Variational methods for second-order elliptic equations 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 47J30: Variational methods [See also 58Exx] 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)


Clapp, Mónica; Ding, Yanheng. Minimal nodal solutions of a Schrödinger equation with critical nonlinearity and symmetric potential. Differential Integral Equations 16 (2003), no. 8, 981--992. https://projecteuclid.org/euclid.die/1356060579

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