## Differential and Integral Equations

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

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

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356060579

**Mathematical Reviews number (MathSciNet)**

MR1989597

**Zentralblatt MATH identifier**

1161.35385

**Subjects**

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.)

#### Citation

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