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Minimal nodal solutions of a Schrödinger equation with critical nonlinearity and symmetric potential. (English) Zbl 1161.35385
Summary: We study the nonlinear Schrödinger equation
$-\Delta u+\lambda a(x)u= \mu u+u^{2^*-1}, \quad u\in\mathbb R^N,$
with critical exponent $$2^*=2N/(N-2)$$, $$N\geq 4$$, 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.

##### MSC:
 35J60 Nonlinear elliptic equations 35B33 Critical exponents in context of PDEs 35J20 Variational methods for second-order elliptic equations 35Q55 NLS equations (nonlinear Schrödinger equations) 47J30 Variational methods involving nonlinear operators