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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
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