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.


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