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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 $$-\Delta u+\lambda a(x)u= \mu u+u^{2^*-1}, \quad u\in\Bbb R^N,$$ with critical exponent $2^*=2N/(N-2)$, $N\ge 4$, where $a\ge 0$ has a potential well and is invariant under an orthogonal involution of $\Bbb R^N$. 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.

35J60Nonlinear elliptic equations
35B33Critical exponents (PDE)
35J20Second order elliptic equations, variational methods
35Q55NLS-like (nonlinear Schrödinger) equations
47J30Variational methods (nonlinear operator equations)