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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

-Δu+λa(x)u=μu+u 2 * -1 ,u N ,

with critical exponent 2 * =2N/(N-2), N4, where a0 has a potential well and is invariant under an orthogonal involution of N . Using variational methods we establish existence and multiplicity of solutions which change sign exactly once. These solutions localize near the potential well for μ small and λ large.

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