Sign Changing Solutions of Superlinear Schrödinger Equations

Abstract

We prove the existence of sign changing solutions in H ¹(ℝ ^N ) for a stationary Schrödinger equation −Δu + a(x)u = f(x, u) with superlinear and subcritical nonlinearity f, and control the number of nodal domains. If f is odd we obtain an unbounded sequence of sign changing solutions u _k , k ≥ 1, so that u _k has at most k + 1 nodal domains. The bound on the number of nodal domains follows from a nonlinear version of Courant's nodal domain theorem which we also prove.

Key Words:

• Nonlinear Schrödinger equations
• Nodal solutions
• Nodal domains

Acknowledgment

Zhaoli Liu was supported by the Alexander von Humboldt foundation in Germany and FUKTME in China.