### Sign-changing solutions for $p$-Laplacian equations with jumping nonlinearity and the Fučik spectrum

#### Abstract

We study the existence of sign-changing solutions for the $p$-Laplacian equation

$$

-\Delta_pu +\lambda g(x)|u|^{p-2}u=f(u),\quad x\in \mathbb{R}^N,

$$

where $\lambda$ is a positive parameter and the nonlinear term

$f$ has jumping nonlinearity at infinity and is superlinear at

zero. The Fučik spectrum plays an important role in the

proof. We give sufficient conditions for the existence of

nontrivial Fučik spectrum.

$$

-\Delta_pu +\lambda g(x)|u|^{p-2}u=f(u),\quad x\in \mathbb{R}^N,

$$

where $\lambda$ is a positive parameter and the nonlinear term

$f$ has jumping nonlinearity at infinity and is superlinear at

zero. The Fučik spectrum plays an important role in the

proof. We give sufficient conditions for the existence of

nontrivial Fučik spectrum.

#### Keywords

Jumping; sign-changing solution; Fučik spectrum

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