LAUSR

Abstract In this paper, we study the multiplicity of solutions for the following problem: $$\begin{equation*} \begin{cases} -\Delta u-\Delta(|u|^{\alpha})|u|^{\alpha-2}u=g(x,u)+\theta h(x,u), \ \ x\in \Omega,\\ u=0, \ \ x\in \partial\Omega, \end{cases} \end{equation*}$$ where α ≥ 2, Ω is a smooth bounded domain in ${\mathbb{R}}$ N , θ is a parameter and g, h ∈ C( $\bar{\Omega}$ × ${\mathbb{R}}$ ). Under the assumptions that g(x, u) is odd and locally superlinear at infinity in u, we prove that for any j ∈ $\mathbb{N}$ there exists ϵ j > 0 such that if |θ| ≤ ϵ j , the above problem possesses at least j distinct solutions. Our results generalize some known results in the literature and are new even in the symmetric situation.