PERTURBATIONS FROM INDEFINITE SYMMETRIC ELLIPTIC BOUNDARY VALUE PROBLEMS

ZHANG, LIANG ; TANG, XIANHUA

Glasgow mathematical journal, Tome 59 (2017) no. 3, pp. 635-648

Résumé

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 εj > 0 such that if |θ| ≤ εj , 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.

Détail

DOI : 10.1017/S0017089516000446

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