Existence of solutions for a class of Kirchhoff type problems in Orlicz–Sobolev spaces

Nguyen Thanh Chung¹

¹ Department of Mathematics Quang Binh University 312 Ly Thuong Kiet Dong Hoi, Quang Binh, Vietnam

Annales Polonici Mathematici, Tome 113 (2015) no. 3, pp. 283-294

Résumé

We consider Kirchhoff type problems of the form $$ \left\{\begin{aligned} {-} M(\rho(u)) (\mathrm{div}(a(|\nabla u|)\nabla u)-a(|u|)u)=K(x)f(u) \quad \text{in } \Omega,\\\textstyle \frac{\partial u}{\partial \nu} = 0 \quad \text{on } \partial\Omega, \end{aligned}\right. $$ where $\Omega \subset \mathbb R^N$, $N \geq 3$, is a smooth bounded domain, $\nu$ is the outward unit normal to $\partial\Omega$, $\rho(u)= \int_\Omega (\Phi (|\nabla u|)+\Phi(|u|) )\, dx$, $M: [0,\infty) \to \mathbb R$ is a continuous function, $K\in L^\infty(\Omega)$, and $f: \mathbb R\to\mathbb R$ is a continuous function not satisfying the Ambrosetti–Rabinowitz type condition. Using variational methods, we obtain some existence and multiplicity results.

Détail

DOI : 10.4064/ap113-3-5

Keywords: consider kirchhoff type problems form begin aligned rho mathrm div nabla nabla a u quad text omega amp textstyle frac partial partial quad text partial omega end aligned right where omega subset mathbb geq smooth bounded domain outward unit normal partial omega rho int omega phi nabla phi infty mathbb continuous function infty omega mathbb mathbb continuous function satisfying ambrosetti rabinowitz type condition using variational methods obtain existence multiplicity results