The Existence and Multiplicity of Solutions for a Class of Quasilinear Elliptic Equations

Let Ω ⊂ \(\mathbb{R} \)N (N ⩾ 3) is a bounded domain with a smooth C2-boundary, 0 ∈ ∂Ω, and n denote the unitoutward normal to ∂Ω. We are concerned with the Neumann boundary problems: \(\left\{\begin{array}{cc}-\operatorname{div}\left(|x|^\alpha \nabla u\right)+\lambda|x|^\gamma u=|x|^\beta u^{p(\alpha, \beta)-1}, & x \in \Omega \\u(x)>0, & x \in \Omega \\\frac{\partial u}{\partial n}=0, & x \in \partial \Omega\end{array}\right.\) where p (α, β) = \(\frac{2(N+β)}{N-2+α}\) > 2, \(\gamma\) > α − 2, α > 0, β < 0. For certain region of the parameters α, β and \(\gamma\), we establish the existence result of least energy solutions. Furthermore we obtain the multiplicity of solutions for a related linear perturbation problem.