On quasilinear elliptic problems with finite or infinite potential wells

Abstract

We consider quasilinear elliptic problems of the form −div(ϕ(∣∇u∣)∇u)+V(x)ϕ(∣u∣)u=f(u),u∈W1,Φ(RN),-{\rm{div}}\hspace{0.33em}(\phi \left(| \nabla u| )\nabla u)+V\left(x)\phi \left(| u| )u=f\left(u),\hspace{1.0em}u\in {W}^{1,\Phi }\left({{\mathbb{R}}}^{N}), where ϕ\phi and ff satisfy suitable conditions. The positive potential V∈C(RN)V\in C\left({{\mathbb{R}}}^{N}) exhibits a finite or infinite potential well in the sense that V(x)V\left(x) tends to its supremum V∞≤+∞{V}_{\infty }\le +\infty as ∣x∣→∞| x| \to \infty . Nontrivial solutions are obtained by variational methods. When V∞=+∞{V}_{\infty }=+\infty , a compact embedding from a suitable subspace of W1,Φ(RN){W}^{1,\Phi }\left({{\mathbb{R}}}^{N}) into LΦ(RN){L}^{\Phi }\left({{\mathbb{R}}}^{N}) is established, which enables us to get infinitely many solutions for the case that ff is odd. For the case that V(x)=λa(x)+1V\left(x)=\lambda a\left(x)+1 exhibits a steep potential well controlled by a positive parameter λ\lambda , we get nontrivial solutions for large λ\lambda .

Keywords

• orlicz-sobolev space
• φ-laplacian
• critical points
• (ps)c sequence
• 35j20
• 35j60
• 35j70