On quasilinear elliptic problems with finite or infinite potential wells

Abstract

Abstract We consider quasilinear elliptic problems of the form − div ( ϕ ( ∣ ∇ u ∣ ) ∇ u ) + V ( x ) ϕ ( ∣ u ∣ ) u = f ( u ) , u ∈ W 1 , Φ ( R N ) , -{\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 f f satisfy suitable conditions. The positive potential V ∈ C ( R N ) 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 W 1 , Φ ( R N ) {W}^{1,\Phi }\left({{\mathbb{R}}}^{N}) into L Φ ( R N ) {L}^{\Phi }\left({{\mathbb{R}}}^{N}) is established, which enables us to get infinitely many solutions for the case that f f is odd. For the case that V ( x ) = λ a ( x ) + 1 V\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 .