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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}),whereϕ\phiandffsatisfy suitable conditions. The positive potentialV∈C(RN)V\in C\left({{\mathbb{R}}}^{N})exhibits a finite or infinite potential well in the sense thatV(x)V\left(x)tends to its supremumV∞≤+∞{V}_{\infty }\le +\inftyas∣x∣→∞| x| \to \infty. Nontrivial solutions are obtained by variational methods. WhenV∞=+∞{V}_{\infty }=+\infty, a compact embedding from a suitable subspace ofW1,Φ(RN){W}^{1,\Phi }\left({{\mathbb{R}}}^{N})intoLΦ(RN){L}^{\Phi }\left({{\mathbb{R}}}^{N})is established, which enables us to get infinitely many solutions for the case thatffis odd. For the case thatV(x)=λa(x)+1V\left(x)=\lambda a\left(x)+1exhibits a steep potential well controlled by a positive parameterλ\lambda, we get nontrivial solutions for largeλ\lambda.

On quasilinear elliptic problems with finite or infinite potential wells