On quasilinear elliptic problems with finite or infinite potential wells
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 conditi...
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| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
De Gruyter
2021
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| Materias: | |
| Acceso en línea: | https://doaj.org/article/d207967a08694081a99e2a46891bbc0b |
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| Sumario: | 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 . |
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