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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De Gruyter
2021
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oai:doaj.org-article:d207967a08694081a99e2a46891bbc0b2021-12-05T14:10:53ZOn quasilinear elliptic problems with finite or infinite potential wells2391-545510.1515/math-2021-0053https://doaj.org/article/d207967a08694081a99e2a46891bbc0b2021-08-01T00:00:00Zhttps://doi.org/10.1515/math-2021-0053https://doaj.org/toc/2391-5455We 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 .Liu ShiboDe Gruyterarticleorlicz-sobolev spaceφ-laplaciancritical points(ps)c sequence35j2035j6035j70MathematicsQA1-939ENOpen Mathematics, Vol 19, Iss 1, Pp 971-989 (2021) |
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EN |
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orlicz-sobolev space φ-laplacian critical points (ps)c sequence 35j20 35j60 35j70 Mathematics QA1-939 |
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orlicz-sobolev space φ-laplacian critical points (ps)c sequence 35j20 35j60 35j70 Mathematics QA1-939 Liu Shibo On quasilinear elliptic problems with finite or infinite potential wells |
| description |
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 . |
| format |
article |
| author |
Liu Shibo |
| author_facet |
Liu Shibo |
| author_sort |
Liu Shibo |
| title |
On quasilinear elliptic problems with finite or infinite potential wells |
| title_short |
On quasilinear elliptic problems with finite or infinite potential wells |
| title_full |
On quasilinear elliptic problems with finite or infinite potential wells |
| title_fullStr |
On quasilinear elliptic problems with finite or infinite potential wells |
| title_full_unstemmed |
On quasilinear elliptic problems with finite or infinite potential wells |
| title_sort |
on quasilinear elliptic problems with finite or infinite potential wells |
| publisher |
De Gruyter |
| publishDate |
2021 |
| url |
https://doaj.org/article/d207967a08694081a99e2a46891bbc0b |
| work_keys_str_mv |
AT liushibo onquasilinearellipticproblemswithfiniteorinfinitepotentialwells |
| _version_ |
1718371630247313408 |