Global boundedness for Generalized Schrödinger-Type Double Phase Problems in $\mathbb{R}^N$ and Applications to Supercritical Double Phase Problems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917426758680576 |
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| author | Ha, Hoang Hai Ho, Ky Quan, Bui The Sim, Inbo |
| author_facet | Ha, Hoang Hai Ho, Ky Quan, Bui The Sim, Inbo |
| contents | We establish two global boundedness results for weak solutions to generalized Schrödinger-type double phase problems with variable exponents in $\mathbb{R}^N$ under new critical growth conditions optimally introduced in [26, 32]. More precisely, for the case of subcritical growth, we employ the De Giorgi iteration with a suitable localization method in $\mathbb{R}^N$ to obtain a-priori bounds. As a byproduct, we derive the decay property of weak solutions. For the case of critical growth, using the De Giorgi iteration with a localization adapted to the critical growth, we prove the global boundedness. As an interesting application of these results, the existence of weak solutions for supercritical double phase problems is shown. These results are new even for problems with constant exponents in $\mathbb{R}^N$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_13434 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global boundedness for Generalized Schrödinger-Type Double Phase Problems in $\mathbb{R}^N$ and Applications to Supercritical Double Phase Problems Ha, Hoang Hai Ho, Ky Quan, Bui The Sim, Inbo Analysis of PDEs 35B45, 35B65, 35J20, 35J60, 46E35 We establish two global boundedness results for weak solutions to generalized Schrödinger-type double phase problems with variable exponents in $\mathbb{R}^N$ under new critical growth conditions optimally introduced in [26, 32]. More precisely, for the case of subcritical growth, we employ the De Giorgi iteration with a suitable localization method in $\mathbb{R}^N$ to obtain a-priori bounds. As a byproduct, we derive the decay property of weak solutions. For the case of critical growth, using the De Giorgi iteration with a localization adapted to the critical growth, we prove the global boundedness. As an interesting application of these results, the existence of weak solutions for supercritical double phase problems is shown. These results are new even for problems with constant exponents in $\mathbb{R}^N$. |
| title | Global boundedness for Generalized Schrödinger-Type Double Phase Problems in $\mathbb{R}^N$ and Applications to Supercritical Double Phase Problems |
| topic | Analysis of PDEs 35B45, 35B65, 35J20, 35J60, 46E35 |
| url | https://arxiv.org/abs/2504.13434 |