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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Online-Zugang: | https://arxiv.org/abs/2601.05959 |
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| _version_ | 1866912812242042880 |
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| author | Ferraz, Diego |
| author_facet | Ferraz, Diego |
| contents | This paper introduces new variational methods centered on the direct application of a profile decomposition theorem for bounded sequences in Sobolev spaces. We employ these methods to prove the existence of ground state solutions for a class of semilinear elliptic equations in $\mathbb{R}^N$ with critical Sobolev growth, set in an asymptotically periodic framework where the coefficients converge to periodic functions at infinity. Our approach successfully addresses highly general nonlinearities, including a subcritical term that does not need to satisfy the classical Ambrosetti-Rabinowitz condition and a critical term that extends far beyond the standard pure power assumption to include functions with oscillatory behavior. We prove the existence of ground states under two alternative conditions: either a strict energy gap between the minimax levels of the original and asymptotic problems or a direct energy comparison between the associated functionals. Some restrictive assumptions, such as specific decay rates for the coefficients or monotonicity properties of the nonlinearities, are not required in our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_05959 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Application of a profile decomposition theorem to elliptic equations with critical growth Ferraz, Diego Analysis of PDEs 35J61, 35B33, 58E05 This paper introduces new variational methods centered on the direct application of a profile decomposition theorem for bounded sequences in Sobolev spaces. We employ these methods to prove the existence of ground state solutions for a class of semilinear elliptic equations in $\mathbb{R}^N$ with critical Sobolev growth, set in an asymptotically periodic framework where the coefficients converge to periodic functions at infinity. Our approach successfully addresses highly general nonlinearities, including a subcritical term that does not need to satisfy the classical Ambrosetti-Rabinowitz condition and a critical term that extends far beyond the standard pure power assumption to include functions with oscillatory behavior. We prove the existence of ground states under two alternative conditions: either a strict energy gap between the minimax levels of the original and asymptotic problems or a direct energy comparison between the associated functionals. Some restrictive assumptions, such as specific decay rates for the coefficients or monotonicity properties of the nonlinearities, are not required in our results. |
| title | Application of a profile decomposition theorem to elliptic equations with critical growth |
| topic | Analysis of PDEs 35J61, 35B33, 58E05 |
| url | https://arxiv.org/abs/2601.05959 |