Abstract
We consider positive solutions of the Dirichlet problem for the fractional Laplace equation with singular nonlinearity
where \(s\in (0,1)\), \(\alpha >0\) and \(\Omega \subset \mathbb R^N\) is a bounded domain with smooth boundary \(\partial \Omega \) and \(N>2s.\) Under some appropriate assumptions of \(\alpha , p, \mu \) and K, we obtain the existence of multiple weak solutions, and among them, including the minimal solution and a ground state solution. Radial symmetry of \( C^{1,1}_{loc}\cap L^{\infty }\) solutions are also established for subcritical exponent p when the domain is a ball. Nonexistence of \( C^{1,1}\cap L^{\infty }\) solutions are obtained for star-shaped domain under a condition of K.
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The second author is supported by the Natural Science Foundation of Hunan Province, China (Grant No. 2022JJ30118).
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Wang, J., Du, Z. Dirichlet Problems for Fractional Laplace Equations with Singular Nonlinearity. Qual. Theory Dyn. Syst. 23, 42 (2024). https://doi.org/10.1007/s12346-023-00900-1
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DOI: https://doi.org/10.1007/s12346-023-00900-1
Keywords
- Fractional Laplacian
- Singular nonlinearity
- Positive solutions
- Existence
- Nonexistence
- Nehari manifold
- Radial symmetry