Multiple Positive Solutions and Estimates of Extremal Values for a Nonlocal Problem with Critical Sobolev Exponent and Concave-Convex Nonlinearities
We are concerned with the following nonlocal problem involving critical Sobolev exponent −a−b∫Ω∇u2dxΔu=λuq−2u+δu2u,x∈Ω,u=0,x∈∂Ω, where Ω is a smooth bounded domain in ℝ4, a,b>0, 1<q<2, δ, and λ are positive parameters. We prove the existence of two positive solutions and obtain uniform esti...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2022-01-01
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| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/2022/1011342 |
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| Summary: | We are concerned with the following nonlocal problem involving critical Sobolev exponent −a−b∫Ω∇u2dxΔu=λuq−2u+δu2u,x∈Ω,u=0,x∈∂Ω, where Ω is a smooth bounded domain in ℝ4, a,b>0, 1<q<2, δ, and λ are positive parameters. We prove the existence of two positive solutions and obtain uniform estimates of extremal values for the problem. Moreover, the blow-up and the asymptotic behavior of these solutions are also discussed when b↘0 and δ↘0. In the proofs, we apply variational methods. |
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| ISSN: | 2314-8888 |