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This paper focuses on the scalar field equation D^{2Î±}_x uâÎ½uâu_{yy}=f(u), where Î±â(0,1), (x,y)âR^NÃ(âL,L)âR^{N+1}, with Nâ¥1 and D^{2Î±}_ x stands for the fractional Laplacian. By using several variational methods, we establish the existence, long behavior, and multiplicity of solutions of this equation under the Dirichlet and Neumann boundary conditions.
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