Existence, Nonexistence, and Symmetry of Positive Solutions for Fractional Laplacian Problems

This paper studies the properties of solutions to a class of elliptic and parabolic problems involving the fractional Laplacian. By applying the mountain pass theorem, we prove the existence of bounded classical positive solutions in the subcritical regime. Moreover, using the method of moving planes, we establish that these solutions are symmetric or monotone in the first variable. In contrast, we show that no such solutions exist in the supercritical or negative exponent cases. An analysis of the asymptotic behavior of solutions at infinity provides further insight into their profiles, which supports applications to real-world problems. The approaches developed in this work can also be extended to a wider range of nonlocal elliptic and parabolic equations, including those with more general operators and nonlinearities.