Strauss's Radial Compactness and Nonlinear Elliptic Equation Involving a Variable Critical Exponent. (17th September 2018) - Hashizume, Masato; Sano, Megumi
Abstract : We study existence of a nontrivial solution of- Δ p u ( x ) + u ( x ) p - 1 = u ( x ) q ( x ) - 1, u ( x ) ≥ 0, x ∈ R N, u ∈ W r a d 1, p ( R N ), under some conditions onq ( x ), especially, l i m i n f | x | → ∞ q ( x ) = p . Concerning this problem, we firstly consider compactness and noncompactness for the embedding fromW r a d 1, p ( R N ) toL q ( x ) ( R N ) . We point out that the decaying speed ofq ( x ) at infinity plays an essential role on the compactness. Secondly, by applying the compactness result, we show the existence of a nontrivial solution of the elliptic equation.