Strauss's radial compactness and nonlinear elliptic equation involving a variable critical exponent (Q1624137)

Summary: We study existence of a nontrivial solution of \(-\Delta_p u(x) + u(x)^{p - 1} = u(x)^{q(x) - 1}\), \(u(x) \geq 0\), \(x \in \mathbb{R}^N\), \(u \in W_{\operatorname{rad}}^{1, p}(\mathbb{R}^N)\), under some conditions on \(q(x)\), especially, \(\liminf_{| x | \rightarrow \infty} q(x) = p\). Concerning this problem, we firstly consider compactness and noncompactness for the embedding from \(W_{\operatorname{rad}}^{1, p}(\mathbb{R}^N)\) to \(L^{q(x)}(\mathbb{R}^N)\). We point out that the decaying speed of \(q(x)\) 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.