On supercritical Sobolev type inequalities and related elliptic equations

Abstract

Sobolev type embeddings for radial functions into variable exponent Lebesgue spaces are studied. In particular, the following inequality is proved: let $$B \subset \mathbb R^N, N \ge 3$$ , be the unit ball, and let $$H_{0,\mathrm rad}^1(B)$$ denote the first order Sobolev space of radial functions, and $$2^* = 2N/(N-2)$$ the corresponding critical Sobolev embeddding exponent. Let $$r = |x|$$ , and $$p(r)=2^*+r^{\alpha }$$ , with $$\alpha > 0$$ ; then 0.1 $$\begin{aligned} \sup \left\{ \int _{B} |u|^{p(r)} \; dx \big | u \in H^1_{0,\mathrm{rad}}(B), \Vert \nabla u \Vert _2=1 \right\} < +\infty . \end{aligned}$$ We point out that the growth of p(r) is strictly larger than $$2^*$$ , except in the origin. Furthermore, we show that for $$p(r) = 2^* + r^\alpha $$ , with $$0< \alpha < \min \{\frac{N}{2};N-2\}$$ , the supremum in (0.1) is attained. Finally, we prove that associated elliptic equations admit nontrivial radial solutions. This is somewhat surprising since the nonlinearities have strictly supercritical growth except in the origin.