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 ⊂ R N , N ≥ 3, be the unit ball, and let H 1 0,rad (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 α , with α > 0; thenWe 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 α , with 0 < α < min{ 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. Mathematics Subject Classification35J20 · 35J25 · 35J50 Communicated by A. Malchiodi. Research supported in part by INCTmat/MCT/Brazil,

Hardy-Sobolev type inequality and supercritical extremal problem

“…This section is devoted to prove the Theorem 1.2. The proof is based in the modified Bliss function introduced by [6] and follows some ideas in [5,13]. Firstly, for each 0 < R ≤ ∞, we define…”

“…For instance, we cite [6,[8][9][10]12,17,18] and references therein. As will be seen later our approach will be more in the line of [13,19]. Then, inspired by the above results, our goals are the following:…”

“…Let B ⊂ R N , N ≥ 3, be the unit ball, and denote H 1 0,rad (B) as the first order Sobolev space of radial functions, and 2 * = 2N/(N − 2) the corresponding critical Sobolev embeddding exponent. In the recent paper [13], J.M doÓ et al investigated Sobolev type embeddings for radial functions into variable exponent Lebesgue spaces proving that sup…”

“…We emphasize that the class of operators in (13) includes, when acting in symmetric function defined in ball B R ⊂ R N , the Laplacian, p-Laplacian and k-Hessian operators. This kind of problem has received attention of some authors in recent years [6, 8-10, 12, 17, 18].…”

“…In [6], Figueiredo et al consider the Brezis-Nirenberg type problems and, in particular, for ϕ(r) = p * the above problem doesn't admit non-trivial solution. In the above problem (13), following [13], we propose a new kind of perturbation which also allows us to produce existence of solution as well as the classical Brezis-Nirenberg's perburbation [5].…”