Sobolev type embedding and quasilinear elliptic equations with radial potentials

Abstract

MSC: 35J05 35J20 35J60 58C20 Keywords: Quasilinear elliptic equations Unbounded or decaying potentials Sobolev type embeddingWe study the existence of nontrivial radial solutions for quasilinear elliptic equations with unbounded or decaying radial potentials. The existence results are based upon several new embedding theorems we establish in the paper for radially symmetric functions.

On the existence of solutions for quasilinear elliptic problems with radial potentials on exterior ball

“…Remark This version of radial inequality was given in and its proof is similar to Lemma 1 in , which was for . The case is known as Ni's inequality .…”

“…We begin with two radial inequalities, which are essential tools in our arguments. Lemma ( ) Let be such that . Then there exists such that for all , the completion of under it holds that …”

“…Proof We follow the arguments in , and give the proof for completeness. We consider the case .…”

Symmetric solutions for a class of singular biharmonic elliptic systems involving critical exponents

“…Besides, we also mention that when μ = s = 0 and the right-hand side term |x| −s u 2 * (s)−1 is replaced by u q−1 (1 < q < 2N N−2 or q = 2N N−2 ) in (1.2), some elegant results of G-symmetric solutions of (1.2) were established in [10][11][12]. Finally, when G = O(N), we remark that Su and Wang [13] proved the existence of nontrivial radial solutions for a class of quasilinear singular equations such as (1.2) with radial potentials by establishing several new embedding theorems.…”

Existence and regularity of positive solutions of a degenerate elliptic problem

“…Such kind of embeddings of weighted Sobolev spaces with different weights have been studied by some authors, see . Especially, some results of embeddings with power weights in or can be found in .…”