some existence results for the equation

Weighted Sobolev embedding with unbounded and decaying radial potentials

“…The proof is finished. 2 Lemma 2 with p = 2 was due to [16], and was reproved by somewhat different arguments in [19,26]. It should be pointed out that when p = q, i.e., c = −p, the inequality (2.2) holds for all u ∈ D 1,p (R N ).…”

Sobolev type embedding and quasilinear elliptic equations with radial potentials

“…Our direct proof here follows the idea of [28] where the case that α = 0 and p = 2 was considered. The above proof is also valid for the case p = q and then a weighted version of the Hardy inequality [12] is obtained.…”

“…The motivation for our study here is twofold. First of all, there are the classical works in [7,18,23,24,28,33,34] which give stronger results for existence of radially symmetric solutions, and classical works in [20][21][22]34] which assert stronger embedding results for functions in radially symmetric class. There has been a revived interest in recent years to further explore along this line of work (e.g., [5,8,13,14,29,31,32,35]).…”

On the inhomogeneous nonlinear Schrödinger equation with harmonic potential and unbounded coefficient

“…When N 3, ω > 0, b 0 and 2 + 2b/(N − 1) < p < 2N/(N − 2) + 2b/(N − 1), one can use Rother's inequality [23] to prove Theorem 4.2 directly following an argument of Sintzoff et al [25]. However, it seems that the method by Sintzoff et al [25] …”