Nonlinear Schrödinger Equations With Unbounded and Decaying Radial Potentials

Abstract

Abstract. We establish some embedding results of weighted Sobolev spaces of radially symmetric functions. The results then are used to obtain ground state solutions of nonlinear Schrödinger equations with unbounded and decaying radial potentials. Our work unifies and generalizes many existing partial results in the literature.

Quasilinear elliptic equations on $${\mathbb{R}^{N}}$$ with singular potentials and bounded nonlinearity

“…We refer to [13,14] and the references therein for more comments and comparisons and some historical results. There are few works concerned elliptic problems on which the embedding of the form similar to Theorem 2.1 is applied.…”

“…When radial potentials are involved, the compactness of the embeddings may be valid for a wider range of q. In [13,14], the authors developed techniques and ideas in establishing weighted Sobolev type embeddings from W 1,p r (R N ; V ) into L q (R N ; Q) with singular radial potentials V and Q for p q. The embedding results in [13,14] include some cases that the embedding …”

“…In [12], the authors further explored the effects of the potentials and extend the techniques and ideas developed in [13,14] to establish the embedding W 1,p r (R N ; V ) → L q (R N ; Q) with 1 < q < p and then to study (P) with sub-p-linear nonlinearity. In this paper, we study (P) with bounded nonlinearity f by applying a compact embedding from W 1,p r (R N ; V ) into L 1 (R N ; Q), which will be stated in Sect.…”

Weighted Sobolev embedding with unbounded and decaying radial potentials

“…When p = 2, these type equations have been studied recently (e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13][14][21][22][23]25,27,28,30,32]), and in [15] a quasilinear problem in bounded domains was considered with Hardy type potentials. The variational framework for (1.1) requires weighted Sobolev type embedding and in [29] we have studied the case for p = 2. There seems to be little work on the case p = 2 (see [20,24]), to the best of our knowledge.…”

“…In this paper, we consider the general case for p by further extending and developing the ideas and techniques in our recent work [29] in which some embedding results was established for the case p = 2. For some cases, our results in the current paper are new even for the case of p = 2.…”