Open Access

Weighted Sobolev spaces and ground state solutions for quasilinear elliptic problems with unbounded and decaying potentials

Boundary Value Problems20132013:189

DOI: 10.1186/1687-2770-2013-189

Received: 13 May 2013

Accepted: 8 August 2013

Published: 23 August 2013

Abstract

In this paper, we prove some continuous and compact embedding theorems for weighted Sobolev spaces, and consider both a general framework and spaces of radially symmetric functions. In particular, we obtain some a priori Strauss-type decay estimates. Based on these embedding results, we prove the existence of ground state solutions for a class of quasilinear elliptic problems with potentials unbounded, decaying and vanishing.

MSC:35J20, 35J60, 35Q55.

Keywords

weighted Sobolev spaces unbounded and decaying potentials quasilinear elliptic problems

1 Introduction

In this paper, we consider the following quasilinear elliptic problems:
{ p u + V ( x ) | u | p 2 u = K ( x ) | u | q 1 u , x R N , | u ( x ) | 0 as  | x | , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ1_HTML.gif
(1.1)

where N > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq1_HTML.gif, 1 < p N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq2_HTML.gif, p u = div ( | u | p 2 u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq3_HTML.gif, V ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq4_HTML.gif and K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq5_HTML.gif are nonnegative measurable functions, and may be unbounded, decaying and vanishing.

Recently, these type elliptic equations have been widely studied. As p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq6_HTML.gif, if V ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq4_HTML.gif and K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq5_HTML.gif satisfied
sup x R N K ( x ) < , inf x R N V ( x ) > 0 and lim | x | + V ( x ) = + , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ2_HTML.gif
(1.2)
Rabinowitz [1] proved the existence of a ground state solution for problem (1.1). Further, when V ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq4_HTML.gif has a positive lower bound and K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq5_HTML.gif is bounded, using critical point theory, del Pino and Felmer [2, 3] obtained that problem (1.1) might also not have a ground state solution. If V ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq4_HTML.gif and K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq5_HTML.gif satisfied
sup x R N ( 1 + | x | ) 2 α V ( x ) > 0 and sup x R N ( 1 + | x | ) β K ( x ) < + , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ3_HTML.gif
(1.3)
where 0 < α < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq7_HTML.gif, β > ( 1 α ) ( N q ( N 2 1 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq8_HTML.gif, Ambrosetti, Felli and Malchiodi [4], Ambrosetti, Malchiodi and Ruiz [5] obtained the ground and bound state solutions for problem (1.1). In fact, condition (1.3) implies that V ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq4_HTML.gif tends to zero at infinity. In particular, when the potentials V ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq4_HTML.gif and K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq5_HTML.gif are neither bound away from zero nor bounded from above, Bonheure and Mercuri [7] proved the existence of the ground state solution for problem (1.1) and obtained the decay estimates by using the Moser iteration scheme. For the radially symmetric space D r a d 1 , 2 ( R N ) = { u ( x ) = u ( | x | ) , u L 2 N N 2 ( R N ) , u L 2 ( R N , R N ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq9_HTML.gif, Strauss [8] obtained the famous Strauss inequality
| u ( x ) | C | x | N 2 2 u D 1 , 2 ( R N ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ4_HTML.gif
(1.4)

for a.e. x R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq10_HTML.gif and u D r a d 1 , 2 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq11_HTML.gif. Berestycki and Lions [9] proved the existence of a ground state solution for some scalar equation. In 2007, as the potentials V ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq4_HTML.gif and K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq5_HTML.gif are radially symmetric, Su, Wang and Willem [10] obtained the existence of a ground state solution for problem (1.1) with V ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq4_HTML.gif and K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq5_HTML.gif unbounded and decaying.

For p 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq12_HTML.gif, to the best of our knowledge, it seems to be little work done. do Ó and Medeiros [11] obtained the existence of a ground state solution for some p-Laplacian elliptic problems in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq13_HTML.gif. Zhang [12] considered a mountain pass characterization of the ground state solution for p-Laplacian elliptic problems with critical growth. When V ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq4_HTML.gif and K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq5_HTML.gif are radially symmetric, Su, Wang and Willem [13] considered the following quasilinear elliptic problem:
{ p u + V ( | x | ) | u | p 2 u = K ( | x | ) | u | q 1 u , x R N , | u ( x ) | 0 as  | x | , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ5_HTML.gif
(1.5)

and proved some embedding results of a weighted Sobolev space for a radially symmetric function, and obtained the existence of ground and bound state solutions for problem (1.5).

In this paper, for the general potentials V ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq4_HTML.gif and K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq5_HTML.gif allowing to be unbounded or vanish at infinity, we obtain some necessary and sufficient conditions about some continuous and compact embeddings for the weighted Sobolev space. Based on variational methods and some compact embedding results, we obtain the existence of ground and bounded state solutions for problem (1.1). On the other hand, for the radial potentials V ( | x | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq14_HTML.gif and K ( | x | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq15_HTML.gif, in [13] various conditions have been considered for V ( | x | ) , K ( | x | ) | x | α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq16_HTML.gif with α R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq17_HTML.gif. Our first purpose is to consider V ( | x | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq14_HTML.gif and K ( | x | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq15_HTML.gif whose behavior can be described by a more general class of functions. Furthermore, we obtain some a priori Strauss-type decay estimates and some continuous and compact embedding results for the radial symmetric weighted Sobolev space. The results then are used to obtain ground and bound state solutions for problem (1.5).

It is worth pointing out that we provide here a unified approach what conditions the potentials V ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq4_HTML.gif and K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq5_HTML.gif should satisfy so that problem (1.1) and problem (1.5) have ground and bound state solutions, respectively. We extend the results in [13] to a large class of weighted Sobolev embeddings and obtain some new embedding theorems for the general potentials and radially symmetric potentials.

The paper is organized as follows. In Section 2, we collect some results. In Section 3, we obtain some embedding results for the general potentials. In Section 4, we focus on radially symmetric potentials and prove the continuous and compact embeddings. Section 5 is devoted to the existence of ground and bound state solutions for problem (1.1) and problem (1.5), respectively.

2 Preliminaries

In this section, let C 0 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq18_HTML.gif denote the collection of smooth functions with compact support. Let D 1 , p ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq19_HTML.gif be the completion of C 0 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq20_HTML.gif under the norm
u D 1 , p ( R N ) = ( R N | u | p d x ) 1 p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ6_HTML.gif
(2.1)

We write C 0 , r ( R N ) = { u C 0 ( R N ) | u ( x ) = u ( | x | ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq21_HTML.gif and D r 1 , p ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq22_HTML.gif is the corresponding subspace of a radial function for D 1 , p ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq19_HTML.gif.

Define, for p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq23_HTML.gif and q 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq24_HTML.gif,
L V p ( R N ) = { u : R N R | u  is measurable,  R N V ( x ) | u | p d x < } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ7_HTML.gif
(2.2)
and
L K q ( R N ) = { u : R N R | u  is measurable,  R N K ( x ) | u | q d x < } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ8_HTML.gif
(2.3)
Then we have W V 1 , p ( R N ) = D 1 , p ( R N ) L V p ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq25_HTML.gif, which is a Banach space under the uniformly convex norm
u W V 1 , p ( R N ) = ( R N ( | u | p + V ( x ) | u | p ) d x ) 1 p = ( u D 1 , p ( R N ) p + u L V p ( R N ) p ) 1 p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ9_HTML.gif
(2.4)

where u L ν p ( R N ) = ( R N V ( x ) | u | p d x ) 1 p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq26_HTML.gif.

Now, we state some Hardy inequalities.

Lemma 2.1 [14]

If N > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq1_HTML.gif, 1 < p < N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq27_HTML.gif, we have
R N | u | p d x ( N p p ) p R N | u | p | x | p d x for every u D 1 , p ( R N ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equa_HTML.gif

Lemma 2.2 [13]

If N > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq1_HTML.gif, 1 < p < N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq27_HTML.gif, p q + 1 < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq28_HTML.gif and q + 1 = p ( N + c ) N p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq29_HTML.gif for some c [ p , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq30_HTML.gif, there exists C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq31_HTML.gif such that
( R N | u | p d x ) C ( R N | x | c | u | q + 1 d x ) p q + 1 for every  u D r 1 , p ( R N ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equb_HTML.gif

Lemma 2.3 [15]

If N = p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq32_HTML.gif, Ω B R ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq33_HTML.gif or Ω B R c ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq34_HTML.gif, then
Ω | u | N d x ( N 1 N ) N Ω | u | N ( | x | log R | x | ) N d x for every  u D 0 1 , N ( Ω ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equc_HTML.gif

where B R ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq35_HTML.gif is the ball in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq36_HTML.gif centered at 0 with radius R, B R c ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq37_HTML.gif denotes the complement of B R ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq35_HTML.gif.

3 Embedding results for general potentials

In this section, we derive a tool giving the embedding results on a piece of the partition. We consider the possible relation between the behavior of V ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq4_HTML.gif and K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq5_HTML.gif.

Lemma 3.1 Let Ω R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq38_HTML.gif be smooth possibly unbounded and
(H) 1 < p < N , p q + 1 < p , p = N p N p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equd_HTML.gif
V ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq4_HTML.gif and K ( x ) : R N R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq39_HTML.gif be measure nonnegative functions. V ( x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq40_HTML.gif a.e. in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq36_HTML.gif.
  1. (a)
    If there exists α [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq41_HTML.gif such that
    N p ( q + 1 ) ( N ( 1 α ) p ) 1 and Ω ( K ( x ) ( V ( x ) ) α ( q + 1 ) p ) N p N p ( q + 1 ) ( N ( 1 α ) p ) d x < , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Eque_HTML.gif
     
then the embedding
W V 1 , p ( Ω ) L K q + 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equf_HTML.gif
is continuous;
  1. (b)
    If there exist α 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq42_HTML.gif and m ( p , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq43_HTML.gif such that
    α 1 ( q + 1 ) p + ( 1 α 1 ) ( q + 1 ) m 1 and K ( x ) ( V ( x ) ) α 1 ( q + 1 ) p L p m p m α 1 m ( q + 1 ) ( 1 α 1 ) p ( q + 1 ) ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equg_HTML.gif
     
and α 2 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq44_HTML.gif such that
N p ( q + 1 ) ( N ( 1 α 2 ) p ) 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equh_HTML.gif
and ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq45_HTML.gif, R ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq46_HTML.gif such that
Ω B R ε [ K ( x ) ( V ( x ) ) α 2 ( q + 1 ) p ] N p N p ( q + 1 ) ( N ( 1 α 2 ) p ) d x < ε , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equi_HTML.gif
then the embedding
W V 1 , p ( Ω ) L K q + 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equj_HTML.gif

is compact.

Proof (a) Since there exists α [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq41_HTML.gif such that N p ( q + 1 ) ( N ( 1 α ) p ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq47_HTML.gif, we have
N p N p ( q + 1 ) ( N ( 1 α ) p ) [ 1 , + ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equk_HTML.gif
By Hölder’s inequality and Ω [ K ( x ) ( V ( x ) ) α ( q + 1 ) p ] N p N ( q + 1 ) ( N ( 1 α ) p ) d x < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq48_HTML.gif, we obtain
Ω K ( x ) | u | q + 1 d x = Ω K ( x ) ( V ( x ) ) α ( q + 1 ) p ( V ( x ) ) α ( q + 1 ) p | u | α ( q + 1 ) | u | ( 1 α ) ( q + 1 ) d x { Ω [ K ( x ) ( V ( x ) ) α ( q + 1 ) p ] N p N ( q + 1 ) ( N ( 1 α ) p ) d x } N ( q + 1 ) ( N ( 1 α ) p ) N p ( Ω V ( x ) | u | p d x ) α ( q + 1 ) p ( Ω | u | N p N P d x ) ( N p ) ( 1 α ) ( q + 1 ) N p C u L V p ( Ω ) α ( q + 1 ) u L p ( Ω ) ( 1 α ) ( q + 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ10_HTML.gif
(3.1)
where ( q + 1 ) ( N ( 1 α ) p ) N p + N p ( q + 1 ) ( N ( 1 α ) p ) N p = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq49_HTML.gif. Since p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq50_HTML.gif is the critical Sobolev exponent, by the Sobolev embedding theorem, we have
Ω K ( x ) | u | q + 1 d x C u W V 1 , p ( Ω ) ( q + 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equl_HTML.gif
Hence, we obtain that the embedding W V 1 , p ( Ω ) L K q + 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq51_HTML.gif is continuous.
  1. (b)
    For any fixed ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq52_HTML.gif, let B R ε ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq53_HTML.gif be the ball in Ω with R ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq54_HTML.gif. Since there exist α 1 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq55_HTML.gif and m ( p , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq43_HTML.gif such that
    ( α 1 p ( q + 1 ) + ( 1 α 1 ) m ( q + 1 ) ) 1 1 and K ( x ) ( V ( x ) ) α 1 ( q + 1 ) p L p m p m α 1 m ( q + 1 ) ( 1 α 1 ) p ( q + 1 ) ( Ω ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equm_HTML.gif
     
arguing as in the proof of (3.1), by the compact embedding of D 0 1 , p ( B R ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq56_HTML.gif into L m ( B R ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq57_HTML.gif, we have
B R ε K ( x ) | u | q + 1 d x C u L V p ( B R ε ) α ( q + 1 ) u L m ( B R ε ) ( 1 α ) ( q + 1 ) C u W V 1 , p ( B R ε ) ( q + 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equn_HTML.gif
Hence, we have
W V 1 , p ( B R ε ) L K q + 1 ( B R ε ) is compact . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ11_HTML.gif
(3.2)
On the domain Ω B R ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq58_HTML.gif, since Ω B R ε [ K ( x ) ( V ( x ) ) α 2 ( q + 1 ) p ] N p N p ( q + 1 ) ( N ( 1 α ) p ) d x < ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq59_HTML.gif, we have
Ω B R ε K ( x ) | u | q + 1 d x C u L V p ( Ω ) α 2 ( q + 1 ) u L p ( Ω ) ( 1 α 2 ) ( q + 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equo_HTML.gif
Assume u n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq60_HTML.gif (weakly) in W V 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq61_HTML.gif, then we have
Ω B R ε K ( x ) | u n | q + 1 d x c ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ12_HTML.gif
(3.3)
Combining (3.2) and (3.3), we have
Ω K ( x ) | u n | q + 1 d x 0 (strongly) as  n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equp_HTML.gif

Hence, we obtain that W V 1 , p ( Ω ) L K q + 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq51_HTML.gif is compact. □

Now, we state our main theorem in this section.

Consider a finite partition M = i { Ω i } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq62_HTML.gif of R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq36_HTML.gif and Ω i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq63_HTML.gif is unbounded.

Theorem 3.2 If condition (H) is satisfied for any i N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq64_HTML.gif and Ω i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq63_HTML.gif, assume that
  1. (a)
    there exist α i [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq65_HTML.gif such that
    N p ( q + 1 ) ( N ( 1 α i ) p ) 1 and Ω i [ K ( x ) ( V ( x ) ) α i ( q + 1 ) p ] N p N p ( q + 1 ) ( N ( 1 α i ) p ) d x < + , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equq_HTML.gif
     
then the embedding W V 1 , p ( R N ) L K q + 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq66_HTML.gif is continuous;
  1. (b)
    there exist α 1 , i [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq67_HTML.gif and m i ( p , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq68_HTML.gif such that
    α 1 , i ( q + 1 ) p + ( 1 α 1 , i ) ( q + 1 ) m i 1 and K ( x ) ( V ( x ) ) α 1 , i p ( q + 1 ) L m i p p m i α 1 , i m i ( q + 1 ) ( 1 α 1 , i ) p ( q + 1 ) ( Ω i ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equr_HTML.gif
     
and α 2 , i ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq69_HTML.gif such that
N p ( q + 1 ) ( N ( 1 α 2 , i ) p ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equs_HTML.gif
and
ε > 0 , R i , ε > 0 , Ω i B R i , ε [ K ( x ) ( V ( x ) ) α 2 , i p ( q + 1 ) ] N p N p ( q + 1 ) ( N ( 1 α 2 , i ) p ) d x < ε , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equt_HTML.gif

then the embedding W V 1 , p ( R N ) L K q + 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq66_HTML.gif is compact.

Proof (a) Arguing as in the proof of (a) in Lemma 3.1, we obtain
R N K ( x ) | u | q + 1 d x = i Ω i K ( x ) | u | q + 1 d x C i u L V p ( Ω i ) α i ( q + 1 ) u L p ( Ω i ) ( 1 α i ) ( q + 1 ) C i u W V 1 , p ( Ω i ) ( q + 1 ) C u W V 1 , p ( R N ) ( q + 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equu_HTML.gif
Hence, we obtain that W V 1 , p ( R N ) L K q + 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq70_HTML.gif is continuous.
  1. (b)
    ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq45_HTML.gif, R i , ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq71_HTML.gif such that
    Ω i B R i , ε [ K ( x ) ( V ( x ) ) α 2 , i ( q + 1 ) p ] N p N p ( q + 1 ) ( N ( 1 α 2 , i ) p ) d x < ε , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equv_HTML.gif
     
then we have
i Ω i B R i ε [ K ( x ) ( V ( x ) ) α 2 , i ( q + 1 ) p ] N p N p ( q + 1 ) ( N ( 1 α 2 , i ) p ) d x < ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ13_HTML.gif
(3.4)
Arguing as in the proof of (b) in Lemma 3.1, when u n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq72_HTML.gif (weakly) in W V 1 , p ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq73_HTML.gif, we have
i Ω i B R i ε K ( x ) | u | q + 1 d x < ε u n W V 1 , p ( R N ) q + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ14_HTML.gif
(3.5)
By (3.5) and the local compactness in B R i , ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq74_HTML.gif, we obtain that
R N K ( x ) | u | q + 1 d x 0 as  n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equw_HTML.gif

Hence, we have W V 1 , p ( R N ) L K q + 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq66_HTML.gif is compact. □

Remark 3.3 (1) Let K ( x ) L ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq75_HTML.gif and q + 1 p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq76_HTML.gif, and q + 1 < p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq77_HTML.gif, we obtain the standard local Sobolev embedding.
  1. (2)

    Let p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq6_HTML.gif, we obtain that if K ( x ) [ V ( x ) ] r L ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq78_HTML.gif with r = q + 1 2 ( N 2 1 ) N 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq79_HTML.gif, the embedding W V 1 , 2 ( R N ) L K q + 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq80_HTML.gif is compact. This has already been obtained in [6].

     

4 Embedding theorem for a radially symmetric function space

Assume that V ( | x | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq14_HTML.gif and K ( | x | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq15_HTML.gif are radial weights. In [13], Su, Wang and Willem considered for potentials V , K r α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq81_HTML.gif with α R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq82_HTML.gif and obtained some embedding theorems. In this section, we extend some results in [13] to a more general class of functions for r + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq83_HTML.gif, 0+. In particular, we also obtain some embedding theorems for the Sobolev space W V , r 1 , N ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq84_HTML.gif. Theorem 4.5 and Theorem 4.6 are new embedding results.

Following [16, 17], we shall refer to this class as the Hardy-Dieudonne comparison class. Define
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equx_HTML.gif
Then we take the set of all the finite products
C 1 ( + ) = { k = 1 n f k : f k C 1 ( + ) , n N } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equy_HTML.gif
Since C 1 ( + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq85_HTML.gif is not closed with respect to the operation f exp f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq86_HTML.gif, we consider
C 1 ( + ) = { exp c f ( x ) : f C 1 ( + ) , f ( + ) = + , c 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equz_HTML.gif

Then we have C 2 ( + ) = C 1 C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq87_HTML.gif and C 2 ( + ) = { k = 1 n f k : f k C 2 ( + ) , n N } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq88_HTML.gif. The process can be iterated, we have the following.

Definition 4.1 The set C ( + ) = n N C n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq89_HTML.gif is called Hardy-Dieudonne class of functions at +∞. C ( x 0 + ) = { f ( x ) = g ( ( x x 0 ) 1 ) , g C ( + ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq90_HTML.gif is called Hardy-Dieudonne class of functions at x 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq91_HTML.gif.

Now, let V ( | x | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq14_HTML.gif, K ( | x | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq15_HTML.gif be continuous nonnegative functions in ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq92_HTML.gif, and
  1. (V)

    lim inf r V ( r ) V ( r ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq93_HTML.gif and lim inf r 0 + V ( r ) V 0 ( r ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq94_HTML.gif;

     
  2. (K)

    lim sup r K ( r ) K ( r ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq95_HTML.gif and lim sup r 0 + K ( r ) K 0 ( r ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq96_HTML.gif,

     

where V 0 , K 0 C ( 0 + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq97_HTML.gif and V , K C ( + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq98_HTML.gif.

By conditions (V) and (K), we obtain that there exist positive constants r 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq99_HTML.gif, r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq100_HTML.gif, a 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq101_HTML.gif, a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq102_HTML.gif, b 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq103_HTML.gif, b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq104_HTML.gif such that
a 0 V 0 ( r ) V ( r ) , r r 0 , and a V ( r ) V ( r ) , r > r , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ15_HTML.gif
(4.1)
b 0 K 0 ( r ) K ( r ) , r r 0 , and b K ( r ) K ( r ) , r > r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ16_HTML.gif
(4.2)
Now, we define the following two radially symmetric Sobolev spaces:
L K , r q + 1 ( R N ) = { u : R N K ( | x | ) | u | q + 1 d x < , u  is radial } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equaa_HTML.gif
and W V , r 1 , p ( R N ) = { u : u D r 1 , p ( R N ) , R N V ( | x | ) | u | p d x < , u is radial } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq105_HTML.gif under the uniformly convex norm
u W V , r 1 , p ( R N ) = ( R N | u | p d x + R N V ( | x | ) | u | p d x ) 1 p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equab_HTML.gif
Lemma 4.2 Assume that u W V , r 1 , p ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq106_HTML.gif and V ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq4_HTML.gif satisfies condition (V), 1 < p < N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq27_HTML.gif. If
  1. (a) (1)
    p ( N 1 ) ( V 0 ( r ) ) + ( p 1 ) r V 0 ( r ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq107_HTML.gif for 0 < r < r 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq108_HTML.gif, then there exists C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq31_HTML.gif such that
    | u ( r ) | C [ r 1 N ( V 0 ( r ) ) 1 p p ] 1 p u W V , r 1 , p ( R N ) a.e. in  ( 0 , r 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ17_HTML.gif
    (4.3)
     
  2. (2)
    p ( N 1 ) ( V ( r ) ) + ( p 1 ) r V ( r ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq109_HTML.gif for r > r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq110_HTML.gif, then there exists C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq31_HTML.gif such that
    | u ( r ) | C [ r 1 N ( V ( r ) ) 1 p p ] 1 p u W V , r 1 , p ( R N ) a.e. in  ( r , + ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ18_HTML.gif
    (4.4)
     
  3. (b) (1)

    p ( N 1 ) ( V 0 ( r ) ) + ( p 1 ) r V 0 ( r ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq111_HTML.gif for 0 < r < r 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq108_HTML.gif and V 0 1 ( r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq112_HTML.gif, V 0 ( r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq113_HTML.gif, r L ( 0 , r 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq114_HTML.gif, then (4.3) holds.

     
  4. (2)

    p ( N 1 ) ( V 0 ( r ) ) + ( p 1 ) r V 0 ( r ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq111_HTML.gif for r > r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq110_HTML.gif and V 1 ( r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq115_HTML.gif, V ( r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq116_HTML.gif, r L ( r , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq117_HTML.gif, then (4.4) holds.

     
Proof (a)(1) By density, it is enough to prove it for u D ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq118_HTML.gif with support in B γ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq119_HTML.gif. We have
| u ( r ) | p ( V 0 ( r ) ) p 1 p r N 1 p r r 0 | u ( s ) | p 1 | u ( s ) | ( V 0 ( s ) ) p 1 p s N 1 d s ( N 1 ) r r 0 | u ( s ) | p ( V 0 ( s ) ) p 1 p s N 2 d s p 1 p r r 0 ( V 0 ( s ) ) 1 p | u ( s ) | p ( V 0 ( s ) ) s N 1 d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equac_HTML.gif
By Hölder’s inequality and (4.1), there exists C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq31_HTML.gif such that
p r r 0 | u ( s ) | p 1 | u ( s ) | ( V 0 ( s ) ) p 1 p s N 1 d s ( r r 0 | u ( s ) | p s N 1 d s ) 1 p ( r r 0 ( V 0 ( s ) ) | u ( s ) | p s N 1 d s ) p 1 p ω N 1 ( B r 0 ( 0 ) B r ( 0 ) | u | p d x ) 1 p ( B r 0 ( 0 ) B r ( 0 ) [ V 0 ( s ) ] | u | p d x ) p 1 p ω N 1 a 0 p 1 p R N ( | u | p + V ( x ) ) | u | p d x C u W V , r 1 , p ( R N ) p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ19_HTML.gif
(4.5)

where ω N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq120_HTML.gif is the volume of the unit sphere in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq36_HTML.gif.

On the other hand, p ( N 1 ) ( V 0 ( r ) ) + ( p 1 ) r V 0 ( r ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq121_HTML.gif for 0 < r < r 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq108_HTML.gif. By a simple computation, we obtain
( N 1 ) r r 0 | u ( s ) | p ( V 0 ( s ) ) p 1 p s N 1 d s p 1 p r r 0 ( V 0 ( s ) ) 1 p | u ( s ) | p ( V 0 ( s ) ) s N 1 d s > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ20_HTML.gif
(4.6)
Combining (4.5) and (4.6), we have
| u ( r ) | C [ r 1 N ( V 0 ( r ) ) 1 p p ] 1 p u W V , r 1 , p ( R N ) a.e. in  ( 0 , r 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equad_HTML.gif
(2) By density, it is enough to prove it for u D ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq118_HTML.gif with support in B R c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq122_HTML.gif, we have
| u ( r ) | p ( V ( r ) ) p 1 p r N 1 p r + | u ( s ) | p 1 | u ( s ) | ( V ( s ) ) p 1 p s N 1 d s ( N 1 ) r + | u ( s ) | p ( V ( s ) ) p 1 p s N 1 d s p 1 p r + ( V ( s ) ) 1 p | u ( s ) | p ( V 0 ( s ) ) s N 2 d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equae_HTML.gif
By a similar computation as for (4.5) and(4.6), we have
| u ( r ) | p ( V ( r ) ) p 1 p r N 1 C u W V , r 1 , p ( R N ) p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equaf_HTML.gif

and this yields (4.4).

(b)(1) If p ( N 1 ) ( V 0 ( r ) ) + ( p 1 ) r V 0 ( r ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq107_HTML.gif for 0 < r < r 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq108_HTML.gif.

By Hölder’s inequality and Lemma 2.1, we have
( N 1 ) r r 0 | u ( s ) | p ( V 0 ( s ) ) p 1 p s N 2 d s = ( N 1 ) r r 0 | u ( s ) | p 1 ( V 0 ( s ) ) p 1 p | u ( s ) | | s | s N 1 d s ( N 1 ) ( r r 0 | u ( s ) | p ( V 0 ( s ) ) s N 1 d s ) p 1 p ( r r 0 | u ( s ) | p | s | p s N 1 d s ) 1 p ω N 1 ( N 1 ) ( B r 0 ( 0 ) B r ( 0 ) V 0 ( | x | ) | u | p d x ) p 1 p ( R N | u | p d x ) ( p N p ) p ( N 1 ) ω N 1 a 0 p 1 p R N ( | u | p + V ( | x | ) ) | u | p d x C 2 u W V , r 1 , p ( R N ) p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ21_HTML.gif
(4.7)
Similarly, we have
p 1 p r r 0 | u ( s ) | p ( V 0 ( s ) ) 1 p ( V 0 ( s ) ) s N 1 d s = p 1 p r r 0 | u ( s ) | p 1 ( V 0 ( s ) ) p 1 p V 0 1 ( s ) V 0 ( s ) s | u ( s ) | s s N 1 d s C 3 V 0 1 ( s ) V 0 ( s ) s L ( 0 , r 0 ) u W V , r 1 , p ( R N ) p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ22_HTML.gif
(4.8)
Combining (4.5), (4.7) and (4.8), we obtain that (4.3) holds.
  1. (2)

    If p ( N 1 ) ( V 0 ( r ) ) + ( p 1 ) r V 0 ( r ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq123_HTML.gif for r > r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq110_HTML.gif and V 1 ( r ) V ( r ) r L ( r , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq124_HTML.gif, we can argue as in the above proof. □

     

Let p = N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq125_HTML.gif, we consider the Sobolev space W V , r 1 , N ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq126_HTML.gif.

Lemma 4.3 Assume that u W V , r 1 , N ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq127_HTML.gif and V ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq4_HTML.gif satisfies condition (V). If
  1. (a)
    N V 0 ( r ) + r V 0 ( r ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq128_HTML.gif for 0 < r < r 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq108_HTML.gif and V 0 ( r ) V 0 1 ( r ) r L ( 0 , r 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq129_HTML.gif, then there exists C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq31_HTML.gif such that
    | u ( r ) | C ( r 1 N ( V 0 ( r ) ) N 1 N | log 1 r | ) 1 N u W V , r 1 , N ( R N ) a.e. in  ( 0 , r 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equag_HTML.gif
     
  2. (b)
    N V ( r ) + r V ( r ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq130_HTML.gif for r > r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq110_HTML.gif and V ( r ) V 1 ( r ) r L ( r , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq131_HTML.gif, then there exists C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq31_HTML.gif such that
    | u ( r ) | C ( r 1 N ( V ( r ) ) N 1 N | log 1 r | ) 1 N u W V , r 1 , N ( R N ) a.e. in  ( r , ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equah_HTML.gif
     
Proof (a) Arguing as in the proof of (2) of (a) in Lemma 4.2, by Hölder’s inequality and Lemma 2.3, we have
( N 1 ) r r 0 | u ( s ) | ( V 0 ( s ) ) N 1 N s N 2 d s = ( N 1 ) | log 1 r | r r 0 | u ( s ) | N ( V 0 ( s ) ) N 1 N | u ( s ) | s | log 1 s | s N 1 d s ( N 1 ) | log 1 r | ( r r 0 | u ( s ) | N ( V 0 ( s ) ) s N 1 d s ) N 1 N ( r r 0 | u ( s ) | N ( s | log 1 s | ) N s N 1 d s ) 1 N ( N 1 ) | log 1 r | ω N 1 ( B r 0 ( 0 ) B r ( 0 ) | u | N ( V 0 ( | x | ) ) d s ) N 1 N ( R N | u | N d x ) ( N N 1 ) N C | log 1 r | u W V , r 1 , N ( R N ) N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ23_HTML.gif
(4.9)
On the other hand, we have
( N 1 ) N r r 0 | u ( s ) | N ( V 0 ( s ) ) 1 N ( V 0 s ) ) s N 1 d s C V 0 1 ( s ) V 0 ( s ) s L ( 0 , r 0 ) | log 1 r | u W V , r 1 , N ( R N ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ24_HTML.gif
(4.10)
Combining (4.9) and (4.10), we obtain
| u ( r ) | C ( r 1 N ( V 0 ( r ) ) N 1 N | log 1 r | ) 1 N u W V , r 1 , N ( R N ) a.e. in  ( 0 , r 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equai_HTML.gif
  1. (b)

    Similarly, we can argue as in the proof of (a) in Lemma 4.3. □

     
Remark 4.4
  1. (1)
    The previous estimates should be compared with Lemma 1 in [13],
    | u ( x ) | C | x | ( N p ) p u D 1 , p ( R N ) for a.e.  x R N , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equaj_HTML.gif
     
for every u D r a d 1 , p ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq132_HTML.gif;
  1. (2)

    Our results extend Lemma 4 and Lemma 5 in [13], and we obtain the general Strauss-type decay estimates;

     
  2. (3)
    Under the conditions of Lemma 4.2 and Lemma 4.3, we obtain that there exist two comparison functions g 1 , g 2 C ( 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq133_HTML.gif such that
    | u ( r ) | C g 1 u W V , r 1 , p ( R N ) a.e. in  ( 0 , r 0 ) and | u ( r ) | C g 2 u W V , r 1 , p ( R N ) a.e. in  ( r , ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ25_HTML.gif
    (4.11)
     

Now, we state our main embedding theorems in this section.

Theorem 4.5 If V ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq4_HTML.gif and K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq5_HTML.gif satisfy (V) and (K), R N { supp ( V ( x ) ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq134_HTML.gif is relatively compact, K L l o c ( R N { 0 } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq135_HTML.gif.
  1. (a)

    If 1 < p < N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq27_HTML.gif and p q + 1 < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq28_HTML.gif, then the embedding W V , r 1 , p ( R N ) L K q + 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq136_HTML.gif is continuous.

     
  2. (1)

    K ( r ) r N p + ( N p ) ( q + 1 ) p L ( 0 , r 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq137_HTML.gif and K ( r ) r N p + ( N p ) ( q + 1 ) p L ( r , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq138_HTML.gif, or

     
  3. (2)

    K 0 ( r ) V 0 1 ( r ) ( g 1 ( r ) ) ( q + 1 ) p L ( 0 , r 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq139_HTML.gif and K ( r ) V 1 ( r ) ( g 2 ( r ) ) ( q + 1 ) p L ( r , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq140_HTML.gif,

     
  4. (b)

    If p = N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq125_HTML.gif, q + 1 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq141_HTML.gif, K 0 ( r ) ( 1 r | log 1 r | ) N ( g 1 ( r ) ) ( q + 1 ) N L ( 0 , r 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq142_HTML.gif and K ( r ) ( 1 r | log 1 r | ) N ( g 2 ( r ) ) ( q + 1 ) N L ( r , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq143_HTML.gif, then the embedding W V , r 1 , N ( R N ) L K q + 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq144_HTML.gif is continuous.

     
Proof (a)(1) If K ( r ) r N p + ( N p ) ( q + 1 ) p L ( 0 , r 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq145_HTML.gif, we obtain
0 r 0 K ( s ) | u | q + 1 s N 1 d s ω N 1 K ( r ) r N p + ( N p ) ( q + 1 ) p L ( 0 , r 0 ) R N | x | N p ( N 1 ) ( q + 1 ) p | u | q + 1 d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equak_HTML.gif
By Lemma 2.2, we obtain
R N | x | N p + ( N 1 ) ( q + 1 ) p | u | q + 1 d x C ( R N | u | p d x ) ( q + 1 ) p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equal_HTML.gif
Hence, we have
0 r 0 K ( s ) | u | q + 1 s N 1 d s C u W V , r 1 , p ( R N ) q + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ26_HTML.gif
(4.12)
If K ( r ) r N p + ( N p ) ( q + 1 ) p L ( r , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq138_HTML.gif, arguing as previously, similarly we obtain
r K ( s ) | u | q + 1 s N 1 d s C ( R N | u | p d x ) ( q + 1 ) p C u W V , r 1 , p ( R N ) q + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ27_HTML.gif
(4.13)
  1. (2)
    If K 0 ( r ) V 0 1 ( r ) ( g 1 ( r ) ) ( q + 1 ) p L ( 0 , r 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq146_HTML.gif, by (4.11) and conditions (V) and (K), we obtain
    0 r 0 K ( s ) | u | q + 1 s N 1 d s ω N 1 K 0 ( r ) V 0 1 ( r ) ( g 1 ( r ) ) ( q + 1 ) p L ( 0 , r 0 ) , ( 0 r 0 K 0 ( s ) u p s N 1 d s ) u W V , r 1 , p ( R N ) ( q + 1 ) p ω N 1 u W V , r 1 , p ( R N ) ( q + 1 ) p 0 r 0 K 0 ( s ) u p s N 1 d s ( 0 r 0 K 0 ( s ) u p s N 1 d s ) u W V , r 1 , p ( R N ) ( q + 1 ) p C u W V , r 1 , p ( R N ) q + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ28_HTML.gif
    (4.14)
     
If K ( r ) V 1 ( r ) ( g 2 ( r ) ) ( q + 1 ) p L ( r , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq140_HTML.gif, we obtain similarly
r K ( s ) | u | q + 1 s N 1 d s C u W V , r 1 , p ( R N ) q + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ29_HTML.gif
(4.15)
(b) If p = N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq125_HTML.gif, K 0 ( r ) ( 1 r | ln 1 r | ) N ( q + 1 ) ( g 1 ( r ) ) ( q + 1 ) N L ( 0 , r 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq147_HTML.gif. By Hölder’s inequality and Lemma 2.3, we have
0 r 0 K ( s ) | u | q + 1 d s C K 0 ( r ) ( r | log 1 r | ) N ( g 1 ( r ) ) q + 1 N L ( 0 , r 0 ) 0 r 0 | u | N ( | x | log 1 x ) N s N 1 d s u W V , r 1 , N ( R N ) ( q + 1 ) N C u W V , r 1 , N ( R N ) ( q + 1 ) N ( 0 r 0 | u | N ( | x | log 1 x ) N s N 1 d s ) C u W V , r 1 , N ( R N ) ( q + 1 ) N ( R N | u | N d x ) N C u W V , r 1 , N ( R N ) q + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ30_HTML.gif
(4.16)
If K 0 ( r ) ( 1 r | ln 1 r | ) N ( g 2 ( r ) ) ( q + 1 ) N L ( r , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq148_HTML.gif, we obtain similarly
r K ( s ) | u | q + 1 d s C u W V , r 1 , N ( R N ) q + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ31_HTML.gif
(4.17)

From Lemma 6 in [13], we have the following.

Under the conditions of Theorem 4.5, for 0 < r < R < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq149_HTML.gif and R 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq150_HTML.gif, the embedding
W V 1 , p ( B R B r ) L K q + 1 ( B R B r ) is compact. https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ32_HTML.gif
(4.18)
Now, we prove that the embedding W V , r 1 , p ( R N ) L K q + 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq151_HTML.gif for 1 < p N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq152_HTML.gif is continuous. It suffices to show
S r ( V , K ) = inf u W V , r 1 , p ( R N ) R N ( | u | p + V ( | x | ) | u | p ) d x ( R N K ( | x | ) | u | q + 1 d x ) p q + 1 > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equam_HTML.gif
Assume to the contrary that S ( V , K ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq153_HTML.gif, then there exists { u n } W V , r 1 , p ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq154_HTML.gif such that
R N K ( | x | ) | u n | q + 1 d x = 1 and R N ( | u n | p + V ( | x | ) | u n | p ) d x 0 as  n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equan_HTML.gif
But from (4.12) and (4.13), or (4.14) and (4.15), or (4.16) and (4.17), and (4.18), let r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq100_HTML.gif be large enough, we obtain
1 = R N K ( | x | ) | u n | q + 1 d x = 0 r 0 K ( | x | ) | u n | q + 1 d x + r K ( | x | ) | u n | q + 1 d x + r 0 r K ( | x | ) | u n | q + 1 d x C u n W V , r 1 , N ( R N ) q + 1 0 as  n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equao_HTML.gif

which yields a contradiction. □

Theorem 4.6 If V ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq4_HTML.gif and K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq5_HTML.gif are nonnegative measurable functions satisfying (V) and (K). K ( x ) L l o c ( R N { 0 } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq155_HTML.gif and R N { supp V ( x ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq156_HTML.gif is relatively compact.
  1. (a)

    If 1 < p < N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq27_HTML.gif, p q + 1 < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq28_HTML.gif. or then the embedding W V , r 1 , p ( R N ) L K q + 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq157_HTML.gif is compact.

     
  2. (1)

    K ( r ) r N p + ( N p ) ( q + 1 ) p = o ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq158_HTML.gif as r 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq159_HTML.gif and K ( r ) r N p + ( N p ) ( q + 1 ) p = o ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq158_HTML.gif as r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq160_HTML.gif,

     
  3. (2)

    K 0 ( r ) V 0 1 ( r ) [ g 1 ( r ) ] q + 1 p = o ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq161_HTML.gif as r 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq159_HTML.gif and K ( r ) V 1 ( r ) [ g 2 ( r ) ] q + 1 p = o ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq162_HTML.gif as r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq160_HTML.gif,

     
  4. (b)

    If p = N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq125_HTML.gif, q + 1 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq141_HTML.gif, K 0 ( r ) ( 1 r | log 1 r | ) N ( q + 1 ) ( g 1 ( r ) ) q + 1 N = o ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq163_HTML.gif as r 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq164_HTML.gif and K ( r ) ( 1 r | log 1 r | ) p ( q + 1 ) ( g 2 ( r ) ) q + 1 p = o ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq165_HTML.gif as r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq160_HTML.gif, then the embedding W V , r 1 , N ( R N ) L K q + 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq166_HTML.gif is compact.

     
Proof (a) Arguing as in the proof of (a) and (b) of Theorem 4.5, we obtain that there exists ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq52_HTML.gif,
0 r 0 K ( | x | ) | u | q + 1 d x ε u W V , r 1 , p ( R N ) q + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equap_HTML.gif

and r K ( | x | ) | u | q + 1 d x ε u W V , r 1 , p ( R N ) q + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq167_HTML.gif.

Assume that u n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq60_HTML.gif (weakly) in W V , r 1 , p ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq168_HTML.gif ( 1 < p N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq152_HTML.gif), we obtain
R N K ( | x | ) | u n | q + 1 d x C ε u n W V , r 1 , p ( R N ) q + 1 C ε , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equaq_HTML.gif

then we have u n L K , r q + 1 ( R N ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq169_HTML.gif (strongly). Hence the embedding is compact. □

5 Ground and bound state solutions

Now, consider problem (1.1) with general potentials
{ p u + V ( x ) | u | p 2 u = K ( x ) | u | q 1 u , x R N , | u ( x ) | 0 as  | x | . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equar_HTML.gif

Theorem 5.1 Under the assumptions of Theorem  3.2, i.e., W V 1 , p ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq73_HTML.gif is compact embedded into L K q + 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq170_HTML.gif, then problem (1.1) has a ground state solution.

Proof Now, we define the functional I ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq171_HTML.gif on the Sobolev space W V 1 , p ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq73_HTML.gif,
I ( u ) = 1 p R N ( | u | p + V ( x ) | u | p ) d x 1 q + 1 R N K ( x ) | u | q + 1 d x = 1 p u W V 1 , p ( R N ) p 1 q + 1 u L K q + 1 ( R N ) q + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equas_HTML.gif

It is obvious that the critical point of the functional I ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq171_HTML.gif is exactly the weak solution of problem (1.1). The existence of a ground state solution follows from the compact embedding W V 1 , p ( R N ) L K q + 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq172_HTML.gif immediately.

Further, consider problem (1.5) with radially symmetric potentials
{ p u + V ( | x | ) | u | p 2 u = K ( | x | ) | u | q 1 u , x R N , | u ( x ) | 0 as  | x | , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equat_HTML.gif

where 1 < p N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq152_HTML.gif. Similarly to Theorem 5.1, we obtain the following theorem. □

Theorem 5.2 Under the assumptions of Theorem  4.6, i.e., W V , r 1 , p ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq173_HTML.gif is a compact embedding into L K , r q + 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq174_HTML.gif, then problem (1.5) has a ground state solution.

For a more general equation than (1.5),
{ p u + V ( | x | ) | u | p 2 u = K ( | x | ) f ( u ) , x R N , | u ( x ) | 0 as  | x | . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equ33_HTML.gif
(5.1)
If f C ( R , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq175_HTML.gif and f ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq176_HTML.gif, | f ( u ) | C ( | u | p 1 + | u | q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq177_HTML.gif and there exists μ > p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq178_HTML.gif such that
0 < μ F ( u ) = μ 0 u f ( s ) d s u f ( u ) , u R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_Equau_HTML.gif

then we have the following theorem.

Theorem 5.3 Under the above conditions and assumptions of Theorem  4.6, problem (5.1) has a positive solution. If, in addition, f is odd in u, then problem (5.1) has infinitely many solutions in W V , r 1 , p ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-189/MediaObjects/13661_2013_Article_443_IEq173_HTML.gif.

Declarations

Acknowledgements

The author gives his sincere thanks to the referees for their valuable suggestions. This paper was supported by Shanghai Natural Science Foundation Project (No. 11ZR1424500) and Shanghai Leading Academic Discipline Project (No. XTKX2012).

Authors’ Affiliations

(1)
College of Sciences, University of Shanghai for Science and Technology

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