In this article, we prove a compact embedding theorem for the weighted Orlicz-Sobolev space of radially symmetric functions

Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 85, pp. 1–18.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

GROUND STATES FOR A MODIFIED CAPILLARY SURFACE EQUATION IN WEIGHTED ORLICZ-SOBOLEV SPACE

GUOQING ZHANG, HUILING FU

Abstract. In this article, we prove a compact embedding theorem for the weighted Orlicz-Sobolev space of radially symmetric functions. Using the em- bedding theorem and critical points theory, we prove the existence of multiple radial solutions and radial ground states for the following modified capillary surface equation

−div

“|∇u|^2p−2∇u p1 +|∇u|^2p

”

+T(|x|)|u|^α−2u=K(|x|)|u|^s−2u, u >0, x∈R^N, u(|x|)→0, as|x| → ∞,

whereN≥3, 1< α < p <2p < N,ssatisfies some suitable conditions,K(|x|) andT(|x|) are continuous, nonnegative functions.

1. Introduction

In this article, we study the following modified capillary surface equation in a weighted Orlicz-Sobolev space,

−div|∇u|^2p−2∇u p1 +|∇u|^2p

+T(|x|)|u|^α−2u=K(|x|)|u|^s−2u, u >0, x∈R^N, u(|x|)→0, as|x| → ∞,

(1.1) whereN ≥3, 1< α < p <2p < N,ssatisfies some suitable conditions,∇udenotes the gradient ofu,T andKare continuous, nonnegative and measurable functions, i.e.,T, K : (0,+∞)→[0,+∞] and may be unbounded, decaying and vanishing.

Recently, these type equations have attracted much attention. As p = 1, the problem (1.1) becomes known as the prescribed mean curvature equation or the capillary surface equation. Peletier and Serrin [15] studied the following problem

−div ∇u p1 +|∇u|²

=−λu+u^q, x∈R^N, u(x)→0, asx→ ∞,

(1.2) where λ >0, q >1 and obtained the existence of radial ground states. As λ= 0, Ni and Serrin [12, 13] established that if 1< q ≤ _N−2^N , no positive solutions exist,

2000Mathematics Subject Classification. 35J65, 35J70.

Key words and phrases. Compact theorem; modified capillary surface equation;

weighted Orlicz-Sobolev space; ground state.

c

2015 Texas State University - San Marcos.

Submitted August 6, 2014. Published March 7, 2015.

1

on the contrary, if q ≥ _N^N+2₋₂, there is a continuum of solutions. del Pino and Guerra [6] proved the existence of large finite number of ground states, provided that q lies below but close enough to the critical exponent ^N_N⁺²₋₂. Moreover, existence, nonexistence and multiplicity of solutions decaying to zero at infinity have been proved by [3, 4, 7, 8, 17].

As p > 1, using minimization sequence method and Mountain Pass Lemma, Narukawa and Suzuki [11] discussed the existence of nonzero solutions for the mod- ified capillary surface equation

−div|∇u|^2p−2∇u p1 +|∇u|^2p

=λf(x, u), u≥0, x∈Ω, u= 0, x∈∂Ω,

(1.3) where Ω is a bounded domain inR^N with smooth boundary,λis a positive param- eter; Liang [9] investigated the following modified capillary equation

−div|∇u|^2p−2∇u p1 +|∇u|^2p

=f(x, u), x∈Ω, u= 0, x∈∂Ω,

(1.4) and obtained a negative and a positive solution by variational methods. In particu- lar, Azzollini, d’Avenia and Pomponio [1] studied the quasilinear elliptic problems

−∇[φ⁰(|∇u|²)∇u] +|u|^α−2u=|u|^s−2u, x∈R^N,

u(x)→0, as|x| → ∞, (1.5)

where φ(t) behaves like t^q2 for small t and t^p2 for large t, 1 < p < q < N, and obtained some existence results in Orlicz-Sobolev space by using critical points theory.

On the other hand, some authors studied the semilinear (quasilinear) elliptic equations with unbounded or decaying radial potentials. Su, Wang and Willem [18, 19] proved some embedding results for the weighted Sobolev spaces of radially symmetric functions. Zhang [20] obtained some Strauss-type decay estimates and obtained some continuous and compact embedding theorems.

In this article, we prove the existence of multiple radial solutions and radial ground states for the problem (1.1). Firstly, we obtain a compact embedding the- orem for the weighted Orlicz-Sobolev space of radially symmetric functions. Sec- ondly, we obtain the existence of radial ground states for the problem (1.1) with unbounded or decaying radial potentials by using this compact embedding theorem and critical points theory.

Consider the functional J(u) = 1

p Z

R^N

(p

1 +|∇u|^2p−1)dx+ 1 α

Z

R^N

T(|x|)|u|^αdx−1 s Z

R^N

K(|x|)|u|^sdx, (1.6) where

p1 +|∇u|^2p−1∼

(|∇u|^p, as|∇u| → ∞,

1

2|∇u|^2p, as|∇u| →0. (1.7) Solutions of (1.1) are, at least formally, critical points of the functionalJ(u). By (1.7), we obtain that this different growth at zero and at infinity of the function p1 +|∇u|^2p−1 and the whole space R^N suggest us not to use classical Sobolev

spaces. Hence, we should define a class of weighted Orlicz-Sobolev space with respect to the functional (1.6) is well defined andC¹. For dealing with the compact properties of the functionalJ(u), we would like to get compactness lies in the fact that the group of translation constitutes an obstruction to compact embedding in R^N, and examine the affects of the unbounded or decaying potentials T(|x|) and K(|x|). Hence, we restrict the domain of the functional J(u) to the suitable Orlicz-Sobolev space.

Now we state our main theorems in this paper. Let |x| =r, T(|x|), K(|x|) be continuous nonnegative functions in (0,∞), and

(T1) There exist real numberaand a₀, such that lim inf_r→∞T(r)/r^a >0, and lim inf_r→0T(r)/r^a0 >0;

(K1) There exist real numberbandb₀, such that lim sup_r→∞K(r)/r^b<∞, and lim sup_r→0K(r/r^b0 <∞,K(r)>0.

The existence and embedding results depend on the potentialsT, K near 0 and

∞. We define the following relations betweenp,2p, anda, bor a₀, b₀:

s∗=











(2p)α(N−1+b)−aα

2p(N−1)+a(2p−1), b≥a >−p,

2p(N+b)

(N−2p), b≥ −p, a≤ −p,

α, b≤max{a,−p},

(1.8)

and

s^∗=











2p(N+b0)

(N−2p) , b0≥ −p, a0≥ −p,

(2p)a(N−1+b0)−a0α

2p(N−1)+a0(2p−1), −p > a₀>−_(2p−1)^(N−1)2p, b₀≥a₀,

∞, a0≤ −^(N_(p−1)⁻¹⁾p, b0≥a0.

(1.9)

Remark 1.1. The idea which for establishing conditions (1.8) and (1.9) comes from Su, Wang and Willem [18, 19]. In this article, we not only develop the methods in [18,19,20] to the modified capillary surface equation, but also improve and extend the results in classical Sobolev space to the Orlicz-Sobolev space.

Theorem 1.2 (Multiplicity Result). Assume that (T1) and (K1) hold, 1 < α <

p < 2p < N, s_∗ < s < s^∗, then there exist infinitely many radially symmetric solutions for (1.1).

Theorem 1.3 (Ground States). Assume that (T1) and(K1) hold, 1 < α < p <

2p < N,s_∗ < s < s^∗, then there exists a radial ground states for (1.1).

This article is organized as follows. In Section 2, we introduce a weighted Orlicz- Sobolev space of radially symmetric function and recall some important lemmas.

In Section 3, we prove some inequalities with radial functions, extending some inequalities in classic Sobolev space to the Orlicz-Sobolev space, and establish a new compact embedding theorem (i.e. Theorem 3.1). Section 4 is devoted to the proof of Theorems 1.2 and Theorem 1.3.

2. Weighted Orlicz-Sobolev spaces

As a first step, we recall some well known facts on the sum of Lebesgue spaces and introduce some notation of function space.

Definition 2.1( [2]). Let 1< p < qand Ω⊂R^N. We denote withL^p(Ω) +L^q(Ω) the completion ofC_c^∞(Ω,R^N) in the norm

kuk_Lp(Ω)+L^q(Ω)= inf

kvkp+kwkq:v∈L^p(Ω), w∈L^q(Ω), u=v+w . (2.1) In this article, we set q= 2pand kukp,2p =kuk_Lp(Ω)+L^2p(Ω). Moreover, from [2], we obtain thatL^p(Ω) +L^2p(Ω) are Orlicz spaces.

Forα >1, s >1, we define L^α(R^N;T) =

u:R^N →R:uis Lebesgue measurable, Z

R^N

T(|x|)|u|^αdx <∞ , and

L^s(R^N;K) =

u:R^N →R:uis Lebesgue measurable, Z

R^N

K(|x|)|u|^sdx <∞ . The corresponding norms inL^α(R^N;T) andL^s(R^N;K) are respectively

kukL^α_T(R^N)=Z

R^N

T(|x|)|u|^αdx^1/α , kukL^s_K(R^N)=Z

R^N

K(|x|)|u|^sdx^1/s .

(2.2)

From [2], we have a list of properties of the Orlicz spacesL^p(Ω) +L^2p(Ω).

Proposition 2.2 ( [2]). Let Ω ⊂ R^N, u ∈ L^p(Ω) +L^2p(Ω) and Λu = {x ∈ Ω| |u(x)|>1}. We have

(i) ifΩ⁰ ⊂Ω is such that|Ω⁰|<+∞, thenu∈L^p(Ω⁰);

(ii) ifΩ⁰ ⊂Ω is such thatu∈L^∞(Ω⁰), then u∈L^2p(Ω⁰);

(iii) |Λu|<+∞;

(iv) u∈L^p(Λ_u)∩L^2p(Λ^c_u);

(v) the infimum in (2.1) is attained;

(vi) L^p(Ω) +L^2p(Ω) is reflexive and(L^p(Ω) +L^2p(Ω))⁰=L^p

0

(Ω)∩L^(2p)

0

(Ω);

(vii) kukL^p(Ω)+L^2p(Ω)≤max{kukL^p(Λ_u),kukL^2p(Λ^c_u)};

(viii) ifB ⊂Ω, thenkuk_Lp(Ω)+L^2p(Ω)≤ kuk_Lp(B)+L^2p(B)+kuk_Lp(Ω\B)+L^2p(Ω\B). LetC_c^∞(R^N,R) denote the collection of smooth functions with compact support and

(C_c^∞(R^N,R))rad={u∈ C_c^∞(R^N,R) :uis radial}.

Definition 2.3. Letα >1,W be the completion of C_c^∞(R^N,R) in the norm kuk_W =kuk_L^α

T(R^N)+k∇ukp,2p, (2.3) Wrad be the completion of (C_c^∞(R^N,R))rad in the norm k · k, namely

Wrad= (C^∞_c (R^N,R))rad k·k.

Lemma 2.4. The space (Wrad,k · k)is a reflexive Banach space.

Proof. Firstly, we prove that (W_rad,k·k) is a Banach space. In fact, sinceL^α(R^N;T) andL^p(R^N) +L^2p(R^N) are completed. Let {un}n be a Cauchy sequence inWrad, then {un}n is a Cauchy sequence in L^α(R^N;T), and there exists u∈L^α(R^N;T), such that kun−uk_L^α

T(R^N)→0, asn→ ∞. Also{∇un}n is a Cauchy sequence in L^p(R^N) +L^2p(R^N), there existsδ∈L^p(R^N) +L^2p(R^N), such thatk∇un−δkp,2p→ 0, asn→ ∞.

Sufficiently, for everyξ∈ C_c^∞(R^N),n∈N, we have

n→∞lim Z

R^N

T(|x|)un∇ξdx= Z

R^N

T(|x|)u∇ξdx, lim

n→∞

Z

R^N

ξ∇undx= Z

R^N

ξδdx.

In fact, by H¨older inequality and Proposition 2.2 (v), by considering (vn,wn) in inL^p(R^N)×L^2p(R^N) such that

∇un−δ=v_n+w_n, k∇un−δkp,2p=kvnkp+kwnk2p, we have

Z

R^N

T(|x|)(un−u)∇ξdx

≤ k∇ξk_α⁰kun−uk_L^α

T(R^N)→0, and

Z

R^N

ξ(∇u_n−δ)dx =

Z

R^N

ξv_ndx+ Z

R^N

ξw_ndx

≤ kξkp⁰kvnkp+kξk_(2p)⁰kwnk2p→0.

Obviously, by the definition of weak derivatives, we have Z

R^N

T(|x|)un∇ξdx=− Z

R^N

T(|x|)ξ∇undx.

Hence, we obtain Z

R^N

T(|x|)u∇ξdx=− Z

R^N

T(|x|)ξδdx;

that is,∇u=δ.

Secondly, we prove that (W_rad,k · k) is reflexive. Indeed, we consider the norm kuk^∗_p,2p= inf{(kvk²_p+kwk²_2p)¹²|v∈L^p(R^N), w∈L^2p(R^N), u=v+w}, and then, onWrad, the norm

kuk^∗_Wrad=kukL^α_T(R^N)+k∇uk^∗_p,2p,

is equivalent to the norm kuk_Wrad. Moreover, by [2, Proposition 2.6], the norm kuk_Lα

T(R^N) and the norm k · k^∗ are uniformly convex. So, on W_rad, we consider uniformly convex normk∇.k^∗_p,2p and the normk · kL^α_T(R^N). By a well known result, also the norm

k · k^]_W

rad=q k · k²_Lα

T(R^N)+ (k∇.k^∗_p,2p)²,

is uniformly convex and then (Wrad,k · k^]) is reflexive. Hence the normk · k^]_W

rad is equivalent tok · k_Wrad. Then, we obtain that (W_rad,k · k) is also reflexive.

Remark 2.5. Similar to [1, Theorem 2.8], we obtain thatWrad coincides with the set of radial functions of W. Hence, using the principle of symmetric criticality in [14], we only consider the functional J(u) in (1.6) restricted to the weighted Orlicz-Sobolev spaceWrad.

3. Embedding theorem

To obtain the compactness of the functionalJ(u), we prove a compact embedding theorem (Theorem 3.1). Denote byB_r the ball inR^N centered at 0 with radiusr.

Theorem 3.1. Let 1 < α < p < 2p < N. Assume(T1) and (K1) hold, then we have the continuous embedding

Wrad,→L^s(R^N;K)

fors∗ ≤s ≤s^∗ when s^∗ <∞, and for s∗ ≤s <∞ whens^∗ =∞. Furthermore, the embedding is compact fors_∗< s < s^∗.

Firstly, we prove some inequalities on radial functions which are interesting.

Lemma 3.2. If1< p <2p < N, there existsM >c 0 such that for everyu∈ W_rad,

|u(x)| ≤ (

Mc|x|^{−(N−2p2p )}k∇ukp,2p, for|x| ≥1,

Mc|x|^{−(N−pp )}k∇ukp,2p, for0<|x|<1. (3.1) The proof of the above lemma is similar to that of [1, Lemma 2.13] and of [19, Lemma 1].

Lemma 3.3. Let 1 < p <2p < N. Assume 2p < s <∞ and write s= ^2p(N+c)_(N−2p), for some−p≤c <∞. Then there exists M >f 0 such that for all u∈ Wrad

Z

R^N

|x|^c|u|^sdx^1/s

≤Mfmax k∇ukp,2p,k∇uk²_p,2p

. (3.2)

Proof. By denseness, it is sufficient to prove that u ∈ (C_c^∞(R^N,R))_rad,(v,w) ∈ L^p(R^N)×L^2p(R^N), such that∇u=v+w. By using Lemma 3.2, ands=^2p(N_(N−2p)^+c), we have

Z

R^N

|x|^c|u|^sdx

=ω_N Z ∞

0

r^(N−1+c)|u(r)|^sdr

=− sωN

(N+c) Z ∞

0

r^(N+c)|u(r)|^(s−2)u(r)u⁰(r)dr

≤ (2p)ω_N (N−2p)

Z ∞ 0

r^(N+c)|u(r)|^(s−1)|u⁰(r)|dr

= 2p

(N−2p) Z

R^N

|x|^(c+1)|u|^(s−1)|∇u|dx

≤ 2p

(N−2p) Z

R^N

|x|^(c+1)|u|^(s−1)|v|dx+ Z

R^N

|x|^(c+1)|u|^(s−1)|w|dx

≤ 2p

(N−2p) hZ

R^N

|v|^pdx1/pZ

R^N

|x|^c|u|^s|x|^(p−1)(p+c)|u|^{(s−p)(p−1)}dx(^p−1_p )

+Z

R^N

|w|^2pdx_2p¹ Z

R^N

|x|^c|u|^s|x|^{(2p+c)(p−1)}|u|^{(s−2p)(2p−1)}dx^(2p−1_2p ⁾i

≤M⁰ 2p (N−2p)

hkvkL^p(R^N)k∇uk⁽

s−p p ) p,2p

Z

R^N

|x|^c|u|^sdx^(p−1_p ⁾

+kwk_L2p(R^N)k∇uk⁽

s−2p 2p ) p,2p

Z

R^N

|x|^c|u|^sdx(^2p−1_2p )i

≤M⁰ 2p

(N−2p)max k∇uk⁽

s−p p ) p,2p ,k∇uk⁽

s−2p 2p ) p,2p

hkvk_Lp(R^N)

Z

R^N

|x|^c|u|^sdx(^2p−1_2p )

+kwk_L2p(R^N)

Z

R^N

|x|^c|u|^sdx(^2p−1_2p )i

≤M⁰ 2p

(N−2p)max k∇uk⁽

s−p p ) p,2p ,k∇uk⁽

s−2p 2p ) p,2p

k∇ukp,2p

Z

R^N

|x|^c|u|^sdx(^2p−1_2p )

≤Mfmax k∇uk

s p

p,2p,k∇uk

s 2p

p,2p

Z

R^N

|x|^c|u|^sdx^(2p−1_2p ⁾ ,

whereω_N is the volume of the unit sphere inR^N. It follows that Z

R^N

|x|^c|u|^sdx^1/s

≤Mfmax

k∇ukp,2p,k∇uk²_p,2p .

Lemma 3.4. Assume (T1)holds,1< α < p <2p < N, anda >−^(N_(2p−1)⁻¹⁾2p. Then there existsMc0>0 such that for all u∈ Wrad,

|u(x)| ≤Mc0|x|⁻⁽2p(N−1)+a(2p−1) α(2p) )

kuk_Wrad, for|x| 1. (3.3) Proof. By assumption (T1), there existsR >1 such that for someM0>0,

T(|x|)≥M0|x|^a, |x|> R >1.

Foru∈ W_rad, as θ >−(N−1), we have d

dr(r^(θ+N−1)|u|^α) =αr^(θ+N−1)|u|^(α−2)udu

dr + (θ+N−1)|u|^αr^(θ+N−2)

≥αr^(θ+N−1)|u|^(α−2)udu dr.

(3.4)

Next we only consider |u| ≥ 1, when |u| ≤ 1, set |u⁰| = _|u|¹ , then |u⁰| ≥ 1. For all u ∈ Wrad, (v,w) ∈ L^p(R^N)×L^2p(R^N), such that ∇u = v+w. Since, a >

−_(2p−1)^(N−1)2p, so take θ = min{^a(p−1)_p ,^a(2p−1)_2p }, then θ > −(N −1). For r > R, 1< α < p <2p < N, we have

|u|^αr^(θ+N−1)≤α Z ∞

r

|u|^(α−1)t^(θ+N−1)|u⁰(t)|dt

= α ω_N

Z

B_r^c

|x|^θ|u|^(α−1)|∇u|dx

≤ α ω_N

Z

B^c_r

|x|^θ|u|^(α−1)|v|dx+ Z

B_r^c

|x|^θ|u|^(α−1)|w|dx

≤ α ω_N

hkvk_Lp(B^c_r)

Z

B^c_r

|x|^(p−1)θp |u|^{((α−1)pp−1 )}dx(^p−1_p )

+kwk_L2p(B_r^c)

Z

B_r^c

|x|^{(2p−1)θ(2p)} |u|^{((α−1)(2p)(2p−1) )}dx(^2p−1_2p )i

≤ α ωN

hkvk_Lp(B^c_r)

Z

B^c_r

|x|^a|u|^{((α−1)pp−1 )}dx^(p−1_p ⁾

+kwk_L2p(B_r^c)

Z

B_r^c

|x|^a|u|^{((α−1)2p(2p−1) )}dx(^2p−1_2p )i

≤M_(α,p,N)h

kvk_Lp(B^c_r)

Z

B^c_r

T(|x|)|u|^{((α−1)pp−1 )}dx(_p(α−1)^p−1 (α−1))

+kwk_L2p(B_r^c)

Z

B_r^c

T(|x|)|u|^{((α−1)2p2p−1 )}dx⁽_2p(α−1)^2p−1 (α−1))

≤M_(α,p,N)k∇ukp,2pkuk^(α−1)

L

(α−1)p (p−1) T (R^N)

≤M_(α,p,N)k∇ukp,2pkuk^(α−1)_Lα T(R^N). By Young inequality, and|x|=r, we obtain

|u| |x|^{(θ+N−1α )}≤M_(α,p,N)^1/α k∇uk^1/α_p,2pkuk⁽_L^α−1α^{α )} T(R^N)

≤Mc0(k∇ukp,2p+kuk_Lα T(R^N));

i.e.,

|u| ≤Mc₀|x|⁻⁽2p(N−1)+a(2p−1) α(2p) )

kukWrad,

where the constantMc0=Mc0(α,p,N).

Lemma 3.5. Assume (T1)holds,1< α < p <2p < N. Then there exist1> r0>

0 andMf0>0 such that for all u∈ Wrad,

|u(x)| ≤Mf0|x|^{−(2p(N−1)+aα(2p)0 (2p−1))}kukW_rad, for0<|x| ≤r0<1, (3.5) whereMf0=Mf0(a0, r0, α, N).

Proof. By assumption (T1), there exists 1 > r0 >0 such that for some constant M₀>0,

T(|x|)≥M₀|x|^a0, 0<|x| ≤r₀<1.

Foru∈ Wrad, we have d

dr(r^(β+N−1)|u|^α) =αr^(β+N−1)|u|^(α−2)udu

dr + (β+N−1)|u|^αr^(β+N−2). Thus, for 0< r≤r₀<1,

r^(β+N−1)|u|^α≤α Z r₀

r

|u|^(α−1)t^(β+N−1)|u⁰(t)|dt+ (β+N−1) Z r₀

r

|u|^αt^(β+N−2)dt.

(3.6) Asβ≥a0+ 1, we have

Z r₀ r

|u|^αt^(β+N−2)dt= Z r₀

r

t^(a0+N−1)|u|^αt^{(β−a0−1)}dt

≤ω⁻¹_N r^(β−a₀ ⁰⁻¹⁾ Z

B_r0(0)\Br(0)

|x|^a0|u|^αdx

≤ω⁻¹_N M₀⁻¹r₀^{(β−a0−1)} Z

B_r0(0)\B_r(0)

T(|x|)|u|^αdx

≤ω⁻¹_N M₀⁻¹r₀^{(β−a0−1)}kuk^α_Lα T.

(3.7)

Let β = max{^(2p−1)_2p a0,^(p−1)_p a0}, for all u ∈ Wrad, (v,w) ∈L^p(R^N)×L^2p(R^N), such that ∇u=v+w, we only consider |u| ≥1. If|u| ≤1, set |u⁰|= 1/|u|, then we have|u⁰| ≥1. Hence, we have

Z r₀ r

|u|^(α−1)t^(β+N−1)|u⁰(t)|dt

=ω⁻¹_N Z

Br0(0)\Br(0)

|x|^β|u|^(α−1)|∇u|dx

≤ω⁻¹_N Z

B_r0(0)\Br(0)

|x|^β|u|^(α−1)|v|dx+ Z

B_r0(0)\Br(0)

|x|^β|u|^(α−1)|w|dx

≤ω⁻¹_N h

kvk_Lp(B_r0(0)\B_r(0))

Z

B_r0(0)\B_r(0)

|x|^(p−1)βp |u|^{(α−1)p(p−1)} dx^(p−1)_p

+kwk_L2p(Br0(0)\Br(0))

Z

Br0(0)\Br(0)

|x|^{(2p−1)β(2p)} |u|^{(α−1)(2p)(2p−1)} dx^(2p−1)_2p i

≤ω⁻¹_N kvkL^p(B_r0(0)\Br(0))

Z

B_r0(0)\Br(0)

|x|^a0|u|^{((α−1)pp−1 )}dx⁽_p(α−1)^p−1 (α−1))

+ω⁻¹_N kwk_L2p(B_r0(0)\B_r(0))

Z

Br0(0)\Br(0)

|x|^a0|u|^{(2p(α−1)2p−1 )}dx(_2p(α−1)^2p−1 (α−1))

≤ω⁻¹_N k∇uk_p,2p(Br

0(0)\Br(0))

Z

Br0(0)\Br(0)

|x|^a0|u|^{((α−1)pp−1 )}dx⁽_p(α−1)^p−1 (α−1))

≤ω⁻¹_N M⁻⁽

p−1 p )

0 k∇uk_p,2p(Br

0(0)\B_r(0))

×Z

Br0(0)\Br(0)

T(|x|)|u|^{((α−1)pp−1 )}dx⁽_p(α−1)^p−1 (α−1))

≤ω⁻¹_N M₁k∇uk_p,2p(Br

0(0)\Br(0))

Z

B_r0(0)\B_r(0)

T(|x|)|u|^αdx(^α−1_α )

=ω⁻¹_N M₁k∇uk_p,2pkuk^(α−1)_Lα

T(B_r0(0)\Br(0)). Since

β+N−1≥0⇐⇒a₀>−(N−1) (2p−1)2p.

It follows thatβ+N−1≤0 impliesβ−a₀−1≥(^N−p_p−1). Hence, from the above arguments, we have

|u(x)| ≤Mf₀|x|^{−(2p(N−1)+aα(2p)0 (2p−1))}kuk_Wrad, 0<|x| ≤r₀<1,

where the constantMf₀=Mf₀(a₀, r₀, α, N).

Lemma 3.6. Let 1< α < p <2p < N,2p < s≤ ∞. Then for any 0< r <1<

R <∞, the following embedding is compact

W_rad(B_R\B_r),→L^s(B_R\B_r;K).

The proof of the above lemma is similar to [19, Lemma 6].

Proof of Theorem 3.1. First we prove that the embedding is continuous. It is suf- ficient to show

Srad(T, K) = inf

u∈Wrad(R^N)

k∇ukp,2p+kuk_L^α

T(R^N)

kukL^s_K(R^N)

>0. (3.8) If not, assume that there exists{un} ⊂ Wradsuch that

k∇ukp,2p+kuk_L^α

T(R^N)=o(1), asn→ ∞, (3.9) kuk_Ls

K(R^N)= 1, for alln∈N. (3.10) It is a contradiction, if we have

kuk_L^s

K(R^N)= 0. (3.11)

By (T1) and (K1), there existR₀>1> r₀>0, for someM₀, K(|x|)≤M₀|x|^b, T(|x|)≥M₀|x|^a, for|x| ≥R₀,

K(|x|)≤M0|x|^b0, T(|x|)≥M0|x|^a0, for 0<|x| ≤r0. (3.12) ForR > R0 and 0< r < r0, we estimate the integrals R

B_rK(|x|)|un|^sdx^1/s and R

B_R^c K(|x|)|un|^sdx^1/s

in different cases according to the definitions ofs^∗ ands_∗, B_R^c denotes the complement ofBR.

Firstly, we estimate the term R

B_rK(|x|)|un|^sdx^1/s .

Case 1.1: For a0 ≥ −p, b0 ≥ −p. Let s = ^2p(N+c)_(N−2p), by s ≤ s^∗, we obtain η1=b0−c≥0. Hence by Lemma 3.3 and (3.9), we have

Z

B_r

K(|x|)|un|^sdx^1/s

≤M₀^1/sZ

B_r

|x|^b0|un|^sdx^1/s

≤M₀^1/sr^(b0s−c)Z

Br

|x|^c|un|^sdx1/s

≤M₀^1/sr^(b0s−c)max

k∇unkp,2p,k∇unk²_p,2p

=r^(b0s−c)o(1), as n→ ∞.

(3.13)

Case 1.2: For−p > a₀>−_(2p−1)^(N−1)2p, b₀≥a₀. Froms≤s^∗, we obtain η₂=b₀−a₀−(s−α)2p(N−1) +a₀(2p−1)

α(2p) ≥0.

We choose a cut-off function φ such that φ = 1 for 0 ≤ |x| ≤ ^r₂⁰, and φ= 0 for

|x| ≥r0. Then by Lemma 3.5, forr < ^r₂⁰, we have Z

Br

K(|x|)|un|^sdx1/s

≤M₀^1/sZ

B_r

|x|^b0|φun|^sdx^1/s

=M₀^1/sZ

Br

|x|^(b0−a0)|φun|^(s−α)|x|^a0|φun|^αdx^1/s

≤M₂kφu_nk⁽_W^{s−αs )}Z

B_r

|x|^{(b0−a0−(s−α)2p(N−1)+aα(2p)0 (2p−1))}T(|x|)|u_n|^αdx1/s

≤M₃r⁽

b0−a0

s −(s−α)^2p(N−1)+a_sα(2p)^{0 (2p−1)})

ku_nk⁽_W^{s−αs )}

rad ku_nk^α/s_Lα T(B_r)

≤M₃⁰r^(b0

−a0

s −(s−α)^2p(N−1)+a_sα(2p)^{0 (2p−1)})

ku_nk_Wrad

=r ^b0

−a0

s −(s−α)^2p(N−1)+a_sα(2p)^{0 (2p−1)}

o(1), asn→ ∞. (3.14)

Case 1.3: For a0 ≤ −^(N_(p−1)⁻¹⁾p, b0 ≥a0, in the cases^∗ =∞. For ∞> s > α, it holds

η3=b0−a0−(s−α)2p(N−1) +a₀(2p−1)

α(2p) ≥0.

With the same functionφgiven in Case 1.2, andr < ^r₂⁰, by Lemma 3.5, we have Z

Br

K(|x|)|un|^sdx^1/s

≤M₀^1/sZ

B_r

|x|^b0|φu_n|^sdx1/s

=M₀^1/sZ

B_r

|x|^(b0−a0)|φun|^(s−α)|x|^a0|φun|^αdx^1/s

≤M4kφunk⁽_W^{s−αs )}

rad

Z

Br

|x|^{(b0−a0−(s−α)2p(N−1)+aα(2p)0 (2p−1))}T(|x|)|un|^αdx1/s

≤M5r⁽

b0−a0

s −(s−α)^2p(N−1)+a_sα(2p)^{0 (2p−1)})

kunk⁽

s−α s ) Wrad kunk_L^αsα

T(Br)

≤M₅⁰r⁽

b0−a0

s −(s−α)^2p(N−1)+a_sα(2p)^{0 (2p−1)})

kunkW_rad

=r^(b0

−a0

s −(s−α)^2p(N−1)+a_sα(2p)^{0 (2p−1)})

o(1), asn→ ∞.

(3.15)

Secondly, we estimate the termR

B_R^c K(|x|)|un|^sdx^1/s . Case 2.1: For−p < a≤b, bys≥s_∗, we obtain

λ1=b−a−(s−α)2p(N−1) +a(2p−1)

α(2p) ≤0.

Hence by Lemma 3.4 and (3.9), forR > R0>1, we have Z

B^c_R

K(|x|)|un|^sdx^1/s

≤M₀^1/sZ

B_R^c

|x|^b|un|^sdx1/s

=M₀^1/sZ

B_R^c

|x|^(b−a)|un|^(s−α)|x|^a|un|^αdx^1/s

≤M6kunk⁽

s−α s ) Wrad

Z

B_R^c

|x|^{(b−a−(s−α)}2p(N−1)+a(2p−1) α(2p) )

T(|x|)|un|^αdx1/s

≤M₇R^{1s(b−a−(s−α)}2p(N−1)+a(2p−1) α(2p) )

kunk⁽

s−α s ) W_rad kunk_L^αsα

T(B^c_r)

≤M₇⁰R^{1s(b−a−(s−α)}

2p(N−1)+a(2p−1) α(2p) )

kunk_Wrad

=R^{1s(b−a−(s−α)}2p(N−1)+a(2p−1) α(2p) )

o(1), as n→ ∞.

(3.16)

Case 2.2: Forb≥ −p, a≤ −p, lets=^2p(N_(N−2p)^+c), bys≥s_∗, we obtainλ2=b−c≤0.

Hence by Lemma 3.3, forR > R₀>1, we have Z

B_R^c

K(|x|)|un|^sdx^1/s

≤M₀^1/sZ

B_R^c

|x|^(b−c)|x|^c|un|^sdx^1/s

≤M8R^{(b−cs )}max k∇unkp,2p,k∇unk²_p,2p

≤M₈⁰R^{(b−cs )}kunk_Wrad =R^{(b−cs )}o(1), as n→ ∞.

(3.17)

Case 2.3: Forb≤max{a,−p},s > α=s_∗. As forR > R₀>1, whena >−p, b≤ a, it always holds

λ₃=b−a−(s−α)2p(N−1) +a(2p−1) α(2p) <0, so similar to Case 2.1, we have

Z

B_R^c

K(|x|)|u_n|^sdx1/s

≤M₇⁰R^{1s(b−a−(s−α)}

2p(N−1)+a(2p−1) α(2p) )

ku_nk_Wrad

=R^{1s(b−a−(s−α)}2p(N−1)+a(2p−1) α(2p) )

o(1), as n→ ∞.

(3.18)

and when a ≤ −p, b≤ −p ≤c, let s = ^2p(N+c)_(N−2p), we obtain (b−c) ≤ 0, we have similar to Case 2.2 that

Z

B^c_R

K(|x|)|un|^sdx^1/s

≤M₈⁰R^{(b−cs )}kunkW_rad =R^{(b−cs )}o(1), asn→ ∞. (3.19) Now we write

Z

R^N

K(|x|)|un|^sdx= Z

B_r

K(|x|)|un|^sdx+ Z

B^c_R

K(|x|)|un|^sdx +

Z

BR\Br

K(|x|)|un|^sdx.

Ass^∗ is finite ands_∗≤s≤s^∗, by (3.13), (3.14), (3.16), (3.17), (3.18) and Lemma 3.6, we obtain that (3.11) holds. As s^∗ is infinite and s_∗ ≤ s < ∞, by (3.15), (3.16), (3.17), (3.18) and Lemma 3.6, we obtain that (3.11) holds. Therefore the embedding is continuous in each case.

Now we show that the embedding obtained above is compact. Let{un} ⊂ Wrad

be such that

ku_nk_Wrad=k∇u_nk_p,2p+ku_nk_Lα

T(R^N)≤M. (3.20) Without loss of generality, we consider

u_n*0, in Wrad as n→ ∞. (3.21)

To obtain the compactness, we only need to show that

n→∞lim Z

R^N

K(|x|)|u_n|^sdx1/s

= 0. (3.22)

As s_∗ < s < s^∗, the exponents ηi of r in the estimates (3.13), (3.14), (3.15) are strictly positive, and the exponentsλj ofR in the estimates (3.16), (3.17), (3.18) are strictly negative, we obtain the following estimates by similar arguments as above

Z

B_r

K(|x|)|u_n|^sdx1/s

≤M r^ηiku_nk_Wrad, i= 1,2,3, (3.23)

Z

B_R^c

K(|x|)|un|^sdx1/s

≤M R^λjkunk_Wrad, j= 1,2,3, (3.24) By (3.20), (3.23), (3.24) and Lemma 3.6, we obtain (3.22). Hence the embedding is compact in each case. In conclusion, the proof of Theorem 3.1 is complete.

4. Proof of Theorems 1.2 and 1.3

In this section, we prove our main theorems. Now, let us define the functional J :Wrad→Ras:

J(u) = 1 p Z

R^N

(p

1 +|∇u|^2p−1)dx+ 1 α

Z

R^N

T(|x|)|u|^αdx−1 s Z

R^N

K(|x|)|u|^sdx.

(4.1) Obviously, by [5, Lemma 2.2], the functional J is well defined and it is of class C¹. We obtain that solutions of (1.1) are critical points of the functional J. By Remark 2.5 and using the standard Palais’ result [14], we infer that W_rad is a natural constraint for the functionalJ.

In the following propositions and lemmas, we show that the functionalJ satisfies the geometrical assumptions Z2-symmetric version of the Mountain Pass Lemma [16]. More precisely, we have the following result.

Proposition 4.1. The functional J satisfies the following properties:

(i) J(0) = 0;

(ii) there existρ, csuch that J(u)≥c, for anyu∈ W_rad with kuk_Wrad=ρ;

(iii) there existsu∈ Wrad such that J(u)≤0.

Proof. (i) Trivially,J(0) = 0. (ii) As there exists a positive constantc such that c|∇u|^p≤p

1 +|∇u|^2p−1, if|∇u| ≥1, c|∇u|^2p≤p

1 +|∇u|^2p−1, if 0≤ |∇u| ≤1.

Then, ifkuk_Wrad is sufficiently small, byα < p <2p < s, Proposition 2.2 (iv), and sinceW_rad ,→L^s(R^N;K), we have that

J(u)≥c₁ Z

Λ^c_∇u

|∇u|^2pdx+c₂ Z

Λ∇u

|∇u|^pdx+ 1 α

Z

R^N

T(|x|)|u|^αdx

−1 s

Z

R^N

K(|x|)|u|^sdx

≥cmaxZ

Λ^c_∇u

|∇u|^2pdx, Z

Λ∇u

|∇u|^pdx + 1

α Z

R^N

T(|x|)|u|^αdx

−1 s

Z

R^N

K(|x|)|u|^sdx

≥c

k∇uk^2p_p,2p+kuk^α_Lα

T(R^N)− kuk^s_Ls K(R^N)

≥c kuk^2p_W

rad− kuk^s_W

rad

≥c.

(iii) Letu∈C_c^∞(R^N,R), as there exists a positive constantC such that ((p

1 +|∇u|^2p−1)≤C|∇u|^p, if|∇u| ≥1,

C|∇u|^2p, if 0≤ |∇u| ≤1;