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Multiple solutions for ( 𝑝, 𝑞)-Laplacian equations 006
007
𝑁
in R with critical or subcritical exponents 008
009
010
Shibo Liu¹* and Kanishka Perera¹ 011
012
¹*Department of Mathematics & Systems Engineering, Florida Institute of 013
Technology, Melbourne, 32901, FL, USA. 014
015
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*Corresponding author(s). E-mail(s): sliu@fit.edu; 017
Contributing authors: kperera@fit.edu; 018
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Abstract 021
In this paper we study the following ( 𝒑, 𝒒)-Laplacian equation with critical exponent 022
023
𝒓 −2 𝒑 ∗ −2 𝑵 024
−𝚫 𝒑 𝒖 − 𝚫𝒒 𝒖 = 𝝀𝒉(𝒙)|𝒖| 𝒖 + 𝒈(𝒙)|𝒖| 𝒖 in R ,
025
∗
where 1 < 𝒒 ≤ 𝒑 < 𝒓 < 𝒑 . After establishing (𝑷𝑺)𝒄 condition for 𝒄 ∈ (0, 𝒄 ) for a ∗ 026
∗
certain constant 𝒄 by employing the concentration compactness principle of Lions, multi- 027
ple solutions for 𝝀 ≫ 1 are obtained by applying a critical point theorem due to Perera [J. 028
Anal. Math., in press; arXiv:2308.07901]. A similar problem with subcritical exponents is 029
also considered. 030
Keywords: ( 𝒑, 𝒒)-Laplacian, Palais-Smale condition, Concentration compactness principle
031
032
MSC Classification: 35J92 , 35J20 , 35J62 033
034
035
036
1 Introduction 037
038
For 𝑝 ∈ (1, 𝑁), we consider the ( 𝑝, 𝑞)-Laplacian equation 039
040
 𝑟 −2 𝑝 ∗ −2
−Δ 𝑝 𝑢 − Δ𝑞 𝑢 = 𝜆ℎ(𝑥)|𝑢| 𝑢 + 𝑔(𝑥)|𝑢| 𝑢, 041
𝑝 𝑁 𝑁 (1.1)
𝑢 ∈ D (R ) ∩ D (R ),
1, 1,𝑞 042
043
∗
where 1 < 𝑞 ≤ 𝑝 < 𝑟 < 𝑝 := 𝑁 𝑝/(𝑁 − 𝑝), and the weight functions ℎ and 𝑔 satisfy 044
045
046

1
047 (A1 ) ℎ ≥ 0, 𝑔 ≥ 0, 𝑚(Ω) > 0 for Ω = {ℎ𝑔 > 0}, where 𝑚 denotes the Lebesgue measure in
048 R𝑁 ,
∗ ∗
049 ℎ ∈ 𝐿 𝑝 /( 𝑝 −𝑟 ) (R 𝑁 ), 𝑔 ∈ 𝐶 (R 𝑁 ) ∩ 𝐿 ∞ (R 𝑁 ). (1.2)
050
As a typical model equation involving the ( 𝑝, 𝑞)-Laplacian operator with critical growth on
051
unbounded domain R 𝑁 , in recent years the equation (1.1) has been investigated by many
052
authors. Chaves et al. [1] obtained a positive solution for the case 𝑔(𝑥) ≡ 1, provided 𝜆 is large
053
enough. The result was then generalized to the case that 𝑔 is not constant by Latifi & Bayat [?
054
] and Baldelli & Filippucci [? ]. In all these papers, the solution is obtained as weak limit of a
055
(𝑃𝑆) 𝑐 sequence generated via the mountain pass theorem [? ].
056
If 1 < 𝑞 < 𝑟 < 𝑝, under the assumption (1.2) on the weight functions ℎ and 𝑔, a sequence
057
of solutions with negative energy were obtained in Baldelli et al. [? ] for 𝜆 ∈ (𝜆 − , 𝜆+ ), where
058
𝜆± > 0. On the other hand, the case 1 < 𝑟 < 𝑞 < 𝑝 was studied in Huang et al. [? ]. Needless
059
to say, for such multiplicity results, the Palais-Smale (𝑃𝑆) 𝑐 condition for 𝑐 < 0 is necessary,
060
which is established in [? , Lemma 8] and [? , Lemma 2.6], respectively.
061
In this paper, as in [1? ? ] we consider the case 1 < 𝑞 ≤ 𝑝 < 𝑟. Thanks to an abstract critical
062
point theorem recently obtained by Perera [? , Theorem 2.1], we will prove the following
063
multiplicity result.
064
065 Theorem 1.1. Let 𝑝 ∈ (1, 𝑁) and 1 < 𝑞 ≤ 𝑝 < 𝑟 < 𝑝 ∗ . Suppose (A1 ) holds. Given 𝑚 ∈ N,
066 there is 𝜆 𝑚 > 0 such that (1.1) has 𝑚 pairs of solutions with positive energy for all 𝜆 > 𝜆 𝑚 .
067
To get multiplicity results like Theorem 1.1, a crucial step is to prove the local Palais-
068
Smale condition, that is the (𝑃𝑆) 𝑐 condition for 𝑐 > 0 small; see Lemma 3.6. This step has
069
been avoided in [1? ? ] because these papers only concern one nontrivial solution. As a price
070
they need to show that the weak limit of the (𝑃𝑆) 𝑐 sequence generated via the mountain pass
071
theorem is nonzero. We will see that this is not less complicated than proving the local Palais-
072
Smale condition, see Remark 3.7.
073
074 One may expect a similar result for the subcritical case in which the exponent 𝑝 ∗ is replaced
075 by some 𝑠 ∈ ( 𝑝, 𝑝 ∗ ). In this case, because 𝑠 = 𝑝 ∗ is the only exponent allowing a con-
076 tinuous embedding D 1, 𝑝 (R 𝑁 ) ↩→ 𝐿 𝑠 (R 𝑁 ), to apply variational method
´ we have to require
∗ ∗
077 𝑔 ∈ 𝐿 𝑝 /( 𝑝 −𝑠) (R 𝑁 ). Then, it is well known that because 𝑢 ↦→ 𝑔|𝑢| 𝑠 is weakly contin-
078 uous on D 1, 𝑝 (R 𝑁 ), the corresponding energy functional satisfies (𝑃𝑆) 𝑐 for all 𝑐 ∈ R, and
079 infinitely many pairs of solutions can be easily obtained by applying the symmetric mountain
080 pass theorem [? ].
081 Therefore, for subcritical case instead of D 1, 𝑝 (R 𝑁 ) we should work on 𝑊 1, 𝑝 (R 𝑁 ) and
082 consider the following equation with two more terms on the left hand side
083 
084 −Δ 𝑝 𝑢 − Δ𝑞 𝑢 + |𝑢| 𝑝−2 𝑢 + |𝑢| 𝑞−2 𝑢 = 𝜆ℎ(𝑥)|𝑢| 𝑟 −2 𝑢 + 𝑔(𝑥)|𝑢| 𝑠−2 𝑢
(1.3)
085 𝑢 ∈ 𝑊 1, 𝑝 (R 𝑁 ) ∩ 𝑊 1,𝑞 (R 𝑁 ),
086
087 where 1 < 𝑞 ≤ 𝑝, {𝑟, 𝑠} ⊂ ( 𝑝, 𝑝 ∗ ). Then we have the following result.
088
Theorem 1.2. Let 𝑝 ∈ (1, 𝑁), 1 < 𝑞 ≤ 𝑝, {𝑟, 𝑠} ⊂ ( 𝑝, 𝑝 ∗ ). Suppose (A1 ) holds. Given
089
𝑚 ∈ N, there is 𝜆 𝑚 > 0 such that (1.3) has 𝑚 pairs of solutions with positive energy for all
090
𝜆 > 𝜆𝑚.
091
092

2
 ∗ ∗
Remark 1.3. In Theorem 1.1 the role of the condition that ℎ ∈ 𝐿 𝑝 /( 𝑝 −𝑟 ) (R 𝑁 ) is two folds: 093
1, 𝑝 𝑁 𝑁
´ is well defined on D (R ) ∩ D (R ), and to ensure the
to ensure that Φ given in (2.2) 1,𝑞
094
weak continuity of 𝜓 : 𝑢 ↦→ ℎ|𝑢| 𝑟 on D 1, 𝑝 (R 𝑁 ). For Theorem 1.2, without this condition 095
the variational functional 𝐼 given in (2.5) may still be well-defined on 𝑊 1, 𝑝 (R 𝑁 ) ∩𝑊 1,𝑞 (R 𝑁 ). 096
Therefore we may replaced this condition by other conditions which ensures the compactness 097
of 𝜓 and 𝜓 ′ on 𝑊 1, 𝑝 (R 𝑁 ). For example, it is well known that if 098
099
lim ℎ(𝑥) = 0, (1.4) 100
| 𝑥 |→∞
101
∗ /( 𝑝 ∗ −𝑟 ) 102
then 𝜓 is weakly continuous and 𝜓 ′ is compact. Hence, replacing ℎ ∈ 𝐿 𝑝 (R 𝑁 ) by
103
(1.4), the conclusion of Theorem 1.2 is still valid.
104
In Theorems 1.1 and 1.2, if 𝑝 = 𝑞 we get similar multiplicity results for 𝑝-Laplacian 105
equations. For example, if 𝑝 = 𝑞, equation (1.1) reduces to 106
107
∗ −2
−Δ 𝑝 𝑢 = 𝜆ℎ(𝑥)|𝑢| 𝑟 −2 𝑢 + 𝑔(𝑥)|𝑢| 𝑝 𝑢, in R 𝑁 . (1.5) 108
109
This equation has been studied by Gonçalve & Alves [? ]; see also Swanson & Yu [? ], where a 110
new term 𝑎(𝑥)|𝑢| 𝑝−2 𝑢 with small coefficient is added to the right hand side of (1.5). A positive 111
solution was obtained in [? ? ]. 112
Elliptic equations driven by ( 𝑝, 𝑞)-Laplacian operator 𝐴 𝑝,𝑞 = Δ 𝑝 + Δ𝑞 come from finding 113
stationary solutions for general reaction-diffusion equations of the form 114
115
𝑢 𝑡 = div(𝐷 (𝑢)∇𝑢) + 𝑐(𝑥, 𝑢), 116
117
where 𝐷 (𝑢) = |∇𝑢| 𝑝−2 + |∇𝑢| 𝑞−2 . This equation has a wide range of applications in physical 118
and related sciences including biophysics, plasma physics, solid state physics, and chemical 119
reaction design. Initiated from Benci et al [? ], such ( 𝑝, 𝑞)-Laplacian equations have captured 120
great interests in the last two decades. For a bounded smooth domain Ω ⊂ R 𝑁 , the following 121
problem  122
∗
−Δ 𝑝 𝑢 − Δ𝑞 𝑢 = 𝜆 |𝑢| 𝑟 −2 𝑢 + |𝑢| 𝑝 −2 𝑢 in Ω, 123
𝑢=0 on 𝜕Ω 124
was studied in [? ], [? ] and [? ] in the cases 1 < 𝑟 < 𝑞, 𝑝 < 𝑟 < 𝑝 ∗ and 𝑞 < 𝑟 < 𝑝, respectively; 125
see also [? ? ] for related results. Results for the case that Ω = R 𝑁 have been mentioned 126
in the paragraphs before the statement of Theorem 1.1. In the papers we have mentioned, 127
the nonlinearity is a combination of power functions of 𝑢. For results on ( 𝑝, 𝑞)-Laplacian 128
equations with general nonlinearity 𝑓 (𝑥, 𝑢), we refer to [? ? ? ]. 129
The paper is organized as follows. In Section 2 we present the variational frameworks, that 130
is the Sobolev function spaces and the variational functionals defined on them corresponding 131
to our problems (1.1) and (1.3). Then we prove Theorems 1.1 and 1.2 by verifying the geomet- 132
ric assumptions required in the critical point theorem of Perera [? ] mentioned above. Due to 133
the lack of compactness of the embedding from Sobolev spaces into 𝐿 𝛼 (R 𝑁 ) for 𝛼 ∈ [ 𝑝, 𝑝 ∗ ], 134
the necessary local Palais-Smale condition is very technical. In Section 3, using the concentra- 135
tion compactness principles of Lions we verify this condition for subcritical case and critical 136
case in the two subsections, respectively. 137
138

3
139 2 Proof of Theorems 1.1 and 1.2
140
141 Let 𝑋 = D 1, 𝑝 (R 𝑁 ) ∩ D 1,𝑞 (R 𝑁 ) be the Banach space equipped with the norm
142
143 ∥𝑢∥ = |∇𝑢| 𝑝 + |∇𝑢| 𝑞 , (2.1)
144
145 where | · | 𝑚 is the 𝐿 𝑚 -norm for 𝑚 ∈ [1, ∞]. Define the 𝐶 1 -functional Φ : 𝑋 → R,
146 ˆ ˆ ˆ ˆ
1 𝑝 1 𝑞 𝜆 𝑟 1 ∗
147 Φ(𝑢) = |∇𝑢| + |∇𝑢| − ℎ|𝑢| − ∗ 𝑔|𝑢| 𝑝 , (2.2)
148 𝑝 𝑞 𝑟 𝑝
149
150 where from now on all integrals are taken over R 𝑁 (with respect to the Lebesgue measure)
151 unless specified differently. Then critical points of Φ are solutions of (1.1). To prove our
152 theorems we will apply the following critical point theorem due to Perera [? ].
153 Let 𝑋 be a Banach space. For a symmetric subset 𝐴 of 𝑋\{0}, we denote by 𝑖( 𝐴) the coho-
154 mological index of 𝐴, which was introduced by Fadell & Rabinowitz [? ], see the paragraph
155 after Remark 1.9 in [? ] for a brief description. If 𝐴 is homeomorphic to the unit sphere 𝑆 𝑚−1
156 in R𝑚 , then 𝑖( 𝐴) = 𝑚.
157 Theorem 2.1 ([? , Theorem 2.1]). Let 𝑋 be a Banach space, 𝐽 : 𝑋 → R be an even 𝐶 1 -
158 functional satisfying (𝑃𝑆) 𝑐 for 𝑐 ∈ (0, 𝑐 ∗ ), where 𝑐∗ is some positive constant. If 0 is a strict
159 local minimizer of 𝐽 and there are 𝑅 > 0 and a compact symmetric set 𝐴 ⊂ 𝜕𝐵 𝑅 , where
160 𝐵 𝑅 = {𝑢 ∈ 𝑋 | ∥𝑢∥ < 𝑅}, such that 𝑖( 𝐴) = 𝑚,
161
162 max 𝐽 ≤ 0, max 𝐽 < 𝑐∗ ,
𝐴 𝐵
163
164
where 𝐵 = {𝑡𝑢 | 𝑡 ∈ [0, 1], 𝑢 ∈ 𝐴}, then 𝐽 has 𝑚 pairs of nonzero critical points with positive
165
critical values.
166
167 Proof of Theorem 1.1. Let
168
169 |∇𝑢| 𝑝𝑝 1 𝑁 / 𝑝 ( 𝑝− 𝑁 )/ 𝑝
𝑆= inf , 𝑐∗ = 𝑆 |𝑔| ∞ .
170 𝑢∈ D 1, 𝑝 (R 𝑁 )\{0} |𝑢| 𝑝𝑝∗ 𝑁
171
172 By Lemma 3.6 below, Φ satisfies (𝑃𝑆) 𝑐 for 𝑐 ∈ (0, 𝑐 ∗ ). Since 𝑞 ≤ 𝑝 < 𝑟 < 𝑝 ∗ , it is clear that
173 𝑢 = 0 is a strict local minimizer of Φ. Let 𝑍 = {𝑢 ∈ 𝑋 | supp 𝑢 ⊂ Ω}, where Ω is as in (A1 ),
174 then 𝑍 is an infinite dimensional subspace of 𝑋. Let 𝑍 𝑚 be an 𝑚-dimensional subspace of 𝑍.
175 On 𝑍 𝑚 ,
176 ˆ  1/𝑟 ˆ  1/ 𝑝∗
𝑟 𝑝∗
177 [𝑢] ℎ = ℎ|𝑢| , [𝑢] 𝑔 = 𝑔|𝑢|
178
are norms of 𝑢 ∈ 𝑍 𝑚 . Since dim 𝑍 𝑚 < ∞, all norms on 𝑍 𝑚 are equivalent. Therefore, for
179
𝑢 ∈ 𝑍 𝑚 we have
180
181 ˆ ˆ ˆ ˆ
1 1 𝜆 1 ∗
182 Φ(𝑢) = |∇𝑢| 𝑝 + |∇𝑢| 𝑞 − ℎ|𝑢| 𝑟 − ∗ 𝑔|𝑢| 𝑝
𝑝 𝑞 𝑟 𝑝
183 ∗
184 ≤ 𝑐 1 ∥𝑢∥ 𝑝 + 𝑐 2 ∥𝑢∥ 𝑞 − 𝜆𝑐 3 ∥𝑢∥ 𝑟 − 𝑐 4 ∥𝑢∥ 𝑝 . (2.3)

4
Take 𝑅 > 0 such that 185
∗
𝑓 (𝑅) := 𝑐 1 𝑅 𝑝 + 𝑐 2 𝑅 𝑞 − 𝑐 4 𝑅 𝑝 < 0. (2.4) 186
Let 𝐴 := 𝑍 𝑚 ∩ 𝜕𝐵 𝑅 , then 𝑖( 𝐴) = 𝑚. For any 𝜆 > 0, if 𝑢 ∈ 𝐴 we have Φ(𝑢) ≤ 𝑓 (𝑅) < 0, see 187
(2.3). Thus 188
max Φ ≤ 0. 189
𝐴 190
Since the function 𝑓 given in (2.4) is continuous and 𝑓 (0) = 0, we may take 𝛿 ∈ (0, 𝑅) such 191
that 𝑓 (𝑠) < 𝑐 ∗ for all 𝑠 ∈ [0, 𝛿]. Let 192
193
𝑓 (𝑠) − 𝑐∗ 194
𝜆 𝑚 = 1 + max .
𝑠∈ [ 𝛿,𝑅] 𝑐 3 𝑠𝑟 195
196
If 𝜆 > 𝜆 𝑚 we have 197
𝑓 (𝑠) − 𝜆𝑐 3 𝑠𝑟 < 𝑐 ∗ for 𝑠 ∈ [𝛿, 𝑅] . 198
Then for 𝑢 ∈ 𝐴, 199
  200
1. if 𝑡 ∈ 𝑅𝛿 , 1 we have ∥𝑡𝑢∥ ∈ [𝛿, 𝑅], hence
201
202
Φ(𝑡𝑢) ≤ 𝑓 (∥𝑡𝑢∥) − 𝜆𝑐 3 · ∥𝑡𝑢∥ 𝑟 < 𝑐 ∗ ;
203
  204
2. if 𝑡 ∈ 0, 𝑅𝛿 then ∥𝑡𝑢∥ ≤ 𝛿, hence Φ(𝑡𝑢) ≤ 𝑓 (∥𝑡𝑢∥) < 𝑐 ∗ .
205
Therefore, for 𝐵 = {𝑡𝑢 | 𝑡 ∈ [0, 1] , 𝑢 ∈ 𝐴} we have 206
207
max Φ < 𝑐 ∗ . 208
𝐵
209
Applying Theorem 2.1, Φ has 𝑚 pairs of nonzero critical points, which are nontrivial solutions 210
of (1.1). □ 211
212
To prove Theorem 1.2, we equip on 𝐸 = 𝑊 1, 𝑝 (R 𝑁 ) ∩ 𝑊 1,𝑞 (R 𝑁 ) the norm 213
ˆ  1/𝑚 214
∥𝑢∥ 𝐸 = ∥𝑢∥ 1, 𝑝 + ∥𝑢∥ 1,𝑞 , where ∥𝑢∥ 1,𝑚 = 𝑚
(|∇𝑢| + |𝑢| ) 𝑚 215
216
217
for 𝑚 ∈ {𝑝, 𝑞}. With this norm, 𝐸 is a Banach space, critical points of 𝐼 : 𝐸 → R defined via 218
ˆ ˆ ˆ ˆ 219
1 1 𝜆 1
𝐼 (𝑢) = (|∇𝑢| 𝑝 + |𝑢| 𝑝 ) + (|∇𝑢| 𝑞 + |𝑢| 𝑞 ) − ℎ|𝑢| 𝑟 − 𝑔|𝑢| 𝑠 (2.5) 220
𝑝 𝑞 𝑟 𝑠 221
222
are solutions of (1.3). Let 223
 224
∥𝑢∥ 1, 𝑝 1 1 𝑠/(𝑠− 𝑝) 𝑝/( 𝑝−𝑠)
𝑆𝑠 = inf , 𝑐𝑠 = − 𝑆 |𝑔| ∞ . 225
𝑢∈𝑊 1, 𝑝 (R 𝑁 )\{0} |𝑢| 𝑠𝑝 𝑝 𝑠 𝑠 226
227
We will show that 𝐼 satisfies (𝑃𝑆) 𝑐 for 𝑐 ∈ (0, 𝑐 𝑠 ) in Lemma 3.4 below. It is clear that 𝐼 pos- 228
sesses similar geometric properties as Φ. Therefore, 𝑚 pairs of critical points can be obtained 229
similarly by applying Theorem 2.1. 230

5
231 3 Local Palais-Smale condition
232
233 To establish local (𝑃𝑆) condition for Φ and 𝐼, we define
234
235 L 𝑚 : D 1,𝑚 (R 𝑁 ) → (D 1,𝑚 (R 𝑁 )) ∗ , ∗
L𝑚 : 𝑊 1,𝑚 (R 𝑁 ) → (𝑊 1,𝑚 (R 𝑁 )) ∗
236
237 for 𝑚 ∈ (1, ∞) via
238 ˆ
239 ⟨L 𝑚 𝑢, 𝜙⟩ = |∇𝑢| 𝑚−2 ∇𝑢 · ∇𝜙, for all 𝜙 ∈ D 1,𝑚 (R 𝑁 ), (3.1)
240 ˆ  
241 ∗
L𝑚 𝑢, 𝜙 = |∇𝑢| 𝑚−2 ∇𝑢 · ∇𝜙 + |𝑢| 𝑚−2 𝑢𝜙 , for all 𝜙 ∈ 𝑊 1,𝑚 (R 𝑁 ). (3.2)
242
243
Remark 3.1. Recall that a nonlinear operator L : 𝑋 → 𝑋 ∗ is of (𝑆)-type if 𝑢 𝑛 ⇀ 𝑢 in 𝑋 and
244
245
⟨L𝑢 𝑛 , 𝑢 𝑛 − 𝑢⟩ → 0
246
247 ∗ are monotone and of (𝑆)-type (see e.g.
imply 𝑢 𝑛 → 𝑢 in 𝑋. It is well known that L 𝑚 and L 𝑚
248
[? , page 181]).
249
250
251
3.1 The subcritical case
252 In this subsection we establish the local Palais-Smale condition in the subcritical case, which
253 is needed for proving Theorem 1.2. The generalized Lebesgue dominated theorem will be
254 needed in the argument. For the reader’s convenience, we recall it below.
255
Proposition 3.2 ([? , page 77]). Let 𝑓𝑛 , 𝑔𝑛 : Ω → R be measurable functions over the
256
measurable set Ω, 𝑓𝑛 → 𝑓 a.e. in Ω, 𝑔𝑛 → 𝑔 a.e. in Ω, | 𝑓𝑛 | ≤ 𝑔𝑛 . Then
257
258 ˆ ˆ ˆ ˆ ˆ
259 𝑓𝑛 → 𝑓, provided 𝑔𝑛 → 𝑔 and 𝑔 < ∞.
Ω Ω Ω Ω Ω
260
261 Remark 3.3. If 𝑔𝑛 = 𝑔, this reduces to the usual Lebesgue dominated theorem.
262
263 Lemma 3.4. The functional 𝐼 satisfies (𝑃𝑆) 𝑐 for all 𝑐 satisfying
264  
1 1 𝑠/(𝑠− 𝑝) 𝑝/( 𝑝−𝑠)
265 0 < 𝑐 < 𝑐 𝑠 := − 𝑆 |𝑔| ∞ . (3.3)
266 𝑝 𝑠 𝑠
267
Proof. Let {𝑢 𝑛 } ⊂ 𝐸 be a (𝑃𝑆) 𝑐 sequence of 𝐼, that is
268
269 ∑ 1 ˆ 𝜆
ˆ
1
ˆ
270 𝐼 (𝑢 𝑛 ) = (|∇𝑢 𝑛 | 𝑚 + |𝑢 𝑛 | 𝑚 ) − ℎ|𝑢 𝑛 | 𝑟 − 𝑔|𝑢 𝑛 | 𝑠 → 𝑐, (3.4)
𝑚 𝑟 𝑠
271 𝑚∈ { 𝑝,𝑞 }
272 ∑ ˆ  
273 ⟨𝐼 ′ (𝑢 𝑛 ), 𝜙⟩ = |∇𝑢 𝑛 | 𝑚−2 ∇𝑢 𝑛 · ∇𝜙 + |𝑢 𝑛 | 𝑚−2 𝑢 𝑛 𝜙
𝑚∈ { 𝑝,𝑞 }
274 ˆ ˆ
275 −𝜆 ℎ|𝑢 𝑛 | 𝑟 −2 𝑢 𝑛 𝜙 − 𝑔|𝑢 𝑛 | 𝑠−2 𝑢 𝑛 𝜙 = 𝑜(∥𝜙∥) (3.5)
276

6
for all 𝜙 ∈ D 1, 𝑝 (R 𝑁 ). Firstly we suppose 𝑟 ≤ 𝑠, noting 𝑞 ≤ 𝑝 < 𝑟, for 𝑛 large we get 277
278
1 ′ 279
1 + 𝑐 + ∥𝑢 𝑛 ∥ 𝐸 ≥ 𝐼 (𝑢 𝑛 ) − ⟨𝐼 (𝑢 𝑛 ), 𝑢 𝑛 ⟩
𝑟 ˆ  ˆ 280
∑ 1 1 1 1 281
𝑚 𝑚
= − (|∇𝑢 𝑛 | + |𝑢 𝑛 | ) + − 𝑔|𝑢 𝑛 | 𝑠
𝑚 𝑟 𝑟 𝑠 282
𝑚∈ { 𝑝,𝑞 }
    283
1 1 𝑝 1 1 𝑞 284
≥ − ∥𝑢 𝑛 ∥ 1, 𝑝 + − ∥𝑢 𝑛 ∥ 1,𝑞 . (3.6)
𝑝 𝑟 𝑞 𝑟 285
286
If 𝑠 ≤ 𝑟, for 𝑛 large we have 287
288
1 ′
1 + 𝑐 + ∥𝑢 𝑛 ∥ 𝐸 ≥ 𝐼 (𝑢 𝑛 ) − ⟨𝐼 (𝑢 𝑛 ), 𝑢 𝑛 ⟩ 289
𝑠 ˆ   ˆ 290
∑ 1 1 1 1
= − (|∇𝑢 𝑛 | 𝑚 + |𝑢 𝑛 | 𝑚 ) + − 𝜆 ℎ|𝑢 𝑛 | 𝑟 291
𝑚 𝑟 𝑠 𝑟 292
𝑚∈ { 𝑝,𝑞 }
    293
1 1 𝑝 1 1 𝑞
≥ − ∥𝑢 𝑛 ∥ 1, 𝑝 + − ∥𝑢 𝑛 ∥ 1,𝑞 . (3.7) 294
𝑝 𝑠 𝑞 𝑠 295
296
From (3.6) or (3.7), it is easy to see that {𝑢 𝑛 } is bounded.
297
To overcome the difficulty that the embedding 𝐸 ↩→ 𝐿 𝑠 (R 𝑁 ) is not compact, we shall
298
apply the first concentration compactness principle of Lions [? ] (see also [? , page 39]). We
299
will process in the spirit of [? ], and [? ? ], where 𝑝-Laplacian and ( 𝑝, 𝑞)-Laplacian equations
300
with critical growth were studied, respectively.
301
Consider the sequence of measure 𝜂 𝑛 = 𝑧 𝑛 d𝑥, where
302
303
𝑧 𝑛 = |∇𝑢 𝑛 | 𝑝 + |∇𝑢 𝑛 | 𝑞 + |𝑢 𝑛 | 𝑝 + |𝑢 𝑛 | 𝑞 + |𝑢 𝑛 | 𝑠 + 𝜆ℎ|𝑢 𝑛 | 𝑟 .
304
´ 305
Firstly, because 𝑧 𝑛 is bounded, up to a subsequence we have
306
ˆ 307
lim 𝑧 𝑛 = Λ > 0. (3.8) 308
𝑛→∞
309
In fact, if Λ = 0 then 310
311
ˆ
𝑝 𝑞 312
∥𝑢 𝑛 ∥ 1, 𝑝 + ∥𝑢 𝑛 ∥ 1,𝑞 = (|∇𝑢 𝑛 | 𝑝 + |∇𝑢 𝑛 | 𝑞 + |𝑢 𝑛 | 𝑝 + |𝑢 𝑛 | 𝑞 ) → 0. 313
314
Hence 𝑢 𝑛 → 0 in 𝐸, and 𝐼 (𝑢 𝑛 ) → 0, contradicting 𝑐 > 0. 315
Because of (3.8), we can apply the first concentration compactness principle mentioned 316
above, and deduce that one of the following three possibilities must occur: 317
318
1. Compactness. 319
320
321
322

7
323 2. Vanishing: for all 𝑅 > 0 there holds
324 ˆ
325 sup 𝑧 𝑛 → 0,
326 𝑦 ∈R 𝑁 𝐵 𝑅 ( 𝑦)
327
328 which can not occur because of (3.8).
329 3. Dichotomy.
330
If dichotomy occurs, there is ℓ ∈ (0, Λ) such that given 𝜀 > 0, there exist 𝑅 > 0, 𝑅𝑛 ↑ ∞
331
and {𝑦 𝑛 } ⊂ R 𝑁 such that
332
333 ˆ ˆ ˆ
334 𝑧 𝑛 − ℓ < 𝜀, 𝑧 𝑛 − (Λ − ℓ) < 𝜀, 𝑧 𝑛 < 𝜀, (3.9)
335 𝐵 𝑅 ( 𝑦𝑛 ) 𝐵c𝑅𝑛 ( 𝑦𝑛 ) 𝐵 𝑅𝑛 ( 𝑦𝑛 )\𝐵 𝑅 ( 𝑦𝑛 )
336
337 here and below, for 𝐴 ⊂ R 𝑁 we denote by 𝐴c its complement.
338 Pick 𝜑 ∈ 𝐶0∞ (R 𝑁 ) such that |𝜑| ∞ = 1,
339
340 𝜑(𝑥) = 1 for 𝑥 ∈ 𝐵1 , 𝜑(𝑥) = 0 for 𝑥 ∈ 𝐵2c , (3.10)
341


342 then set 𝑢 1𝑛 = 𝜑1𝑛 𝑢 𝑛 and 𝑢 2𝑛 = 1 − 𝜑2𝑛 𝑢 𝑛 , with
343
𝑥 − 𝑦   
344 𝑛 𝑥 − 𝑦𝑛
345 𝜑1𝑛 (𝑥) = 𝜑 , 𝜑2𝑛 (𝑥) = 𝜑 .
𝑅 𝑅𝑛 /2
346
347 Then |∇𝜑1𝑛 | ∞ ≤ 𝑅1 |∇𝜑| ∞ , |∇𝜑2𝑛 | ∞ ≤ because 𝑅𝑛 ≥ 𝑅 for 𝑛 large. For 𝑚 ∈ {𝑝, 𝑞},
𝑅 |∇𝜑| ∞
2
348 using (3.9) we have
349
ˆ
350
351 |𝜑1𝑛 ∇𝑢 𝑛 + 𝑢 𝑛 ∇𝜑1𝑛 | 𝑚
𝐵 𝑅𝑛 ( 𝑦𝑛 )\𝐵 𝑅 ( 𝑦𝑛 )
352 ˆ !
 
353 ≤2 𝑚
|𝜑1𝑛 ∇𝑢 𝑛 | 𝑚 + |𝑢 𝑛 ∇𝜑1𝑛 | 𝑚
354 𝐵 𝑅𝑛 ( 𝑦𝑛 )\𝐵 𝑅 ( 𝑦𝑛 )
355  ˆ
1
356 ≤ 2𝑚 1 + |∇𝜑| ∞
𝑚
(|∇𝑢 𝑛 | 𝑚 + |𝑢 𝑛 | 𝑚 )
𝑅 𝐵 𝑅𝑛 ( 𝑦𝑛 )\𝐵 𝑅 ( 𝑦𝑛 )
357  ˆ  
358 𝑚 1 𝑚 𝑚 1 𝑚
≤ 2 1 + |∇𝜑| ∞ 𝑧 𝑛 ≤ 2 1 + |∇𝜑| ∞ 𝜀. (3.11)
359 𝑅 𝐵 𝑅𝑛 ( 𝑦𝑛 )\𝐵 𝑅 ( 𝑦𝑛 ) 𝑅
360
361 Consequently, noting
362 supp 𝑢 1𝑛 ⊂ 𝐵2𝑅 (𝑦 𝑛 ) ⊂ 𝐵 𝑅𝑛 (𝑦 𝑛 )
363 for 𝑛 large, and 𝜑1𝑛 = 1 in 𝐵 𝑅 (𝑦 𝑛 ), we have
364
ˆ ˆ
365
366 |∇𝑢 1𝑛 | 𝑚 = |𝜑1𝑛 ∇𝑢 𝑛 + 𝑢 𝑛 ∇𝜑1𝑛 | 𝑚
𝐵 𝑅𝑛 ( 𝑦𝑛 )
367
368

8
 ˆ ˆ
= |∇𝑢 𝑛 | 𝑚 + |𝜑1𝑛 ∇𝑢 𝑛 + 𝑢 𝑛 ∇𝜑1𝑛 | 𝑚 369
𝐵 𝑅 ( 𝑦𝑛 ) 𝐵 𝑅𝑛 ( 𝑦𝑛 )\𝐵 𝑅 ( 𝑦𝑛 ) 370
ˆ
𝑚
371
= |∇𝑢 𝑛 | + 𝑜 𝜀 (1). (3.12) 372
𝐵 𝑅 ( 𝑦𝑛 )
373
Similar arguments yield 374
375
ˆ ˆ
376
|∇𝑢 2𝑛 | 𝑚 = |𝜑2𝑛 ∇𝑢 𝑛 + 𝑢 𝑛 ∇𝜑2𝑛 | 𝑚 377
𝐵c𝑅 (𝑦 )
/2 𝑛
ˆ 𝑛
ˆ 378
𝑚
= |∇𝑢 𝑛 | + |𝜑2𝑛 ∇𝑢 𝑛 + 𝑢 𝑛 ∇𝜑2𝑛 | 𝑚 379
𝐵c𝑅𝑛 ( 𝑦𝑛 ) 𝐵 𝑅𝑛 ( 𝑦𝑛 )\𝐵 𝑅𝑛 /2 ( 𝑦𝑛 ) 380
ˆ ˆ
381
≤ |∇𝑢 𝑛 | 𝑚 + |𝜑2𝑛 ∇𝑢 𝑛 + 𝑢 𝑛 ∇𝜑2𝑛 | 𝑚 382
𝐵c𝑅𝑛 ( 𝑦𝑛 ) 𝐵 𝑅𝑛 ( 𝑦𝑛 )\𝐵 𝑅 ( 𝑦𝑛 )
ˆ 383
= |∇𝑢 𝑛 | 𝑚 + 𝑜 𝜀 (1) 384
𝐵c𝑅𝑛 ( 𝑦𝑛 ) 385
386
because 𝐵 𝑅 (𝑦 𝑛 ) ⊂ 𝐵 𝑅𝑛 /2 (𝑦 𝑛 ), and 387
ˆ ˆ ˆ ˆ 388
|𝑢 1𝑛 | 𝑝 = |𝑢 𝑛 | 𝑝 + 𝑜 𝜀 (1), |𝑢 2𝑛 | 𝑝 = |𝑢 𝑛 | 𝑝 + 𝑜 𝜀 (1), 389
𝐵 𝑅 ( 𝑦𝑛 ) 𝐵c𝑅𝑛 ( 𝑦𝑛 ) 390
ˆ ˆ ˆ ˆ 391
ℎ|𝑢 1𝑛 | 𝑟 = ℎ|𝑢 𝑛 | 𝑟 + 𝑜 𝜀 (1), ℎ|𝑢 2𝑛 | 𝑟 = ℎ|𝑢 𝑛 | 𝑟 + 𝑜 𝜀 (1), 392
𝐵 𝑅 ( 𝑦𝑛 ) 𝐵c𝑅𝑛 ( 𝑦𝑛 )
ˆ ˆ ˆ ˆ 393
𝑔|𝑢 1𝑛 | 𝑠 = 𝑠
𝑔|𝑢 𝑛 | + 𝑜 𝜀 (1), 𝑔|𝑢 2𝑛 | 𝑠 = 𝑔|𝑢 𝑛 | 𝑠 + 𝑜 𝜀 (1). 394
𝐵 𝑅 ( 𝑦𝑛 ) 𝐵c𝑅𝑛 ( 𝑦𝑛 ) 395
396
Using these and (3.9), as well as the boundedness of {𝑢 𝑖𝑛 }, we deduce 397
398
2 
∑ ˆ ˆ  399
1 𝑝 1 𝑖 𝑞 𝜆 1
∥𝑢 𝑖𝑛 ∥ 1, 𝑝 + ∥𝑢 𝑛 ∥ 1,𝑞 − ℎ|𝑢 𝑖𝑛 | 𝑟 − 𝑔|𝑢 𝑖𝑛 | 𝑠 = 𝑐 + 𝑜 𝑛 (1) + 𝑜 𝜀 (1), (3.13) 400
𝑝 𝑞 𝑟 𝑠
𝑖=1 401
ˆ ˆ
𝑝 𝑖 𝑞 402
∥𝑢 𝑖𝑛 ∥ 1, 𝑝 + ∥𝑢 𝑛 ∥ 1,𝑞 − 𝜆 ℎ|𝑢 𝑖𝑛 | 𝑟 − 𝑔|𝑢 𝑖𝑛 | 𝑠 + 𝑜 𝜀 (1) = 𝑜 𝑛 (1) (3.14)
403
404
from (3.4) and (3.5) with 𝜙 = 𝑢 𝑖𝑛 and supp 𝑢 1𝑛 ∩ supp 𝑢 2𝑛 = ∅. 405
Up to a subsequence, we may assume |𝑢 𝑖𝑛 | 𝑠𝑝 → 𝛾𝑖 and 406
ˆ ˆ 407
408
𝜆 ℎ|𝑢 𝑖𝑛 | 𝑟 → 𝛼𝑖 , 𝑔|𝑢 𝑖𝑛 | 𝑠 → 𝛽𝑖 .
409
410
Letting 𝑛 → ∞ in ˆ ˆ 411
𝑔|𝑢 𝑖𝑛 | 𝑠 ≤ |𝑔| ∞ |𝑢 𝑖𝑛 | 𝑠 , 412
413
414

9
415 we obtain
416 𝛾𝑖 ≥ 𝛽𝑖𝑝/𝑠 |𝑔| ∞
− 𝑝/𝑠
. (3.15)
417 From (3.14) we have
418
𝑝 𝑖 𝑞
419 ∥𝑢 𝑖𝑛 ∥ 1, 𝑝 + ∥𝑢 𝑛 ∥ 1,𝑞 = 𝛼𝑖 + 𝛽𝑖 + 𝑜 𝑛 (1) + 𝑜 𝜀 (1). (3.16)
420
421 Since 𝑞 ≤ 𝑝 ≤ 𝑟, (3.13) and (3.16) yield
422
423 2 
∑   
𝛼𝑖 + 𝛽𝑖 𝛼𝑖 𝛽𝑖 1 1
424 𝑐≥ − − ≥ − (𝛽1 + 𝛽2 ) . (3.17)
425 𝑝 𝑟 𝑠 𝑝 𝑠
𝑖=1
426
427 If {𝑦 𝑛 } is bounded, then {𝑦 𝑛 } ⊂ 𝐵𝜌 for some 𝜌 > 0. Because supp 𝑢 2𝑛 ⊂ 𝐵c(𝑅𝑛 /2) −𝜌 , and
428
ˆ
429
430 𝜓 : D 1, 𝑝 (R 𝑁 ) → R, 𝜓(𝑢) = ℎ|𝑢| 𝑟 (3.18)
431
432 is weakly continuous (see [? , Lemma 1], note that 𝑢 𝑛 ⇀ 𝑢 in D 1, 𝑝 ), we deduce
433 ˆ ˆ
434
ℎ|𝑢 2𝑛 | 𝑟 · 𝜒𝑛c →0 a.e. in R , 𝑁
ℎ|𝑢 2𝑛 | 𝑟 · 𝜒𝑛c ≤ ℎ|𝑢 𝑛 | ,𝑟 𝑟
ℎ|𝑢 𝑛 | → ℎ|𝑢| 𝑟 ,
435 R𝑁 R𝑁
436
437 where 𝜒𝑛c is the indicator function of 𝐵c(𝑅𝑛 /2) −𝜌 . Applying the generalized Lebesgue dominated
438 theorem (Proposition 3.2) we get
439 ˆ ˆ ˆ
440 ℎ|𝑢 2𝑛 | 𝑟 = ℎ|𝑢 2𝑛 | 𝑟 = ℎ|𝑢 2𝑛 | 𝑟 · 𝜒𝑛c → 0.
441 𝐵c(𝑅 R𝑁
𝑛 /2) −𝜌
442
443 It follows that 𝛼2 = 0. Using (3.16) with 𝑖 = 2 and then (3.15), we get
444
445 𝑝 2 𝑝 𝑝/𝑠 − 𝑝/𝑠
𝛽2 ≥ lim ∥𝑢 2𝑛 ∥ 1, 𝑝 ≥ 𝑆 𝑠 lim |𝑢 𝑛 | 𝑠 = 𝑆 𝑠 𝛾2 ≥ 𝑆 𝑠 𝛽2 |𝑔| ∞ . (3.19)
446 𝑛→∞ 𝑛→∞

447
448 Thus 𝛽2 ≥ 𝑆 𝑠𝑠/(𝑠− 𝑝) |𝑔| ∞
𝑝/( 𝑝−𝑠)
. Now using (3.17) we get a contradiction with (3.3):
449    
450 1 1 1 1 𝑠/(𝑠− 𝑝) 𝑝/( 𝑝−𝑠)
𝑐≥ − 𝛽2 ≥ − 𝑆 |𝑔| ∞ .
451 𝑝 𝑠 𝑝 𝑠 𝑠
452
453 If {𝑦 𝑛 } is unbounded, up to a subsequence we may assume |𝑦 𝑛 | → ∞. Denoting by 𝜒𝑛 the
454 indicator function of 𝐵 𝑅 (𝑦 𝑛 ) we have
455
456 ℎ|𝑢 1𝑛 | 𝑟 · 𝜒𝑛 → 0 a.e. in R 𝑁 .
457
458 By similar argument 𝛼1 = 0, 𝛽1 ≥ 𝑆 𝑠𝑠/(𝑠− 𝑝) |𝑔| ∞
𝑝/( 𝑝−𝑠)
, and we get a contradiction with (3.3)
459 similarly. This completes the proof that dichotomy can not occur.
460

10
 Therefore, the bounded sequence 𝜂 𝑛 = 𝑧 𝑛 d𝑥 is compact: there is a sequence {𝑦 𝑛 } in R 𝑁 461
such that given 𝜀 > 0, there is 𝑅 𝜀 ∈ (0, ∞) such that 462
ˆ 463
𝑧 𝑛 ≤ 𝜀. (3.20) 464
𝐵c𝑅 𝜀 ( 𝑦𝑛 ) 465
466
We claim that {𝑦 𝑛 } is bounded. Otherwise we may assume |𝑦 𝑛 | → ∞. Given 𝜌 > 0, 467
𝐵𝜌 ⊂ 𝐵c𝑅 𝜀 (𝑦 𝑛 ) for 𝑛 large, thus 468
469
ˆ ˆ ˆ
𝑟 𝑟
470
ℎ|𝑢 𝑛 | ≤ ℎ|𝑢 𝑛 | ≤ 𝑧 𝑛 ≤ 𝜀. 471
𝐵𝜌 𝐵c𝑅 𝜀 ( 𝑦𝑛 ) 𝐵c𝑅 𝜀 ( 𝑦𝑛 )
472
Letting 𝑛 → ∞ and then 𝜌 → ∞, by the weak continuity of the functional 𝜓 given in (3.18), 473
474
we deduce ˆ ˆ
475
lim ℎ|𝑢 𝑛 | 𝑟 = ℎ|𝑢| 𝑟 = 0. (3.21) 476
𝑛→∞
Consequently, let 𝜙 = 𝑢 𝑛 in (3.5), we get 477
478
ˆ
𝑝 𝑞 479
∥𝑢 𝑛 ∥ 1, 𝑝 +
∥𝑢 𝑛 ∥ 1,𝑞 = 𝑔|𝑢 𝑛 | 𝑠 + 𝑜(1). (3.22) 480
481
Up to a subsequence we have 482
ˆ 483
 
𝑝
lim ∥𝑢 𝑛 ∥ 1, + 𝑞
∥𝑢 𝑛 ∥ 1,𝑞 = lim 𝑔|𝑢 𝑛 | 𝑠 =: 𝛽. (3.23) 484
𝑛→∞ 𝑝 𝑛→∞ 485
486
We have 𝛽 > 0, otherwise 𝑢 𝑛 → 0 in 𝐸 and 𝐼 (𝑢 𝑛 ) → 0, a contradiction. Using estimate 487
similar to (3.19) we deduce 488
𝑝/( 𝑝−𝑠)
𝛽 ≥ 𝑆 𝑠𝑠/(𝑠− 𝑝) |𝑔| ∞ . 489
Now, using (3.21), (3.22) and (3.23), we get the following contradiction with (3.3): 490
  491
1   1ˆ 492
𝑝 𝑞 𝑠
𝑐 = lim 𝐼 (𝑢 𝑛 ) ≥ lim ∥𝑢 𝑛 ∥ 1, 𝑝 + ∥𝑢 𝑛 ∥ 1,𝑞 − 𝑔|𝑢 𝑛 |
𝑛→∞ 𝑛→∞ 𝑝 𝑠 493
    494
1 1 1 1 𝑠/(𝑠− 𝑝) 𝑝/( 𝑝−𝑠)
= − 𝛽≥ − 𝑆 |𝑔| ∞ . 495
𝑝 𝑠 𝑝 𝑠 𝑠
496
Having proved that {𝑦 𝑛 } is bounded, we may assume |𝑦 𝑛 | ≤ 𝜌. From (3.20) we deduce 497
498
ˆ ˆ 499
𝑠
|𝑢 𝑛 | ≤ 𝑧𝑛 ≤ 𝜀 500
𝐵𝜌+𝑅
c
𝜀
𝐵c𝑅 𝜀 ( 𝑦𝑛 )
501
502
for 𝑛 large. From the compact embedding 𝑊 1, 𝑝 (𝐵𝜌+𝑅 𝜀 ) ↩→ 𝐿 𝑠 (𝐵𝜌+𝑅 𝜀 ), we have 503
ˆ ˆ 504
𝑠
|𝑢 𝑛 | → |𝑢| 𝑠 . 505
𝐵𝜌+𝑅 𝜀 𝐵𝜌+𝑅 𝜀 506

11
507 Therefore
508 ˆ ˆ ˆ !
509 𝑠 𝑠 𝑠
lim |𝑢 𝑛 | = lim |𝑢 𝑛 | + |𝑢 𝑛 |
510 𝑛→∞ 𝑛→∞ 𝐵𝜌+𝑅
c 𝐵𝜌+𝑅 𝜀
𝜀
511 ˆ ˆ
𝑠
512 ≤𝜀+ |𝑢| ≤ 𝜀 + |𝑢| 𝑠 .
513 𝐵𝜌+𝑅 𝜀
514
515 Letting 𝜀 → 0 yields |𝑢 𝑛 | 𝑠 → |𝑢| 𝑠 . Noting 𝑢 𝑛 ⇀ 𝑢 in 𝐿 𝑠 (R 𝑁 ), we get 𝑢 𝑛 → 𝑢 in 𝐿 𝑠 (R 𝑁 ).
516 Consequently ˆ
517 𝑔|𝑢 𝑛 | 𝑠−2 𝑢 𝑛 (𝑢 𝑛 − 𝑢) → 0.
518
519 Because 𝜓 ′ is also compact (see [? , Lemma 1]), we deduce
520 ˆ
521 ℎ|𝑢 𝑛 | 𝑟 −2 𝑢 𝑛 (𝑢 𝑛 − 𝑢) = 𝜓(𝑢 𝑛 ) − 𝑟 ⟨𝜓 ′ (𝑢 𝑛 ), 𝑢⟩
522
523 → 𝜓(𝑢) − 𝑟 ⟨𝜓 ′ (𝑢), 𝑢⟩ = 0. (3.24)
524
525 Combining these with 𝑢 𝑛 ⇀ 𝑢 in 𝐸, we have
526
527 ⟨L ∗𝑝 𝑢 𝑛 − L ∗𝑝 𝑢, 𝑢 𝑛 − 𝑢⟩ + ⟨L𝑞∗ 𝑢 𝑛 − L𝑞∗ 𝑢, 𝑢 𝑛 − 𝑢⟩
528 = ⟨𝐼 ′ (𝑢 𝑛 ) − 𝐼 ′ (𝑢), 𝑢 𝑛 − 𝑢⟩ + 𝑜 𝑛 (1) → 0, (3.25)
529
530 where for 𝑚 ∈ {𝑝, 𝑞}, L 𝑚 ∗ : 𝑊 1,𝑚 (R 𝑁 ) → (𝑊 1,𝑚 (R 𝑁 )) ∗ is the operator given in (3.2).
531 Because 𝑢 𝑛 ⇀ 𝑢 in both 𝑊 1, 𝑝 (R 𝑁 ) and 𝑊 1,𝑞 (R 𝑁 ), and as is well known the two terms
532 on the left hand side of (3.25) are nonnegative (the monotonicity of L ∗𝑝 and L𝑞∗ ), we deduce
533
534 ⟨L ∗𝑝 𝑢 𝑛 , 𝑢 𝑛 − 𝑢⟩ → 0, ⟨L𝑞∗ 𝑢 𝑛 , 𝑢 𝑛 − 𝑢⟩ → 0.
535
536
Hence 𝑢 𝑛 → 𝑢 in both 𝑊 1, 𝑝 (R 𝑁 ) and 𝑊 1,𝑞 (R 𝑁 ), see Remark 3.1. Therefore, 𝑢 𝑛 → 𝑢 in
537
𝐸. □
538
539
540 3.2 The critical case
541 To study the critical case, we need convergence of measures on R 𝑁 . Let 𝐶b (R 𝑁 ) be the space of
542 all bounded continuous functions 𝑢 : R 𝑁 → R equipped with the 𝐿 ∞ -norm |·| ∞ , and 𝐶0 (R 𝑁 )
543 be the closure of 𝐶c (R 𝑁 ) in 𝐶b (R 𝑁 ), where 𝐶c (R 𝑁 ) is the set of all compactly supported
544 continuous functions.
545 It is known that M (R 𝑁 ), the space of regular finitely additive signed measures on R 𝑁 ,
546 can be identified as the dual space of 𝐶b (R 𝑁 ). For a sequence {𝜇 𝑛 } ⊂ M (R 𝑁 ), we say that
547 {𝜇 𝑛 } converges to a measure 𝜇, denoted by 𝜇 𝑛 ⇀ 𝜇, if
548
ˆ ˆ
549
550 𝜑 d𝜇 𝑛 → 𝜑 d𝜇 (3.26)
551
552

12
 ∗
for all 𝜑 ∈ 𝐶0 (R 𝑁 ). If (3.26) holds for all 𝜑 ∈ 𝐶b (R 𝑁 ), we write 𝜇 𝑛 ⇀ 𝜇 and say that the 553
convergence is tight. 554
It is known that if {𝜇 𝑛 } is bounded, then it has a convergent subsequence. Since tight 555
convergence is stronger than convergence, to get tight convergence we need the following 556
result. 557
558
Proposition 3.5 ([? , Theorem 1.208]). Let {𝜈𝑛 } be a sequence of bounded Borel measures
559
on R 𝑁 . Assume that for any 𝜀 > 0, there exists a compact set 𝐾 𝜀 ⊂ R 𝑁 such that
560
561
sup 𝜈𝑛 (R 𝑁 \𝐾 𝜀 ) ≤ 𝜀.
𝑛 562
563
∗ 564
Then up to a subsequence 𝜈𝑛 ⇀ 𝜈 for some Borel measure 𝜈.
565
Lemma 3.6. The functional Φ satisfies (𝑃𝑆) 𝑐 for all 𝑐 satisfying 566
567
1 𝑁 / 𝑝 ( 𝑝− 𝑁 )/ 𝑝
0 < 𝑐 < 𝑐 ∗ := 𝑆 |𝑔| ∞ . (3.27) 568
𝑁 569
570
Proof. Let {𝑢 𝑛 } be a (𝑃𝑆) 𝑐 sequence of Φ, that is
571
ˆ ˆ ˆ ˆ 572
1 1 𝜆 1 ∗
Φ(𝑢 𝑛 ) = |∇𝑢 𝑛 | 𝑝 + |∇𝑢 𝑛 | 𝑞 − ℎ|𝑢 𝑛 | 𝑟 − ∗ 𝑔|𝑢 𝑛 | 𝑝 → 𝑐, (3.28) 573
𝑝 𝑞 𝑟 𝑝
ˆ ˆ 574
⟨Φ′ (𝑢 𝑛 ), 𝜙⟩ = |∇𝑢 𝑛 | 𝑝−2 ∇𝑢 𝑛 · ∇𝜙 + |∇𝑢 𝑛 | 𝑞−2 ∇𝑢 𝑛 · ∇𝜙 575
ˆ ˆ 576
∗
− 𝜆 ℎ |𝑢 𝑛 | 𝑟 −2 𝑢 𝑛 𝜙 − 𝑔 |𝑢 𝑛 | 𝑝 −2 𝑢 𝑛 𝜙 = 𝑜(∥𝜙∥) (3.29) 577
578
579
for all 𝜙 ∈ D 1, 𝑝 (R 𝑁 ). Similar to (3.6) we have 580
    581
1 1 1 1
1 + 𝑐 + ∥𝑢 𝑛 ∥ ≥ − |∇𝑢| 𝑝𝑝 + − |∇𝑢| 𝑞𝑞 582
𝑝 𝑟 𝑞 𝑟 583
584
for 𝑛 large, where ∥·∥ is the norm on 𝑋 defined in (2.1). It follows that {𝑢 𝑛 } is bounded in 𝑋. 585
Therefore, the two sequences of bounded measures {𝜇 (𝑛) } and {𝜈 (𝑛) } with 586
∗ 587
𝜇 (𝑛) = (|∇𝑢 𝑛 | 𝑝 + |∇𝑢 𝑛 | 𝑞 ) d𝑥, 𝜈 (𝑛) = |𝑢 𝑛 | 𝑝 d𝑥 588
589
are bounded. 590
It has been shown in [? , Lemma 7] using the first concentration compactness principle 591
and Proposition 3.5 that, for some bounded measures 𝜈 on R 𝑁 there holds 592
∗ ∗
593
|𝑢 𝑛 | 𝑝 d𝑥 ⇀ 𝜈. (3.30) 594
595
Because {𝜇 (𝑛) } is bounded, there is also a bounded nonnegative 𝜇 measures on R 𝑁 , such that 596
597
(|∇𝑢 𝑛 | 𝑝 + |∇𝑢 𝑛 | 𝑞 ) d𝑥 ⇀ 𝜇. (3.31) 598

13
599 By the second concentration compactness principle of Lions [? ] (see also [? , page 44]), there
600 exist 𝐽 ⊂ N and
601
602 {𝑥 𝑗 } 𝑗 ∈ 𝐽 ⊂ R 𝑁 , {𝜇 𝑗 } 𝑗 ∈ 𝐽 ⊂ (0, ∞) , {𝜈 𝑗 } 𝑗 ∈ 𝐽 ⊂ (0, ∞)
603
∗
604 with 𝜇 𝑗 ≥ 𝑆𝜈 𝑝/
𝑗
𝑝
, such that
605
606 ∑ ∗
∑
𝜇 ≥ |∇𝑢| 𝑝 + |∇𝑢| 𝑞 + 𝜇 𝑗 𝛿𝑥𝑗 , 𝜈 = |𝑢| 𝑝 + 𝜈 𝑗 𝛿𝑥𝑗 , (3.32)
607
𝑗∈𝐽 𝑗∈𝐽
608
609 where 𝛿 𝑥 𝑗 is the Dirac-mass of mass 1 concentrated at 𝑥 𝑗 . Since 𝑞 ≤ 𝑝 < 𝑟, let 𝑛 → ∞ in
610
611 1 ′
612 Φ(𝑢 𝑛 ) − ⟨Φ (𝑢 𝑛 ), 𝑢 𝑛 ⟩
𝑝
613 ˆ ˆ ˆ ˆ
1 𝑝 1 𝑞 𝜆 𝑟 1 ∗
614 = |∇𝑢 𝑛 | + |∇𝑢 𝑛 | − ℎ|𝑢 𝑛 | − ∗ 𝑔|𝑢 𝑛 | 𝑝
615 𝑝 𝑞 𝑟 𝑝
ˆ ˆ ˆ ˆ 
616 1 𝑝 𝑞 𝑟 𝑝∗
− |∇𝑢 𝑛 | + |∇𝑢 𝑛 | − 𝜆 ℎ|𝑢 𝑛 | − 𝑔|𝑢 𝑛 |
617 𝑝
618  ˆ ˆ
1 1 𝑝∗ 1 ∗
619 ≥ − 𝑔|𝑢 𝑛 | = 𝑔|𝑢 𝑛 | 𝑝 ,
𝑝 𝑝∗ 𝑁
620
621 we deduce
622 ˆ ∑
1 © ª
623 𝑐≥ ­ 𝑔 d𝜈 +
𝑔(𝑥 𝑗 )𝜈 𝑗 ® . (3.33)
𝑁
624 « 𝑗∈𝐽 ¬
625 If 𝐽 ≠ ∅, take 𝑗 ∈ 𝐽 and set 𝑥 − 𝑥𝑗 
626 𝜙 𝜀 (𝑥) = 𝜑 ,
627 𝜀
628 where 𝜑 is the function given in (3.10). Then for 𝑚 ∈ {𝑝, 𝑞} we have
629 ˆ
630 lim lim 𝑢 𝑛 |∇𝑢 𝑛 | 𝑚−2 ∇𝑢 𝑛 · ∇𝜙 𝜀 = 0, (3.34)
631 𝜀→0 𝑛→∞
632
633 see [? , Claim 1]. From ⟨Φ′ (𝑢 𝑛 ), 𝜙 𝜀 𝑢 𝑛 ⟩ → 0, (3.30) and (3.31), we get
634 ˆ ˆ   ˆ ˆ
𝑝−2 𝑞−2 𝑟
635 𝜙 𝜀 d𝜇 + lim 𝑢 𝑛 |∇𝑢 𝑛 | + |∇𝑢 𝑛 | ∇𝑢 𝑛 · ∇𝜙 𝜀 − 𝜆 ℎ|𝑢| 𝜙 𝜀 − 𝑔𝜙 𝜀 d𝜈 = 0.
𝑛→∞
636
637 ∗

638 Thanks to (3.34), letting 𝜀 → 0 we conclude 𝜇 𝑗 = 𝑔(𝑥 𝑗 )𝜈 𝑗 . From this and 𝜇 𝑗 ≥ 𝑆𝜈 𝑝/

𝑗
𝑝
, we
639 have   −𝑁/𝑝
640 𝜈 𝑗 ≥ 𝑆 𝑁 / 𝑝 𝑔(𝑥 𝑗 ) .
641 Now using (3.33) we get a contradiction with (3.27):
642
1 1   1− 𝑁 / 𝑝 1 ( )/
643 𝑐≥ 𝑔(𝑥 𝑗 )𝜈 𝑗 ≥ 𝑆 𝑁 / 𝑝 𝑔(𝑥 𝑗 ) ≥ 𝑆 𝑁 / 𝑝 |𝑔| ∞𝑝− 𝑁 𝑝 .
644 𝑁 𝑁 𝑁

14
 ∗
So, 𝐽 = ∅ and the equality in (3.32) becomes 𝜈 = |𝑢| 𝑝 , which implies |𝑢 𝑛 | 𝑝∗ → |𝑢| 𝑝∗ . 645
∗ ∗
Noting 𝑢 𝑛 ⇀ 𝑢 in 𝐿 𝑝 (R 𝑁 ), we get 𝑢 𝑛 → 𝑢 in 𝐿 𝑝 (R 𝑁 ). Hence 646
ˆ 647
∗ 648
𝑔|𝑢 𝑛 | 𝑝 −2 𝑢 𝑛 (𝑢 𝑛 − 𝑢) → 0.
649
650
Because similar to (3.24) there holds 651
ˆ 652
ℎ|𝑢 𝑛 | 𝑟 −2 𝑢 𝑛 (𝑢 𝑛 − 𝑢) → 0, 653
654
655
we deduce
656
657
⟨L 𝑝 𝑢 𝑛 − L 𝑝 𝑢, 𝑢 𝑛 − 𝑢⟩ + ⟨L𝑞 𝑢 𝑛 − L𝑞 𝑢, 𝑢 𝑛 − 𝑢⟩
658
= ⟨Φ′ (𝑢 𝑛 ) − Φ′ (𝑢), 𝑢 𝑛 − 𝑢⟩ + 𝑜 𝑛 (1) → 0, (3.35) 659
660
where for 𝑚 ∈ {𝑝, 𝑞}, L 𝑚 : D 1,𝑚 (R 𝑁 ) → (D 1,𝑚 (R 𝑁 )) ∗ is the operator given in (3.1). 661
Because 𝑢 𝑛 ⇀ 𝑢 in both D 1, 𝑝 (R 𝑁 ) and D 1,𝑞 (R 𝑁 ), and as is well known the two terms 662
on the left hand side of (3.35) are nonnegative (the monotonicity of L 𝑝 and L𝑞 ), we deduce 663
664
⟨L 𝑝 𝑢 𝑛 , 𝑢 𝑛 − 𝑢⟩ → 0, ⟨L𝑞 𝑢 𝑛 , 𝑢 𝑛 − 𝑢⟩ → 0. 665
666
Hence 𝑢 𝑛 → 𝑢 in both D 1, 𝑝 (R 𝑁 ) and D 1,𝑞 (R 𝑁 ), see Remark 3.1. Therefore, 𝑢 𝑛 → 𝑢 in 667
𝑋. □ 668
Remark 3.7. It has been shown in [? , Lemma 9] that if 𝜆 is large enough then 669
670
𝑐 𝜆 := inf max Φ𝜆 (𝑡𝑢) < 𝑐 ∗ , 671
𝑢∈𝑋\{0} 𝑡 ≥0 672
673
here we write Φ𝜆 for our functional Φ to emphasize its dependence on 𝜆. Since 𝑐 𝜆 is precisely 674
the mountain pass level [? , Remark 4] and (𝑃𝑆) 𝑐𝜆 holds (by our Lemma 3.6), the mountain 675
pass theorem gives rise to a nontrivial solution 𝑢 𝜆 with Φ𝜆 (𝑢 𝜆 ) = 𝑐 𝜆 > 0. 676
For getting a mountain pass solution (which can easily be shown to be nonnegative, thus 677
is not included in the solutions given by Theorem 1.1), this approach is simpler than [1? ? 678
], because in these papers to show that the weak limit 𝑢 of the Palais-Smale sequence {𝑢 𝑛 } 679
produced by mountain pass theorem is a nontrivial solution, it is crucial to show ∇𝑢 𝑛 → ∇𝑢 680
a.e. in R 𝑁 , see e.g. [? , Lemma 8], which is much complicated than proving our Lemma 3.6. 681
Moreover, it is not clear whether the nontrivial solution they found has positive energy. 682
683
Conflict of interest 684
The authors declare that they have no conflict of interests. 685
686
Ethics approval 687
688
The research does not involve humans and/or animals. The authors declare that there are no 689
ethics issues to be approved or disclosed. 690

15