半线性椭圆方程

µ − a + β . K•§ (0.12) – k˜‡
ë•©zµ [1] L. Caffarelli, R. Kohn, L. Nirenberg, First order interpolation inequality with weights, Compos. Math. 53 (1984) 259–275. [2] K. Chou, C. Chu, On the best constant for a weighted Sobolev–Hardy inequality, J. Lond. Math. Soc. (2) 48 (1993) 137–151. [3] F. Catrina, Z. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of external functions, Comm. Pure Appl. Math. 54 (2) (2001) 229–258. [4] X.J. Huang, X.P. Wu, C.L. Tang, Multiple positive solutions for semilinear elliptic equations with critical weighted Hardy-Sobolev exponents, Nonlinear Anal. 74 (2011) 2602–2611 [5] L. Ding, C.L. Tang, Existence and multiplicity of positive solutions for a class of semilinear elliptic equations involving Hardy term and Hardy-Sobolev critical exponents, J. Math. Anal. Appl. 339 (2008) 1073–1083. [6] L. Huang, X.P. Wu, C.L. Tang, Existence and multiplicity of solutions for semilinear elliptic equations with critical weighted Hardy-Sobolev exponents, Nonlinear Anal. 71 (2009) 1916–1924. [7] P. H. Rabinowitz. Minimax methods in critical point theory with applications to differential equations, Conference Board of the Mathematical Sciences, vol. 65 Providence, RI: American Mathematical Society, 1986. [8] W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987. [9] H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (4) (1983) 437–477. [10] D. Kang, G. Li, S. Peng, Positive solutions and critical dimensions for the elliptic problems involving the Caffarelli-Kohn-Nirenberg inequalities, J. Jilin. Univ. Sci. 46 (2008) 423–427.
|u|p−2 u |x|bp
+ f (x, u),
x∈Ω x ∈ ∂Ω
(0.6) √ √ µ, 0 ≤ µ < ( µ − a)2 a = b ž, ¼ê, ½ ˜‡
k.«•, …k 0 ∈ Ω. d
2N N −2(1+a−b)
,k0≤a<
(N −2)2 , Ú a ≤ b < a + 1. p = p(a, b) 4 2N p(a, a) = N −2 = 2∗ ´ Sobolev .•ê. , t • F (x, t) = 0 f (x, s)ds Ù¥ x ∈ Ω, t ∈ R.
p(pq −q +1) 1+(q −p)(p−1) …λ 2
> 0;
1
•§ −∆ u = λ|u|r−2 u + µ |u|q−2 u p |x|s u=0 x ∈ ∂Ω 1,e2 ≤ p < q = p∗ (s), r < p∗ (1) p = r = p∗ , n > p3 − p2 + p, µ > 0, 0 < λ < λ1 ¶ (2) p < r < p∗ , µ > 0, λ¿©Œ¶ (3) p < r < p∗ , n >
∞ C0 (Ω)
1 © PH = H0 (Ω, |x|−2a ), = L «
~‰êe
| 8, to denote the
with respect to the norm, u
Ω
|x|
−2a
|∇u| − µ
2
u2 |x|2(1+a)
1 2
dx
,
2
Šâªf (0.8), d‰ê ê•:
1 du˜m H0 (Ω, |x|−2a ) ¥
©znã
‘M“‘9M“•ê•§ 1,•§ −∆u = µ u + |u|2∗ −2 u + λu |x|2 u=0 x ∈ ∂Ω
+2 2 ) , λ > 0,•§£1.1¤k˜š²…)¶ e0 ≥ µ < µ − ( NN
x∈Ω
(0.1)
eN ≥ 7, µ ∈ [0, µ − 4)…λ ∈ (0, λ1 (µ)),•§£1.1¤kCÒ)¶ e0 < µ < µ − 1, λ ∈ (0, λ1 (µ)),•§£1.1¤k˜ )¶ eµ − 1 < µ < µ, Ω = B1 (0), λ∗ ∈ (0, λ1 (µ)), …= λ ∈ (λ∗ , λ1 (µ))ž§•§£1.1¤k˜ )" 2,•§ ∗ −∆u − µ u = |u|2 (s)−2 u + λ|u|r−2 u |x|2 |x|s u=0 x ∈ ∂Ω
a = b = µ = 0 ù˜AÏœ¹ •§(„©z [9]), Xe 2∗ −1 + f (x, u), −∆ u = u u > 0, u = 0, •§ (0.9)k˜‡ ). on Ω on Ω on ∂ Ω (0.9)
Brezis
©Ù¥•Ñ§é·
¼ê f
6 xÚ$?Ø
´ Hardy-Sobolev
.•ê, …
f ∈ C (Ω × R, R), P F (x, t) •¼ê f
µ = 0 ž, ¯K (0.6) |x|
RN −bp p
9 Ͷ
2 p
Caffarelli-Kohn-Nirenberg Ø ª(„©z [1])
∞ |x|−2a |∇u|2 dx, for all u ∈ C0 (RN ),
p(pq −q +p) p+(q −p)(p−1) …λ p2 …λ ∈ (0, λ1 );
N −(p2 +s−ps) N −s p−1 ( p ) p
(0.4)
> 0;
þã^‡÷vÙ˜§•§£1.4¤k )¶ (4)ep = q, N > p(p − p + 1)…λ ∈ (0, λ1 ); þã^‡÷vÙ˜§•§£1.4¤k»•)"
0 ≤ s < 2, 0 ≤ µ < µ, λ > 0, Ú √ N −2 µ−µ N √ √ , < q < 2∗ . max 2, √ µ+ µ−µ µ
1 K•§ (0.10) 3˜m H0 (Ω) k˜‡ ).
7 ©z [17] ¥, ¶Ú/ïÄ Xe¹k˜„‘¼ê •§ ∗ )−2 −∆u − µ u 2 = |u|2 (s u + f (x, u), | x|s |x| u = 0, Ù¥ 0 ≤ µ < µ, 0 ≤ s < 2. ¶Ú/ ½ n 2.(©z [17] ¥ (JXe
x ∈ Ω \ {0} x ∈ ∂Ω
(0.11)
√ ½ n 1) b N ≥ 3, 0 ≤ µ < µ, 0 ≤ s < 2, f (x, t) f (x, t) (f1 ) f ∈ C (Ω × R+ , R), lim = 0, Ú lim 2∗ −1 = 0, x ∈ Ω ž˜—¤á; t→+∞ t t t→0+ (f2 ) •3˜‡~ê ρ, ρ > 2, ¦ 0 < ρF (x, t) ≤ f (x, t)t é¤k x ∈ Ω, t ∈ R+ \ {0} ¤á. N √ √ N −2 µ−µ √ µ−µ µ ,
•§¥kÄ–¼ê f (x, t), …¹ Hardy-Sobolev a=0Ú b=0
ÛɯK. ©z [17] ¥, ¶Ú/?Ø
, ©z [4] ¥, ‘, ÇÚ/q?Ø
¹, = a = 0 … b = 0. ¨‚ïÄ 5 Brezis ïÄ
´¼ê f (x, t) äk]àš‚55Ÿœ¹ •§.
a = 0, b = 0 Ú µ = 0 ž •§(„©z[19]), Xe ∗ )−2 −∆u − µ u 2 = |u|2 (s u + λ|u|q−2 u, x∈Ω |x|s |x| u = 0, x ∈ ∂Ω Xe(J