2017 : msjmeeting-2017sep-05i002 ( ) 1.. u = g(u) in R N, u > 0 in R N, u H 1 (R N ). (1), N 2, g C 1 g(0) = 0. g(s) = s + s p. (1), [8, 9, 17],., [15] g. (1), E(u) := 1 u 2 dx G(u) dx : H 1 (R N ) R 2 R N R N G(s) := s., 0 g(t) dt. [8, 9, 17] inf{e(u) ; u (1) }. (1),.,,. (1) g(s) = s + s p [12, 19], g(s) [7, 14, 21, 22, 23, 25]. g(s), g(s). g(s). g(s).,. 2010 Mathematics Subject Classification: 35J61, 35A02, 35B05,, 432-8561 3-5-1 e-mail: adachi@shizuoka.ac.jp web: https://wwp.shizuoka.ac.jp/adachi/
(g1) g C 1 ([0, )), g(0) = 0, g (0) < 0. (g2) b > 0, g(s) < 0 on (0, b), g(s) > 0 on (b, ), g (b) > 0. (g3) K g (s) := sg (s) g(s) (g4) K g (s) 1 on (0, b).. (b, ) K g (s) > 1 K g(s) < 0. 1 ([4]). N 2, (g1) (g4). (1). 2. (1) u, u L := + g (u) : H 2 (R N ) L 2 (R N ), φ H 2 rad (RN ), Lφ = 0 φ 0. 3. (g2), 1. (g2 ) b > 0, b (b, ], g(s) < 0 on (0, b), g(s) > 0 on (b, b), g(s) < 0 on ( b, ), g (b) > 0, g ( b) < 0. (g2 ), Allen-Cahn g(s) = s(s b)(s b) 1., g(s) = s + s p + s q (1 < p q) (g2) (g3), 1. K g (s) = sg (s) growth function, g(s) g(s)., g(s) = s + s p (p > 1) K g (s) = psp 1 1 s p 1 1, K g ( ) = p., > 1 (s > 1) K g(s) = (p 1)2 s p 2 < 0 (s > 1) (s p 1 1) 2, g(s) = s + s p (p > 1) (g1) (g4). g(s) = s + s p (p > 1) (1) [19], [21] [19], (g1), (g2), (g4) g(s) (1). (g5) K g (s) > 1 on (b, ). (g6) K g(s) < 0 on (b, ). (g5), (g6), [7]., (g5) g(s) K g ( ) 1, [19] g(s) K g ( ) < 1. (g3) (g5), (g6), g(s)., [25] N 3, (g1), (g2), (g6) (1), (g6).
2. 1 2.1.. u + N 1 u + g(u) = 0, r u(0) = d > 0, u (0) = 0. (2) d > 0 (2) u(r; d), d > 0 3. N = {d > 0 ; R d > 0 u (r; d) < 0 on (0, R d ], u(r d ; d) = 0}, G = {d > 0 ; u (r; d) < 0 on (0, ), lim r u(r; d) = 0}, P = {d > 0 ; u(r; d) > 0 on (0, ), lim r u(r; d) 0}. d G R d =. N, G, P. 4.. (i) N, G, P, N G P = (0, )., P N (0, ). d 0 > 0, (0, d 0 ) P. (ii) d 0 G. (iii) ϵ > 0 (d 0, d 0 + ϵ) N., G = {d 0 }, d > d 0 > 0 u(r; d) R d > 0., lim R d = d 1 > 0, d 1 G d d1,. (2) u = u(r; d) δ + N 1 δ + g (u)δ = 0, r δ(0) = 1, (3) δ (0) = 0. δ(r; d) := d u(r; d) δ(r; d) (3), δ(r d; d) R d > 0. 5.. (i) δ(r d ; d) > 0, 0 = u(r d ; d) < u(r d ; d + ϵ), R d < R d+ϵ. (ii) δ(r d ; d) < 0, 0 = u(r d ; d) > u(r d ; d + ϵ), R d > R d+ϵ.
(iii) d [d 0, d 0 + ϵ) δ(r; d) (0, R d ) 1, δ d (R d ; d) < 0., R d > 0, δ(r; d). 2. 6. d G, δ(r; d) 1. 7 (strictly admissible). d N G. δ(r; d) (0, R d ) 1., (i) d N, δ(r d ; d) < 0. (ii) d G, lim r δ(r; d) =. 8. 6, 7 v λ (r; d) := ru (r; d) + λu(r; d) (λ > 0) δ(r; d) Sturm. [21], [21] 6, 7 (g5), (g6). (g3) v λ (r; d),., 5 7,. d (d 0, ) d 0, δ(r d ; d) = 0, 5 (iii). Case A: δ(r; d) (0, R d ) 1, δ(r d ; d) = 0. Case B: δ(r; d) > 0 on (0, R d ), δ(r d ; d) = 0., Case A 7, Case A., Case B., R d., d 1 > 0, lim d d1 R d = d 1 G, δ(r; d 1 ) > 0 on (0, )., 6., (d 0, ) d 1 G d 1 > d 0, G = {d 0 }. 2.2. 7. (1) u H 1 (R N ) φ + N 1 φ + g (u)φ = 0, φ (0) = 0, r φ Hrad 2 (RN )., a R, φ(r) = aδ(r; d 0 ). 7 (ii) lim δ(r; d 0 ) =, φ Hrad 2 (RN ). r 9. u, [7], { } u u Ker (L) = span,..., x 1 x N, L.
3. 1 1. u κu α 1 (u α ) = u + u p in R N, (4) u > 0 in R N, u H 1 (R N )., N 2, κ > 0, α > 1, 1 < p < α2 1. 2 Sobolev 2N (N 3), 2 := N 2 (N = 2). (4) [10, 11, 18]. (4) N 2, κ > 0, α > 1, 1 < p < α2 1 [13, 20]),, κ p, N [1, 2, 3, 5, 6, 16, 24]., 1, N 2, κ > 0, α > 1, 1 < p < α2 1,. 10. N 2, κ > 0, α > 1, 1 < p < α2 1. (4)., 1 10. 3.1. 10 (4) I(u) = 1 u 2 dx + κ (u α ) 2 dx + 1 u 2 dx 1 u p+1 dx 2 R 2α N R 2 N R N = 1 (1 + ακu 2α 2 ) u 2 dx + 1 u 2 dx 1 u p+1 dx 2 R 2 N R { } N : X := u H 1 (R N ) ; (u α ) 2 dx < R R N, I(u) (4)., u = f(v), I(f(v)) = 1 (1 + ακf(v) 2α 2 ) f(v) 2 dx + 1 f(v) 2 dx 1 f(v) p+1 dx 2 R 2 N R N = 1 (1 + ακf(v) 2α 2 )f (v) 2 v 2 dx + 1 f(v) 2 dx 1 f(v) p+1 dx 2 R 2 N R N, (1 + ακf(v) 2α 2 )f (v) 2 1 f, f J(v) := I(f(v)) = 1 v 2 dx + 1 f(v) 2 dx 1 f(v) p+1 dx 2 R 2 N R N
. J(v). v = f(v)f (v) + f(v) p f (v) in R N, v > 0 in R N, v H 1 (R N ). (5), f, (4) (5) 1 1, 10 (5). f (4) (5), [13, 20]. f. f (s) = 1 on s [0, ), 1 + ακf(s) 2α 2 (6) f(0) = 0. (6) f(s), g(s) := f(s)f (s) + f(s) p f (s), b := f 1 (1) (g1) (g4), 1 (5) v,, (4) u = f(v). f(s). 11.. (i) f(s) s, 0 < f (s) 1, f (s) 0 on [0, ). (ii) f(s) αsf (s) (α 1)f(s)f (s) 2 on [0, ). (iii) sf (s) f(s) on [0, ). f(s) (iv) lim s s 1 α (v) lim s 0 f(s) s = = 1. ( α κ ) 1 2α, lim s sf (s) f(s) = 1 α. 11 (ii) (g3). 12. g(s) = f(s)f (s) + f(s) p f (s) K g (s) = sg (s), 11 g(s) lim K g(s) = p α + 1 s α., p α + 1 α 1 p 2α 1, g(s) p 2α 1.
3.2. (4),,. div (a(u) u) + 1 2 a (u) u 2 = h(u) in R N, (7) u > 0 in R N, u H 1 (R N )., h(s), a(s). (h1) h C 1 ([0, )), h(0) = 0, h (0) < 0. (h2) b > 0, b (b, ], h(s) < 0 on (0, b), h(s) > 0 on (b, b), h(s) < 0 on ( b, ), h (b) > 0, h ( b) < 0. (h3) K h (s) (b, b) K h (s) > 1 K h (s) < 0. (h4) K h (s) 1 on (0, b). (h5) s 0 > 0, H(s 0 ) = lim sup s (h6) l > 0, lim sup s (a1) a C 2 ([0, )), inf a(s) > 0. s 0 (a2) a (s) 0 on (0, ). s0 0 h(s) s (l+1)n+2 N 2 h(t) dt > 0. (a3) K a (s) K a (b) on (0, b), K a(s) 0 on [b, b). a(s) (a4) a > 0, lim = a s s l. 0 if N 3, h(s) e βsl+2 0 for any β > 0 if N = 2. 13 ([4]). (h1) (h6), (a1) (a4). (7),. 14. h(s) = s + s p (1 < p < α2 1), a(s) = 1 + κs 2α 2, (h1) (h6), (a1) (a4) (b = 1, b =, l = 2α 2), (7) (4). [1] S. Adachi, M. Shibata, T. Watanabe, Asymptotic behavior of positive solutions for a class of quasilinear elliptic equations with general nonlinearities, Commun. Pure Appl. Anal. 13 (2014), 97 118. [2] S. Adachi, M. Shibata, T. Watanabe, Asymptotic behavior of positive solutions for a class of quasilinear elliptic equations in R 2, Funkcialaj Ekvacioj 57 (2014), 297 317. [3] S. Adachi, M. Shibata, T. Watanabe, Global uniqueness results for ground states for a class of quasilinear elliptic equations, Kodai Math. J. 40 (2017), 117 142.
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