| Summary: | PROJECT RESULTS: Describe the results of your research in reference to its original and/or modified Project objectives. The maximum length for this section is 5 pages (Arial or Verdana font, size 10). 1 Introduction The main goal of this research project is the study of the existence, properties of extremals, and the best constant to the minimization problem (P) Sp,A,B(⌦) = inf { ˆ ⌦ ��xAru ��p dx : u 2 D1,p,A(⌦), ˆ ⌦ ��xBu ��p⇤ dx = 1 } , where ⌦ = (R+)k ⇥ RN−k for k 2 { 1, 2, . . . , N }, and D1,p,A(⌦) is the completion of C1 c (⌦) under the norm kuk1,p,A = ✓ˆ ⌦ ��xAru ��p dx ◆ 1 p . Here A = (a1, . . . , ak, 0), B = (b1, . . . , bk, 0) 2 Rk ⇥ RN−k, p, p⇤ ≥ 1 are parameters that are related by the equation (1.1) 1 p⇤ + b + 1 N = 1 p + a N , where a = a1 + . . . + ak, b = b1 + . . . + bk, and satisfying ai > 0, 0 ai − bi 1, ai p⇤ + bi p0 > 0, 8 i 2 { 1, . . . , k } , and (1.2) 0 a − b 1, (1.3) where p0 is the H¨older conjugate exponent of p. A secondary goal for this project is to investigate the asymptotic behavior as p % 2⇤ of the least energy solutions to (1.4) 8 > < > : −div(x2Aru) = xBpup−1 in ⌦ u > 0 in ⌦ u = 0 on @⌦, where now xA is the monomial weight, and A, B 2 RN satisfy (1.1)-(1.3) for p = 2 and p⇤ = 2⇤. In the following sections I will show the results obtained regarding these problems. 2 Extremals for Hardy-Sobolev type inequalities with monomial weights published at Journal of Mathematical Analysis and Applications Regarding the main goal of this research project, and if we denote by Sp the best constant in Sobolev inequality (that is, when A = B = 0) we have obtained Theorem 1. Suppose p > 1, A, B, p⇤ 2 RN satisfy the aforementioned conditions. If (i) either p < p⇤ < Np N−p, or (ii) p⇤ = Np N−p and Sp,A,A(⌦) < Sp. then Sp,A,B(⌦) is attained in D1,p,A(⌦). 1 This theorem completely answer the question of existence in the region of parameters satisfying p 0 and N ≥ 4, then S2,A,A(⌦) < S2, and extremals do exist.(ii) If ai ≥ 1 for all i 2 { 1, . . . , k } ...
|
|---|