Consider the following inequalities due to Caffarelli, Kohn, and Nirenberg [6]:. We shall answer some fundamental questions concerning these inequalities such as the best embedding constants, the existence and nonexistence of extremal functions, and their qualitative properties. While the case a ≥ 0 has been studied extensively and a complete solution is known, little has been known for the case a < 0. Our results for the case a < 0 reveal some new phenomena which are in striking contrast with those for the case a ≥ 0. Results for N = 1 and N = 2 are also given.
We consider positive solutions ofwhere for N 2: a < (N − 2)/2, a < b < a + 1, and p = 2N/(N − 2(1 + a − b)). Ground state solutions are the extremal functions of the Caffarelli-Kohn-Nirenberg inequalities [6]. In [10] the authors have observed symmetry breaking phenomena for ground state solutions in a subregion of the parameters. In this paper, we continue our study on the structure of bound state solutions and construct bound state solutions having prescribed symmetry.
We find positive rapidly decaying solutions for the equationwhere N 3, the nonlinearity is given by the critical Sobolev exponent 2 * = 2N/(N − 2), the weight is K(x) = exp( 1 4 |x| α ), α 2 and λ is a parameter.
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