In this paper we consider Dirichlet boundary value problems for a semilinear equation involving the k-order polyharmonic operator, depending on a real parameter λ. We are interested in the range of admissible λ's for which the problem under consideration has a nontrivial radial solution. The problem in the case of k=1 was studied by H. Brézis and L. Nirenberg [Comm. Pure Appl. Math. 36 (1983), 437-477], who showed that the lower bound of the range of admissible λ's is zero for n≥4, while for n=3 the lower bound is positive. We extend this famous result by showing that for k≥2 there are "critical dimensions'' for which the lower bound is positive. For example, if k=2 then n=5,6,7 are critical dimensions. We also conjecture that the critical dimensions are precisely those in the range 2k<n<4k.
Critical exponents and critical dimensions for polyharmonic operators
PUCCI, Patrizia;
1990
Abstract
In this paper we consider Dirichlet boundary value problems for a semilinear equation involving the k-order polyharmonic operator, depending on a real parameter λ. We are interested in the range of admissible λ's for which the problem under consideration has a nontrivial radial solution. The problem in the case of k=1 was studied by H. Brézis and L. Nirenberg [Comm. Pure Appl. Math. 36 (1983), 437-477], who showed that the lower bound of the range of admissible λ's is zero for n≥4, while for n=3 the lower bound is positive. We extend this famous result by showing that for k≥2 there are "critical dimensions'' for which the lower bound is positive. For example, if k=2 then n=5,6,7 are critical dimensions. We also conjecture that the critical dimensions are precisely those in the range 2kI documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
