asymptotic behavior of positive solutions of a singular nonlinear dirichlet problem abstract let be a c 1 1 bounded domain in r n for n 2 in this paper we are concerned with the asymptotic behavior of the unique positive classical solution to the singular boundary value problem u a x u 0 in u 0 where 0 a is a nonnegative function in c loc 0 1 and there exists c 0 such that 1 c a x x k 1 m log k x k c here 2 k r is a positive constant and x dist x

asymptotic behavior of positive solutions of a semilinear dirichlet problem abstract in this paper we study the asymptotic behavior of the unique positive classical solution to the following semilinear boundary value problem u a x u 0 x u 0 in u 0 here is a bounded c 1 1 domain 1 and the function a is in c l o c 0 1 such that there exists c 0 satisfying for each x 1 c a x x exp x z t t d t c where 2 d d i a m x d i s t x and z is a continuous function on 0 with z 0 0

internal duality for resolution of rings the following example provided the motivation for this paper let f be 8 the field with 8 elements and let f be its group of units which acts on f 8 8 u u via left multiplication the cohomology ring h f i f f 8 8 2 w xzr7 w x f x x x is of much interest to topologists a cs m it is a 2 1 2 3 straightforward exercise to construct minimal sets of homogeneous generators and relations for this ring there are 13 generators and 54 relations the surprising fact is that the minimal relations exhibit a certain internal duality

indecomposablea module summands inh v which are unstable algebras letp be a prime and denote bya the modp steenrod algebra we determine the indecomposablea module summands ofh p d f p which admit the structure of an unstablea algebra in fact it turns out that this is equivalent to the problem of determining those indecomposablea module summands which arise as the modp cohomology of a space or even a classifying space of a finite group we reduce this problem to one in modular representation theory namely for whichd andp is the projective cover of the trivial one dimensional gl d f p representationf p a permutation module our solution of this latter problem makes use of the classification of subgroups of gl d f p acting transitively on f p d 0 and hence depends on the classification of finite simple groups on feit thompson s odd order theorem ifp 2

existence and estimates of solutions for singular nonlinear elliptic problems for both bounded and unbounded domain in r d 3 with smooth boundary the function 0 0 is measurable and satisfies some appropriate hypotheses detailed below the problem 1 has been studied extensively for the special nonlinearity x t p x t 0 indeed in 7 edelson proved the existence of a positive solution u c2 loc r decaying as x 2 n as x provided that p c loc r 0 0 for x 0 and satisfying the growth condition

asymptotic behavior of positive solutions of a semilinear polyharmonic problem in the unit ball abstract let m be a positive integer we investigate the existence and the asymptotic behavior of positive continuous solutions to the following semilinear polyharmonic boundary value problem in the unit ball b of r n n 2 m u a x u lim x 1 u x 1 x m 1 0 where 1 1 and a is a positive measurable function in b such that there exists c 0 satisfying for each x b 1 c a x 1 x exp 1 x z s s d s c 1 m 1 1 and z is a continuous function on 0 with z 0 0

numerical radius attaining compact linear operators abstract we show that there are compact linear operators on banach spaces which cannot be approximated by numerical radius attaining operators

the asymptotic behaviour of the unique solution for the singular lane emden fowler equation abstract by karamata regular variation theory and constructing comparison functions we show the exact asymptotic behaviour of the unique classical solution u c 2 c near the boundary to a singular dirichlet problem u k x g u u 0 x u 0 where is a bounded domain with smooth boundary in r n g c 1 0 0 lim t 0 g t g t for each 0 for some 0 and k c loc for some 0 1 is nonnegative on which is also singular near the boundary

positivity for equations involving polyharmonic operators with dirichlet boundary conditions cranston fabes and zhao 26 5 established the uniform bound sup x y 2 x 6 y r g1 n x z g1 n z y dz g1 n x y m 1 1 where g1 n x y is the green function for the laplacian with dirichlet boundary conditions on a lipschitz domain rn with n 3 see 27 for n 2 this estimate was used in 23 and 18 to obtain positivity uniformly with respect to f 0 for noncooperative elliptic systems as

exact asymptotic behavior near the boundary to the solution for singular nonlinear dirichlet problems abstract we take up the existence and the exact asymptotic behavior near the boundary of the unique classical solution to a singular dirichlet problem u a x g u x u 0 in u 0 where is a c 1 1 bounded domain in r n n 2 and the function a is singular near the boundary and belongs to the kato class k

norm attaining compact operators abstract we show examples of compact linear operators between banach spaces which cannot be approximated by norm attaining operators this is the negative answer to an open question posed in the 1970s actually any strictly convex banach space failing the approximation property serves as the range space on the other hand there are examples in which the domain space has a schauder basis

simultaneously continuous retraction and bishop phelps bollobas type theorem abstract the dual space x of a banach space x is said to admit a uniformly simultaneously continuous retraction if there is a retraction r from x onto its unit ball b x which is uniformly continuous in norm topology and continuous in weak topology we prove that if a banach space resp complex banach space x has a normalized unconditional schauder basis with unconditional basis constant 1 and if x is uniformly monotone resp uniformly complex convex then x admits a uniformly simultaneously continuous retraction it is also shown that x admits such a retraction if x x i c 0 or x x i l 1 where x i is a family of separable banach spaces whose duals are uniformly convex with moduli of convexity i e with inf i i e 0 for all 0 e 1 let k be a locally compact hausdorff space and let c 0 k be the real banach space consisting of all real valued continuous functions vanishing at infinity as an application of simultaneously continuous retractions we show that a pair x c 0 k has the bishop phelps bollobas property for operators if x admits a uniformly simultaneously continuous retraction as a corollary c 0 s c 0 k has the bishop phelps bollobas property for operators for every locally compact metric space s