We study the existence and nonexistence of positive (super-) solutions to a singular semilinear elliptic equation −∇ · (|x|∇u)−B|x|u = C|x|u in cone–like domains of R (N ≥ 2), for the full range of parameters A,B, σ, p ∈ R and C > 0. We provide a complete characterization of the set of (p, σ) ∈ R such that the equation has no positive (super-) solutions, depending on the values of A,B and the p...

Semilinear elliptic equations which give rise to solutions blowing up at the boundary are perturbed by a Hardy potential μ/δ(x, ∂Ω). The size of this potential effects the existence of a certain type of solutions (large solutions): if μ is too small, then no large solution exists. The presence of the Hardy potential requires a new definition of large solutions, following the pattern of the asso...

Most of arithmetically significant Dirichlet series such as the Riemann zeta-function ζ(s), Dirichlet L-functions, and Hecke L-functions associated with various cusp forms satisfy Riemannian functional equations connecting values at s = σ + it and 1 − s of respective functions. Essentially best possible estimates for these functions near the line σ = 1 and σ = 0 can usually be deduced from the ...

In this paper we study the spatial behavior of solutions to the equations obtained by taking formal Taylor approximations to the heat conduction dual-phase-lag and three-phaselag theories, reflecting Saint-Venant’s principle. Depending on the relative order of derivation with respect to the time we propose different arguments. One is inspired by the arguments for parabolic problems and the othe...

We break the convexity bound in the t–aspect for L–functions attached to cuspforms f for GL2(k) over arbitrary number fields k. The argument uses asymptotics with error term with a power saving, for second integral moments over spectral families of twists L(s, f ⊗χ) by grossencharacters χ, from our previous paper [Di-Ga]. §0. Introduction In many instances, for cuspidal automorphic forms f on r...

Ordinary Differential Equations (ODEs) are ubiquitous in physical applications of mathematics. The Picard-Lindelöf theorem is the first fundamental theorem in the theory of ODEs. It allows one to solve differential equations numerically. We provide a constructive development of the Picard-Lindelöf theorem which includes a program together with sufficient conditions for its correctness. The proo...

We consider the cardinal invariant CG(X) of the minimal number of weakly compact subsets which generate a Banach space X. We study the behavior of this index when passing to subspaces, its relation with the Lindelöf number in the weak topology and other related questions. A Banach space is weakly compactly generated if there is a weakly compact subset which is linearly dense and weakly Lindelöf...