Necessary and sufficient conditions for existence of blow‐up solutions for elliptic problems in Orlicz–Sobolev spaces

Abstract

This paper is principally devoted to revisit the remarkable works of Keller and Osserman and generalize some previous results related to the those for the class of quasilinear elliptic problem { div ϕfalse(false|∇ufalse|false)∇u=afalse(xfalse)ffalse(ufalse)inΩ,u≥0inΩ,u=∞on∂Ω,where either Ω⊂boldRN with N≥1 is a smooth bounded domain or Ω=boldRN. The function ϕ includes special cases appearing in mathematical models in nonlinear elasticity, plasticity, generalized Newtonian fluids, and in quantum physics. The proofs are based on comparison principle, variational methods and topological arguments on the Orlicz–Sobolev spaces.

A Stochastic production planning problem

“…Theorem 3.1. Given the positive constant α, there exists a unique positive radially symmetric solution uα ∈ C 2 [0, R], to the problem (10) subject to the Dirichlet boundary condition ( 11). Moreover, the solution is convex and increasing, and the following holds true…”

“…The interest in studying the above equation comes, for instance, from various physical situations, such as quantum mechanics, quantum optics, nuclear physics and reaction-diffusion processes (cf. [1,9,10,11]). For instance, a basic preoccupation for the study of problem (26) is the timeindependent Schrödinger equation (single non-relativistic particle)…”

“…In general, the existence of the solutions and numerical approximation of the elliptic problem ( 26) is widely open. See the paper of Santos, Zhou and Santos [11], which includes a nice survey and recent progresses for Eq. ( 26).…”

“…Theorem 4.2. (see [11]) The problem (10) with BR (0) replaced with R N , admits a sequence of symmetric radial solutions u…”

“…Moreover, reasoning in the same manner we think that similar solutions can be constructed for the total (non relativistic) and potential energies of a particle of mass m in (27). Next, we posit the following open problems inspired by the two solutions and [11].…”