Alan V. Lair

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We show that large positive solutions exist for the equation (P±) :∆u±|∇u|q = p(x)uγ in Ω ⊆ RN(N ≥ 3) for appropriate choices of γ > 1,q > 0 in which the domain Ω is either bounded or equal to RN . The nonnegative function p is continuous and may vanish on large parts of Ω. If Ω = RN , then p must satisfy a decay condition as |x| →∞. For (P+), the decay(More)
The author proves that the abstract differential inequality [[u’ (t) A(t)u(t),,2[[ 7 (t) + ()d in which the linear operator A(t) M(t) + 0 N(t), M symmetric and N antisymmetric, is in general unbounded, w(t) t-2(t)[[u(t)[[ 2 + [[M(t)u(t)[[ [[u(t)[[ and 7 is a positive constant has a nontrivial solution near t-0 i which vanishes at t-0 if and only if(More)
We show that the reaction-diffusion system ut = ∆φ(u) + f (v), vt = ∆ψ(v) + g(u), with homogeneous Neumann boundary conditions, has a positive global solution on Ω× [0,∞) if and only if ∫∞ds/ f (F−1(G(s)))=∞ (or, equivalently, ∫∞ds/g(G−1(F(s)))=∞), where F(s) = ∫ s 0 f (r)dr and G(s) = ∫ s 0 g(r)dr. The domain Ω ⊆ RN (N ≥ 1) is bounded with smooth boundary.(More)
We consider the semilinear elliptic equation 4u = p(x)uα + q(x)uβ on a domain Ω ⊆ Rn, n ≥ 3, where p and q are nonnegative continuous functions with the property that each of their zeroes is contained in a bounded domain Ωp or Ωq, respectively in Ω such that p is positive on the boundary of Ωp and q is positive on the boundary of Ωq. For Ω bounded, we show(More)
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