By Marius Ghergu, Steven D. Taliaferro

During this monograph, the authors current a few strong equipment for facing singularities in elliptic and parabolic partial differential inequalities. right here, the authors take the original process of investigating differential inequalities instead of equations, the reason is that the best technique to research an equation is frequently to review a corresponding inequality; for instance, utilizing sub and superharmonic features to review harmonic capabilities. one other strange characteristic of the current e-book is that it's according to necessary illustration formulae and nonlinear potentials, that have now not been broadly investigated to this point. This method is usually used to take on larger order differential equations. The booklet will attract graduate scholars drawn to research, researchers in natural and utilized arithmetic, and engineers who paintings with partial differential equations. Readers would require just a easy wisdom of useful research, degree conception and Sobolev areas.

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N If p(n−2) < n2 then u ∈ L∞ ( ). Otherwise repeat the argument a finite number of times until u ∈ L∞ ( ). 7), − u ∈ L∞ ( ). Thus N , and hence u, has a C 1 extension to the origin. 6), u(0) > 0. 1 is essentially optimal. 2 Let f : (0, ∞) → (0, ∞) be a continuous function satisfying lim f (t) t→∞ t n/(n−2) = ∞. 29) 28 Isolated singularities of nonlinear elliptic inequalities such that u(x) = O(ϕ(|x|)) as x → 0. 2 relies on the following lemma which will be used on several occasions in this book.

82) of δ that u is asymptotically radial. 63) and the positivity of u in B2r0 (0) \ {0} we see that n 0 ≤ − u¯ ≤ K u¯ n−2 in \ {0}. ¯ → 0 as r → 0+ . 8 holds with replaced with K. 67) with replaced with + ε. 67) holds as stated. 8. 8 is too weak to imply u is asymptotically harmonic. 1) are asymptotically harmonic. 8. We use the following notation: log1 := log, log2 := log ◦ log, log3 := log ◦ log ◦ log, etc. 83) in a punctured neighborhood of the origin in Rn (n ≥ 3) where β ∈ (1, ∞) and q is a positive integer.

103) which are arbitrarily large at the origin, regardless of how large f is at ∞. 14 Let f : (0, ∞) → (0, ∞) be a continuous function. 103) in satisfying u(x) = O(ϕ(|x|)) as x → 0. 14 that depend on f is necessary. 105) has a C 2 positive solution where r0 is a positive constant and f : (0, ∞) → (0, ∞) is a continuous function such that f (u) ≥ au for u > 0. Then a ≤ C where C = C(r0 , n) is a positive constant. 15 eigenvalue for Let = x ∈ Rn : 1 r 4 0 < |x| < 34 r0 . Let λ1 be the first − w = λw, in w = 0, on ∂ and let w1 (x) be a corresponding positive eigenfunction.