Abstract
We prove that the nonlinear partial differential equation urn:x-wiley:20903332:media:ista823105:ista823105-math-0001 , with u(0) > 0, where f and g are continuous, f(u) > 0 and g(|x|, u) > 0 for u > 0, and urn:x-wiley:20903332:media:ista823105:ista823105-math-0002 , has no positive or eventually positive radial solutions. For g(|x|, u) ≡ 0, when n/(n − 2) ≤ q < (n + 2)/(n − 2) the same conclusion holds provided 2F(u) ≥ (1 − 2/n)uf(u), where . We also discuss the behavior of the radial solutions for f(u) = u3 + u5 and f(u) = u4 + u5 in ℝ3 when g(|x|, u) ≡ 0.
Highlights
In recent years, numerous authors have given substantial attention to the existence of positive solutions of semilinear elliptic equations involving critical exponents
Where f and g are continuous functions, with f(u) > 0 and g(Ix I,u) > 0 whenever u > 0. Such equations arise in many areas of applied mathematics; Ix Ic solutions that exist in n and satisfy u(x)O as are called ground states
In this paper we show that the reversal of the inequality in (1.3) together with the assumption that uf(u) > 0 and ug(r, u) > 0 for all u 0, and mild conditions on the order of the growth of f(u) to zero as u--,0, lead to the nonexistence of positive solutions to (.1.2) with u0 > 0
Summary
-with u(0)> 0, where f and g are continuous, f(u)> 0 and g(Ix I,u)> 0 for u > 0, and lim u--0.
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