A Sobolev-Hardy Inequality with¶Applications to a Nonlinear Elliptic Equation¶arising in Astrophysics

A nonlinear elliptic equation with singular potential and applications to nonlinear field equations

“…Accordingly, in this section we write x = (y, z) ∈ R 2 × R N−2 with N > 2. As a result, this proves Proposition 5 and allows us to conclude that Theorem 4 actually provides a weak solution to equation (6), in the sense of Definition 2.…”

“…We will also consider the linear subspace X s (R N ) := {u ∈ X(R N ) : u(y, z) = u(|y|, z)} where, with a slight abuse of notation, writing u(y, z) = u(|y|, z) we naturally mean u(y, z) = u(gy, z) for any rotation g : R k → R k and almost every (y, z) ∈ R N . Our main result is Theorem 3, which states the existence of weak solutions to equation (6) in the sense of the following definition.…”

“…In this section we study asymptotic and summability properties of nonnegative weak solutions of equation (6), which amounts to proving Proposition 6 and thus, by Theorem 4 and Proposition 5, to concluding the proof of Theorem 3. Accordingly, we assume that f ∈ C(R; R) satisfies condition (f 1 ) and u ≥ 0 is a weak solution in the sense of Definition 2.…”

“…Now we set z := (0, z) for any z ∈ R N −k and observe that u(· − z) ≥ 0 is still a weak solution to equation (6). Hence, by repeating the above argument, one infers that for every z ∈ R N−k there exist ρ z , C z > 0 such that u(· − z) ≤ C z almost everywhere in B (N ) ρ z , i.e., u ≤ C z almost everywhere in B (N ) ρ z ( z).…”

“…To overcome this difficulty, we set the equation in the exact space on which the Euler functional of the linear part of the equation is well defined, namely the weighted Hilbert space, say X(R N ), consisting of the functions having finite L 2 gradient norm and such that R N |y| −2 u 2 dx < ∞. By the results of [6], the latter condition is redundant if k > 2 (see Remark 7), whereas it is necessary in the case k = 2. Since our assumptions ensure that |f (t)| ≤ const · |t| 2 * −1 (see (f 1 )), also the functional related to the nonlinearity makes sense on X(R N ).…”