Existence theorems for entire solutions of stationary Kirchhoff fractional p-Laplacian equations

Entire solutions for critical $p$-fractional Hardy Schrödinger Kirchhoff equations

“…We refer e.g. to [2,10,16,29,30,32,36,37] and the references therein for details. But the equations treated here contain also Hardy terms, which make the analysis more delicate and quite interesting.…”

“…But the equations treated here contain also Hardy terms, which make the analysis more delicate and quite interesting. For related problems we just mention [5,10,11,21] and the references cited in there.…”

“…For this reason we assume that (1.1) is non-degenerate, that is (1.3) inf Stationary non-degenerate Kirchhoff problems have been extensively studied in the last decades, but usually under the request that M is nondecreasing in R + 0 , as in [22,33,35] and the reference therein. As in [2,10,36] the monotonicity assumption is replaced by (M). In particular, as shown in [2,36], the Kirchhoff function M (t) = (1 + t) k + (1 + t) −1 , k ∈ (0, 1), verifies both (M) and (1.3), but is not monotone.…”

“…(G) g : R N × R → R is a Carathéodory function and there exist exponents r and µ in (θp, p * s ) such that for all ε > 0 there exists C ε > 0 and |g(x, t)| ≤ θpε|t| θp−1 + rC ε |t| r−1 for a.a. x ∈ R N and all t ∈ R, and either (i) θp < µ < q and µG(x, t) ≤ t g(x, t) for a.a. x ∈ R N and all t ∈ R, where G(x, t) = t 0 g(x, τ ) dτ , or (ii) q ≤ µ < p * s and 0 ≤ µG(x, t) ≤ t g(x, t) for a.a. x ∈ R N and all t ∈ R. For examples of subcritical nonlinear terms which satisfy conditions (F) and (G) we refer to [10]. The condition, assumed in [36], namely inf{G(x, t) : x ∈ R N , |t| = 1} > 0, is no longer required here and in [10] thanks to the possible presence of the nontrivial nonlinearity f .…”

“…The condition, assumed in [36], namely inf{G(x, t) : x ∈ R N , |t| = 1} > 0, is no longer required here and in [10] thanks to the possible presence of the nontrivial nonlinearity f .…”