On a semilinear elliptic equation with inverse-square potential

A note on a critical problem with natural growth in the gradient

“…Finally, by (9) and using the same ideas as in [2] (see also [4]), we conclude that u satisfies the equation in the renormalized sense (see [11]). …”

Sum of weighted Lebesgue spaces and nonlinear elliptic equations

“…More recently, the zero mass case of equations (1.1) with noncritical nonlinearities behaving as a single power has been widely studied in both the autonomous and nonautonomous cases (see e.g. [10,12,22,25,26,37] and [1,3,21,31,37] respectively, and the references therein), showing essentially that the existence of solutions relies on suitable compatibility conditions between the power of u and the growth and decaying rates of V (x) (and possibly of the nonlinearity) at zero and infinity.…”

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

“…For instance (other references can be found in [31]), the case V (x) = λ + µ|x| −2 is studied in [24], [27], [32] (see also [18]), where the solvability of the equation is examined in connection with the sign and size of the parameters λ, µ. In the presence of more general nonlinearities of the form f (x, u) = u p + λb(x), inverse-square potentials V (x) = −A|x| −2 with 0 < A ≤ (N − 2) 2 /4 are considered in [16] (case λ = 0) and [20] (case λ > 0), where compatibility conditions on A, p, λ and the space dimension are exhibited in order to ensure the existence of solutions. The results of [20] are extended in [21] to a larger class of potentials and nonlinearities.…”