Existence of Solutions for Singular Critical Growth Semilinear Elliptic Equations

Abstract

A semilinear elliptic problem containing both a singularity and a critical growth term is considered in a bounded domain of R n : existence results are obtained by variational methods. The solvability of the problem depends on the space dimension n and on the coefficient of the singularity; the results obtained describe the behavior of critical dimensions and nonresonant dimensions when the BrezisNirenberg problem is modified with a singular term.

Ground state and multiple solutions for a critical exponent problem

“…The proof of (a) can be found in [14] and the proof of (b) can be found in [9]. In fact they proved some similar conclusions with Hardy potential …”

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.…”

“…Note that C ε > 0 does not depend on n, since {T n u n } is bounded in L 2 * . Recalling (24), this implies a n ≤ C ε λ (N−2)(p−2 * )/2 n + C 1 ε 1/2 * for all n so that lim sup n→∞ a n ≤ C 1 ε 1/2 * since λ n → ∞ and (N − 2)(p − 2 * )/2 < 0. Therefore, letting ε → 0, one obtains lim n→∞ a n = 0, which contradicts (23).…”

Asymptotic behavior of solutions to semilinear elliptic equations with Hardy potential

“…A. Ferrero and F. Gazzola [8] also obtained some results for problem (1.1). For other relevant papers see [1,3,6,7,10,12], and the references therein.…”