Singular equation with positive solution based on the perturbation of domains method

A singular Liouville equation on planar domains

“…Equations with 𝑙𝑜𝑔-singular terms were investigated in [35] by means of sub-super-solutions, by variational methods in [34] and by the method of perturbation of domains in [37]. Replacing the 𝑙𝑜𝑔-term by −1∕𝑢 𝛽 , similar equations like −Δ𝑢 = (−𝑢 −𝛽 + 𝜆𝑢 𝑝 )𝜒 {𝑢>0} were studied in [7,18,19,22,36,37,40] where 0 < 𝛽 < 1 and either 0 < 𝑝 < 1 or 1 ≤ 𝑝 ≤ (𝑁 + 2)∕(𝑁 − 2). Free boundary properties were observed in [18] and [19] for small values of 𝜆, in this respect see also [24] and [39].…”

“…Equations with $$log$$‐singular terms were investigated in [35] by means of sub‐super‐solutions, by variational methods in [34] and by the method of perturbation of domains in [37]. Replacing the $$log$$‐term by $$-1/u^\beta$$, similar equations like $$-\Delta u = (- u^{-\beta } + \lambda u^{p} )\chi _{\lbrace u&gt;0\rbrace }$$ were studied in [7, 18, 19, 22, 36, 37, 40] where $$0&lt;\beta &lt;1$$ and either $$0&lt;p&lt;1$$ or $$1 \le p \le (N+2)/(N-2)$$. Free boundary properties were observed in [18] and [19] for small values of λ, in this respect see also [24] and [39].…”

A class of singular elliptic equations

“…On the other hand, in Chapters 2-6 we consider superlinear nonlinearities, and we obtain nonnegative (but not strictly positive) solutions thanks to the fact that these nonlinearities satisfy a version of the Ambrosetti-Rabinowitz condition (this condition is not satisfied for sublinear terms). Some of the results of this chapter were published in [55].…”