Let $\Omega \subset {\mathbb{R }}^{N}$ be a bounded regular domain of dimension $N\ge 3,\;h$ a positive $L^{1}$ function on $\Omega .$ Elliptic equations of singular growth like 

$$\begin{aligned} -\Delta u=\frac{h(x)}{u^{p}}\quad \mathrm{in}\;\Omega , \quad u>0\quad \mathrm{in}\;\Omega , \quad u=0\quad \mathrm{on}\;\partial \Omega , \end{aligned}$$

have been the target of investigation for decades. A very nice result for existence of solutions of such an equation is due to Lazer–McKenna (Proc AMS 111:720–730, 1991) when $h$ is a positive continuous function on $\overline{\Omega }.$ In that paper the Lazer–McKenna obstruction was first presented: the equation has a $H_{0}^{1}$-solution if and only if $p<3.$ In this paper we provide an extension of the classical Lazer–McKenna obstruction and reveal the role of 3. Moreover, we give a local description of the solution set.

The role of the power 3 for elliptic equations with negative exponents

Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity

In the present paper, we study the following singular Kirchhoff problem M ∬ R 2 N | u ( x ) − u ( y ) | 2 | x − y | N + 2 s d x d y ( − Δ ) s u = λ f ( x ) u − γ + g ( x ) u 2 s ∗ − 1 in Ω , u > 0 in Ω , u = 0 in R N ∖ Ω , where Ω ⊂ R N is an open bounded domain, dimension N > 2 s with s ∈ ( 0 , 1 ) , 2 s ∗ = 2 N ∕ ( N − 2 s ) is the fractional critical Sobolev exponent, parameter λ > 0 , exponent γ ∈ ( 0 , 1 ) , M models a Kirchhoff coefficient, f ∈ L 2 s ∗ 2 s ∗ + γ − 1 ( Ω ) is a positive weight, while g ∈ L ∞ ( Ω ) is a sign-changing function. Using the idea of Nehari manifold technique, we prove the existence of at least two positive solutions for a sufficiently small choice of λ . This approach allows us to avoid any restriction on the boundary of Ω .

The Nehari manifold for fractional Kirchhoff problems involving singular and critical terms

We derive a compatible condition between coefficients and negative exponents, which is optimal for H01-solutions of strongly singular problems. This allows us to establish the existence of H01-solutions for p ≥ 3. Our proof is based on a variational argument.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S030821051100117X

Compatibility phenomena in singular problems

Abstract The aim of this paper is to study the existence and multiplicity of solutions for a class of fractional Kirchho problems involving Choquard type nonlinearity and singular nonlinearity. Under suitable assumptions, two nonnegative and nontrivial solutions are obtained by using the Nehari manifold approach combined with the Hardy-Littlehood-Sobolev inequality.

Combined effects of Choquard and singular nonlinearities in fractional Kirchhoff problems

An exact estimate result for a class of singular equations with critical exponents

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An exact estimate result for a class of singular equations with critical exponents