AN ELLIPTIC PROBLEM DRIVEN BY A SINGULARITY

The study of elliptic equation driven by singular nonlinearities: −∆u =p(x)u^{−γ} in Ω, u =0 on ∂Ω, by Lazer and McKenna has opened a new avenue of mathematical research in the field of elliptic PDEs. In this discourse we will try to see the existence of multiple solutions to an elliptic problem driven by a singular nonlinearity and an exponential nonlinearities as follows:
−∆u =λ|u|^{−γ−1} u exp(βu^{2}) in Ω ⊂ R^{2}, u =0 on ∂Ω, λ, β > 0, 1 < γ < 3.
The issues here are not only the presence of nonlinearities of two different natures but also the order of singularity being larger than 1 which makes the associated functional discontinuous at u = 0. There are theories in the literature to handle the case of 0 < γ < 1, however, when γ > 1 we are compelled to restrict the dimension to 2. We will try to prove the existence of arbitrary number of multiple pairs of solution to this problem using an abstract critical point theory for sufficiently large values of λ (see attachment too)