Construction of singular limits for four-dimensional elliptic problems with exponentially dominated nonlinearity

AbstractThe authors consider the existence of singular limit solution for a family of nonlinear elliptic problems with exponentially dominated nonlinearity and Navier boundary condition

Singular limiting solutions to 4-dimensional elliptic problems involving exponentially dominated nonlinearity and nonlinear terms

Let $\Omega \in \mathbb{R}^4$ be a bounded open regular set, $x_1, x_2, \dots, x_m \in \Omega$, $\lambda, \rho >0$ and $Q_\lambda$ be a non linear operator (which will be defined later). We prove that the problem $$\Delta^2u +Q_\lambda(u)= \rho^4 e^u$$ has a positive weak solution in $\Omega$ with $u=\Delta u=0$ on $\partial \Omega$, which is singular at each $x_i$ as the parameters $\lambda$ and $\rho$ tends to 0