Construction of singular limits for a strongly perturbed four-dimensional Navier problem with exponentially dominated nonlinearity and nonlinear terms

Abstract

Abstract Given a bounded open regular set Ω∈R4,x1,x2,…,xm∈Ω,λ,ρ>0,γ∈(0,1) $\varOmega \in \mathbb{R}^{4}, x_{1}, x_{2}, \ldots, x_{m} \in \varOmega, \lambda, \rho > 0, \gamma \in (0,1)$, and Qλ ${\mathscr{Q}}_{\lambda }$ some nonlinear operator (which will be defined later), we prove that the problem Δ2u+Qλ(u)=ρ4(eu+eγu) $$ \Delta ^{2}u +{\mathscr{Q}}_{\lambda }(u)= \rho ^{4} \bigl(e^{u} + e^{\gamma u}\bigr) $$ has a positive weak solution in Ω with u=Δu=0 $u = \Delta u=0$ on ∂Ω, which is singular at each xi $x_{i}$ as the parameters λ and ρ tend to 0.

Keywords

• Biharmonic operator
• Nonlinear operator
• Singular limits
• Green’s function
• Nonlinear domain decomposition method