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

Résumé

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

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Statistiques

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