Asymptotic expansion of integrals of the form $\int_{\mathcal{D}} \exp(\lambda\, \phi(x))\, g(x) \,dx$ for small $\lambda.$

Question

In the limit $\lambda\to\infty$ the asymptotic expansion of integrals of the form $\int_{\mathcal{D}}\exp(\lambda\,\phi(x))\,g(x)\,dx$ (where $\mathcal{D}\subseteq \mathbb{R^n}$ denotes the domain of integration and $dx$ is the Lebesgue measure) can be evaluated using the Laplace's method or the saddle point method. Are there any methods to find asymptotic expansion of such integrals in the limit of small $\lambda$ ? I am specifically interested in the case where the integrals $\int_{\mathcal{D}}(\phi(x))^n\,g(x)\,dx$ do not converge, so that simple expansion in $\lambda$ doesn't work.