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We use some tools from nonlinear analysis to study two examples of singularstochastic elliptic PDEs that cannot be solved by the contraction principle orthe Schauder fixed point theorem. Let $\xi$ stand for a spatial white noise ona closed Riemannian surface $S$. We prove the existence of a solution to theequation $$ (-\Delta + a)u = f(u) + \xi u $$ with a potential $a\in L^p(S)$ and$p>1$, and $f$ subject to growth conditions. Under an additional paritycondition on $f$ -- met for instance when $f(u) = u\vert u\vert^\ell$, with$\ell$ an even innteger, we further prove that this equation has infinitelymany solutions, in stark contrast with all the well-posedness results that havebeen proved so far for singular stochastic PDEs under a small parameterassumption. This kind of results is obtained by seeing the equation ascharacterizing the critical points of an energy functional based on theAnderson operator $H=\Delta + \xi$ and by resorting to variants of the mountainpass theorem. There are however some interesting equations that cannot becharacterized as the critical points of an energy functional. Such is the caseof the singular Choquard-Pecard equation on $\mathbb{T}^2$ $$ (-\Delta + a)u =\big(w\star f(u)\big)g(u) + \xi u $$ One can use Ghoussoub's machinery ofself-dual functionals to prove the existence of a solution to that equation asthe minimum of a self-dual strongly coercive functional under properassumptions on the coefficients $a,w,f$ and $g$.

Variational methods for some singular stochastic elliptic PDEs