Potential Theory for semilinear PDEs (with applications to the Ginsburg-Landau Equation)

Let $f$ be a function from $\mathbb{R}$ to $\mathbb{R}$ with $f(0)=0$, continuous, increasing, and differentiable at zero. Let ${\mathbb{R}}^d$ with $d\ge 2$. For every open set $U\subset X$, we set: ${\mathcal{H}}_f(U)=\{u\in \mathcal{C}(U):\mathrm{\Delta }u=f(u)\}$ in the distributional sense. We define a function in ${\mathcal{H}}_f(U)$ tending to infinity at the regular boundary of $U$ to be a regular Evans function associated with $f$, $U$, and $\mathrm{\Delta }$. We'll say that $f$ has the KO (Keller-Osserman) property if there exists some natural number $d\ge 2$ such that a regular Evans function associated with $f$, $B$, and $\mathrm{\Delta }$ for every ball $B\subset {\mathbb{R}}^d$. We'll give an explicit characterisation of that property and examine the relationship between the KO condition, the Harnack principle, and the Brelot convergence property. We prove that, in the nonlinear case --- and in contrast to the linear case, we do not have the equivalence between the latter two properties. We continue the investigation of regular Evans functions in the case of uniformly elliptic or uniformly parabolic operators where we replace the function f by a function $\psi$ from ${\mathbb{R}}^d\times \mathbb{R}$ to $\mathbb{R}$ which, in contrast to many other authors, we do not suppose to be convex or locally Lipschitz. We then show that the previous investigations lead in a simple and natural way to results for the following generalized Ginzburg-Landau equation: $$Lu-u(|u{|}^{2\alpha }-1)=0inthedistributionalsenseon{\mathbb{R}}^d$$ where $\alpha$ is a positive real number and L is a strongly elliptic operator with bounded, uniformly Hölder-continuous coefficients, admitting an adjoint $L^{\ast }$ in the distributional sense. We shall obtain Hervé and Hervé-Harnack inequalities and discuss the solvability of the Dirichlet problem for real and complex valued solutions for the Ginzburg-Landau equation.