$I_0(\mu)=+\infty$ for any $\mu\in \mathcal{M}_1(K)$ and we say that a Borel set $E$ is log-polar if every compact subset of $E$ is log-polar. Moreover, we say that an assertion holds **quasi-everywhere** (q.e.) on $A\subset {\mathbb R}^2$ if it holds on $A\backslash P$ where $P$ is log-polar.

We recall that the Lebesgue measure of a log-polar set is zero.

Now we recall results about the existence, the uniqueness and the characterization of the equilibrium measure $\mu_V$ proved in [@Frost; @SaffTotik] for the classical growth assumption , and by Hardy and Kuijlaars [@Hardy; @Hardy2] (for existence and uniqueness) and Bloom, Levenberg and Wielonsky [@Bloom:2014qy] (for Frostman type variational inequalities) for weak growth assumption .

\[equi\][([@Frost; @SaffTotik; @Hardy; @Hardy2; @Bloom:2014qy])]{.nodecor} Let $V$ be a lower semicontinuous function on ${\mathbb R}^2$ such that $\{x\in {\mathbb R}^2 ; V(x)<+\infty \}$ is a non log-polar subset of ${\mathbb R}^2$ satisfying $$\liminf_{|x|\to +\infty}\{V(x)-\log(1+|x|^2) \}>-\infty.$$ Then we have :

1.  $\displaystyle \inf_{\mu \in \mathcal{M}_1({\mathbb R}^2)}I_V(\mu)$ is finite, where $I_V$ is given by .

2.  There exists a unique equilibrium measure $\mu_V\in \mathcal{M}_1({\mathbb R}^2)$ with $$I_V(\mu_V)=\inf_{\mu \in \mathcal{M}_1({\mathbb R}^2)}I_V(\mu)$$ and the logarithmic energy $I_0(\mu_V)$ is finite.

3.  The support $\Sigma_V$ of $\mu_V$ is contained in $\{x\in {\mathbb R}^2 ; 