overlap with the results in [@seo0; @seo1; @seo2]. Indeed, from one hand [@seo0; @seo1; @seo2] consider more general potentials; on the other hand we obtain here a *strong* unique continuation and a unique continuation from sets of positive measure, which are stronger results than the weak unique continuation obtained in [@seo0; @seo1; @seo2]. We also observe that we assume that equation is satisfied only on the set $\Omega$ and not in the whole ${{\mathbb R}}^N$.

The paper is organized as follows. In section \[sec:monot-argum\] we develop the monotonicity argument, proving in particular the existence of a finite limit for the frequency function as $r\to0^+$. In section \[sec:blow-up-analysis\] we carry out a careful blow-up analysis for scaled solutions, which allows proving Theorem \[t:asym\] and, as a consequence, Theorem \[t:sun-ext\]. Finally section \[sec:appl-fourth-order\] is devoted to applications of Theorem \[t:asym\] to fourth order problems and higher order fractional problems , with the proofs of Theorems \[t:asym-ho\] and \[t:frac\].

The monotonicity argument {#sec:monot-argum}
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For all $r\in(0,R)$ we define the functions $$\label{eq:D(r)}
D(r)=r^{-N+1} \left[\int_{B_r^+} \left(|\nabla U|^2+|\nabla
    V|^2+UV\right)dz
-\int_{B_r'} h(x)u(x)v(x)\, dx  \right]$$ and $$\label{eq:H(r)}
H(r)=r^{-N}\int_{S_r^+} (U^2+V^2)\, dS.$$ We define the space $\mathcal
D^{1,2}({{\mathbb R}}^{N+1}_+)$ as the completion of the space $C^\infty_c(\overline{{{\mathbb R}}^{N+1}_+})$ with respect 