Norm of union for dual Morrey spaces with applications to nonlinear elliptic PDEs

For Banach spaces, the union as a general construction is a nonsence – it is not even a linear set. For a purposes of analysis, the construction of a sum of spaces is sufficient in (almost) all situations. For nolinear PDEs it is not so. Even minor simpleness of a construction $\inf_j \|f\|_{X_j}$ in comparison with $\inf_{\sum_j f_j =f} \sum_j \|f_j\|_{X_j}$ can be crucial.
For $p\in (1,\infty)$, $a\in (0,n(p-1))$ we let define spaces $L_{p,a}=L_{p,a}(R^n)$ by the (quasi)norm
$$ \|f\|_{p,a}= \inf_\sigma \|f; L_p(R^n;\omega)\| , $$
$L_p(R^n;\omega)$ – weighted Lebesgue spaces with
$$ \omega(x) = \biggl( \int_{R^{n+1}_+} r^{a/(1-p)} 1_{\{|x-y|<r\}}\, d\sigma(y,r) \biggr)^{1-p} $$
inf is taken over nonnegative Borel measures $\sigma$ on $R^{n+1}_+ =\{(y,r): y\in R^n, r>0 \}$ with normalization $\sigma (R^{n+1}_+)=1$.
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N.B. The dual for $L_{p,a}$ with $a>0$ are classical Morrey spaces.
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We consider nonlinear elliptic equations of the form
$$ \operatorname{div}^m A(x, D^m u) = f(x) $$
in $R^n$ with the natural energetic space $W^m_p$, $p\in (1,\infty)$, and standard structure conditions (e.g. $m,p$-Laplacian).
We establish the existence of very weak solution $u \in W^m_{p,a}$ for some range of $a \in (0, a^*)$ where $a^* >0$ depends on $n, m, p$ and a modulus of ellipticity of equation.
Key difference from spaces $W^m_q$ with $q\ne p$ (a priori estimates in $W^m_{p-\varepsilon}$ are known since 1993) is that weighted spaces $W^m_{p,\omega}$ allow to establish pseudomonotonicity of our nonlinear operator.