## A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces.(English)Zbl 1113.35062

Summary: We present a new general method to obtain regularity and a priori estimates for solutions of semilinear elliptic systems in bounded domains. This method is based on a bootstrap procedure, used alternatively on each component, in the scale of weighted Lebesgue spaces $$L^p_\delta(\Omega)=L^p(\Omega\delta(x)\,dx)$$, where $$\delta(x)$$ is the distance to the boundary. Using this method, we significantly improve the known existence results for various classes of elliptic systems.

### MSC:

 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 35J50 Variational methods for elliptic systems 35J60 Nonlinear elliptic equations
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### References:

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