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Integrability for solutions to quasilinear elliptic systems. (English) Zbl 1224.35139
The authors give structural conditions on coefficients of quasilinear elliptic systems in divergence form $-\sum _{i=1}^n D_i\left (\sum _{j=1}^n \sum _{\beta =1}^N a_{ij}^{\alpha \beta }(x,u(x))D_j u^{\beta }(x)=0\right )\;\;\text{on $$\Omega$$ for $$\alpha = 1,...,N$$}$ which guarantee the higher integrability of solution $$u$$. The structural conditions require decay of non-diagonal coefficients for $$u \to \infty$$ described by $g^{\gamma }(L)= \max _{i,j}\max _{\beta \neq \gamma }\sup _{| y^{\gamma }| >L} \sup _{x}| a_{ij}^{\gamma \beta }(x,y)| .$ With no extra assumptions $$g^{\gamma }$$ is nonnegative, bounded and decreasing function. Then the authors prove the existence of a constant $$C$$ such that $$| \{x \in \Omega; | u^{\gamma }(x)| > 2L\} | \leq C\big (g^{\gamma }(L)/L\big )^{2^{*}}.$$ Here $$2^{*}= 2n/(n-2)$$ is Sobolev’s embedding exponent. If non-diagonal coefficients decay polynomially, i.e., for a positive $$q$$ $$| a_{ij}^{\gamma \beta }(x,y)| \leq c/| y^{\gamma }| ^{-q}$$ and $$u^{\gamma }$$ in bounded on $$\partial \Omega$$ then $$u \in L^{2^*(1+q)}_{weak}(\Omega )$$. Both results are obtained under weaker ellipticity condition: For given $$\gamma$$ there are positive constants $$\nu , \theta$$ so that $\theta ^{\gamma } \leq | y^{\gamma }| \Rightarrow \nu | \xi | ^2 \leq \sum _{i,j=1}^n a_{ij}^{\gamma \gamma }(x,y)\xi _i \xi _j.$

##### MSC:
 35J62 Quasilinear elliptic equations 35J47 Second-order elliptic systems 35D30 Weak solutions to PDEs 35B45 A priori estimates in context of PDEs
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