## Existence and uniqueness for elliptic equations with lower-order terms.(English)Zbl 1232.35051

The author considers the Dirichlet problem
$\left\{\begin{matrix} \text{div}(\mathcal A(x)\nabla u+E(x)u)=\text{div}F & \text{in}\;\;\Omega,\\ u=0 & \text{on}\;\;\partial\Omega,\end{matrix} \right.\tag{1}$
where $$\Omega$$ is a bounded regular domain in $$\mathbb R^N$$, $$N>2$$. Here $$\mathcal A(x)=(a_{ij}(x))$$ is a symmetric positive definite $$N\times N$$ matrix and $$x\to E(x):\Omega\to\mathbb R^N$$ is a vector field. It is also assumed that the entries of $$\mathcal A(x)$$ satisfy \begin{aligned} &\alpha|\xi|^2\leq\sum_{i,j=1}^Na_{ij}(x)\xi_i\xi_j,\\ &\|\mathcal A(x)\|\leq\beta, \end{aligned} with $$0<\alpha<\beta$$, for every $$\xi\in\mathbb R^N$$ and for $$x\in\Omega$$ a.e. Under various assumptions on $$E(x)$$, uniqueness or/and existence (of some weak solutions) in Sobolev spaces $$W_0^{1,p}(\Omega)$$, $$p>1$$, is/are proved for problem (1). The corresponding spaces for $$F$$ are described for different cases.

### MSC:

 35J25 Boundary value problems for second-order elliptic equations 35J50 Variational methods for elliptic systems 35D30 Weak solutions to PDEs
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### References:

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