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

Moscariello, Gioconda

Adv. Calc. Var. 4, No. 4, 421-444 (2011).

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.

Reviewer: Yakov Yakubov (Tel-Aviv)

MSC:

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

Keywords:

Dirichlet problem; elliptic equation; infinite energy solution; interpolation spaces; existence; uniqueness; weak solution

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