A priori estimates, existence theorems, and the behavior at infinity of solutions of quasielliptic equations in $\mathbf{R}^n$

L. A. Bagirov

Sbornik. Mathematics, Tome 38 (1981) no. 4, pp. 437-452

Résumé

The equation $$ A(x,D)u(x)=\sum_{\langle\alpha\cdot\theta\rangle\leqslant m}a_\alpha(x)D^\alpha u(x)=f(x),\qquad x\in\mathbf R^n, $$ is studied in this paper. Here $\theta=(\theta_1,\dots,\theta_n)$ is the index of quasihomogeneity of the operator $A$ and $\langle\alpha\cdot\theta\rangle=\alpha_1\theta_1+\dots+\alpha_n\theta_n$. The quasiellipticity condition $$ \biggl|\sum_{\langle\alpha\cdot\theta\rangle=m}a_\alpha(x)\xi^\alpha\biggr|\geqslant\delta\sum_{k=1}^n|\xi_k|^{m_k},\qquad\delta>0,\quad\xi\in\mathbf R^n,\quad x\in\mathbf R^n,\quad\frac{m_k}m=\theta_k^{-1}, $$ is assumed to hold. Theorems on the Noether property of $A$ in weighted spaces are proved under two types of conditions on the behavior of the coefficients $a_\alpha(x)$ at infinity. Bibliography: 18 titles.

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