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Let $a^{ij}, b_{i}, c,f$ are smooth function. Suppose $\Lambda I\geq (a^{ij})\geq \lambda I$, where $I$ is identity matrix, $\lambda, \Lambda$ is positive constant. Is there solution of the following equation $$ \sum_{i,j}^{n}a^{ij}u_{i}u_{j}+\sum_{i}b_{i}u_{i}+cu=f$$

on a domain $\Omega\subset\mathbb{R}^{n}$ or a closed $n-$dimension manifold, where $u_{i}=\frac{\partial u}{\partial x_{i}}$? If not, are there known condidtions ensuring that this equation admits a soulution or locally solution?

Let $a^{ij}, b_{i}, c$ and $f>0$ are smooth function. Suppose $\Lambda I\geq (a^{ij})\geq \lambda I$, where $I$ is identity matrix, $\lambda, \Lambda$ is positive constant. Is there solution of the following equation $$ \sum_{i,j}^{n}a^{ij}u_{i}u_{j}+\sum_{i}b_{i}u_{i}+cu=f$$

on a domain $\Omega\subset\mathbb{R}^{n}$ or a closed $n-$dimension manifold, where $u_{i}=\frac{\partial u}{\partial x_{i}}$? If not, are there known condidtions ensuring that this equation admits a soulution or locally solution?

Let $a^{ij}, b_{i}, c,f$ are smooth function. Suppose $\Lambda I\geq (a^{ij})\geq \lambda I$, where $I$ is identity matrix, $\lambda, \Lambda$ is positive constant. Is there solution of the following equation $$ \sum_{i,j}^{n}a^{ij}u_{i}u_{j}+\sum_{i}b_{i}u_{i}+cu=f$$

on a domain $\Omega\subset\mathbb{R}^{n}$ or a closed $n-$dimension manifold, where $u_{i}=\frac{\partial u}{\partial x_{i}}$?

Let $a^{ij}, b_{i}, c,f$ are smooth function. Suppose $\Lambda I\geq (a^{ij})\geq \lambda I$, where $I$ is identity matrix, $\lambda, \Lambda$ is positive constant. Is there solution of the following equation $$ \sum_{i,j}^{n}a^{ij}u_{i}u_{j}+\sum_{i}b_{i}u_{i}+cu=f$$

on a domain $\Omega\subset\mathbb{R}^{n}$ or a closed $n-$dimension manifold, where $u_{i}=\frac{\partial u}{\partial x_{i}}$? If not, are there known condidtions ensuring that this equation admits a soulution or locally solution?

Let $a_{ij}, b_{i}, c,f$ are smooth function. Suppose $\Lambda I\geq (a^{i,j})\geq \lambda I$, where $I$ is identity matrix, $\lambda, \Lambda$ is positive constant. Is there solution of the following equation $$ \sum_{i,j}^{n}a^{ij}u_{i}u_{j}+\sum_{i}b_{i}u_{i}+cu=f$$

on a domain $\Omega\subset\mathbb{R}^{n}$ or a closed $n-$dimension manifold, where $u_{i}=\frac{\partial u}{\partial x_{i}}$?

Let $a^{ij}, b_{i}, c,f$ are smooth function. Suppose $\Lambda I\geq (a^{ij})\geq \lambda I$, where $I$ is identity matrix, $\lambda, \Lambda$ is positive constant. Is there solution of the following equation $$ \sum_{i,j}^{n}a^{ij}u_{i}u_{j}+\sum_{i}b_{i}u_{i}+cu=f$$

on a domain $\Omega\subset\mathbb{R}^{n}$ or a closed $n-$dimension manifold, where $u_{i}=\frac{\partial u}{\partial x_{i}}$?