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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?

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    $\begingroup$ Is $u_i$ the partial derivative with respect to $x_i$, or something else? $\endgroup$
    – David Roberts ♦
    Commented Jul 22, 2020 at 6:51
  • $\begingroup$ Yes ,it is the partial derivative with respet to $x_{i}$. Thank you and D. Tampieri for pointing out it. $\endgroup$
    – liding
    Commented Jul 22, 2020 at 7:24
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    $\begingroup$ There's not always a solution. Just look at $(u')^2 = -1$ when $n=1$. Are you asking for sufficient conditions that a solution exist? Would existence of local solutions suffice for what you want? Perhaps you can make your question more precise. $\endgroup$
    – Robert Bryant
    Commented Jul 22, 2020 at 9:18
  • $\begingroup$ I have a PDE similar with this problem on closed manifold. I just want to find a solution. Both the sufficient conditions and the local solution will help me. $\endgroup$
    – liding
    Commented Jul 22, 2020 at 14:43
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    $\begingroup$ This isn't a general first order scalar PDE, but of a particular sort: If $M$ is a Riemannian manifold with metric $a$, $b$ is a vector field on $M$, and $c$ and $f$ are functions on $M$, the equation becomes $$|\nabla u|_a^2 + 2b\,\cdot\nabla u+c\,u - f =0.$$ (In the flat case, the inequalities on $a$ ensure that the metric is complete.) Writing it as $$|\nabla u+b|_a^2 + c\,u - (f+|b|^2_a) = 0,$$ makes it clearer when solutions exist in a neighborhood of a point. For example $c(x)\not=0$ or $f(x)+ |b(x)|^2_a>0$ would guarantee local solutions near $x$, by the method of characteristics. $\endgroup$
    – Robert Bryant
    Commented Jul 24, 2020 at 10:18

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