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Let $\Omega$ be a smooth bounded domain in Euclidean space. Let $u \in C_c^1(\Omega)$. The subscript indicates compact support. Let $1 < p < \infty$.

Can a $v \in C_c^1(\Omega)$ (or in its closure in the Sobolev space $W^{1,q}(\Omega)$, where $1/p + 1/q = 1$) be found such that $\nabla u \cdot \nabla v = |\nabla u|^p$ (a.e.) in $\Omega$?

Or merely $\int_\Omega \nabla u \cdot \nabla v = \int_\Omega |\nabla u|^p$?

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Given a function $u\in C_c^1(\Omega)$, and a number $p\in (1,\infty)$, you want to find some partial global classical solution $v\in C_c^1(\Omega)$ of a linear inhomogeneous first order PDE with continuous variable coefficients $$\nabla\,u\cdot\nabla\,v=|\nabla\,u|^{p}\quad {\rm in}\;\,\Omega\subset\mathbb{R}^n,\;\;n\geqslant 2. $$ You want too much!  This is a classical two-and-a-half-century-old problem, generally known to be solved only locally. Whether or not its global solution exist depends completely on the given function $u$.  A simple, seemingly obvious partial global "solution" $\nabla\,v=\nabla\,u|\nabla\,u|^{p-2}\;$ is not a solution at all, excluding the case of spherical symmetry $u(x)=\varphi(|x|)$ and some other trivial cases when the vector field $\,\nabla\,u|\nabla\,u|^{p-2}\,$ proves to be potential. So all you need do is find characteristic curves for the ODE system $$\frac{dx_1}{a_1(x)}=\dots=\frac{dx_n}{a_n(x)}=\frac{dv}{b(x)}\,,\quad a_j(x)\overset{def}{=}\frac{\partial u(x)}{\partial x_j}\,,\,j=1,\dots,n,\,\;\,b(x)\overset{def}{=}|\nabla\,u(x)|^{p-2}. $$ But what you will find will be a solution corresponding to a certain given $u$.

As to your second question, follow the abvice of Tomás, i.e. take $v=\mu u$ choosing the number $$\mu\overset{def}{=}\,\frac{\int_{\Omega}|\nabla\,u(x)|^{p}dx} {\int_{\Omega}|\nabla\,u(x)|^{2}dx}\,. $$

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  • $\begingroup$ I don't understand the motivation for the ODE system. $\endgroup$ – user66081 Jan 27 '14 at 10:31
  • $\begingroup$ uer66081: General solution of a 1st PDE can be represented here by an implicit function of $n$ independent integrals of the system of characteristic ODEs, writen down here in a standard form suitable for memorizing. For the details, see textbooks of PDEs. Most often, it is just a chapter in a textbook. For a brief superficial introduction to the topic, follow, say, the link stanford.edu/class/math220a/handouts/firstorder.pdf or read Ch.1B in the textbook bookza.org/book/539085/ae3797 $\endgroup$ – mkl314 Jan 27 '14 at 16:06
  • $\begingroup$ user66081: Here are some other helpful links. Besides the classical hanbook in German by E.Kamke, now there many others, e.g., bookza.org/book/1049863/b9e883 You can as well visit the web-page eqworld.ipmnet.ru/en/solutions/firstpde.htm $\endgroup$ – mkl314 Jan 27 '14 at 17:59
  • $\begingroup$ user66081: Of course, it must be $n+1$ independent integrals. Just a slip of the keyboard. $\endgroup$ – mkl314 Jan 27 '14 at 21:17
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Paraphrasing V. Barbu, Nonlinear differential equations of monotone types in Banach spaces, 2010, Springer, p.4:

Let $1 < p < \infty$ and $\Omega$ a bounded and open subset of $\mathbb{R}^n$. Write $X := W_0^{1,p}(\Omega)$. For $u \in X$ define $J(u) \in X'$ by $$J(u) = -\|u\|_X^{2-p} \sum_{i=1}^n \partial_i (|\partial_i u|^{p-2} \partial_i u).$$ Then $J : X \to X'$ is the duality mapping (it is indeed single-value by uniform convexity of $X$), for which $$\langle J(u), u \rangle = \|u\|_X^2 = \|J(u)\|_{X'}^2.$$ This (almost) answers the second part of the question.

As an aside, the duality mapping is, in general, a set-valued mapping $J : X \to 2^{X'}$ on a Banach space defined by $$J(x) = \{ f \in X' : f(x) = \|x\|_X^2 = \|f\|_{X'}^2 \},$$ and this is always non-empty by the Hahn-Banach theorem (analytic version).

End of paraphrase.

Let $u \in C_c^1(\Omega)$ be nonzero. Then $J(u)$ is a bounded linear functional on $H_0^1(\Omega)$. Let $v \in H_0^1(\Omega)$ solve $\int_\Omega \nabla v \cdot \nabla \varphi = \langle J(u), \varphi \rangle$ for all $\varphi \in C_c^1(\Omega)$. Then $\|u\|_X^2 = \langle J(u), u \rangle = \int_\Omega \nabla v \cdot \nabla u$.

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