# Euler functional associated to a $p-$laplacian bvp

I have this BVP: \begin{cases} -\Delta_{p}(u)=\lambda_1 |x|^{\theta}|u|^{q-2} u+f(x,u)-h ~~\text{in}~\Omega,\\ u\in W^{1,p}_0(\Omega). \end{cases}

where $\Delta_p$ is denoting the $p-$Laplacian operator, i.e., $\Delta_p u= \text{div}~ (|\nabla u|^{p-2}\nabla u),$

What is the Euler functional associated to this problem please.

The first step is:

$-\int_{\Omega} \Delta_p(u) v dx-\lambda_1 \int_{\Omega} |x|^{\theta} |u|^{q-2} u v dx-\int_{\Omega} f(x,u) v dx+\int_{\Omega} h v dx=0$

The second step is to do apply the Green formula,then we obtain:

$$(J'(u),v)=\int_{\Omega}|\nabla u|^{p-2}\nabla u \nabla v dx-\lambda_1 \int_{\Omega} |u|^{q-2} uv dx -\int_{\Omega} f(x,u)v dx+\int_{\Omega} hv dx$$

My question is what is $J(u)$ ?

For a problem like this:

\begin{cases} -\Delta_{p}(u)=\lambda_1 |u|^{q-2} u+f(x,u)-h ~~\text{in}~\Omega,\\ u\in W^{1,p}_0(\Omega). \end{cases}

$J(u)=\frac1p \int_{\Omega} |\nabla u|^p dx -\frac{\lambda_1}{q} \int_{\Omega} |u|^q dx +\int_{\Omega} F(x,u) dx+\int_{\Omega} h u dx$

But what happen when there is $|x|^{\theta}$ in the problem ?

thank you.

The derivative of $$J_0(u)= \frac1{q}\int_\Omega |x|^\theta |u|^{q} dx$$ with respect to $u$ is $$J_0'(u)v = \int_\Omega |x|^\theta |u|^{q-2}uv dx.$$ When deriving the Euler-Lagrange equation, you do not need to differentiate with respect to the coordinate $x$.
• As the energy is minimized with respect to $u$, the term $|x|^\theta$ acts like a constant in this setting.
• Please an other question if i have $J(u)=\frac1q \int_{\Omega}(|x|^{\theta} |u|)^q dx$ then $J'(u)v=\int |x|^{q\theta} |u|^{q-2}uv dx$ right ? Commented Mar 25, 2015 at 11:16