# Seeking corresponding functional to show existence of weak solution to PDE

$$-\Delta u = \lambda u \space, \space\space\space\ x∈Ω$$ $$u(x) = 0\space, \space \space\space\space\space x∈\partial Ω$$

What functional would form a good correspondence, and whose minimization could show that the above function, $u$, has a weak solution in $H^1_0(\Omega)$? Is this analogous solving the case of:

$$-\Delta u = f(x) \space, \space\space\space\ x∈Ω$$ $$u(x) = 0\space, \space \space\space\space\space x∈\partial Ω$$ taking $f(x)=\lambda u$, or is the there a better way to solve this specific case?

• It's more complicated than that. Without an $f$ on the right hand side it becomes an eigenvalue problem and you would need two functionals. See mathreview.uwaterloo.ca/archive/voli/2/nica.pdf May 23, 2013 at 2:12
• In this case, we are more interested in what $\lambda$'s values are, rather than $u$ itself. May 23, 2013 at 4:25
• Just to be clear, I'm only working right now on showing the existence of a weak solution in ${H^1_0}$, not on identifying it (yet, at least).
– BBB
May 23, 2013 at 5:09

If $\lambda\leq 0$ and is known, then the equation is nothing but a regular Poisson equation with an $L^2$-term.
If $\lambda >0$ is is unknown, then this is an eigenvalue problem, the link toypajme's comment is pretty nice and readable. Here I just add a bit more about motivation. Normally for eigenvalue problem, we are minimizing the Rayleigh quotient: $$F(u) = \frac{\| \nabla u\|_{L^2(\Omega)}^2}{\| u\|_{L^2(\Omega)}^2},\tag{1}$$ whereas $$\int_\Omega u w_i = 0\quad \text{for eigenfunction } w_i, i=1,\ldots,n-1,$$ and the minimizer $u = w_n$ is another eigenfunction while $F(w_n)$ is the $n$-th eigenvalue. Notice problem (1) can be written as the same constraint minimization problem in the link in the toypajme's comment.
If $\lambda = \omega^2$ is known and is not one of the eigenvalues for the operator $-\Delta$, also we have a right hand side $f$, then this is a Helmholtz equation, and it weak formulation $$\int_{\Omega}\nabla u\cdot \nabla v - \omega^2 \int_{\Omega}uv = \int_\Omega fv$$ has a unique solution in $H^1_0$ by Fredholm alternative.