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I'm looking for existence results for the equation $$\lambda u-\frac{1}{(1+(u')^2)^2} \, \Delta u = f \quad \text{on the domain $[a,b]$}$$ for $u:[a,b] \to \mathbb{R}$, with either zero Dirichlet or Neumann BCs.

Here $f$ is some data, but preferably some results on $f=f(u)$ nonlinear and smooth would be good.

The coefficient of the Laplacian doesn't seem to satisfy the requirements of the theory in Gilbarg and Trudinger nor LSU. Does anyone know if one can get existence of solutions for this PDE?

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  • $\begingroup$ If f=f(u), why not multiply the equation by u' and integrate once? $\endgroup$
    – Michael Renardy
    Commented Dec 7, 2017 at 13:39
  • $\begingroup$ In other words, you want to know if there are stationary points of $$ I(u)=\int_a^b\left[\lambda\frac {u^2}2+G(u')-F(x,u)\right]\,dx $$ where $G(t)=\int_0^t\frac{1}{(1+s^2)^2}\,ds$ and $F(x,t)$ is an antiderivative of $f(x,s)$ with respect to $s$. In what regime do you have trouble with the existence of both the maximizer and the minimizer? $\endgroup$
    – fedja
    Commented Dec 7, 2017 at 20:14

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