We have the operator: $$\Phi_p(x)= x\lvert x \rvert^{p-2}$$ as I think is the one-dimensional p-laplacian. We also have the differential equation :
$$ (\Phi_p(x'(t)))'+\lambda \Phi_p (x(t)) = 0$$
where ' mean $\frac{d}{dt}$. I need to solve the equation for p =2 , that give me : $$\Phi_2(x'(t)))' +\lambda x = 0 $$
That I read : $$\Phi'_2(\frac{dx}{dt}) \frac{d^2x}{dt^2} = -\lambda x$$
I need to solve for $\lambda$, but honestly , I don't know where to start.
Does somoene have a hint on how to solve this equation ?
Note that $\Phi_2(x) = x$ so $(\Phi_2(x^\prime))^\prime = (x^\prime)^\prime$. Thus, you need to solve $$ x^{\prime\prime} = -\lambda x $$ Here, $\prime$ is for $\frac{d}{dt}$. This is a basic second order ODE. There are many resources to se how to solve it. For instance, Paul's online notes (problem 4) has a solution with $\lambda =16$. See here. The solution on the real line depends whether $-\lambda$ has real or complex roots.