I am struggling to prove that my EL equation gives a local maximum. (if possible, I want to show it is a global maximum as well).
It satisfies the Legendre necessary condition for maximum, but I cannot show for sufficient conditions. I checked Gelfand and Fomin, sufficient conditions for a strong extremum. However, "no conjugate point" condition seems not holding and even my objective function has no terminal point constraint. u is a strictly concave function here and initial condition is just enough small scalar.
$$\text{fun}:=\int_0^\infty e^{-rt}\left[u(c'(t))+w-p\,c'(t)\right]e^{-b\,c(t)}\,dt. $$
$$\text{EL}:= u'(c'(t))-p=\frac{b\left[u(c'(t))+w-p\,c'(t)\right]+u''(c'(t))\,c''(t)}{r+b\,c'(t)}$$
This isn't a full answer, but it's too long for a comment.
In terms of the Lagrangian $L:=e^{-(r+bc)}(u(c^\prime)+w-pc^\prime)$, so$$\partial_c L=-bL,\,\partial_{c^\prime}L=e^{-(r+bc)}(\partial_{c^\prime}u-p),$$the EL equation is
$$\frac{d}{dt}\partial_{c^\prime}L=e^{-(r+bc)}(c^{\prime\prime}\partial_{c^\prime}^2u-bc^\prime(\partial_{c^\prime}u-p))$$
$$0=\partial_cL-\frac{d}{dt}\partial_{c^\prime}L=e^{-(r+bc)}\left[-b(u+w-pc^\prime)-c^{\prime\prime}\partial_{c^\prime}^2u+bc^\prime(\partial_{c^\prime}u-p)\right].$$I realise you managed that part, but don't be tempted to rearrange this into a different equation just yet. You can't classify stationary points without second derivatives, obtained from differentiating the original first derivative rather than something proportional to it. I think that was why you were struggling.
When we derive an EL equation from an action of the form $S:=\int_a^bL(c,\,c^\prime)dt$ as $0=\frac{\delta S}{\delta c}(c,\,c^\prime,\,c^{\prime\prime})=\partial_cL-\frac{d}{dt}\partial_{c^\prime}L$, we need to check whether the second functional derivative is negative for a putative maximum. In other words, we need to check$$0>\left(\frac{\partial}{\partial c}-\frac{d}{dt}\frac{\partial}{\partial c^\prime}+\frac{d^2}{dt^2}\frac{\partial}{\partial c^{\prime\prime}}\right)\times\\\left[e^{-(r+bc)}\left[-b(u+w-pc^\prime)-c^{\prime\prime}\partial_{c^\prime}^2u+bc^\prime(\partial_{c^\prime}u-p)\right]\right].$$I'll leave evaluating the right-hand side to you; you can then see whether the result follows from your parameter constraints.