First order conditions with exponentials in Mathematica

Question

I have tried using the 'Solve' and 'NSolve' functions in Mathematica to find the value of p that maximizes (p-1)(1+exp(u-p)+exp(v-q)) - exp(u-q) = 0, where u,v,p,q are greater than 0, to no avail. What other functions can I use?

Answer 1

Solve[(p - 1) (1 + Exp[(u - p)] + Exp[(v - q)]) == 0, p]

{{p -> 1}, {p -> q + u - Log[-E^q - E^v]}}

In the second result, Log[-E^q - E^v] with q > 0 and v > 0 cannot produce a real result because it would require taking the log of a negative number. Therefore maximum is p = 1.