The proof follows Tonelli's direct method in four steps.

Step 1: Boundedness below and minimizing sequence. Coercivity gives $J[y] \geq \alpha\int_a^b|y'|^r\,dx + \beta(b-a)$, so $J$ is bounded below. Let $m = \inf J$ and choose a minimizing sequence $\{y_n\}$ with $J[y_n] \to m$.

Step 2: Uniform bounds. From $J[y_n] \leq m + 1$ (for large $n$): $\alpha\|y_n'\|_{L^r}^r \leq m + 1 - \beta(b-a)$, giving uniform bounds on $\|y_n'\|_{L^r}$. Combined with the boundary condition $y_n(a) = A$, the Poincare inequality yields $\|y_n\|_{W^{1,r}} \leq C$ (uniform bound in Sobolev norm).

Step 3: Weak compactness. Since $W^{1,r}([a,b])$ is reflexive for $r > 1$ (as a closed subspace of $L^r$), the bounded sequence $\{y_n\}$ has a weakly convergent subsequence: $y_{n_k} \rightharpoonup y^*$ in $W^{1,r}$. By the Rellich-Kondrachov compactness theorem, $y_{n_k} \to y^*$ strongly in $C([a,b])$ (and in $L^r$), so $y^*(a) = A$ and $y^*(b) = B$.

Step 4: Lower semicontinuity. The key step: convexity of $L$ in $p$ implies weak lower semicontinuity of $J$. For each fixed $(x,y)$, convexity gives: $L(x,y,p) \geq L(x,y,q) + L_p(x,y,q)(p-q)$

Integrating with $q = (y^*)'$ and $p = y_n'$: $J[y_n] \geq \int_a^b L(x,y_n,(y^*)')\,dx + \int_a^b L_p(x,y_n,(y^*)')\cdot(y_n'-(y^*)')\,dx$

As $n\to\infty$: the first integral converges to $J[y^*]$ (by strong convergence of $y_n$), and the second integral converges to $0$ (by weak convergence of $y_n'$ tested against the $L^{r'}$ function $L_p(x,y^*,(y^*)')$). Therefore: $m = \lim_{n\to\infty} J[y_n] \geq J[y^*] \geq m$

giving $J[y^*] = m$. $\square$