Suppose we are given a sequence $\phi_k$ of traces (i.e. functions defined on boundary $\partial B_1$) such that $$ \phi_k \rightarrow 0 \;\mbox{in $L^{\infty}(\partial B_1)$} $$ (one can consider $C^{\alpha}$ convergence if required).

Now we consider affine subspaces Sobolev space $W^{1,p}(B_1)$ $$ W^{1,p}_{\phi_k}(B_1):= \Big \{ v\in W^{1,p}(B_1): \text{Trace}(v) =\phi_k \Big \} $$ Let $v_k\in W^{1,p}_{\phi_k}(B_1)$ be such that $v_k$ has least $W^{1,p}(B_1)$ norm in the space $W^{1,p}_{\phi_k}(B_1)$. That is $v_k$ is the minimizer of the following convex functional $$ J_k(v):= \int_{B_1}(|\nabla v|^p +|v|^p)\,dx,\;\;v\in W^{1,p}_{\phi_k}(B_1). $$ We can see that $v_k$ satisfies the following Euler-Lagrange equation of $J_k$ : $$ \begin{cases} -\Delta_p v_k = |v_k|^{p-2}v_k\;\;\mbox{in $B_1$}\\ v_k =\phi_k\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mbox{ on $\partial B_1$}. \end{cases} $$

Question is since the traces $\phi_k$ are tending to Zero in $L^{\infty}(\partial B_1)$, is there any result which shows that the smallest elements in $W^{1,p}_{\phi_k}(B_1)$ also tend to zero in **strong $W^{1,p}(B_1)$** topology?
We have to show that
$$
v_k \rightarrow 0 \;\;\mbox{in $W^{1,p}(B_1)$}.
$$

Isn't it the case that the smallest possible Sobolev norm for an element in $W^{1,p}_{\phi_k}(B_1)$ has to be controlled by some norm of $\phi_k$.

The above question can also be seen in terms of $\Gamma$ convergence where we try to show that $J_k \xrightarrow{\Gamma} 0$ in weak $W^{1,p}(B_1)$ topology.

Moreover, we can also see it as eigenvalue problem for $p$-Laplacian.

Thank you in advance.