Sorry for having been so\dots elliptic in my answer.I should have added the following details: first we choose $\varphi \geq 0$, so that the critical points $u_{k}$ are nonnegative. Then according to Gidas-Ni-Nirenberg's result $u_{k}$ has a spherical symmetry, that is $u_{k}$ is radial. Next, since $$\Vert \nabla u_{k}\Vert^2 + \Vert u_{k}\Vert^2 = \Vert u_{k}\Vert_{p}^{p} \lesssim \Vert \nabla u_{k}\Vert^{\theta p}\Vert u_{k}\Vert^{(1-\theta)p},$$(using Gagliardo-Nirenberg inequality, for some $0<\theta<1$) then one deduces that there is $R_{0} >0$ such that for all $k \geq1$ we have $\Vert u_{k}\Vert_{p}^{p} = \Vert \nabla u_{k}\Vert^2 + \Vert u_{k}\Vert^2 \geq R_{0}^2$. Using the fact that the imbedding $H^1_{\rm rad}({\Bbb R}^n) \subset L^{p}({\Bbb R}^n)$ is compact, one infers that $u_{k}\to {\overline u}$ strongly in $L^{p}$, and weakly in $H^1$. In particular $\Vert {\overline u}\Vert_{p}^{p}\geq R_{0}^2$, and thus ${\overline u}\not\equiv 0$. Using the equation satisfied by $u_{k}$ and the strong convergence in $L^{p}$, one checks easily that ${\overline u}$ is solution to (E), and hence $$\Vert \nabla {\overline u}\Vert^2 + \Vert {\overline u}\Vert^2 = \Vert {\overline u}\Vert_{p}^{p},$$yielding also that the convergence in $H^1$ is strong.
If I am not grossly mistaken, with your notations, set $c_k := \inf_{\gamma \in \Gamma_k}\max_{u \in \gamma([0,1])} J_k(u)$ (where $\Gamma_k$ is the set of paths $\gamma : [0,1] \to H^1_0(B_k(0))$ such that $\gamma(0) = 0$ and $\gamma(1) = \varphi$ for some fixed $\varphi \in H^1_0(B_1)$ such that $J(\varphi) <0$ and $\|\varphi\|{H^1{0}}$ \Vert\varphi\Vert_{H^1_{0}}$is large enough. Denote also by$c_\infty$the analogous mountain pass value for$k=\infty$Since upon extending a function by zero one can consider that$H^1_0(B_k(0)) \subset H^1_0(B_{k+1}(0))$, one has also$\Gamma_k \subset \Gamma_{k+1}$and thus$0 < c_\infty \leq c_{k+1} \leq c_k$. From this one may conclude that indeed$u_k \to {\overline u} \not\equiv 0$and that$J({\overline u})=c_{\infty}$, so that${\overline u}$is a non trivial solution to equation (E). 1 If I am not grossly mistaken, with your notations, set$c_k := \inf_{\gamma \in \Gamma_k}\max_{u \in \gamma([0,1])} J_k(u)$(where$\Gamma_k$is the set of paths$\gamma : [0,1] \to H^1_0(B_k(0))$such that$\gamma(0) = 0$and$\gamma(1) = \varphi$for some fixed$\varphi \in H^1_0(B_1)$such that$J(\varphi) <0$and$\|\varphi\|{H^1{0}}$is large enough. Denote also by$c_\infty$the analogous mountain pass value for$k=\infty$Since upon extending a function by zero one can consider that$H^1_0(B_k(0)) \subset H^1_0(B_{k+1}(0))$, one has also$\Gamma_k \subset \Gamma_{k+1}$and thus$0 < c_\infty \leq c_{k+1} \leq c_k$. From this one may conclude that indeed$u_k \to {\overline u} \not\equiv 0$and that$J({\overline u})=c_{\infty}$, so that${\overline u}\$ is a non trivial solution to equation (E).