2$. Regarding the diffusion parameter $\lambda$, once $\lambda_n$ depends on $(p,q)$, the question arises if the connected components of $E_\lambda(p,q)$ and $E_\lambda(p,p)$ have similar cardinality properties when $q$ is close to $p$, whether when $p\neq q$ the equilibrium components of $E_\lambda(p,q)$ that have natural correspondence with some component $E_{\lambda}^n(p,p)$ when $p=q$ are discrete or continuum according to the respective cardinality of $E_{\lambda}^n(p,p)$. The answer to this query is no, as detailed in Section 4. There is but one situation described in Case 3, Section 4, in which this fact must be addressed, but the continuity of $E_\lambda(p,q)$ is not affected. Similarly, despite the fact that sequence $\lambda_{n}^*(p,q)$ depends on $q$, the same maximum value $M$ for the amount of zeros allowed to an equilibrium in $E_{\lambda}(p,p)$ and $E_{\lambda}(p,q)$, $p0$ there exists $\widetilde{K}_{1}>0$ such that $\|u_{pq}(t)\|_2 < \widetilde{K}_1$ for $t\geq T_0$ and $(p,q) \in R$. Furthermore, given $B\subset L^2(0,1)$, $B$ bounded, there exists ${K}_{1}>0$ such that $\|u_{pq}(t)\|_2 < K_1$ for $t\geq 0$, $(p,q) \in R$ and $u_{0} \in B$. The positive constants $\widetilde{K}_1, K_1$ are independent of $(p,q) \in R$, $r>0$ and $\lambda >0$. \end{lemma} \begin{remark} \label{rmk2.2} \rm We note that the constant $\widetilde{K}_1$ gives us a $L^2(0,1)$ estimate after some time has elapsed from the origin, and it is uniform on $(p, q) \in R$, completely independent of the initial data and uniform on bounded sets with respect to the parameter $r$. The constant $K_1$, which estimates solutions since the origin, carries, as expected, a dependence on the initial data which is uniform however on bounded subsets of $L^2(0,1)$. \end{remark} To establish the estimates on $W_0^{1,p}(0,1)$ we introduce the following notation: $\varphi^1_{pq},\varphi^2_q:L^2(0,1)\to {\mathbb R}$ given by $$ \varphi^1_{pq} (u) \doteq \begin{cases} \frac{\lambda}{p}\int_0^1 |u_x(x)|^p dx +\frac{1}{q+r}\int_0^1 |u(x)|^{q+r} dx,& u \in W^{1,p}_0(0,1)\\ +\infty, &\text{otherwise}, \end{cases} $$ and $$ \varphi^2_{q} (u) \doteq\ \begin{cases} \frac{1}{q}\int_0^1 |u(x)|^{q} dx, & u \in L^q(0,1)\\ +\infty, &\text{otherwise} \end{cases} $$ It is advantageous to rewrite the equation in \eqref{eqTY} in an abstract way involving the difference of two subdifferential operators. Thus the existence of global solutions is easily obtained as a consequence of \cite{otani} and new estimates can be obtained, this time in a stronger norm. \begin{equation}\label{formaabstrata} \frac{du}{dt}(t) + \partial\varphi^1_{pq}(u(t))-\partial\varphi^2_q(u(t)) = 0 \end{equation} where $\partial\varphi^1_{pq}$ and $\partial\varphi^2_{q}$ are subdifferential of $\varphi^1_{pq}$ and $\varphi^2_{q}$ respectively. \begin{remark} \label{diferencadesubdif} \rm Given $c_0$ and $q_M$, $0 < c_0 < 1$, $q_M>2$, there exists $c > 0$ depending only on $r$ and $c_0$ such that $$ \varphi^2_{q}(u) \leq c_0\varphi^1_{pq}(u) + c $$ for each $u \in W^{1, p}_0(0,1)$, $\lambda > 0$ and $2\leq q\leq q_M$. In fact, if $\eta> 0$, \begin{align*} \varphi^2_q(u) &= \frac{1}{q}\| u \|^q_{L^q(0,1)} \\ &\leq \frac{r}{(q + r)(\eta q)^{\frac{q + r}{r}}} + q \eta^{\frac{q + r}{q}} \Big( \frac{\lambda}{p} \| u \|^p_{W^{1, p}_0(0,1)} + \frac{1}{q + r}\| u \|^{q + r}_{{L^{q+r}(0,1)}}\Big). \end{align*} Let $g(q)= (\frac{c_0}{q})^{\frac{q}{q + r}} = e^{\frac{q}{q+r}\ln(c_0/q)}$, then $g'(q) <0$ and it is enough to choose $\eta$ such that $0 < \eta < (\frac{c_0}{q_M})^{\frac{q_M}{ q_M+ r}}$ for $2\leq q\leq q_M$. Then $q\eta^{\frac{q+r}{q}} \leq c_0$ and \[ c \doteq \frac{r}{(2 + r)2^{\frac{2+r}{2}}\eta^{\frac{q_M +r}{r}}}\,. \] \end{remark} The following lemmas show the estimates we have in $W_0^{1,p}(0,1)$ norm. Note that, even if the initial data are taken into $L^2(0,1)$, since the flow is governed by a subdifferential (so it has good smoothing properties), we can ensure strong estimates in $W_0^{1,p}(0,1)$ from any positive time elapsed from the origin. \begin{lemma} \label{lem2.4} Given $\delta >0 $ there exists $\tilde{K}_2>0$ such that $\|u_{pq}(t)\|_{W_0^{p,q}(0,1)} \leq \tilde{K}_2$ for $t \geq \delta$ and for all initial data $u_0$ in $L^2(0,1)$ \end{lemma} \begin{remark} \label{rmk2.5} \rm The above lemma is a direct consequence of \cite[Lemma 2.1]{TY} and the first assertion of Lemma \ref{estunifh}. The constant $\tilde{K}_2$ carries the same dependence of $\tilde{K _1}$, that means, it is uniform on $ (p, q) \in R$, completely independent of the initial data and uniform on bounded sets with respect to the parameter $r$. \end{remark} However, if we are interested in estimates since the beginning of evolution, so we naturally find upper bounds dependent on the initial data. The demonstration is exactly the same as \cite[Lemma 2.2]{BGP1}. \begin{lemma} \label{estunifwup} Let $u_{pq}$ be a solution of \eqref{eqTY} with $u_{pq}(0)=u_{0}\in W_0^{1,p}(0,1)$. Given $M>0$ there exists a positive constant $K_{2}>0$ such that $\|u_{pq}(t)\|_{_{W_0^{1,p}(0,1)}} < K_2$ for $t\geq 0$ and $(p,q)\in R$. Furthermore, the positive constant $K_2$ can be uniformly chosen for $(p,q)\in R$, and $\|u_{0}\|_{W_0^{1,p}(0,1)} \leq M$. \end{lemma} Finally, from the above lemma we conclude our set of uniform estimates of $\{u_{pq}\}$, giving bounds to the solutions of the problem \eqref{eqTY} in $L^\infty(0,1)$. \begin{lemma}\label{normaLinfinito} Let $u_{pq}$ be a solution of \eqref{eqTY} with $u_{pq}(0)=u_{0}\in W_0^{1,p}(0,1)$ and $\|u_{0}\|_{W_0^{1,p}(0,1)} \leq M$. From Lemma \ref{estunifwup} we obtain $$ \|u_{pq}(t)\|_{\infty}\leq{K_{3}(M)},\quad t\geq 0. $$ \end{lemma} \begin{remark} \label{rmk2.8} \rm If the initial data are in $L^2(0,1)- W_0^{1,p}(0,1)$, for each $\delta>0$ we find $\tilde{K}_3$ depending on $\delta$ and $p$, with $$ \|u_{pq}(t)\|_{\infty}\leq{\tilde{K}_{3}(\delta,p)},\quad t\geq \delta. $$ \end{remark} The existence of the global attractor in $L^2(0,1)$ is a simple consequence of Lemma \ref{estunifh} and Lemma \ref{estunifwup}, as it is claimed in \cite[Corollary 2.3]{BGP1}. It is also very simple to obtain the existence of global attractor to the restriction of the semigroup to the space $W_0^{1,p}(0,1)$. In fact, for each $(p,q) \in R$, let us denote by $\{S_{pq}(t)\}$ the semigroup associated with problem \eqref{eqTY} in $W_0^{1,p}(0,1)$. We prove below that $\{S_{pq}(t)\}$ is a continuous semigroup of compact operators. The following result will be necessary. \begin{lemma}\label{estimaut} Let $B \subset W_0^{1,p}(0,1)$ be a bounded set and let $T>0$, $q_M>2$. There is a constant $K_4$ such $\| \frac{\partial}{\partial t}u_{pq}(t) \|_{L^2(0,1)} \leq K_4$ for any $p> 2$, $20$ the mapping $S_{pq}(t):W_0^{1,p}(0,1) \to W_0^{1,p}(0,1)$ is continuous and compact. \end{theorem} \begin{proof} Let $T>0$, $00$, we can choose $\theta \in A$ close to $t$ and $n$ large enough to obtain $|\varphi^1_{pq} (u_n(t)) - \varphi^1_{pq} (u(t))| \leq \eta$. Therefore we conclude that $u_n(t)\to u(t)$ in $W_0^{1,p}(0,1)$. We observe that, in fact, this proof shows that $S_{pq}(t)$ is continuous from $L^2(0,1)$ to $ W_0^{1,p}(0,1)$. To prove the second statement, let $B \subset W_0^{1,p}(0,1)$ a bounded subset. Let us prove that $S_{pq}(t)B$ is relatively compact in $W_0^{1,p}(0,1)$. As $W_0^{1,p}(0,1)$ is compactly immersed in $L^2(0,1)$, given any sequence $\{u_{0_n}\}\subset B$, there is $u_0$ such that $u_{0_n} \to u_0 \in L^2(0,1)$ and so $S_{pq}(t)u_{0_n} \to S_{pq}(t)u_{0}$ in $ W_0^{1,p}(0,1)$, which concludes the proof. \end{proof} The existence of a global attractor $\mathcal{A}_{pq}$ for $S_{pq}(t)$ in $ W_0^{1,p}(0,1)$ is a consequence of Lemma \ref{estunifwup}, Theorem \ref{continuidade}, and \cite[Theorem 2.2]{LD}. \begin{proposition} \label{prop2.11} Given $(p,q) \in R$, let $S_{pq}(t):W_0^{1,p}(0,1) \to W_0^{1,p}(0,1)$ the semigroup determined by problem \eqref{eqTY}. Then $\{S_{pq}(t)\}$ has a global attractor, which is compact and invariant. \end{proposition} \section{Continuity of flows and upper semicontinuity of the attractors} In this section we proof that, given $T>0$ and $(p_0,q_0) \in R$, the solutions $\{u_{pq}\}$ of \eqref{eqTY} go to the solution $u_{p_0q_0}$ of \eqref{eqTY} in $C([0,T];L^2(0,1))$ , when $p \to p_0$ and $q \to q_0$. After that, we will obtain the upper semicontinuity of the family of global attractors $$\{\mathcal{A}_{pq}\subset W_0^{1,p}(0,1);\ (p,q) \in R \}$$ of \eqref{eqTY} at $(p_0,q_0)$ in the topologies of $L^2(0,1)$ and $C([0,1])$. Furthermore, when $p=p_0$ we will prove the upper semicontinuity in $W_0^{1,p_0}(0,1)$. First of all we observe that from Section 2, there exists a positive constant $M$, independent of $t\geq 0$ and $(p,q) \in R$, such that $$ \|u_{pq}(t) \|_{W_0^{1,p}(0,1)} \leq M$$ for all $t\geq 0 $ and $(p,q) \in R$. Following exactly the same steps in Section 3 of \cite{BGP1} we obtain an adapted version of Baras'Theorem, \cite{V}, as we state bellow. \begin{lemma}\label{lema3.1} Given $T>0$, the set \begin{align*} M^{pq} :=\big\{& u_{pq}\subset W_0^{1,p}(0,1) : (p,q) \in R,\ u_{pq} \text{ is a solution of \eqref{eqTY} with } \\ &u_{pq}(0)= u_{0\,pq} \in W_0^{1,p}(0,1),\; u_{0\,pq}\to u_0\text{ as } (p,q) \to (p_0,q_0) \text{ in } L^2(0,1)\\ &\text{and } \|u_{0\,pq}\|_{W_0^{1,p}(0,1)}\leq M, \; \forall (p,q) \in R\big\}, \end{align*} is relatively compact in $C([0,T];L^2(0,1))$. \end{lemma} \begin{theorem} \label{teo3.1} For each $(p,q)\in R$, let $\{u_{pq}\}\subset W_0^{1,p}(0,1)$ be a solution of \begin{gather*} \frac{d}{dt} u_{pq}(t) - \lambda (|(u_{pq}(t))_x|^{p-2} (u_{pq}(t))_ x)_x = |u_{pq}(t)|^{q-2}u_{pq}(t)(1+|u(t)|_{pq}^r), \quad t>0\\ u_{pq}(0) = u_{0\,pq} \in W_0^{1,p}(0,1). \end{gather*} Suppose that $\|u_{0\,pq}\|_{W_0^{1,p}(0,1)}\leq M$ for every $(p,q) \in R$ and $u_{0\,pq}\to u_0$ as $ (p,q) \to (p_0,q_0)$ in $ L^2(0,1)$. Then, for each $T>0$, $u_{pq}\to u$ in $C([0,T];L^2(0,1))$ as $(p,q)\to (p_0,q_0)$, where $u$ is a solution of \begin{gather*} \frac{d}{dt} u(t) - \lambda (|u_x(t)|^{p_0-2} u_x(t) )_x =|u(t)|^{q_0-2}u(t)(1+|u(t)|^r),\quad t>0\\ u(0) = u_0 \in L^{2}(0,1). \end{gather*} \end{theorem} \begin{proof} Throughout this proof we denote $$ f_q( v (t))=|v(t)|^{q-2}v(t)(1+|v(t)|^r). $$ Since $$ \|u_{pq}(t) \|_{W_0^{1,p}(0,1)} \leq M $$ for all $t>0 $, $(p,q) \in R$ with $M$ independent of $t\geq 0$ and $(p,q) \in R$, we obtain that $\{u_{pq}(t)\}$ is uniformly bounded in $L^\infty (0,1)$ for $(p,q) \in R$ and $t\in [0,T]$. Furthermore, from Lemma \ref{lema3.1}, $\{u_{pq}\}$ converges in $C([0,T]; L^2(0,1))$ to a function $u: [0,T] \to L^2(0,1)$, when $p\to p_0$ and $q \to q_0$. Since $f_q(u_{pq})$ is uniformly integrable in $ L^1([0,T]; L^2(0,1))$ and \begin{align*} \|f_q( u_{pq} (t)) - f_{q_0}(v(t))\| &\leq \|f_q( u_{pq} (t)) - f_{q_0}(u_{pq}(t))\| + \|f_{q_0}( u_{pq} (t)) - f_{q_0}(u(t))\| \\ &\leq \bar{K}|q-q_0| + \tilde{K}|u_{pq}(t)-u(t)|, \end{align*} we obtain $f_q( u_{pq} (t)) \to f_{q_0}(u(t))$ in $L^2(0,1)$ for each $t>0$ when $p \to p_0$ and $q \to q_0$. Now, with the same arguments used in \cite{BGP1} we obtain that $u$ is a weak solution of \begin{align*} \frac{d}{dt} u(t) - \lambda (|u_x(t)|^{p_0-2} u_x(t) )_x =|u(t)|^{q_0-2}u(t)(1+|u(t)|^r),\quad t>0\\ u(0) = u_0 \in L^{2}(0,1). \end{align*} and we obtain the desired result. \end{proof} \begin{corollary}\label{coro3.1} The family of global attractors $\{\mathcal{A}_{pq}\subset W_0^{1,p}(0,1)): (p,q) \in R\}$ of problem \eqref{eqTY} is upper semicontinuous at $(p_0,q_0)$ in the $L^2(0,1)$ topology. \end{corollary} \begin{proof} The results in Section 2 imply that there exists a bounded set $\mathcal{B}\subset L^2(0,1)$ such that $\mathcal{A}_{pq}\subset \mathcal{B}$, for every $(p,q) \in R$. Since $\mathcal{A}_{p_0,q_0}$ attracts bounded sets of $L^2(0,1)$, for every $\delta>0$, there is $T_{1}>0$ in such way that $$ \sup_{\psi_{pq}\in \mathcal{A}_{pq},\, (p,q) \in R}\operatorname{dist}_{L^2(\Omega)} (u_{p_0,q_0}(T_{1};\psi_{pq}),\mathcal{A}_{p_0,q_0}) \leq \frac{\delta}{2}, $$ where $u_{p_0q_0}(t;\psi_{pq})$ is a solution of problem \eqref{eqTY} when $p=p_0$ and $q=q_0$ with initial condition $\psi_{pq}$. Now, the previous results in this section imply that there exist $\delta_0>0$ and $\epsilon>0$ such that $$ \|u_{pq}(t;\psi_{pq})-u_{p_0q_0}(t;\psi_{pq})\|_{L^2(0,1)} <\frac{\delta}{2}, $$ for $|p-p_0|<\delta_0$, $|q-q_0|<\epsilon$ and $T \geq t\geq T_{1}$. Thus, for $|p-p_0|<\delta_0$, we obtain \begin{align*} &\operatorname{dist}_{ L^2(0,1)}(u_{pq}(T_{1};\psi_{pq}),\mathcal{A}_{p_0q_0}) \\ & \leq \|u_{pq}(T_{1},\psi_{pq})-u_{p_0q_0}(T_{1};\psi_{pq})\|_{L^2(0,1)} + \operatorname{dist}_{ L^2(0,1)}(u_{p_0q_0}(T_{1},\psi_{pq}),\mathcal{A}_{p_0q_0}) < \delta. \end{align*} On the other hand, it follows from the invariance of the attractors that $$ \operatorname{dist}_{L^2(0,1)}(\mathcal{A}_{pq},\mathcal{A}_{p_0q_0})\leq \delta, $$ for every $|p-p_0|<\delta_0$ and $q$ such that $|q-q_0|\leq \epsilon$ showing the upper semicontinuity desired. \end{proof} \begin{remark} \label{rmk3.4} \rm It follows from Theorem \ref{estunifwup}, Corollary \ref{coro3.1}, \cite[Lemma 1.1]{piskarev} and the compact immersion of $W_0^{1,2}(0,1)$ in $C([0,1]$ that the family $\{\mathcal{A}_{pq}\}$ is upper semicontinuous at $(p_0,q_0)$ in the topology of $C([0,1])$. \end{remark} Now we are interested in obtaining the upper semicontinuity of global attractors of \eqref{eqTY} in a stronger topology. To do that, we consider $p$ fixed, and $q \to q_0$. \begin{theorem} \label{teo3.2} For each $(p_0,q)\in R$, let $\{u_{p_0q}\}\subset W_0^{1,p_0}(0,1)$ be a solution of \begin{gather*} \begin{aligned} &\frac{d}{dt} u_{p_0q}(t) - \lambda (|(u_{p_0q}(t))_x|^{p_0-2} (u_{p_0q}(t))_ x)_x \\ &=|u_{p_0q}(t)|^{q-2}u_{p_0q}(t)(1+|u(t)_{p_0q}|^r), \quad t>0 \end{aligned}\\ u_{p_0q}(0) = u_{0\,p_0q} \in W_0^{1,p_0}(0,1). \end{gather*} Suppose that $\|u_{0\,p_0q}\|_{W_0^{1,p_0}(0,1)}\leq M$ for every $(p_0,q) \in R$ and $u_{0\,p_0q}\to u_0$ in $ L^2(0,1)$ as $ q \to q_0$. Then, for each $T>0$, $u_{p_0q}\to u$ in $C([0,T];W_0^{1,p_0}(0,1))$ as $q\to q_0$, where $u$ is a solution of \begin{gather*} \frac{d}{dt} u(t) - \lambda (|(u(t))_x|^{p_0-2}(u(t))_ x)_x =|u(t)|^{q_0-2}u(t)(1+|u(t)|^r),\quad t>0\\ u(0) = u_0 \in W_0^{1,p_0}(0,1). \end{gather*} \end{theorem} The above theorem is a simple consequence of Tartar's Inequality, Lemma \ref{normaLinfinito} and Lemma \ref{estimaut}. With the same argument as in the proof of Corollary \ref{coro3.1} we can prove the next result. \begin{corollary}\label{coro3.2} The family of global attractors $\{\mathcal{A}_{pq}\subset W_0^{1,p_0}(0,1)): (p_0,q) \in R\}$ of problem \eqref{eqTY} is upper semicontinuous at $(p_0,q_0)$ in the topology of $W_0^{1,p_0}(0,1)$. \end{corollary} \section{Continuity of equilibrium sets } \label{continuidadedosequilibrios} In this section, considering $p$ fixed, we prove the continuity of the family of equilibrium points of the equation \eqref{eqTY} when $q$ goes to $p$. To analyze the continuity of the equilibrium sets it is interesting to remember how the stationary solutions are obtained in \cite{TY}. Let $\phi_{\alpha q}$ be a solution of \begin{equation}\label{auxiliarequilibriumequation} \begin{gathered} \lambda (\psi)_x + f_q(\phi_{\alpha q}) =0 ,\quad \text{in } (0,\infty)\\ \phi_{\alpha q}(0)=0, \\ \psi(0)=\alpha \end{gathered} \end{equation} where $\alpha $ is a parameter, $\psi=|(\phi_{\alpha q})_x|^{p-2}(\phi_{\alpha q})_x$ and $f_q(\phi) =|\phi|^{q-2}\phi(1-|\phi|^r)$. We observe that, in order to a solution of \eqref{auxiliarequilibriumequation} be an equilibrium point of \eqref{eqTY}, $\alpha$ must be such that $\phi_{\alpha q}(1)=0$. We denote by $X(\alpha,p,q)$ the function that measure the $x$-time that the solution $\phi_{\alpha q}$ of \eqref{auxiliarequilibriumequation} takes to reach the first maximum point. Because of the symmetry we have that $\phi(2X(\alpha,p,q))=0$ or, more generally, $\phi_{\alpha,q}(2kX(\alpha, p, q))=0$, $k=1,2,\ldots$. Also, we have that $2n X(\alpha,p,q)=1$ is a sufficient condition to $\phi_{\alpha q}$ be an equilibrium point of \eqref{eqTY} with ${n-1}$ zeros in $(0,1) \subset {\mathbb R}$. Due to the symmetry of the problem, we can only consider $\alpha >0$. The function $X$ is $$ X(\alpha,p,q) = \Big(\frac{\lambda(p-1)}{p}\Big)^{1/p} I(p,q,\tilde{\phi}_{\alpha,q}), $$ where $\tilde{\phi}_{\alpha,q}$ is the maximum value of $\phi_{\alpha,q}$ and $$ I(p,q,a)= \int_0^a (F_q(a)-F_q(\phi))^{-1/p}d\phi, $$ with $F_q(\phi)=F(\phi,q) = \int_0^\phi f_q(s)ds ={ \frac{\phi^q}{q} -\frac{\phi^{q+r}}{q+r}} \in C^1((0,\infty)\times(2,\infty))$. In \cite{TY}, the authors studied the behavior of the function $Y(p,q)$, which describes the distance between two consecutive zeros of an equilibrium and, analyzing their graphs for $p>q$, $p=q$ and $p q$ there exists a decreasing sequence $\lambda_{n}(p,q)$, $\lambda_{n}(p,q) \to 0$ when $n \to \infty$ such that the equilibrium set $E=\{0\} \cup \cup_{i=0}^{\infty} E_i^\pm$ where $E_i^\pm$ denote the equilibrium sets within the equilibria with $i$ zeros in $(0,1)$ and if $\lambda < \lambda_{n}(p,q)$, the set $E_i^\pm$ is diffeomorphic to $[0,1]^i$, for $1\leq i \leq n$. We observe that in this case there are equilibrium points with any amount of zeros in $(0,1)$. If $p\leq q$ there exist decreasing sequences $\lambda_n(p,q)\to 0$ and $\lambda^*_{n}(p,q)\to 0$ such that $\lambda^*_n(p,q) > \lambda_{n}(p,q)$. If $p=q$, for $\lambda_{M+1}\leq \lambda < \lambda_M$, the equilibrium set is given by $E=\{0\} \cup \cup_{i=0}^{M} E_i^\pm$. If $pq$ or $p=q$ and $\lambda< \lambda_0$. The equilibrium $\phi_0^+$ is asymptotically stable if $\lambda> \lambda_0^*$ and attractive for $\lambda \leq \lambda_0^*$, and if $q>p$, $\psi_0$ is unstable for $\lambda \leq \lambda_0$. Since we deal with the dependence on the parameter $q$ and there are qualitative changes in the equilibrium sets depending on the relation between $p$ and $q$, if necessary, we will exhibit explicitly the parameters $p$ and $q$. To prove the continuity of the equilibrium set, we take a sequence of equilibria in $E_i^\pm$ with a fixed number of zeros and, analyzing the initial slopes of such stationary solutions, we conclude through the continuity properties of problem \eqref{auxiliarequilibriumequation}, that this sequence must converge to an equilibrium point of the limit problem with the same amount of zeros in $ (0,1)$ or, when it is not possible, the sequence converges to the null stationary solution. We also prove that any sequence of equilibria taken in $\{\psi_i\}\subset F_i^\pm$ converges to zero. We start with the analysis of the dependence of $\tilde{\phi}_{\alpha, q}$ on $q$ and $\alpha$. We know that $\tilde{\phi}_{\alpha, q}$ is strictly increasing and $C^1$ in $\alpha$, $\alpha \in [0, \alpha_0)$ (see \cite{BGP1}). With respect to $q$, since $\tilde{\phi}_{\alpha,q}$ is the maximum value of $\phi_{\alpha,q}$, then $\tilde{\phi}_{\alpha,q}$ satisfies $$ F(\tilde{\phi}_{\alpha,q},q) = \lambda \frac{(p-1)}{p} |\alpha|^{\frac{p}{p-1}}. $$ Calculating \[ \frac{\partial}{\partial q}F(\phi,q) ={\frac{\phi^q(q\ln \phi -1)}{q^2} -\frac{\phi^{q+r}((q+r)\ln \phi -1)}{(q+r)^2}} =\beta(q) -\beta(q+r), \] where $\beta(\theta) = \frac{\phi^\theta(\theta \ln \phi -1)}{\theta^2}$, for $\theta \geq 2$. As $\beta' (\theta) > 0$, thus $\frac{\partial}{\partial q}F(\phi,q) < 0$. Also $\frac{\partial}{\partial \phi}F(\phi,q) = \phi^{q-1}-\phi^{q+r-1} > 0$, if $\phi \in (0,1)$. Using the Implicit Function Theorem, we obtain that the map $\tilde{\phi}_{\alpha q}$ is $C^1$ on $(\alpha,q)$ . Also, $$ \frac{\partial}{\partial q}\tilde{\phi}_{\alpha q} = - \frac{\frac{\partial}{\partial q} F(\tilde{\phi}_{\alpha q},q)} {\frac{\partial}{\partial \phi} F(\tilde{\phi}_{\alpha q},q)} > 0, $$ then $\tilde{\phi}_{\alpha q}$ is strictly increasing on $q$. Now we analyze the function $I(p,q,a)$. In \cite{TY}, the authors rewrite $I(p,q,a)$ as $$ I=I(p,q,a)=\int_{0}^{a} (F_q(a)-F_q(\phi))^{-1/p}d\phi= a^{1-q/p} \int_0^1\Phi_q(s,a)^{-1/p}ds, $$ where $\Phi_q(s,a)=\frac{1-s^q}{q}-\frac{1-s^{q+r}}{q+r}a^r$. Then we obtain $I(p,q,a)$ is $C^2$ on $(2,\infty)\times[2,\infty)\times(0,1]$. For each $p$ fixed, we analyze the behavior of $I(p,q,a)$ with respect the parameter $q$. We study the behavior of $I(p,q,a)$ with respect to $q$ for $a$ close to zero because $I(p,q,a)$ is $C^2$ on $(2,\infty)\times[2,\infty)\times(0,1]$ and the major difference in the cases occurs close to zero. We prove that $I(p,q, a)$ is increasing with respect to $q$ for $a$ near to zero. \begin{lemma} \label{lem4.1} For $0 \leq a0$, for $(p,q) \in (2,\infty) \times [2,\infty)$. \end{lemma} \begin{proof} In fact, since $I(p,q,a)= \int_0^a (F_q(a)-F_q(\phi))^{-1/p}d\phi$, it follows that \begin{align*} \frac{\partial}{\partial q} I(p,q,a) &= \int_0^a \frac{\partial}{\partial q}(F_q(a)-F_q(\phi))^{-1/p}d\phi\\ & = \int_0^a -\frac{1}{p} (F_q(a)-F_q(\phi))^{-1/p -1} \frac{\partial}{\partial q} (F_q(a)-F_q(\phi)) d\phi \end{align*} Since $ (F_q(a)-F_q(\phi))^{-1/p -1}>0$, we only consider \begin{align*} \frac{\partial}{\partial q} (F_q(a)-F_q(\phi)) &= \frac{a^q\ln(a)}{q} + \frac{-a^q}{q^2} - \frac{a^{q+r}\ln(a)}{(q+r)} + \frac{a^{q+r}}{(q+r)^2} \\ &\quad - \big[ \frac{\phi^q\ln(\phi)}{q} + \frac{-\phi^q}{q^2} - \frac{\phi^{q+r}\ln(\phi)}{(q+r)} + \frac{\phi^{q+r}}{(q+r)^2}\big] \end{align*} Now we define ${ \varphi(\theta) = \frac{a^\theta}{\theta^2}-\frac{\phi^\theta}{\theta^2}}$. Then $\varphi'(\theta) \leq 0$, thus $\varphi(q+r) -\varphi(q) < 0$. Define also ${ \psi(\theta)= \frac{\theta^q \ln(\theta)}{q}-\frac{\theta^{q+r} \ln (\theta)}{(q+r)}}$. Then, for $\theta < e^{-1/2}$ $$ \psi'(\theta) \leq [\theta^{q-1} -\theta^{q+r-1}] \Big(\ln \theta + \frac{1}{q+r}\Big) < 0, $$ thus $\psi(a)-\psi(\phi) < 0$ for $0< \phi< a< e^{-1/2}$. Therefore, \begin{equation} \frac{\partial}{\partial q} (F_q(a)-F_q(\phi)) = \varphi(q+r) -\varphi(q) + \psi(a)-\psi(\phi) < 0 \end{equation} Finally, we obtain \begin{equation} \frac{\partial I}{\partial q} (p,q,a) = \int_0^a -\frac{1}{p} (F_q(a)-F_q(\phi))^{-1/p -1} \frac{\partial}{\partial q} (F_q(a)-F_q(\phi)) d\phi > 0. \end{equation} \end{proof} Now we consider $p>q$. In \cite{TY}, the authors show that $\frac{\partial I}{\partial a}(p,q,a) >0$ for $q 0, \quad p > q. $$ Fixed $p$ and $n$, for each $q

0$, $\frac{\partial I}{\partial a} >0 $ for $q

0$, $\frac{\partial\tilde{\phi}_{\alpha q}}{\partial q}>0$ then we conclude that $\frac{d\alpha}{dq}<0$. We summarize the previous results in the following lemma. \begin{lemma} \label{lem4.2} If $p>q$, let $\alpha(q)$ be such that $X(\alpha(q),p,q)$ remains constant. Then $\alpha(q)$ is decreasing with respect to $q$. \end{lemma} Now we can prove the following result. \begin{theorem}\label{teo4.3} Suppose $p>2$ fixed. Let $M$ be the maximum number of zeros of an equilibrium when $q=p$. Let $\phi_n(q) \in E_n^\pm $ for $p> q$. If $n\leq M$, then $\phi_n(q)$ converges to another stationary solution, with the same amount of zeros when $q\to p^-$. If $n$ is greater than $M$, then $\|\phi_n(q)\|_{C^1(0,1)}$ goes to zero when $q \to p^-$. \end{theorem} \begin{proof} We rewrite \eqref{auxiliarequilibriumequation} in the form \begin{equation} \label{sistemasimples} \dot{z} = h(z,q), \end{equation} where $z=[\phi,\psi]$ and $h((\phi,\psi),q)=(sign(\psi)|\psi|^{1/(p-1)},-f_q(\phi)/\lambda)$. We have that the map $h$ depends continuously on $q$ and its local Lipschitz constant with respect to $z$ is independent of $q$ for $q\in (q_0,p]$, where $q_0$ is close enough to $p$. As it is done in \cite{BGP1}, if $\alpha^n_q$ is such that $X(\alpha^n_q,p,q)=\frac{1}{2n}$, there is an open set $U \subset {\mathbb R}^2$ such that $(\alpha,q) \in U$ and $\alpha$ is a $C^1$ function of $q$. Then, once we have that the solution $z_q$ of \eqref{sistemasimples} depends continuously on $q$ and on $(\phi(0),\psi(0))= (0,\alpha_{q})$, (see \cite{Hale2}), $z_q$ converges to $z_p$ when $q \to p$. If $n> M$, we obtain $\alpha^n_q \to 0$ when $q \to p^-$. In fact, since $\alpha^n_q $ is decreasing and bounded, given a sequence $q_j$, $q_j \to p^-$, and $\alpha^n_j = \alpha^n(q_j)$, there exists $\alpha^n$ such that $\alpha^n_j \to \alpha^n$. If $\alpha^n >0$, by continuity, we obtain that when $p=q$ there exists an equilibrium point of \eqref{eqTY} with $n$ zeros in $(0,1)$. Since $n >M$, it is not possible, then $\alpha^n =0$. Therefore, from the continuous dependence of initial data and parameters, we obtain $\|\phi_j(q)\|_{C^1(0,1)}$ goes to zero when $q \to p^-$. \end{proof} Regarding the case $q>p$, since $\frac{\partial I}{\partial a}(p,q,a) <0$ when $q>p$ and $a$ is close to zero it is not possible analyze the sign of $\frac{\partial X}{\partial q}$. In \cite{TY}, it was proved that for each $q$, $q>p$ there is only one $a^*(q)$ such that $a^*(q)$ is the minimum point of $I(p,q,a)$, that means, $\frac{\partial I}{\partial a}(q,a^*(q))=0$ and $\frac{\partial ^2 I}{\partial a^2}(q,a^*(q))>0$. We will prove that $a^*(q)$ goes to zero when $q$ goes to $p^+$. First of all, using the Implicit Function Theorem for $\frac{\partial I}{\partial a}(q,a)=0$, we obtain that $a^*(q)$ is a $C^1$ function. Then we have the following theorem. \begin{theorem} \label{teo4.4} Suppose $p>2$ fixed. Let $\phi_i(q) \in E_i^\pm $ for $q > p$. Then $\phi_i(q)$ converges to another stationary solution, with the same amount of zeros when $q\to p^+$. If $\psi_i(q) \in F_i^\pm $, then $\|\psi_i(q)\|_{C^1(0,1)}$ goes to zero when $q \to p^+$. \end{theorem} \begin{proof} The first part of the statement follows as in the previous theorem. Let $q_n$ be a sequence that $q_n \to p^+$ and $a^*_n=a^*(q_n)$. Since $a^*_n$ is a bounded sequence it contains a convergent subsequence $a^*_{n_k}$. Suppose that $a^*_{n_k} \to a^* >0$. Then $I(p,p,a^*) = \lim_{k\to \infty} I(p,q_{n_k}, a^*_{n_k}) $ and $\frac{\partial I}{\partial a }(p,p,a^*) = \lim_{k \to \infty} \frac{\partial I}{\partial a }(p,q_{n_k},a^*_{n_k}) =0$, that means, $a^*$ is a critical point of $I(p,p,a)$. But, in \cite{TY} the authors have proved that $I(p,p,a)$ is strictly increasing in $[0,1)$. Then, it is only possible $a^* =0$ for any sequence $a^*_n$. Thus, we conclude that $a^*(q_n)$ goes to $0$ when $q_n \to p^+$. Therefore, since that each equilibrium point $\psi_i(q)$ of \eqref{eqTY} is a solution of \eqref{auxiliarequilibriumequation} with initial date $\phi(0)=0$ and $\psi(0) = {\tilde{\alpha}}_{nq}$, where ${\tilde{\alpha}}_{nq}$ is the $\alpha$ such that ${\tilde{\alpha}}_{nq} < a^*(q)$, from the continuous dependence with respect initial data and parameter $q$, we have that $\psi_i(q_n) $ converges to zero when $q_n \to p^+$ in $C^1[0,1]$. \end{proof} Now we join some results about $I(p,q,a)$ for $q>p$ in the following lemma. \begin{lemma} \label{I_min_q>p} If $q>p$, then \begin{itemize} \item[(i)] $a^*(q) \to 0$ when $q \to p^+$, \item[(ii)] $\tilde{I}(q) = I(p,q, a^*(q))$ is increasing with respect to $q$, \item[(iii)] $\tilde{I}(q) \to I_0=I(p,p,0)$, when $q \to p^+$. \end{itemize} \end{lemma} \begin{proof} Item (i) follows from the prior discussion. (ii) $\tilde{I}(q) = I(p,q, a^*(q))$, with $p$ fixed. We obtain \[ \frac{d\tilde{I}}{dq}(q) = \frac{\partial I}{\partial a }(q,a^*(q)) \frac{d a^*}{dq}(q) + \frac{\partial I}{\partial q }(q,a^*(q)) = \frac{\partial I}{\partial q }(q,a^*(q))>0, \] which means that the minimum value of $I$ is increasing with $q$. (iii) It follows by using (ii) and the continuity of $I(p,p,a)$ in $a=0$ and $I(p,q,a)$ for $a>0$. \end{proof} Since the sequences $\lambda_n(p,q)$ and $\lambda_n^*(p,q)$ depends on $(p,q)$, even if $\lambda $ is fixed it is possible to occur changes in the relation between $\lambda$ and $\lambda^*_n(p,q)$ and $\lambda_n(p,q)$ when $q \to p$. Then, before proving the continuity of equilibrium sets $E(p,q)$ in $q=p$ we analyze that the possibilities among $\lambda$, $\lambda_n(p,q)$ and $\lambda_n^*(p,q)$. Let $\{\lambda_n\}$, $\{\lambda_n^*\}$, $\{\lambda_{n}(p,q)\}$ and $\{\lambda^*_{n}(p,q)\}$ be defined as follows: \begin{gather*} \lambda^*_n \doteq \frac{p}{p-1}(2(n+1)I_0)^{-p},\\ \lambda^*_{n}(p,q) \doteq \frac{p}{p-1}(2(n+1)I^*(q))^{-p}, q>p, \end{gather*} where $I^*(q)= I(p,q,a^*(q))$ denotes the minimum value of $I(p,q,a)$ with relation to $a$, $I_0=lim_{a\to 0^+} I(p,p,a)$, and \begin{gather*} \lambda_n \doteq \lambda_n(p,p)= \frac{p}{p-1}(2(n+1)I(p,p,1))^{-p}; \lambda_n(p,q)= \frac{p}{p-1}(2(n+1)I(p,q,1))^{-p}. \end{gather*} Here $\{\lambda^*_n\}$, $\{\lambda^*_{n}(p,q)\}$ are the sequence that determine the number of zeros allowed to a stationary solution of \eqref{eqTY} when $p=q$ and $p

j$. \end{enumerate} We have the following: \smallskip \noindent\textbf{Case 1.} Let $j_0$ be the least index such that $\lambda > \lambda_{j_0}(p,p)$. Since $I(p,q,1)$ behaves continuously on $q$, if $q$ is close enough to $p$, than $\lambda > \lambda_{j_0}(p,q)$. By Lemma \ref{I_min_q>p}, we also have that $I^*(q) = \min I(p,q,a)$ is increasing with $q$ if $q>p$ and $I^*(q) \to I_0$ when $q \downarrow p$. So, if $\lambda > \lambda^*_{i_0}$ for some given ${i_0}$, then $\lambda > \lambda^*_{i_0}(p,q)$, if $q$ is close enough to $p$. Therefore, if $ p < q $, $q$ can be chosen in a neighborhood of $p$ in such way that the maximum number of zeros of any equilibrium in $E_{\lambda}(p,q)$ is $M$ and, in both case $p>q$ or $pp$. By Lemma \ref{I_min_q>p}, $I^*(q)$ is increasing with $q$ if $q>p$ and $I^*(q) \to I_0$ when $q \downarrow p$, then $\lambda^*_j(p,q)< \lambda^*_j$. Thus, $\lambda = \lambda^*_j$ implies $\lambda > \lambda^*_j(p,q)$. This allows us to conclude that if there exist stationary solutions in $E_\lambda(p,q)$ having $n$ zeros in $(0,1)$, then there is also solutions in $E_\lambda(p,p)$ having $n$ zeros in $(0,1)$. In other words, once $\lambda = \lambda_j^*$ there is no solution with $j$ zeros in $(0,1)$ and, as $\lambda_j^*(p,q) < \lambda_j^* = \lambda$ there is no solution with $j$ zeros in $(0,1)$ for $q>p$. Finally, if $\lambda = \lambda_j^*$ then $\lambda < \lambda_k^*$, for $0\leq k\leq j-1$ the analysis follows the Case 1, for solutions with $k$ zeros in $(0,1)$. \smallskip \noindent\textbf{Case 3.} Once $\lambda_i(p,q)= \frac{p}{p-1}(2(i+1)I(p,q,1))^{-p}$, using the continuity of $I(p,q,1)$ we obtain $\lambda_i(p,q) \to \lambda_i$ when $q \to p$. If $I(p,q,1) < I(p,p,1)$ there exists a continuum of solutions with $i$ zeros for $(p,q)$. In despite of this, we know that, if $X_j(q)$ is the ``$x$-time'' that an equilibrium $\phi_j(q) \in E_\lambda^j(p,q)$ needs to reach its first maximum, then $X_j(q) \to \frac{1}{2(j-1)}$ as $q \to p$. So we obtain that all sequence of stationary solutions in the continuum sets $E_\lambda^j(p,q)$ converges to the same equilibrium in $E_\lambda^j(p,p)$, when $q \to p$. If $I(p,q,1) > I(p,p,1)$ the solutions with $i$ zeros do not reach the maximum value equal 1 for $(p,q)$. In this case, by the continuity of $I$ the maximum value goes to 1. \smallskip \noindent\textbf{Case 4.} This case follows from Cases 2 and 3. \begin{remark} \label{rmk4.6} \rm Regarding the equilibria $\pm\psi_n$ that appear when $q>p$ it is known that with respect to parameter $\lambda$ they arise as spontaneous bifurcations, \cite{TY}, but our analysis shows that with respect to $q$, $\pm\psi_n$ bifurcate from trivial solution. \end{remark} Now we are ready to state our main result concerning to the continuity on $q$ of the equilibrium sets $E(p,q)$. The upper semicontinuity in $L^2 (0,1)$ and $W_0^{1,p}(0,1)$ follows easily from Theorem \ref{teo3.1} and Corollaries \ref{coro3.1} and \ref{coro3.2}. From the prior analysis presented in this section we can conclude the upper and lower semicontinuity in $C^1[0,1]$. \begin{theorem} \label{teo4.7} The family $E(p,q)$ is upper and lower semicontinuous on $q$ as $q$ goes to $p$ in $C^1[0,1]$. \end{theorem} \begin{proof} If $q\downarrow p$, given any sequence $\{\varphi_q\}$, $\varphi_q \in E(p,q)$ for each $q$, there is a subsequence of $\{\varphi_q\}$ containing only equilibria with the same amount of zeros in $(0,1)$. Then we know from Theorem \ref{teo4.4} that this subsequence converges to an equilibrium in $E(p,p)$. If $q\uparrow p$, given a sequence $\{\phi_q\}$, $\phi_q \in E(p,q)$ for each $q$, which contains a subsequence with the same amount of zeros, then we know from Theorem \ref{teo4.3} that this subsequence converges to an equilibrium in $E(p,p)$. But in this case it is also possible to find a sequence $\phi_q \in E(p,q)$ in such way that the number of zeros of $\phi_q$ goes to infinity with $q$. In this case, we observe that this sequence goes to the null solution. So we conclude from \cite[Lemma 1.1]{piskarev} that $E(p,q)$ is upper semicontinuous at $q=p$. To prove the lower semicontinuity, let $\phi_p \in E(p,p)$. We have three possible situations. If the maximum value of $\phi_p$ is less than 1 and $n$ is the amount of zeros of $\phi_p$ in $(0,1)$, the sequence $\phi_q \in E(p,q)$ containing only equilibria with $n$ zeros converges to $\phi_p$ according with Theorems \ref{teo4.3} and \ref{teo4.4}. If $\phi_p$ achieves 1 but does not have flat cores we can repeat the prior argument (observe that it is possible only if $\lambda =\lambda_n$ and this situation was discussed in the Case 3). When $\phi_p$ presents flat cores, then $\lambda < \lambda_n$ and, from the continuity of $\lambda_n(p,q)$ on $q$, we conclude that equilibria with $n$ zeros in $E(p,q)$ present flat cores as well (we have used an analogous argument in Case 1). In this case, we construct the approaching sequence. Let $f_i$ be the length of the $i$-th flat core, for $i=1, \ldots, n+1$. For $q$ close to $p$, let $X(p,q)$ the $x$-time spent to an equilibrium in $E_n(p,q)$ achieve the maximum value equals to 1. If $X(p,q) > X(p,p)$ we pick in $E(p,q)$ an equilibrium $\phi_q$ with $n$ zeros in $(0,1)$ such that the length of $i$-th flat core is $f_i -2(X(p,q)-X(p,p))$. If $X(p,q) < X(p,p)$ we choose an equilibrium $\phi_q$ with $n$ zeros in $(0,1)$ such that the length of $i$-th flat core is $f_i +2(X(p,p)-X(p,q))$. In any case $\phi_q \to \phi_p$ as $q \to p$. The lower semicontinuity follows from \cite[Lemma 1.1]{piskarev}. \end{proof} \subsection*{Acknowledgments} S. M. 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