\documentclass{amsart} \begin{document} {\noindent\small {\em Electronic Journal of Differential Equations}, Vol.~2000(2000), No.~33, pp.~1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu \quad ejde.math.unt.edu (login: ftp)} \thanks{\copyright 2000 Southwest Texas State University and University of North Texas.} \vspace{1cm} \title[\hfilneg EJDE--2000/33\hfil Existence of solutions ] { Existence of solutions for a sublinear \\ system of elliptic equations } \author[Carlos Cid \& Cecilia Yarur \hfil EJDE--2000/33\hfilneg] { Carlos Cid \& Cecilia Yarur } \address{Carlos Cid \hfill\break Departamento de Ingenier\'{\i}a Matem\'atica, Universidad de Chile \hfill\break Casilla 170/3, Correo 3, Santiago, Chile} \email{ccid@dim.uchile.cl} \address{Cecilia Yarur \hfill\break Departamento de Matem\'aticas, Universidad de Santiago de Chile \hfill\break Casilla 307, Correo 2, Santiago, Chile} \email{cyarur@fermat.usach.cl} \date{} \thanks{Submitted January 21, 2000. Published May 9, 2000.} \thanks{The second author was partially supported by FONDECYT grant 1990877, \hfill\break\indent FONDAP de Matem\'aticas Aplicadas, and by DICYT} \subjclass{ 35A20, 35J60 } \keywords{Semilinear elliptic systems, sub-harmonic functions, \hfill\break\indent super-harmonic functions } \begin{abstract} We study the existence of non-trivial non-negative solutions for the system $$ \displaylines{ -\Delta u = |x|^av^p \cr \Delta v = |x|^bu^q\,, }$$ where $p$ and $q$ are positive constants with $pq<1$, and the domain is the unit ball of ${\mathbb R}^N$ ($N>2$) except for the center zero. We look for pairs of functions that satisfy the above system and Dirichlet boundary conditions set to zero. Our results also apply to some super-linear systems. \end{abstract} \maketitle \newtheorem{thm}{Theorem}[section] \newtheorem{cor}[thm]{Corollary} \theoremstyle{remark} \newtheorem{rem}{Remark}[section] \numberwithin{equation}{section} \section{Introduction} The purpose of this paper is to study the existence of non-trivial non-negative solutions to the Dirichlet problem \begin{equation}\label{poten} \begin{gathered} -\Delta u = |x|^av^p \quad\text{in $B'$},\\ \Delta v = |x|^bu^q \quad\text{in $B'$},\\ u=v= 0\quad\text{on $\partial B$}\,, \end{gathered} \end{equation} where $p>0$, $q>0$, $pq<1$, $B$ is the unit ball of ${\mathbb R}^N$ ($N>2$) centered at $0$, and $B'=B\setminus\{0\}$. By a non-negative solution of \eqref{poten} we mean a pair of functions $u$, $v$ in $ C^2(B')$ such that $u\ge 0$, $v\ge 0$, and $(u, v)$ satisfies \eqref{poten}. Note that $u$ is super-harmonic whereas $v$ is sub-harmonic in $B'$. In \cite{cy}, we proved the existence of solutions for \eqref{poten} in the super-linear case, $pq>1$. Bidaut-Veron and Grillot \cite{bg} studied the behavior of solutions near zero and the non-existence of non-negative solutions without boundary conditions. A non-negative non-trivial solution $(u, v)$ is said to be {\em singular} at zero (or just singular) if $$\limsup_{x\to 0} (u(x)+v(x))=+\infty\,. $$ Note that since $v$ is sub-harmonic it must be singular at zero, and thus any non-trivial non-negative solution to \eqref{poten} is singular at zero. Let $$ L:=\limsup_{x\to 0} |x|^{N-2}(u(x)+v(x)). $$ If $0 0 \quad \text{in $B'$},\\ u= 0 \quad\text{on $\partial B$}, \end{gathered} \end{equation} solutions that are singular and non-negative exist if $$q <\frac{N+2}{N-2}. $$ In such a case, the solution $u$ with a singularity at zero satisfies $$ 0\le \limsup_{x \to 0} |x|^{N-2}u(x)< +\infty, $$ and thus the singularity is of fundamental type. See Lions \cite{li}, Ni and Sacks \cite{nsa}, Lin \cite{lin} and the references therein. Br\'ezis and Veron \cite{bv} showed that for $q\ge N/(N-2)$ solutions of \begin{equation} \label{bv} \Delta u= |u|^{q-1}u \quad \text{in $B'$} \end{equation} are bounded near zero. For $q < N/(N-2)$, Veron \cite{v} proved the existence of non-negative singular solutions of \eqref{bv} with either a strong or a fundamental singularity at zero. Next, we state our main result for Problem \eqref{poten}. \begin{thm} \label{noexis} Let $p>0$, $ q>0$ and $pq<1$. Then there exists a non-trivial non-negative solution to \eqref{poten} if and only if \begin{equation}\label{gamma2} p< \frac{N+a}{N-2} \ \text{and \ $ N+a + \beta p > 0$}, \end{equation} where \begin{equation}\label{defbeta} \beta:= b+2-(N-2)q\,. \end{equation} Moreover, if $(a, b, p, q)$ satisfies \eqref{gamma2}, then for any $c > 0$, there exists a non-negative solution $(u, v)$ such that $$\lim_{x\to 0} |x|^{N-2}u(x) = c\,. $$ If in addition $$ q \ge \frac{N+b}{N-2}\,,$$ the above solution has a singularity of strong type at zero. \end{thm} In Section 2, we shall prove the existence of singular non-negative solutions for a system more general than \eqref{poten}; see Theorems~\ref{exis4} and \ref{weak} below. As for \eqref{bv}, under additional assumptions for \eqref{poten}, we find both fundamental and strong types of singularities. In Section 3, we prove Theorem \ref{noexis}, and give some applications of our result for bi-harmonic equations. \section{Existence results for general systems} In this section, we prove the existence of singular non-negative radially symmetric solutions to \begin{equation} \label{potengeneralradial} \begin{gathered} -(r^{N-1}u'(r))' = r^{N-1}f(r,v(r)) \quad \text{ in $(0,1)$,}\\ (r^{N-1} v'(r))' = r^{N-1}g(r,u(r)) \quad \text{ in $(0,1)$,}\\ u(1)=v(1)= 0, \end{gathered} \end{equation} without sub-linear type conditions. In particular the results in this section apply to \eqref{poten} with $pq\not=1$. When $pq<1$, our results are optimal as stated in Theorem \ref{noexis}. When $pq >1$, our results extend some results in \cite{cy} to the inequality case. Throughout this section we will assume that $f$ and $g$ are non-negative continuous functions from $(0, 1)\times{\mathbb R}^+$ to $\mathbb R$ and satisfying \begin{equation} \label{hipotesisradial} 0 \le f(r, s) \le f_1(r, s), \quad 0\le g(r, s) \le g_1(r, s), \end{equation} where $f_1$ and $g_1$ are continuous functions that are non-decreasing as functions of $s$. Set $u_0(r):=r^{2-N}-1$, and fixed positive values $\alpha$ and $d$, define \begin{equation} \label{defv1} v_{\alpha}(r):= du_0(r)+ \int_r^1 s^{1-N}\int_s^1 t^{N-1}g_1(t, \alpha t^{2-N}) \,dt\, ds\,. \end{equation} To state the main result of this section, we assume that $$\displaylines{ \rlap{(H1)}\hfill \Lambda_{\alpha}:= (N-2)^{-1}\int_0^1 t^{N-1}f_1(t, v_{\alpha}(t)) dt <\infty\,.\hfill }$$ \begin{thm} \label{exis4} Assume that $f$ and $g$ are two non-negative continuous functions satisfying \eqref{hipotesisradial}. Assume that there exists $\alpha>0$ such that (H1) is satisfied and $\Lambda_{\alpha} < \alpha$. Then there exist infinitely many positive solutions to \eqref{potengeneralradial}. Moreover, for any $c \in [0,\alpha- \Lambda_{\alpha})$ there exists a solution $(u,v)$ such that \begin{equation}\label{compotamientodesolucionu} \lim_{r\to 0^+}r^{N-2}u(r)=c\,. \end{equation} \end{thm} \paragraph{\bf Proof} Let $c$ be such that $0\leq c < \alpha-\Lambda_{\alpha}$. Consider the the system of integrals \begin{equation}\label{integral2} \begin{aligned} u(r)&= cu_0(r)+\int_r^1s^{1-N}\int_0^s t^{N-1}f(t,v(t))\,dt\,ds\,,\\ v(r)&= du_0(r)+\int_r^1s^{1-N}\int_s^1 t^{N-1}g(t,u(t))\,dt\,ds\,.\\ \end{aligned} \end{equation} Define the operator $T=(T_1, T_2)$, where \begin{equation}\label{operador2} \begin{aligned} T_1(u, v)(r)&=cu_0(r)+\int_r^1s^{1-N}\int_0^s t^{N-1}f(t,v(t))\,dt\,ds\,,\\ T_2(u, v)(r)&= du_0(r)+ \int_r^1s^{1-N}\int_s^1 t^{N-1}g(t,u(t))\,dt\,ds\,. \end{aligned} \end{equation} Then a non-negative fixed point $(u,v)$ of the operator $T$ is is a non-negative solution to \eqref{potengeneralradial}. To apply the Schauder fixed point Theorem to $T$, we do the following three steps. First construct an invariant set ${\mathcal M}$ under $T$. Second transform the set ${\mathcal M}$ into a set ${\mathcal A}$, and thus the operator $T$ into an operator $W$. Third prove the continuity and compactness of $W$ on ${\mathcal A}$. \bigskip \noindent{\bf Step 1.} Let ${\mathcal M}$ be a subset of $(C(0,1])^2$ defined by \begin{equation}\label{cotas4} {\mathcal M}:=\{(u,v): 0\le u(r) \le \alpha r^{2-N}, \quad 0\le v(r) \le v_{\alpha}(r)\}. \end{equation} Next, we show that $T({\mathcal M})\subset {\mathcal M}$. Let $(u, v) \in {\mathcal M}$, and thus $v(r) \le v_{\alpha}(r)$. Therefore, from the definition of $T_1$ and \eqref{hipotesisradial} we have $$ \begin{aligned} T_1(u,v)(r)\leq& c u_0(r)+\int_r^1s^{1-N}\int_0^st^{N-1}f_1(t,v_{\alpha}(t))dtds\\ \leq& cu_0(r)+ (N-2)\Lambda_{\alpha}\int_r^1 s^{1-N}ds\\ \leq & \alpha r^{2-N},\\ \end{aligned} $$ where we used the choice of $c$. Now, we show that $T_2(u,v)(r)\leq v_{\alpha}(r)$. Since $(u, v)\in {\mathcal M}$, and from the definition of $v_{\alpha}$ given by \eqref{defv1} $$ T_2(u,v)(r)\leq du_0(r)+ \int_r^1s^{1-N}\int_s^1t^{N-1}g_1(t,\alpha t^{2-N})\,dt\,ds =v_{\alpha}(r). $$ \bigskip \noindent{\bf Step 2.} Let $\varepsilon > 0$, and let $\vartheta\in C^1((0, 1))\cap C([0, 1])$ be a non-negative function such that $$\vartheta(r):= \begin{cases} 0 & \text{if $r=0$},\\ v^{-1-\varepsilon}_{\alpha}(r) & \text{if $r\in (0, 1/2)$}, \\ 1 & \text{if $r\in(3/4, 1]$}. \end{cases}$$ Since $v_{\alpha}(r) \ge d r^{2-N}$ near zero, the continuity of $\vartheta$ at zero follows. Let ${\mathcal A}$ be the subset of $(C[0,1])^2$ defined by $$ {\mathcal A}= \left\{ (y,z): 0\le y(r) \le \alpha r^{\varepsilon}, \ 0\le z(r) \le \vartheta(r)v_{\alpha}(r) \right\}. $$ Define in ${\mathcal A}$ the operator \begin{equation} \label{definicionoperadorTtilde122A} W(y,z)(r)=(W_1(y,z)(r), W_2(y,z)(r)), \end{equation} where \begin{equation} \label{definicionoperadorTtilde122B} \begin{aligned} W_1(y,z)(r)&= r^{N-2+\varepsilon}T_1\left(r^{2-N-\varepsilon}y(r),\vartheta^{-1}(r)z(r) \right),\\ W_2(y, z)(r)&=\vartheta(r)T_2\left(r^{2-N-\varepsilon}y(r),\vartheta^{-1}(r)z(r) \right), \end{aligned} \end{equation} and thus \begin{equation}\label{definicionoperadorTtilde122} \begin{aligned} W_1(y,z)(r)&= r^{N-2+\varepsilon}\Bigl(cu_0(r)+\int_r^1s^{1-N}\int_0^s t^{N-1}f\left(t,\vartheta^{-1}(t)z(t)\right)\,dt\,ds\Bigr),\\ W_2(y, z)(r)&= \vartheta(r)\Bigl(d u_0(r)+ \int_r^1s^{1-N}\int_s^1 t^{N-1}g\left(t,t^{2-N-\varepsilon}y(t)\right)\,dt\,ds\Bigr). \end{aligned} \end{equation} By \eqref{definicionoperadorTtilde122A} and \eqref{definicionoperadorTtilde122B} we have that $(y, z)$ is a fixed point of $W$ if and only if $( u, v)=(r^{2-N-\varepsilon}y,\vartheta^{-1}z)$ is a fixed point of $ T$. Moreover, from Step 1 we have that $W({\mathcal A})\subset {\mathcal A}$. Furthermore, ${\mathcal A}$ is a closed convex bounded subset of $(C[0,1])^2$. In order to show existence of a fixed point, via Schauder fixed point theorem, to $W$ in ${\mathcal A}$ it is enough to prove that $W$ is a continuous and compact operator, which is done in the next step. \bigskip \noindent{\bf Step 3.} First, we show that $W({\mathcal A})$ is a relatively compact subset of $(C[0,1])^2$. Since $W({\mathcal A})$ is bounded, by Ascoli-Arzela theorem, it is enough to prove that $W({\mathcal A})$ is an equicontinuos subset of $(C[0,1])^2$. This can be done by proving the existence of two functions $\psi,\varphi\in L^1(0,1)$ and a positive constant $C$ such that for any $r\in [0,1]$, \begin{equation}\label{condicionsobrederivadadeW1} \left|\frac{d}{dr}{W}_1(y,z)(r)\right|\leq C\psi(r) \end{equation} and \begin{equation}\label{condicionsobrederivadadeW2} \left|\frac{d}{dr}{W}_2(y,z)(r)\right|\leq C\varphi(r)\,. \end{equation} From \eqref{definicionoperadorTtilde122} and with \ $' = d/dr $ we have \begin{align*} {W'}_1(y,z)(r)=& (N-2+\varepsilon)r^{-1}W_1(y, z)(r) \\ &-c(N-2) r^{\varepsilon-1}- r^{\varepsilon-1}\int_0^r t^{N-1}f\left(t,\vartheta^{-1}(t)z(t)\right)\,dt\,. \end{align*} Thus, using invariance property of $W$ in ${\mathcal A}$ and the definition of $\Lambda_{\alpha}$ we obtain $$ \left|\frac{d}{dr}{W}_1(y,z)(r)\right|\leq \Bigl((N-2)(\alpha +c + \Lambda_{\alpha}) +\varepsilon \alpha \Bigr) r^{\varepsilon-1}. $$ Hence, $W_1$ satisfies \eqref{condicionsobrederivadadeW1} with $\psi(r)=r^{\varepsilon-1}$. Similarly, by \eqref{definicionoperadorTtilde122} we obtain \begin{align*} W'_2(y,z)(r)=&\frac{\vartheta'(r)}{\vartheta(r)} W_2(y,z)(r) -d(N-2)r^{1-N}\vartheta(r) \\ &-\vartheta(r)r^{1-N}\int_r^1 t^{N-1}g(t,t^{2-N-\varepsilon}y(t))dt\,. \end{align*} Using again the invariance property of $W$ in ${\mathcal A}$ we obtain $$ |W'_2(y,z)(r)| \leq |\vartheta'(r)|v_{\alpha}(r) + \vartheta(r)r^{1-N}\bigl( (N-2)d + \int_r^1 t^{N-1}g_1(t,\alpha t^{2-N})dt\bigr), $$ and by definition \eqref{defv1} of $v_{\alpha}$ we have $$ |W'_2(y,z)(r)|\leq |\vartheta'(r)|v_{\alpha}(r)+ \vartheta(r)|v'_{\alpha}(r)|=\varphi(r). $$ The function $\varphi \in L^1(0,1)$, since it is bounded for $r>1/2$ and for $r$ near zero $$\varphi(r)= -(2+\varepsilon)v'_{\alpha}(r)v^{-1-\varepsilon}_{\alpha}(r).$$ Finally, we prove the continuity of $W$ in ${\mathcal A}$. Let $(y_n,w_n)$ be any sequence converging on ${\mathcal A}$ to $(y,w)$ and let $r\in[0,1]$ be fixed. From the definition of $W$ given by \eqref{definicionoperadorTtilde122} and the continuity of $u\mapsto f(t,u)$, $u\mapsto g(t,u)$, uniform convergence of $(y_n,w_n)$ to $(y,w)$ and the Lebesgue dominated convergence theorem we easily deduce that \begin{equation}\label{convergenciapuntualdeW} \lim_{n\to\infty}W(y_n,w_n)(r)=W(y,w)(r) \end{equation} for all $r\in[0,1]$. Moreover, since ${\mathcal A}$ is closed and $W({\mathcal A})$ is equicontinuous, then $\{W(y_n,w_n):n\geq 1\}\cup\{W(y,w)\}$ is an equicontinuous family. Thus from pointwise convergence \eqref{convergenciapuntualdeW} we obtain the uniform convergence, that is, $W(y_n,w_n)$ converges to $W(y,w)$ uniformly. Therefore $W$ is a continuous operator. Thus by Schauder fixed point theorem, there exists $(u,v)\in {\mathcal M}$ satisfying $$ \begin{aligned} u(r)&= cu_0(r)+\int_r^1s^{1-N}\int_0^s t^{N-1}f(t,v(t))dt ds,\\ v(r)&= du_0(r)+\int_r^1s^{1-N}\int_s^1 t^{N-1}g(t,u(t))\,dt\,ds\,.\\ \end{aligned} $$ Hence there exists a positive solution to \eqref{potengeneralradial}. The behavior of $u$ at zero is a consequence of L'H\^opital rule. $$ \begin{aligned} \lim_{r\to 0^+}r^{N-2}u(r)&= c +\lim_{r\to 0^+}\frac{ \int_r^1s^{1-N}\int_0^s t^{N-1}f(t,v(t))\,dt\,ds}{r^{2-N}},\\ &=c+\lim_{r\to 0^+}\frac{ r^{1-N}\int_0^r t^{N-1}f(t,v(t))dt } {(N-2)r^{1-N}},\\ &=c+\frac{1}{N-2}\lim_{r\to 0^+}\int_0^r t^{N-1}f(t,v(t))dt, \\ &=c. \end{aligned}$$ \quad\hfill$\square$ \smallskip As a consequence of the construction of non-negative solutions given in the above theorem, we have the following result about existence of positive solutions with a strong singularity. \begin{cor} \label{existenciadesingfuerte} Assume that the hypotheses in Theorem \ref{exis4} hold and $g(r, s)$ is non decreasing in $s$. Then, \begin{enumerate} \item[(i)] If \begin{equation} \int_0^1 t^{N-1}g(t, \alpha t^{2-N})dt=+\infty \quad \text{for any $\alpha >0,$} \label{condicionsobrearadial} \end{equation} there exists a non-negative solution $(u, v)$ to \eqref{potengeneralradial} with a strong singularity. \item[(ii)] If $$\int_0^1 t^{N-1}g(t, \alpha t^{2-N})dt <+\infty \quad \text{for any $\alpha >0$, }$$ any non-negative non-trivial radially symmetric solution has fundamental singularity.\end{enumerate} \end{cor} \paragraph{\bf Proof} Assume first that \eqref{condicionsobrearadial} is satisfied. Let $(u,v)$ be a solution to \eqref{potengeneralradial} constructed in Theorem \ref{exis4} with $c>0$. Thus, $$ v(r)= du_0(r)+ \int_r^1s^{1-N}\int_s^1 t^{N-1}g(t,u(t))\,dt\,ds\,. $$ By a generalize version of L'H\^opital rule (see Proposition 7.1 in \cite{GaMaMiYa}) we have $$ \begin{aligned} \liminf_{r\to 0^+}r^{N-2}v(r)& \ge \liminf_{r\to 0^+}\frac{ \int_r^1s^{1-N}\int_s^1 t^{N-1}g(t,u(t))\,dt\,ds}{r^{2-N}},\\ & \ge \liminf_{r\to 0^+}\frac{ r^{1-N}\int_r^1 t^{N-1}g(t,u(t))dt }{(N-2)r^{1-N}},\\ &=\frac{1}{N-2}\lim_{r\to 0^+}\int_r^1 t^{N-1}g(t,u(t))dt=+\infty, \\ \end{aligned} $$ where the last equality holds by \eqref{condicionsobrearadial} and since $\lim_{r \to 0^+}r^{N-2}u(r) = c$. Assume now, that $\int_0^1 t^{N-1}g(t, \alpha t^{2-N})dt <+\infty$, and let $(u, v)$ be a non-negative solution to \eqref{potengeneralradial}. Since $-r^{N-1}u'(r)$ is non decreasing, we easily obtain that $u(r) \le \alpha r^{2-N}$, where $\alpha= -u'(1)/(N-2)$. Moreover, from the second in \eqref{potengeneralradial} $v$ satisfies $$v(r)= du_0(r)+ \int_r^1 s^{1-N} \int_s^1 t^{N-1} g(t, u(t))\,dt\,ds\,$$ and thus $$v(r)\le \Bigl(d + (N-2)^{-1}\int_0^1 t^{N-1} g(t, \alpha t^{2-N} ) dt \Bigr) u_0(r),$$ and the conclusion follows. \hfill$\square$ \smallskip Next, we will show a general existence result of fundamental singular solutions which is included in \cite{cy3}, Theorem 4.3. We give an idea of the proof for the sake of completeness . For this purpose, let $\alpha >0 $ and let $$ u_{\alpha}(r):= \int_r^1 s^{1-N}\int_0^s t^{N-1} f_1(t, \alpha t^{2-N}) \,dt\,ds\,.$$ \begin{thm} \label{weak} Assume that $f$ and $g$ are two non negative functions satisfying \eqref{hipotesisradial}. Assume that $$\int_0^1 t^{N-1}f_1(t, \alpha t^{2-N})dt <\infty,$$ and $$ \lambda_{\alpha}:= \frac{1}{N-2}\int_0^1 t^{N-1}g_1(t, u_{\alpha}(t))dt <\infty.$$ Moreover, suppose that for some $\alpha>0$, we have $\lambda_{\alpha} <\alpha$. Then, for any $d \in ( \lambda_{\alpha}, \alpha]$, there exists a non-negative solution $(u, v)$ to \eqref{potengeneralradial} such that $$\lim_{r \to 0^+}r^{N-2}(u,v)(r)= (0, d).$$ \end{thm} \paragraph{\bf Proof} The proof of this result is similar to the proof of Theorem \ref{exis4}. Let $d\in ( \lambda_{\alpha}, \alpha]$, and let $F= (F_1, F_2)$ be given by \begin{align*} F_1(u, v)(r) &= \int_r^1 s^{1-N}\int_0^s t^{N-1} f(t,v(t))dt,\\ F_2(u, v)(r) &= du_0(r) -\int_r^1 s^{1-N}\int_0^s t^{N-1} g(t,u(t))dt. \end{align*} Define ${\mathcal N}$ as the subset of $C((0,1])^2$ such that $$ {\mathcal N}:= \{(u, v) \ | \ 0\le u\le u_{\alpha}, \ 0\le v\le \alpha u_0 \}. $$ Under the assumptions of the theorem, it is not difficult to prove that $F({\mathcal N}) \subset {\mathcal N}$. The rest of the proof follows the ideas of Theorem \ref{exis4}. \hfill$\square$ \smallskip Next, we will apply Theorem \ref{exis4} to problem \eqref{potengeneralradial} with \begin{equation} \label{potencia} 0\le f(r, s)\le r^a s^p,\quad 0\le g(r, s)\le r^b s^q. \end{equation} \begin{thm}\label{h1} Let $p>0$ and $q>0, $ with $pq\not=1$ and suppose that $(a, b, p, q)$ satisfies \eqref{gamma2}. Assume that $f$ and $g$ are two non negative functions satisfying \eqref{potencia}. Then, there exist $c_0 > 0$ such that for any $c\in [0, c_0)$ there exists $(u, v)$ a non-negative singular solution to \eqref{potengeneralradial} such that $$ \lim_{r \to 0^+}r^{N-2}u(r) =c.$$ Moreover, if $pq <1$ then $c_0= +\infty$. \end{thm} \paragraph{\bf Proof} Let $$ f_1(r, s)= r^a s^p, \quad g_1(r, s)= r^b s^q.$$ The function $v_{\alpha}$ defined by \eqref{defv1} is given now by \begin{equation}\label{defv1poten} v_{\alpha}(r):= du_0(r) + \alpha^q\int_r^1 s^{1-N} \int_s^1 t^{N-1+b-(N-2)q}\,dt\,ds\,. \end{equation} Next, we show that (H1) is equivalent to $$N+a+ \min\{\beta, 2-N\}p >0,$$ where $\beta$ is defined by \eqref{defbeta}. Let \begin{equation} \label{defw1} w_1(r):= \int_r^1 s^{1-N} \int_s^1 t^{N-1+b-(N-2)q}\,dt\,ds\,.\end{equation} Thus, by setting $\rho:=\beta +N-2$, $$w_1(r)=\begin{cases} \frac{u_0(r)}{\rho(N-2) }+ \frac{r^{\beta}-1}{\rho \beta} &\text{if $\beta\not=0 $ and $\rho\not=0$, }\\[5pt] \frac{u_0(r)}{(N-2)^2}+\frac{\log(r)}{N-2} & \text{if $\beta=0$},\\[5pt] \int_r^1s^{1-N}|\log(s)| ds &\text{ if $ \rho =0$.} \end{cases}$$ Moreover, if $\rho = 0$, $$\lim_{r\to 0^+} r^{N-2}| \log(r)|^{-1} w_1(r)= (N-2)^{-1}.$$ Now, the proof of the equivalence to (H1) follows easily. To prove the existence of a non-negative solution it is sufficient to find $d$ and $\alpha$ positive constants such that \begin{equation} \label{desigualdad} \Lambda_{\alpha} = (N-2)^{-1}\int_0^1 t^{N-1+a} v^p_{\alpha}(t)dt < \alpha\,. \end{equation} Since $v_{\alpha}= du_0 + \alpha^q w_1$, where $w_1$ is given by \eqref{defw1}, and using the inequality $(x+y)^p\le C (x^p+ y^p)$, for any non-negative numbers $x$ and $y$, and where $C=\max\{1,2^{p-1}\}$, we see that \eqref{desigualdad} is satisfied if \begin{equation} \label{desigualdad1} A d^p + B\alpha^{pq}<(N-2)\alpha\,, \end{equation} where $$A:= \int_0^1 t^{N-1+a}u^p_0(t)dt, \quad B:= \int_0^1 t^{N-1+a}w^p_1(t)\, dt\,.$$ If we choose, for instance, $d$ such that $Ad^p = B \alpha^{pq}$, \eqref{desigualdad1} is satisfied for any $\alpha$ such that $$ 2B\alpha^{pq-1} < N-2\,.$$ Moreover, by Theorem \ref{exis4} there exists $(u, v)$ non-negative singular solution such that $\lim_{r\to 0^+}r^{N-2}u(r)=c$, for any $c\in [0, \alpha-\Lambda_{\alpha})$, and thus if $pq<1$ and since $\alpha-\Lambda_{\alpha} $ tends to infinity as $\alpha $ does, the existence in the sub-linear case is for any $c>0$. \hfill$\square$ \smallskip The following result is an application of Theorem \ref{weak} to problem \eqref{potengeneralradial}. \begin{thm}\label{weakpotencia} Assume that $f$ and $g$ are two non-negative functions satisfying \eqref{potencia}, with $p>0$ and $q>0$ and $pq\not=1$. Also assume that $(a, b, p, q)$ satisfies $$ p < (N+a)/(N-2) \ \text{ and \ $ N+b + \mu q>0$} ,$$ where $\mu:=\min\{ a+2-(N-2)p, 0\}$. Then, there exist $d_0\ge 0$ and $d_1>0$, with $d_0< d_1$, such that for any $d\in (d_0, d_1)$ there exists $(u, v)$ a non-negative singular solution to \eqref{potengeneralradial} such that $$ \lim_{r \to 0^+}r^{N-2}(u, v)(r) =(0, d).$$ Moreover, if $pq <1$, $d_1= +\infty $ and if $pq> 1$, $d_0=0$. \end{thm} \begin{rem} In \cite{cy} we proved existence of solutions for \eqref{poten} in the super-linear case, that is when $pq >1$. In the super-linear case, Theorem \ref{h1} and Theorem \ref{weakpotencia} do not give the optimal region of the values $(a, b, p, q)$ of existence of non-negative solutions to \eqref{poten}. However, we will show later that for the sub-linear case Theorem \ref{h1} is optimal, see Theorem 1.1. \end{rem} As a consequence of Theorem \ref{h1}, Corollary \ref{existenciadesingfuerte} and Theorem \ref{weakpotencia} we have the following. \begin{cor} Let $p>0$ and $q>0$, and consider \begin{equation}\label{particular1} \begin{gathered} -\Delta u = v^p \quad\text{in $ B'$}, \\ \Delta v = u^q \quad\text{in $ B'$}, \\ u=v= 0\quad\text{on $\partial B$}, \end{gathered} \end{equation} with $q \ge N/(N-2)$ and $N+ \bigl(2-(N-2)q\bigr) p>0$. Then, there exist fundamental and strongly singular non-negative solutions of \eqref{particular1}. \end{cor} \section{Proof of main theorem \ref{noexis}} In this section we prove our main result and we give some applications to bi-harmonic equations. \paragraph{\bf Proof of Theorem \ref{noexis} } Let $p>0$, $q>0$ and $pq<1$. Assume that $(a, b, p, q)$ satisfies \eqref{gamma2}. The existence of a non-negative singular solution to \eqref{poten} follows from Theorem \ref{h1}. Assume that $(a, b, p, q)$ does not satisfies \eqref{gamma2}, with $p>0$, $q>0$ and $pq<1$ and let $(u, v)$ be a non-negative solution to \eqref{poten}. We will show that $(u, v)$ must be the trivial solution. First, when $N+a +\beta p \le 0$, the conclusion follows form Theorem 1.2 in \cite{bg}. Now, assume that $p \ge (N+a)/(N-2)$. Since $u$ is a non-negative super-harmonic function, from Theorem 1 in \cite{bl}, we obtain that $$|x|^{a }v^p\in L_{loc}^1(B). $$ Moreover, since $v$ is sub-harmonic there exists a non-negative constant $c$ (possible $c=\infty$) such that $\lim_{r\to 0^+}r^{N-2}\overline{v}(r)=c$, where $\overline{v}$ is the spherical average of $v$. Assume first that $c=0$. Let $w(s):=s{\overline v}(r)$ with $s=r^{N-2}$. We easily obtain that $w$ is a convex function satisfying $w(0)=w(1)=0$, and thus $v=0$. On the other hand, if $c\neq 0$, we have for some positive constant $C$ such that for all $r$ near $0$, \begin{equation} \label{funda} \overline{v}(r)\ge Cr^{2-N}\,. \end{equation} By Remark 3.1 in \cite{bg} and \eqref{funda} we deduce that $\overline{v^p}(r)\ge C {\overline{v}}^p(r)\ge Cr^{(2-N)p}$, and thus $$\infty>\int_{B_{\epsilon}(0)}|x|^{a }v^p(x) dx \ge C \int_{0}^{\epsilon} r^{a+N-1 }{\overline{v}}^p(r)dr\ge C \int_{0}^{\epsilon} r^{a+N-1-p(N-2) }dr, $$ contradicting $p\ge \frac{N+a}{N-2}$. The last assertion on the theorem follows from Corollary \ref{existenciadesingfuerte}. \hfill$\square$ \smallskip The following two results are applications of Theorem \ref{noexis} to the bi-harmonic equation. \begin{cor} Let $N>2$, and let $0 < q < 1$. Then there exist positive solutions of \begin{equation} \label{bihar} \begin{gathered} \Delta^2 u +|x|^b u^q=0 \quad \hbox{in }B'_1(0), \\ u=\Delta u= 0 \quad \hbox{on } \partial B_1(0), \end{gathered} \end{equation} such that $-\Delta u\ge 0$, if and only if $$ q < \frac{N+b+2}{N-2}. $$ \end{cor} \begin{cor} Let $N>2$, and $0 < q < 1$. Then there exist positive solutions of {\rm (\ref{bihar})} such that $\Delta u\ge 0$, if and only if $$ q < \frac{N+b}{N-2}. $$ \end{cor} As a consequence of the results in \cite{cy} and the two corollaries above, we obtain the following. \begin{cor} Let $N>2$, and $0 < q \not= 1$. Then \begin{enumerate} \item[(i)] There exist positive solutions of {\rm (\ref{bihar})} with $b=0$ such that $-\Delta u\ge 0$, if and only if $$ (N-4)q < N. $$ \item[(ii)] There exist positive solutions of {\rm (\ref{bihar})} with $b=0$ such that $\Delta u\ge 0$, if and only if $$ q < \frac{N}{N-2}. $$ \end{enumerate} \end{cor} \begin{thebibliography}{00} \bibitem{bl} H. 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