\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 102, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2018/102\hfil Schr\"odinger quasilinear elliptic problems] {Unbounded solutions for Schr\"odinger quasilinear elliptic problems with perturbation by a positive non-square diffusion term} \author[C. A. Santos, J. Zhou \hfil EJDE-2018/102\hfilneg] {Carlos Alberto Santos, Jiazheng Zhou} \address{Carlos Alberto Santos \newline Universidade de Bras\'ilia, Departamento de Matem\'atica, 70910-900, Bras\'ilia - DF, Brazil} \email{csantos@unb.br} \address{Jiazheng Zhou \newline Universidade de Bras\'ilia, Departamento de Matem\'atica, 70910-900, Bras\'ilia - DF, Brazil} \email{jiazzheng@gmail.com} \dedicatory{Communicated by Claudianor O. Alves} \thanks{Submitted October 10, 2017. Published May 3, 2018.} \subjclass[2010]{35J10, 35J62, 35B08, 35B09, 35B44} \keywords{Schr\"odinger equations; blow up solutions; quasilinear problem; \hfill\break\indent non-square diffusion; multiplicity of solutions} \begin{abstract} In this article, we present a version of Keller-Osserman condition for the Schr\"odinger quasilinear elliptic problem \begin{gather*} -\Delta u+\frac{k}{2}u\Delta u^2=a(x)g(u)\quad\text{in } \mathbb{R}^N,\\ u>0\quad \text{in }\mathbb{R}^N,\quad \lim_{|x|\to \infty} u(x)= \infty\,, \end{gather*} where $a:\mathbb{R}^N \to [0,\infty)$ and $g:[0,\infty) \to [0,\infty)$ are suitable continuous functions, $N \geq 1$, and $k>0$ is a parameter. By combining a dual approach and this version of Keller-Osserman condition, we show the existence and multiplicity of solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} In this article, we consider the problem \begin{equation}\label{P} \begin{gathered} -\Delta u+\frac{k}{2}u\Delta u^2=a(x)g(u)\quad \text{in } \mathbb{R}^N,\\ u>0\quad \text{in }\mathbb{R}^N,\quad \lim_{|x|\to \infty} u(x)\to \infty, \end{gathered} \end{equation} where $\Delta $ is the $Laplacian$ operator, $a(x)$ is a nonnegative continuous function, $g:[0,\infty) \to [0,\infty)$ is a nondecreasing continuous function that satisfies $g(s)>0$, $s>0$, $N\geq 1$ and $k>0$ is a parameter. Equation \eqref{P} is a modified nonlinear Schr\"odinger equation by the quasilinear and nonconvex term $u\Delta u^2$, which is called of square diffusion. A solution of \eqref{P} is related to standing wave solutions for the quasilinear Schr\"odinger equation \begin{equation}\label{i} iz_t+\Delta z-\omega(x)z+\kappa\Delta(h(|z|^2))h'(|z|^2)z+\eta(x,z)=0,\quad x\in\mathbb{R}^N, \end{equation} where $\omega$ is a potential function, $h$ and $\eta$ are real functions and $\kappa$ is a real constant. This connection is established by the fact that $z(t,x)=e^{-i\beta t}u(x)$ is a solution to \eqref{i} if and only if $u$ satisfies the equation in \eqref{P} for suitable constants $\omega$, $h$, $\eta$ and $\kappa$. This kind of equations appears in several applications: superfluid film in plasma physics \cite{kurihara}; in models of the self-channeling of a high-power ultrashort laser in matter \cite{kosevich} and \cite{quispel}; in the theory of Heidelberg ferromagnetism and magnus \cite{hasse}; in dissipative quantum mechanics \cite{adachi}; and in condensed matter theory \cite{byeon}. Even for bounded solutions, there are only a few results in the literature studying existence and multiplicity of such solutions to the equation in \eqref{P} with positive perturbation; that is, $k>0$. One important result, that shows the existence of solutions for a related operator of the equation in \eqref{P}, is due to Alves, Wang and Shen \cite{eg100} who showed the existence of bounded solutions satisfying $$ \sup_{x\in \mathbb{R}^N}| u(x) | \leq \sqrt{1/k} $$ for each $0< k 0$. In fact, they considered the equation $$ -\Delta u+V(x)u+\frac{k}{2}u\Delta u^2=a(x)g(u)\quad \text{in } \mathbb{R}^N $$ for some appropriate potential $V$. For more references on this direction, we refer the reader to \cite{aw,as, wg,sgs} and references therein. On the other hand, after these papers, we wondered whether it is possible to exist unbounded solutions for \eqref{P}; that is, solutions $u(x)\to \infty$ as $| x | \to \infty$. Surprisingly, under an appropriate version of Keller-Osserman condition to this operator, we were able to show the existence of an infinite number of solutions to \eqref{P}. Our solutions satisfy $$ \inf_{x\in \mathbb{R}^N}| u(x) | \geq \sqrt{1/k} $$ for a given $k>0$. Research about existence of explosive solutions (or unbounded solutions) is motivated principally by its applications in models of population dynamical, subsonic motion of a gas, non-Newtonian fluids, non-Newtonian filtration as well as in the theory of the electric potential in a glowing hollow metal body. Remarkable work about unbounded solutions was done by Keller \cite{K} and Osserman \cite{O}, both in 1957. They established necessary and sufficient conditions for the existence of solutions and sub solutions to the semilinear and autonomous problem (that is, $a \equiv 1$) \begin{equation} \label{P1} \begin{gathered} \Delta u=a(x)g(u)\quad \text{in } \mathbb{R}^N,\\ u\geq 0\quad \text{in } \mathbb{R}^N,\; \lim_{|x|\to \infty} u(x)\to \infty, \end{gathered} \end{equation} where $g$ is a non-decreasing continuous function. This is done under the condition \begin{equation} \label{eG} \int_1^{+\infty} \frac{dt}{G(t)^{1/2}}=\infty,\quad \text{where } G(t)=\int_0^tg(s)ds, \quad t>0. \end{equation} After these works, a function $g$ satisfies the well-known Keller-Osserman condition, for the Laplacian operator, if $$ \int_1^{+\infty} \frac{dt}{G(t)^{1/2}}<\infty\,. $$ Recently, there have been a number of papers trying to obtain ``Keller-Osserman conditions'' for various operators. The authors have also considered this question for $\phi$-Laplacian operator in \cite{santosz}. For \eqref{P1} non-autonomous, it has arisen an important issues on existence of solutions, namely, ``how radial" is $a(x)$ at infinity; that is, how big is the function $$ a_{\rm osc}(r):= \overline{a}(r) - \underline{a}(r),\quad r\geq 0, $$ where \begin{equation}\label{defa} \underline{a}(r)=\min\{a(x): |x|= r\},\quad \overline{a}(r) =\max\{a(x): |x| = r\},\quad r\geq 0. \end{equation} As a consequence of this, we have that $a_{\rm osc}(r)= 0$, $r\geq r_0$ ,if and only if, $a$ is symmetric radially for $| x | \geq r_0$, for some $r_0\geq 0$. In particular, if $r_0=0$ we say that $a(x)$ is radially symmetric. Considering $a_{\rm osc}\equiv 0$, Lair and Wood in \cite{lairwood} proved that \begin{equation}\label{conda} \int_1^{\infty} r a(r) dr = \infty \end{equation} is a sufficient condition for \eqref{P1} to have radial solution. They considered $g(u)=u^{\gamma}$, $u\geq 0$ with $0 < \gamma \leq 1$, that is, $g$ satisfies \eqref{eG}. In 2003, Lair \cite{lair3} allowed $a(x)$ to be not necessarily radial in the whole space, but he did not allow $a(x)$ to have $a_{\rm osc} $ too big. More exactly, he assumed $$ \int_0^{\infty}r a_{\rm osc}(r) \exp(\underline{A}(r)) dr< \infty, \quad \text{where } \underline{A}(r)= \int_0^r s\underline{a}(s) ds,\; r\geq 0 $$ and proved that \eqref{P1}, with suitable $g$ that includes $u^{\gamma}$ for $0< \gamma \leq 1$, admits a solution, if and only if, \eqref{conda} holds with $\underline{a}$ in place of $a$. In this way, Mabroux and Hansen \cite{hansen} in 2007 improved the above results, considering the hypothesis $$ \int_0^{\infty}r a_{\rm osc}(r) (1 + \underline{A}(r))^{\gamma/(1-\gamma)} dr < \infty\,. $$ For a more general operator, Rhouma and Drissi \cite{drissi} in 2014 proved similar results. Before stating our main results, we dfine a solution of \eqref{P} as a positive function $u\in C^1(\mathbb{R}^N)$ such that $u \to \infty$ as $| x | \to \infty$, and \begin{align*} &\int_{\mathbb{R}^N}(1-ku^2)\nabla u\nabla\varphi dx - k\int_{\mathbb{R}^N}|\nabla u|^2u\varphi dx \\ &=\int_{\mathbb{R}^N}a(x)g(u)\varphi dx\quad \text{for all } \varphi\in C_0^{\infty}(\mathbb{R}^N). \end{align*} Throughout this article we assume that $g:[0,\infty) \to [0,\infty)$ is a nondecreasing continuous function with $g(s)>0$ for $s>0$. Also we use the condition \begin{equation} \label{eg} \liminf_{t\to\infty}\frac{g(t)}{t}>0. \end{equation} Our first result reads as follows. \begin{theorem}\label{thm1} Assume that \eqref{eg} is satisfied and \begin{equation} \label{eG'} \int_1^{+\infty} \frac{dt}{G_0(t)^{1/2}}=\infty, \quad \text{where } G_0(t)=\int_0^tg(\sqrt{s})ds,\; t>0\,. \end{equation} If $a_{\rm osc}\equiv 0$ and \begin{equation}\label{a} \int_0^\infty\Big(s^{1-N}\int_0^st^{N-1}a(t)dt\Big)ds=\infty, \end{equation} then for each $\sigma>1$ and $k>0$, there exists a solution $u=u_{\sigma,k} \in C^1(\mathbb{R}^N)$ to problem \eqref{P}. Furthermore, $$ \inf_{x \in \mathbb{R}^N} u(x) \geq \sqrt{\sigma/k}\,. $$ \end{theorem} For non-radial potentials $a(x)$, we need to control the size of this non-radiality. So, for each $\sigma>1$, let us assume that $\mathcal{G}=\mathcal{G}_\sigma:(0,\infty) \to (0,\infty)$, defined by $$ \mathcal{G}(t)=\frac{{(\sigma-1)\sqrt{k}}}{8 \sqrt{\sigma}}t^2/ g(t),\quad t>0, $$ is non-decreasing and is invertible; such that \begin{equation} \label{eGcal} \begin{aligned} 0\leq\overline{H}&:=\frac{1}{\sqrt{\sigma-1}}\int_0^\infty \Big(s^{1-N}\int_0^s t^{N-1}a_{\rm osc}(t)dt \Big) \\ &\quad\times \Big [g\Big(\mathcal{G}^{-1} \Big(s\Big(\int_0^s\overline{a}(t)dt\Big)\Big)\Big)\Big]ds<\infty\,. \end{aligned} \end{equation} Our second result reads as follows. \begin{theorem}\label{thm2} Assume $g$ satisfies \eqref{eG'} and that for $t>0$ the function $g(t)/t^\delta$ is non-decreasing for some $\delta\geq{\sigma}/{(\sigma-1)}$. Also suppose that $a(x)$ is such that $\underline{a }$ satisfies \eqref{a} and $\overline{a}$ satisfies \eqref{eGcal}. Then there exists a solution $u=u_{\alpha,\sigma,k,\varepsilon}\in C^1(\mathbb{R}^N)$ of problem \eqref{P} satisfying $$ \inf_{x\in \mathbb{R}^N}u(x) \geq \sqrt{\sigma/k}\quad\text{and}\quad \alpha\leq u(0) \leq (\alpha + \varepsilon)+ \overline{H} $$ for each $\sigma>1$ and $\alpha,k, \varepsilon>0$ given so that $\alpha > \sqrt{\sigma/k}$. \end{theorem} We note that this work contributes to the literature of quasilinear Schr\"odinger equation in at least two aspects: Firstly, as far as we know, there are no results considering this kind of operators (a positive perturbation) in the context of unbounded solutions. We mention the authors have already considered in \cite{santosz1} a negative perturbation, that is, $k<0$ in the problem \eqref{P}. Secondly, we present a version of ``Keller-Orsemann condition'' for this kind of operator that ``captures'' the influence of the perturbation term. We organized this article the following way: in section 2, we establish an equivalent problem to the \eqref{P}, via a very specific change of variable. In the last section we complete the proof of Theorems \ref{thm1} and \ref{thm2}. \section{Auxiliary results} In this section, a change of variables allows us to transform problem \eqref{P} into a new problem. In the new problem, we establish a version of Keller-Orsemann condition and show the existence of an entire solution that is unbounded. First, we note that the problem \eqref{P} is equivalent to the modified quasilinear Schr\"odinger problem \begin{equation}\label{l} \begin{gathered} \operatorname{div}(l^2(u)\nabla u)-l(u)l'(u)|\nabla u|^2=a(x)g(u),\quad x\in\mathbb{R}^N,\\ u>0\quad \text{in }\mathbb{R}^N,\quad \lim_{|x|\to \infty} u(x)\to \infty, \end{gathered} \end{equation} whenever $u(x)>\sqrt{\sigma/k}$ for $x \in\mathbb{R}^N$, where $l(t)=\sqrt{kt^2-1}$ for $t>\sqrt{{\sigma}/{k}}$ for each $k>0$ and $\sigma>1$ given. In these situations, we conclude that the solutions obtained to \eqref{l} are solutions of the original problem \eqref{P}. So, we look for by a positive function $u\in C^1(\mathbb{R}^N)$ that satisfies $u \to \infty$ as $| x | \to \infty$ and \[ -\int_{\mathbb{R}^N}l^2(u)\nabla u\nabla \varphi dx -\int_{\mathbb{R}^N}l(u)l'(u)|\nabla u|^2\varphi dx = \int_{\mathbb{R}^N}a(x)g(u)\varphi dx \] for all $\varphi\in C_0^\infty(\mathbb{R}^N)$; that is, this $u\in C^1(\mathbb{R}^N)$ will be an unbounded solution to \eqref{l}. To do this, first let us define $l:[1/\sqrt{k\sigma},\infty) \to [0,\infty)$ by $$ l(t)=l_{\sigma,k}(t) =\begin{cases} \frac{\sqrt{k\sigma}}{\sqrt{\sigma-1}}t-\frac{1}{\sqrt{\sigma-1}} &\text{if } \frac{1}{\sqrt{k\sigma}}\leq t\leq\sqrt{\frac{\sigma}{k}},\\ \sqrt{kt^2-1} &\text{if } t>\sqrt{\frac{\sigma}{k}}, \end{cases} $$ for each $\sigma>1$, and set \begin{equation}\label{L} L(t)=L_{\sigma,k}(t) =\int_{1/\sqrt{k\sigma}}^tl(s)ds\quad \text{for } t\geq1/\sqrt{k\sigma}. \end{equation} It is a consequence of the above definitions that the function $L:[1/\sqrt{k\sigma}, \infty) \to [0,\infty)$ is a $C^2$-injective function; that is, the inverse function $L^{-1}:[0,\infty) \to [1/\sqrt{k\sigma}, \infty)$ is well-defined and $L^{-1}$ is a $C^2$-function as well. After this, we are able to prove more Lemmas. The first lemma follows from the definitions and properties of $l$ and $L$. \begin{lemma}\label{lem1} Under the above conditions, the functions $l$ and $L^{-1}$ satisfy: \begin{itemize} \item[(1)] $0\leq l(t)\leq\frac{\sqrt{k\sigma}}{\sqrt{\sigma-1}}t$ for all $t\in[1/\sqrt{k\sigma},\sqrt{\sigma/k}]$ and $\frac{\sqrt{(\sigma-1)k}}{\sqrt{\sigma}}t\leq l(t)\leq \sqrt{k}t$ for all $t>\sqrt{\sigma/k}$, \item[(2)] $0\leq L(t)\leq \frac{\sqrt{k\sigma}}{\sqrt{\sigma-1}}t^2$ for all $t\in[1/\sqrt{k\sigma},\sqrt{\sigma/k}]$ and \[ \frac{1}{2}\frac{\sqrt{(\sigma-1)k}}{\sqrt{\sigma}}t^2 -\frac{1}{2}\frac{\sqrt{(\sigma-1)\sigma}}{\sqrt{k}}\leq L(t)\leq\sqrt{k}t^2, \] for all $t>\sqrt{\sigma/k}$, \item[(3)] $L^{-1}(t)\leq\sqrt{\frac{2t\sqrt{\sigma}}{\sqrt{(\sigma-1)k}}+ \frac{\sigma}{k}}$ for $t>0$, and $$ L^{-1}(t)\geq \begin{cases} \sqrt[4]{\frac{\sigma-1}{k\sigma}}\sqrt{t} &\text{for } \frac{1}{\sqrt{k\sigma(\sigma-1)}} \leq t \leq \frac{\sqrt{\sigma^3}}{\sqrt{k(\sigma-1)}},\\ \sqrt{\frac{t}{\sqrt{k}}} &\text{for } t \geq \frac{\sigma}{\sqrt{k}}, \end{cases} $$ \item[(4)] for $t>\sqrt{\sigma/k}$, the function $\frac{t^\delta}{l(t)}$ is nondecreasing for each $\delta\geq \frac{\sigma}{\sigma-1}$. \end{itemize} \end{lemma} In the sequel, we use the definitions and properties of $l,L$ and $L^{-1}$ to provide solutions \eqref{l} by establishing solutions to \eqref{PP} below. \begin{lemma}\label{lem2} Assume $u=L^{-1}(w)$, where $w\in C^1(\mathbb{R}^N)$ is a solution of the problem \begin{equation}\label{PP} \begin{gathered} \Delta w=a(x)\frac{g(L^{-1}(w))}{l(L^{-1}(w))}\quad \text{in } \mathbb{R}^N,\\ w>\sigma/\sqrt{k}\quad\text{in }\mathbb{R}^N,\quad \lim_{|x|\to \infty} w(x) = \infty. \end{gathered} \end{equation} Then $u\in C^1(\mathbb{R}^N)$ is a solution of problem \eqref{P} that satisfies $u(x)\geq \sqrt{\sigma/k}$ for all $x \in \mathbb{R}^N$. \end{lemma} \begin{proof} First, note that $u\geq \sqrt{\sigma/k}$ is a consequence of $w\geq \sigma/\sqrt{k}$. By the regularity of $L$, we obtain $u\in C^1(\mathbb{R}^N)$, because $w\in C^1(\mathbb{R}^N)$. Besides this, it follows from the behavior of $L$ and $L^{-1}$ that $w(x)\to+\infty$ as $|x|\to+\infty$ if and only if $u(x)\to+\infty$ as $|x|\to+\infty$. Since $$ w=L(u)=\int_{1/\sqrt{k\sigma}}^ul(s)ds, $$ it follows that $$ \nabla w=l(u)\nabla u=(ku^2-1)^{1/2}\nabla u; $$ that is, $$ \nabla u=(ku^2-1)^{-1/2}\nabla w. $$ Thus, for each $\varphi\in C_0^1(\mathbb{R}^N)$, we have \begin{equation}\label{k12} (1-ku^2)\nabla u\nabla\varphi=-(ku^2-1)^{1/2}\nabla w\nabla\varphi. \end{equation} On the other hand, since $w \in C^1(\mathbb{R}^N)$ is a solution of problem \eqref{PP}, we have \begin{align*} \int_{\mathbb{R}^N}(ku^2-1)^{1/2}\nabla w\nabla\varphi &= \int_{\mathbb{R}^N}\nabla w\nabla\{(ku^2-1)^{1/2} \varphi\}-\int_{\mathbb{R}^N}\frac{ku}{ku^2-1}|\nabla w|^2\varphi\\ &= -\int_{\mathbb{R^N}}a(x)\frac{g(u)}{l(u)}(ku^2-1)^{1/2}\varphi -\int_{\mathbb{R^N}}ku|\nabla u|^2\varphi\\ &= -\int_{\mathbb{R^N}}a(x)g(u)\varphi-\int_{\mathbb{R^N}}ku|\nabla u|^2\varphi. \end{align*} Then using \eqref{k12}, we have $u\in C^1(\mathbb{R}^N)$ is a solution to \eqref{P}. This completes the proof. \end{proof} \section{Proof of main results} Below, we show the existence of a solution to \eqref{PP} and after this, by using the second Lemma above, we are able to show the existence of a solution to Problem \eqref{P1}. This arguments relies on ideas found in \cite{mahammed}. \begin{proof}[Proof of Theorem \ref{thm1}] Since $a_{\rm osc}\equiv 0$, that is, $a(x)=a(|x |)$ for all $x \in \mathbb{R}^N$, we have that a radial solution for \eqref{PP} can be obtained by solving the problem \begin{equation}\label{w} \begin{gathered} (r^{N-1}w')'=r^{N-1}a(r)\frac{g(L^{-1}(w))}{l(L^{-1}(w))}\quad\text{in } (0,\infty),\\ w'(0)=0,\quad w(0)=\alpha\geq 0, \end{gathered} \end{equation} where $r=| x | \geq 0$ and $\alpha>\sigma/\sqrt{k}$ is a real number, for each $\sigma>1$ and $k>0$. Since $a$, $g$, $l$ and $L^{-1}$ are continuous functions, we can follow the approach in \cite{haitao} to show that there exist a right maximal extreme $\Gamma(\alpha)>0$, and a function $w_\alpha\in C^2(0,\Gamma(\alpha))\cap C^1([0,\Gamma(\alpha)))$ solution of the problem \eqref{w} on $(0,\Gamma(\alpha))$, for each $\alpha>\sigma/\sqrt{k}$ given. If we assumed that $\Gamma(\alpha)<\infty$ for some $\alpha>\sigma/\sqrt{k}$, then we would obtain, by standard arguments of ordinary differential equations, that either $w_\alpha(r)\to\infty$ as $r\to\Gamma(\alpha)^-$ or $w'_\alpha(r)\to\infty$ as $r\to\Gamma(\alpha)^-$; that is, $w_\alpha(|x|)$ would satisfy the problem \begin{equation} \label{112} \begin{gathered} (r^{N-1}w')'=r^{N-1}a(r)\frac{g(L^{-1}(w))}{l(L^{-1}(w))}\quad \text{in } (0,\Gamma(\alpha)),\\ w'(0)=0,\ w(0)=\alpha>0,\\ \lim_{r\to \Gamma(\alpha)^-} w_\alpha(r)= \infty\quad \text{or}\quad \lim _{r\to \Gamma(\alpha)^-} w'_\alpha(r)= \infty. \end{gathered} \end{equation} So, using that a solution $w$ of \eqref{112} is non-decreasing and $l(t)\geq\sqrt{\sigma-1}$ for all $t\geq\sqrt{\sigma/k}$, we obtain that $w$ satisfies \begin{equation}\label{ww} \begin{gathered} (r^{N-1}w')'\leq \frac{a_\infty}{\sqrt{\sigma-1}} r^{N-1}g(L^{-1}(w)) \quad\text{in } (0,\Gamma(\alpha)),\\ w(0)=\alpha>0,\quad w'(0)=0,\\ \lim_{r\to \Gamma(\alpha)^-} w_\alpha(r)= \infty\quad\text{or}\quad \lim_{r\to \Gamma(\alpha)^-} w'_\alpha(r) = \infty, \end{gathered} \end{equation} where $a_\infty=\max_{\bar{B}_{\Gamma(\alpha)}}a(x)$. By integrating the inequality above over $(0,r)$ with $00$, we obtain $$ \limsup_{r\to \Gamma(\alpha)^-}w'(r)\leq \Gamma(\alpha)^{1-N} \int_0^{\Gamma(\alpha)}t^{N-1}g(L^{-1}(w(t)))dt<\infty $$ by the continuity of all involved functions. So, from now on, we assume that $w_\alpha(x){\to} \infty$ as ${r\to \Gamma(\alpha)^-}$. By using $w'\geq 0$ again, we can rewrite the inequality in \eqref{ww} as $$ w'' \leq \frac{a_{\infty}}{\sqrt{\sigma-1}} (g\circ L^{-1})(w)\quad \text{for all } 0\sigma/\sqrt{k}; $$ that is, $$ \int_\alpha^t(g\circ L^{-1})(s)ds\leq\int_\alpha^tg \Big(\sqrt{\frac{2t\sqrt{\sigma}}{\sqrt{(\sigma-1)k}}+ \frac{\sigma}{k}}\Big)ds~\text{for all }t>\alpha. $$ As a consequence of this and \eqref{comp1}, we have $$ \int_{\alpha}^\infty\Big\{\int_\alpha^tg\ Big(\sqrt{\frac{2s\sqrt{\sigma}}{\sqrt{(\sigma-1)k}}+ \frac{\sigma}{k}}\Big)ds\Big\}^{-1/2}dt \leq \int_{\alpha}^\infty\Big\{\int_\alpha^t(g\circ L^{-1})(s)ds\Big\}^{-1/2}dt. $$ So, by estimating in the last inequality and using \eqref{comp1} again, we obtain $$ \int_{1}^\infty G_0(t)^{-{1}/{2}}dt \leq C\Big(\sqrt{a_\infty/\sqrt{\sigma-1}} \Big)\Gamma(\alpha) < \infty, $$ for some real constant $C>0$. This is impossible, because we are assuming that $g$ satisfies \eqref{eG'}. It follows from Lemma \ref{lem1}-(1), hypothesis \eqref{eg} and $L^{-1}(w_\alpha)\geq \sqrt{\sigma/k}$, that \begin{equation} \label{115} \frac{g(L^{-1}(w_\alpha(r)))}{l(L^{-1}(w_\alpha(r)))} \geq \frac{g(L^{-1}(w_\alpha(r)))}{\sqrt{k}L^{-1}(w_\alpha(r))} \geq M>0\quad \text{for all }r>0, \end{equation} and for each $\alpha>\sigma/\sqrt{k}$ given and for some $M>0$, because $w_\alpha(r) \geq \alpha$ for all $r\geq 0$. Since, $w_\alpha$ satisfies \begin{equation} \label{116} w_\alpha(r)=\alpha+\int_0^r\Big(t^{1-N}\int_0^ts^{N-1}a(s) \frac{g(L^{-1}(w_\alpha))}{l(L^{-1}(w_\alpha))}ds\Big)dt,\quad r\geq 0, \end{equation} it follows from \eqref{115}, that \begin{equation}\label{117} w_\alpha(r)\geq\alpha+M\int_0^r\Big(t^{1-N}\int_0^ts^{N-1}a(s)ds\Big)dt\to\infty, \quad \text{as } r\to\infty. \end{equation} This completes the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2}] Set $\beta>\alpha>\sigma/\sqrt{k}$. Since the hypothesis $g(t)/t^\delta$ being non-increasing implies \eqref{eg}, from Theorem \ref{thm1} there exist positive and radially symmetric solutions $w_\alpha,w_\beta \in C^1(\mathbb{R}^N)$ to the problems \begin{gather*} \Delta w_\alpha=\overline{a}(|x|)\frac{g(L^{-1}(w_\alpha))}{l(L^{-1}(w_\alpha))} \quad\text{in }\mathbb{R}^N,\\ w_\alpha(0)=\alpha,\quad \lim_{|x|\to \infty} w_\alpha(x)= \infty, \end{gather*} and \begin{gather*} \Delta w_\beta=\underline{a}(|x|)\frac{g(L^{-1}(w_\beta))}{l(L^{-1}(w_\beta))} \quad\text{in }\mathbb{R}^N,\\ w_\beta(0)=\beta,\quad \lim_{|x|\to \infty} w_\beta(x)= \infty, \end{gather*} respectively, where $\underline{a}$ and $\overline{a}$ were defined in \eqref{defa}. Besides this, from \eqref{116}, \eqref{117}, $w_\alpha,g ,L^{-1}$ be non-decreasing and Lemma \ref{lem1}-(3), it follows that \begin{align*} w_\alpha(r) &\leq 2\int_0^r\Big(\int_0^t \overline{a}(s) \frac{g(L^{-1}(w_\alpha))}{l(L^{-1}(w_\alpha))}ds\Big)dt\\ &\leq 2g(L^{-1}(w_\alpha(r)))\int_0^r\Big(\int_0^t \frac{\overline{a}(s)}{l(L^{-1}(w_\alpha))}ds\Big)dt\\ &\leq 2g\Big(\sqrt{2\sqrt{\frac{\sigma}{(\sigma-1)k}}{w_\alpha (r)} +\frac{\sigma}{k}}\Big)\int_0^r\Big(\int_0^t \frac{\overline{a}(s)}{l(L^{-1} (w_\alpha))}ds\Big)dt\\ &\leq 2g\Big(2\sqrt[4]{\frac{\sigma}{(\sigma-1)k}} \sqrt{w_\alpha}\Big)\int_0^r \Big(\int_0^t \frac{\overline{a}(s)}{l(L^{-1}(w_\alpha))}ds\Big)dt\\ &\leq \frac{2}{\sqrt{\sigma-1}}g\Big(2\sqrt[4]{\frac{\sigma}{(\sigma-1)k}} \sqrt{w_\alpha}\Big)\Big[r\Big(\int_0^r\overline{a}(t)dt\Big) -\int_0^rt\overline{a}(t)dt\Big]\\ &\leq \frac{2}{\sqrt{\sigma-1}}g\Big(2\sqrt[4]{\frac{\sigma}{(\sigma-1)k}} \sqrt{w_\alpha}\Big)r \int_0^r\overline{a}(t)dt. \end{align*} for all $r>0$ sufficiently large. That is, it follows from the definition of $\mathcal{G}$, that $$ 2\sqrt[4]{\frac{\sigma}{(\sigma-1)k}}\sqrt{w_\alpha} \leq \mathcal{G}^{-1}\Big(r\int_0^r \overline{a}(t)dt\Big)\quad \text{for all }r>>0. $$ Now, setting $$ 00 :w_\alpha(r)\alpha + \overline{H}$ and for each $\alpha>\sigma/\sqrt{k}$ given. In fact, by assuming this is not true, then there exists a $\beta_0>\alpha + \overline{H}$ such that $w_\alpha(S(\beta_0))=w_\beta(S(\beta_0))$. So, by using that ${g(t)}/{t^\delta}$ is non-decreasing for $\delta>\sigma/(\sigma -1)$, Lemma \ref{lem1} and $w_{\alpha} \leq w_{\beta}$ on $[0,S(\beta_0)]$, we obtain \begin{equation}\label{wh} \begin{aligned} &\beta_0 \\ &= \alpha +\int_0^{S(\beta_0)}t^{1-N}\Big[\int_0^ts^{N-1} \Big(\overline{a}(s)\frac{g(L^{-1}(w_\alpha))}{l(L^{-1}(w_\alpha))} -\underline{a}(s)\frac{g(L^{-1}(w_\beta))}{l(L^{-1}(w_\beta))}\Big)ds\Big]dt\\ &= \alpha +\int_0^{S(\beta_0)}t^{1-N}\Big[\int_0^ts^{N-1} \Big(\overline{a}(s)\frac{g(L^{-1}(w_\alpha))}{l(L^{-1}(w_\alpha))} \\ &\quad -\underline{a}(s)\frac{g(L^{-1}(w_\beta))}{L^{-1}(w_\beta)^\delta} \frac{L^{-1}(w_\beta)^\delta}{l(L^{-1}(w_\beta))}\Big)ds\Big]dt\\ &\leq \alpha +\int_0^{S(\beta_0)}t^{1-N}\Big[\int_0^ts^{N-1} \Big(\overline{a}(s)\frac{g(L^{-1}(w_\alpha))}{l(L^{-1}(w_\alpha))} -\underline{a}(s)\frac{g(L^{-1}(w_\alpha))}{l(L^{-1}(w_\alpha))}\Big)ds\Big]dt\,. \end{aligned} \end{equation} On the other hand, from $g,l$ and $w_\alpha$ being non-decreasing, it follows that \begin{align*} 0 &\leq t^{1-N}\Big[\int_0^ts^{N-1}\Big(\overline{a}(s) \frac{g(L^{-1}(w_\alpha))}{l(L^{-1}(w_\alpha))} -\underline{a}(s)\frac{g(L^{-1}(w_\alpha))}{l(L^{-1}(w_\alpha))}\Big)ds\Big] \chi_{[0,S(\beta)]}(t)\\ &= t^{1-N}\Big[\int_0^ts^{N-1}a_{\rm osc}(s) \frac{g(L^{-1}(w_\alpha))}{l(L^{-1}(w_\alpha))}ds\Big]\\ &\leq \frac{1}{\sqrt{\sigma-1}}\Big(t^{1-N} \int_0^ts^{N-1}a_{\rm osc}(s)ds\Big) g\Big(\mathcal{G}^{-1} \Big(t\int_0^t\overline{a}(s)ds\Big)\Big): =\mathcal{H}(t) , \end{align*} for $t\gg0$, where $\chi_{[0,S(\beta)]}$ stands for the characteristic function of $[0,S(\beta)]$. So, from the hypothesis \eqref{eGcal} and \eqref{wh}, it follows that $$ \beta_0 \leq \alpha +\int_0^\infty \mathcal{H}(s)ds\leq \alpha+ \overline{H}, $$ but this is impossible. Now, by setting $\beta=(\alpha + \epsilon) + \overline{H}$, for each $\alpha>\sigma/\sqrt{k}$ and $\epsilon>0$ given, and by considering the problem \begin{equation}\label{Wn} \begin{gathered} \Delta w={a}(x)\frac{g(L^{-1}(w))}{l(L^{-1}(w))}\quad \text{in } B_n(0),\\ w\geq 0 \text{ in }B_n(0),\quad w=w_\alpha \quad \text{on } \partial B_n(0), \end{gathered} \end{equation} we can infer by standard methods of sub and super solutions that there exists a $w_n = w_{n,\alpha} \in C^1(\overline{B}_n)$ solution of \eqref{Wn} satisfying $\sigma/\sqrt{k}<\alpha\leq w_\alpha\leq w_n\leq w_\beta$ in $B_n$ for all $n \in \mathbb{N}$. So, by defining $w^n_m=w_m|_{B_n}$ for $m>n$ and for each $n \in \mathbb{N}$ given, where $w_m$ is a solution of Problem \eqref{Wn} in the ball $B_m(0)$, we obtain that $\{w^n_m\}$ is a bounded $m$-sequence in $C^{1,\nu_n}(\bar{B}_n)$ for some $0<\nu_n \leq 1$ by Regularity theory. Hence, we can extract subsequences of $\{w^n_m\}$ such that \begin{gather*} w^1_2, w^1_3, w^1_4, \dots \xrightarrow{C^1(\bar{B}_1)} w^1,\\ w^2_3, w^2_4, w^2_5, \dots \xrightarrow{C^1(\bar{B}_2)} w^2,\\ w^3_4, w^3_5, w^3_6, \dots \xrightarrow{C^1(\bar{B}_3)} w^3, \\ \dots \end{gather*} So, the function $w:\mathbb{R}^N \to (0,\infty)$ given by $w(x)=w^n(x)$ for $ x\in B_n$ is well-defined and the sequence $\{w^n_{2n}\}$ satisfies $w^n_{2n} \to w$ in $C^1(K)$ for any compact set $K \subset \mathbb{R}^N$ with $\sigma/\sqrt{k}<\alpha\leq w_\alpha\leq w\leq w_\beta$; that is, $w\in C^1(\mathbb{R}^N)$ and is a solution of \eqref{P}. \end{proof} \subsection*{Acknowledgements} Carlos Alberto Santos was supported by CAPES/Brazil Proc. no. 2788/2015-02. Jiazheng Zhou was supported by CNPq/Brazil Proc. no. 232373/2014-0. \begin{thebibliography}{00} \bibitem{adachi} S. Adachi, T. 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