\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 102, pp. 1--25.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/102\hfil Existence of bounded solutions] {Existence of bounded solutions for nonlinear fourth-order elliptic equations with strengthened coercivity and lower-order terms with natural growth} \author[M. V. Voitovich \hfil EJDE-2013/102\hfilneg] {Michail V. Voitovich} % in alphabetical order \address{Michail V. Voitovich \newline Institute of Applied Mathematics and Mechanics, Rosa Luxemburg Str. 74, 83114 Donetsk, Ukraine} \email{voytovich@bk.ru} \thanks{Submitted April 5, 2013. Published April 24, 2013.} \subjclass[2000]{35B45, 35B65, 35J40, 35J62} \keywords{Nonlinear elliptic equations; strengthened coercivity; \hfill\break\indent lower-order term; natural growth; Dirichlet problem; bounded solution; $L^{\infty}$-estimate} \begin{abstract} In this article, we consider nonlinear elliptic fourth-order equations with the principal part satisfying a strengthened coercivity condition, and a lower-order term having a ``natural'' growth with respect to the derivatives of the unknown function. We assume that there is an absorption term in the equation, but we do not assume that the lower-order term satisfies the sign condition with respect to unknown function. We prove the existence of bounded generalized solutions for the Dirichlet problem, and present some a priori estimates. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Skrypnyk \cite{Skr78} introduced a class of nonlinear elliptic equations of the form \begin{equation} \label{1intr} \sum_{|\alpha|\leq m}(-1)^{|\alpha|} D^\alpha \mathcal{A}_\alpha (x,u,\dots,D^{m}u)= 0 \quad \text{in } \Omega, \end{equation} where $m>1$ and $\Omega$ is a bounded domain of $\mathbb{R}^{n}$. All generalized solutions to this equation are bounded and H\"older continuous. This class of equations is characterized by a strengthened coercivity condition on coefficients $\mathcal{A}_\alpha$, $1\leq|\alpha|\leq m$. In a typical case this condition means that for every $x\in\Omega$ and every $\xi=\{\xi_{\alpha}\in\mathbb{R}:|\alpha|\leq m\}$, the following inequality holds: \begin{equation} \label{strcoer} \sum_{1\leq|\alpha|\leq m}\mathcal{A}_\alpha (x,\xi)\xi_{\alpha}\geq C \big\{\sum_{|\alpha|=1} |\xi_\alpha|^q + \sum_{|\alpha|=m} |\xi_\alpha|^{p} \big\} \end{equation} where $p\geq 2$, $mp0$. At the same time, in \cite{Skr78} it was assumed that the lower-order term $\mathcal{A}_0$ may have the growth of a rate less than $nq/(n-q)-1$ with respect to the function $u$ and the growth rates are definitely less than $q$ and $p$ with respect to the derivatives $D^\alpha u$, $|\alpha|=1$, and the derivatives $D^\alpha u$, $|\alpha|=m$, accordingly. We observe that the proof of the boundedness of generalized solutions in \cite{Skr78} uses a modification of Moser's method \cite{Moser}. Using an analogue of Stampacchia's method (see \cite{Stam61}, \cite{Stam66} and \cite{KindSt}), a weaker (exact) condition on integrability of data was established in \cite{KovVoi} to guarantee the boundedness of generalized solutions of nonlinear fourth-order equations with a strengthened coercivity. Moreover, a dependence of summability of generalized solutions of these equations on integrability of data was described in \cite{KovVoi}. Analogous results for nonlinear high-order equations with a strengthened coercivity were obtained in \cite{Voi07}. In the present article, we consider a class of nonlinear fourth-order equations of type \eqref{1intr} with the principal part satisfying a strengthened coercivity condition like \eqref{strcoer}, where $m=2$, and with the lower-order term $\mathcal{A}_0$ admitting, unlike \cite{KovVoi, Skr78, Voi07}, the growth of the rate $q$ with respect to the derivatives $D^\alpha u$, $|\alpha|=1$, and the growth of the rate $p$ with respect to the derivatives $D^\alpha u$, $|\alpha|=2$. The main result of the article is a theorem on the existence and $L^{\infty}$-estimate of bounded generalized solutions of the Dirichlet problem for the equations under investigation. We note that in the case under consideration, $q$ and $p$ are exponents of an energy space corresponding to the given problem. Similar results for nonlinear fourth-order equations with strengthened coercivity and a lower-order term of natural growth were established in \cite{Voi11} in the case where the lower-order term satisfies the sign condition $\mathcal{A}_0(x,u,Du,D^{2}u)u\geq 0$ and admits an arbitrary growth with respect to $u$. In the given article, we do not assume that the sign condition is satisfied. At the same time the presence of an absorption term in the left-hand side of the equation is required. Existence and $L^{\infty}$-estimate of bounded solutions of nonlinear elliptic second-order equations with natural growth lower-order terms were established for instance in \cite{BocMurPl92}--\cite{DrNic}. At the same time, in \cite{BocMurPl92, DlglGiachPl02} it is not assumed that the lower-order terms satisfy the sign condition. Observe that in order to obtain an $L^{\infty}$-estimate of a solution by Stampacchia's method, in \cite{BocMurPl92, DlglGiachPl02} superpositions of the solution and the functions \begin{equation} \label{2intr} (\exp(\lambda|s-T_{k}(s)|)-1)\operatorname{sign}(s-T_{k}(s)), \quad k>0, \; s\in\mathbb{R}, \end{equation} were used as test functions. Here $T_{k}(s)=\max\{\min\{s,k\},-k\}$ is the standard cut-off function. The use of the function $(\exp(\lambda|s|)-1)\operatorname{sign}s$ with a suitable $\lambda>0$ in the test functions (superpositions) leads to the absorption of the lower-order term of natural growth by the coercive principal part of the equation (see \cite{BocMurPl92, DlglGiachPl02}). In this article, for obtaining $L^{\infty}$-estimates, we modify the method of \cite{KovVoi} and use the functions $$ |s-h_{k}(s)|^{\lambda k}\exp(\lambda|s-h_{k}(s)|) \operatorname{sign}(s-h_{k}(s)), \quad k>0, \;s\in\mathbb{R}, $$ which play a role similar to that of functions \eqref{2intr} in the case of elliptic second-order equations with lower-order terms of natural growth. Here $h_{k}$ is an odd "cut-off " function of the class $C^2(\mathbb{R})$ such that $h_{k}(s)=s$ if $|s|\leq k$, and $h'_{k}(s)=0$ if $|s|\geq 2k$. We remark that a theory of existence and properties of solutions of nonlinear elliptic fourth-order equations with coefficients satisfying a strengthened coercivity condition and $L^1$-right-hand sides was developed in \cite{Kov01, Kov09}. \section{Preliminaries and the statement of the main result} Let $n\in\mathbb{N}$, $n>2$, and let $\Omega$ be a bounded open set of $\mathbb{R}^n$. We shall use the following notation: $\Lambda$ is the set of all $n$-dimensional multi-indices $\alpha$ such that $|\alpha|=1$ or $|\alpha|=2$; $\mathbb{R}^{n,2}$ is the space of all mappings $\xi: \Lambda\to\mathbb{R}$; if $ u\in W^{2,1}(\Omega)$, then $\nabla_2 u: \Omega\to\mathbb{R}^{n,2}$, and for every $ x\in\Omega$ and for every $ \alpha\in\Lambda$, $(\nabla_2u (x))_\alpha = D^\alpha u(x)$. If $r\in[1,+\infty]$, then $\|\cdot\|_{r}$ is the norm in $L^{r}(\Omega)$ and $r'=r/(r-1)$. For every measurable set $E\subset\Omega$ we denote by $\text{meas} E$ $n$-dimensional Lebesgue measure of the set $E$. Let $p\in(1, n/2)$ and $q\in(2p, n)$. We denote by $W^{1,q}_{2,p}(\Omega)$ the set of all functions in $W^{1,q}(\Omega)$ that have the second-order generalized derivatives in $L^p(\Omega)$. The set $W^{1,q}_{2,p}(\Omega)$ is a Banach space with the norm $$ \|u\| = \|u\|_{W^{1,q}(\Omega)} + \Big(\sum_{|\alpha|=2} \int_\Omega |D^\alpha u|^p dx\Big)^{1/p}. $$ We denote by ${\mathaccent"7017 W}^{1,q}_{2,p}(\Omega)$ the closure of the set $ C^\infty_0(\Omega)$ in $ W^{1,q}_{2,p}(\Omega)$. We set $q^\ast = nq/(n-q)$. As is known (see for instance \cite[Chapter 7]{GlbTr}), \begin{equation} {\mathaccent"7017 W}^{1,q}(\Omega)\subset L^{q\ast}(\Omega),\label{1} \end{equation} and there exists a positive constant $c$ depending only on $n$ and $q$ such that for every function $u\in{\mathaccent"7017 W}^{1,q}(\Omega)$, \begin{equation} \Big(\int_{\Omega} |u|^{q^{\ast}} dx\Big)^{1/q^{\ast}} \leq c\Big( \sum_{|\alpha|=1} \int_{\Omega} |D^\alpha u|^q dx\Big)^{1/q}. \label{2} \end{equation} Next, let $c_0, c_1, c_2, c_3, c_4, c_5>0$, let $g_1, g_2, g_3, g_4, g_5$ be nonnegative summable functions on $ \Omega$, $g_5 \in L^{q'}(\Omega)$, and let $A_0:\Omega\times \mathbb{R} \to \mathbb{R}$, $B:\Omega\times \mathbb{R}\times \mathbb{R}^{n,2} \to \mathbb{R}$ and $A_\alpha : \Omega\times \mathbb{R}^{n,2}\to\mathbb{R}$, $\alpha\in\Lambda$, be Carath\'{e}odory functions. We assume that for almost every $x\in\Omega$, for every $s\in\mathbb{R}$ and for every $\xi\in\mathbb{R}^{n,2}$ the following inequalities hold: \begin{gather} \sum_{|\alpha|=1}|A_\alpha (x,\xi)|^{q/(q-1)} \leq c_1 \big\{\sum_{|\alpha|=1}|\xi_\alpha|^q + \sum_{|\alpha|=2}|\xi_\alpha|^{p} \big\} + g_1(x), \label{3} \\ \sum_{|\alpha|=2}|A_\alpha (x,\xi)|^{p/(p-1)} \leq c_2 \big\{\sum_{|\alpha|=1}|\xi_\alpha|^q + \sum_{|\alpha|=2}|\xi_\alpha|^{p} \big\} + g_2(x), \label{4} \\ \sum_{\alpha\in\Lambda}A_\alpha (x,\xi)\xi_\alpha \geq c_3 \big\{\sum_{|\alpha|=1}|\xi_\alpha|^q + \sum_{|\alpha|=2}|\xi_\alpha|^{p} \big\} - g_3(x), \label{5} \\ |B(x,s,\xi)|\leq c_4\big\{\sum_{|\alpha|=1}|\xi_\alpha|^q + \sum_{|\alpha|=2}|\xi_\alpha|^{p} \big\}+ g_4(x), \label{7} \\ A_0(x,s)s\geq c_0|s|^q, \label{8} \\ |A_0(x,s)|\leq c_5|s|^{q-1}+ g_5(x). \label{grA} \end{gather} Further, let \begin{equation} f\in L^{q^\ast/(q^\ast-1)} (\Omega). \label{9} \end{equation} We consider the Dirichlet problem \begin{gather} \sum_{\alpha\in\Lambda}(-1)^{|\alpha|}D^\alpha A_\alpha (x,\nabla_2 u)+A_0(x,u)+B(x,u,\nabla_2 u) = f \quad \text{in } \Omega, \label{10} \\ D^\alpha u = 0, \quad |\alpha| = 0,1, \quad \text{on } \partial\Omega. \label{11} \end{gather} Observe that, by \eqref{3} and \eqref{4}, for every $u, v\in{\mathaccent"7017 W}^{1,q}_{2,p}(\Omega)$ and for every $\alpha\in\Lambda$ the function $A_\alpha(x,\nabla_2 u) D^\alpha v$ is summable on $\Omega$. By \eqref{grA}, for every $u, v\in{\mathaccent"7017 W}^{1,q}_{2,p}(\Omega)$ the function $A_0(x,u)v$ belongs to $L^{1}(\Omega)$, and by \eqref{7}, for every $u\in{\mathaccent"7017 W}^{1,q}_{2,p}(\Omega)$ and for every $v\in{\mathaccent"7017 W}^{1,q}_{2,p}(\Omega)\cap L^{\infty}(\Omega)$ the function $B(x,u,\nabla_2 u) v$ is summable on $\Omega$. Moreover, it follows from \eqref{1} and \eqref{9} that for every $v\in{\mathaccent"7017 W}^{1,q}_{2,p}(\Omega)$ the function $fv$ is summable on $\Omega$. \begin{definition} \rm A generalized solution of problem \eqref{10}, \eqref{11} is a function $u\in{\mathaccent"7017 W}^{1,q}_{2,p}(\Omega)$ such that for every function $v\in{\mathaccent"7017 W}^{1,q}_{2,p}(\Omega)\cap L^{\infty}(\Omega)$, \begin{equation}\label{12} \int_\Omega \big\{\sum_{\alpha\in\Lambda} A_\alpha(x,\nabla_2 u) D^\alpha v+A_0(x,u)v+B(x,u,\nabla_2 u) v\big\} dx = \int_\Omega f v dx. \end{equation} \end{definition} The following theorem is the main result of the present article. \begin{theorem} \label{th2.2} Let $r>n/q$, let the functions $g_2$, $g_3$, $g_4$ and $f$ belong to $L^r(\Omega)$, and let for almost every $x\in \Omega$ and for every $\xi, \xi' \in \mathbb{R}^{n,2}$, $\xi\neq \xi'$, the following inequality holds: \begin{equation} \sum_{\alpha\in\Lambda} [A_\alpha (x,\xi)- A_\alpha (x,\xi') ](\xi_{\alpha}- \xi'_{\alpha})>0. \label{6} \end{equation} Then there exists a generalized solution $u_0$ of problem {\rm\eqref{10}, \eqref{11}} such that $u_0\in L^{\infty}(\Omega)$ and \begin{equation} \|u_0\|_{\infty} \leq C_1 \label{121} \end{equation} where $C_1$ is a positive constant depending only on $n$, $p$, $q$, $r$, $\operatorname{meas} \Omega$, $c$, $c_0$, $c_2$, $c_3$, $c_4$ and the functions $g_2$, $g_3$, $g_4$ and $f$. \end{theorem} Let us give an example of functions satisfying conditions \eqref{3}--\eqref{grA} and \eqref{6}. \begin{example}\rm Let for every $n$-dimensional multiindex $\alpha$, $|\alpha|=1$, $A_\alpha:\Omega\times\mathbb{R}^{n,2}\to\mathbb{R}$ be the function defined by $$ A_\alpha(x,\xi) =\Big(\sum_{|\beta|=1}\xi^{2}_{\beta}\Big)^{(q-2)/2}\xi_{\alpha}, \quad (x,\xi)\in \Omega\times\mathbb{R}^{n,2}, $$ and let for every $n$-dimensional multiindex $\alpha$, $|\alpha|=2$, $A_\alpha:\Omega\times\mathbb{R}^{n,2}\to\mathbb{R}$ be the function defined by $$ A_\alpha(x,\xi)= \Big(\sum_{|\beta|=2}\xi^{2}_{\beta}\Big)^{(p-2)/2}\xi_{\alpha}, \quad (x,\xi)\in \Omega\times\mathbb{R}^{n,2}. $$ Then the functions $A_\alpha$, $\alpha\in\Lambda$, satisfy inequalities \eqref{3}--\eqref{5} and \eqref{6}. Next, let \begin{gather*} B(x,s,\xi)=b(x)\big\{\sum_{|\alpha|=1} |\xi_\alpha|^q + \sum_{|\alpha|=2} |\xi_\alpha|^{p}\big\}, \quad (x, s, \xi)\in \Omega\times \mathbb{R}\times \mathbb{R}^{n,2}, \\ A_0(x,s)=c_0|s|^{q-2}s, \quad (x, s)\in \Omega\times \mathbb{R}, \end{gather*} where $c_0>0$ and $b\in L^{\infty}(\Omega)$. Then the function $B$ satisfies inequality \eqref{7}, and the function $A_0$ satisfies inequalities \eqref{8} and \eqref{grA}. \end{example} Observe that the coefficients of the biharmonic operator $\Delta^{2}u$ do not satisfy condition \eqref{5}. We will prove Theorem \ref{th2.2} in Section \ref{proof1}. The key point of its proof is obtaining a priori energy- and $L^{\infty}$-estimates for bounded generalized solutions of problem \eqref{10}, \eqref{11}. These estimates are contained in the following two theorems which will be established in Sections \ref{proof2} and \ref{proof3} respectively. \begin{theorem} \label{th2.4} Let the functions $g_2$, $g_3$, $g_4$ and $f$ belong to $L^{n/q}(\Omega)$, and let $u$ be a generalized solution of problem {\rm \eqref{10}, \eqref{11}} such that $u\in L^{\infty}(\Omega)$. Then for every $\lambda>c_4/c_3$ we have \begin{equation} \int_\Omega \Big(\sum_{|\alpha|=1} |D^\alpha u|^q + \sum_{|\alpha|=2} |D^\alpha u|^p\Big)\exp(\lambda|u|) dx \leq C_2 \label{West} \end{equation} where $C_2$ is a positive constant depending only on $n$, $p$, $q$, $\operatorname{meas} \Omega$, $c$, $c_0$, $c_2$, $c_3$, $c_4$, $\lambda$ and the functions $g_2$, $g_3$, $g_4$ and $f$. \end{theorem} \begin{theorem}\label{th2.5} Let $r>n/q$, let the functions $g_2$, $g_3$, $g_4$ and $f$ belong to $L^r(\Omega)$, and let $u$ be a generalized solution of problem \eqref{10}, \eqref{11} such that $u\in L^{\infty}(\Omega)$. Then \begin{equation}\label{1213} \|u\|_{\infty} \leq C_1 \end{equation} where $C_1$ is the positive constant from Theorem \ref{th2.2}. \end{theorem} \begin{remark}\label{rem2.6}\rm The condition $r>n/q$ in the statements of Theorems \ref{th2.2} and \ref{th2.5} coincides with the condition of boundedness of generalized solutions of the Dirichlet problem considered in \cite{KovVoi} for equation \eqref{10} with $A_0\equiv 0$ and $B \equiv 0$. \end{remark} Before proving Theorems \ref{th2.2}, \ref{th2.4} and \ref{th2.5}, we give several auxiliary results. \section{Auxiliary results}\label{auxresults} By analogy with \cite[Lemma 2.2]{Kov01}, we establish the following result. \begin{lemma} \label{lem3.1} Let $u\in{\mathaccent"7017 W}^{1,q}_{2,p}(\Omega)\cap L^{\infty}(\Omega)$, $h\in C^{2}(\mathbb{R})$ and $h(0)=0$. Then $h(u)\in{\mathaccent"7017 W}^{1,q}_{2,p}(\Omega)\cap L^{\infty}(\Omega)$ and the following assertions hold: \begin{itemize} \item[(a)] for every $n$-dimensional multi-index $\alpha$, $|\alpha|=1$, $$ D^{\alpha}h(u)=h'(u)D^{\alpha}u \quad \text{a. e. in } \Omega, $$ \item[(b)] for every $n$-dimensional multi-index $\alpha$, $|\alpha|=2$, $$ D^{\alpha}h(u)=h'(u)D^{\alpha}u+h''(u)D^{\beta}u D^{\gamma}u \quad \text{a. e. in } \Omega, $$ where $\alpha=\beta+\gamma$, $|\beta|=|\gamma|=1$. \end{itemize} \end{lemma} \begin{lemma} \label{lem3.2} Let $h$ be an odd function on $\mathbb{R}$ such that $h\in C^{1}(\mathbb{R})$, $h\in C^{2}(\mathbb{R}\setminus \{0\})$ and $h''$ has a discontinuity of the first kind at the origin. Let \begin{equation}\label{18} u\in{\mathaccent"7017 W}^{1,q}_{2,p}(\Omega)\cap L^{\infty}(\Omega). \end{equation} Then $h(u)\in{\mathaccent"7017 W}^{1,q}_{2,p}(\Omega)\cap L^{\infty}(\Omega)$ and the following assertions hold: \begin{itemize} \item[(i)] for every $n$-dimensional multi-index $\alpha$, $|\alpha|=1$, $$ D^{\alpha}h(u)=h'(u)D^{\alpha}u \quad \text{a. e. in } \Omega ; $$ \item[(ii)] for every $n$-dimensional multi-index $\alpha$, $|\alpha|=2$, $$ D^{\alpha}h(u)=\begin{cases} h'(u)D^{\alpha}u+h''(u) D^{\beta}u D^{\gamma}u & \text{a. e. in } \{u\neq 0\},\\ h'(0)D^{\alpha}u & \text{a. e. in } \{u=0\} \end{cases} $$ where $\alpha=\beta+\gamma$, $|\beta|=|\gamma|=1$. \end{itemize} \end{lemma} \begin{proof} Let $u\in{\mathaccent"7017 W}^{1,q}_{2,p}(\Omega)\cap L^{\infty}(\Omega)$. We define the function $H:\mathbb{R}\to \mathbb{R}$ by \begin{equation} H(s)=h(s)-h'(0)s, \quad s\in \mathbb{R}. \label{21} \end{equation} Let \begin{equation} w_\alpha=H'(u)D^{\alpha}u \quad \text{if } |\alpha|=1, \label{22} \end{equation} and let \begin{equation} w_\alpha = \begin{cases} H'(u)D^{\alpha}u+H''(u)D^{\beta}u D^{\gamma}u & \text{in }\{u\neq 0\},\\ 0 &\text{in }\{u=0\} \end{cases} \label{23} \end{equation} if $|\alpha|=2$ and $\alpha=\beta+\gamma$, $|\beta|=|\gamma|=1$. Clearly, \begin{equation} w_\alpha\in L^q(\Omega) \text{ if } |\alpha|=1 \quad\text{and}\quad w_\alpha\in L^{p}(\Omega) \text{ if } |\alpha|=2. \label{20} \end{equation} We fix $\varepsilon>0$. Let $H_{\varepsilon}:\mathbb{R}\to \mathbb{R}$ be the function such that $$ H_\varepsilon(s) = \begin{cases} H(s)+(\frac{1}{2}\varepsilon H''(\varepsilon) -H'(\varepsilon))(s-\varepsilon) +\frac{1}{6}\varepsilon^{2}H''(\varepsilon)-H(\varepsilon) &\text{if } s>\varepsilon, \\ H''(\varepsilon)s^{3}/(6\varepsilon) & \text{if } |s|\leq \varepsilon, \\ H(s)+(\frac{1}{2}\varepsilon H''(\varepsilon)-H'(\varepsilon))(s+\varepsilon) -\frac{1}{6}\varepsilon^{2}H''(\varepsilon)+H(\varepsilon) &\text{if } s<-\varepsilon. \end{cases} $$ We have \begin{gather} H_{\varepsilon}\in C^{2}(\mathbb{R}), \label{13} \\ H'_{\varepsilon}(s) = \begin{cases} H'(s)+\varepsilon H''(\varepsilon)/2-H'(\varepsilon) &\text{if } |s|>\varepsilon, \\ H''(\varepsilon)s^{2}/(2\varepsilon) & \text{if } |s|\leq \varepsilon, \end{cases} \nonumber \\ H''_{\varepsilon}(s) = \begin{cases} H''(s) &\text{if } |s|>\varepsilon, \\ H''(\varepsilon)s/\varepsilon & \text{if } |s|\leq \varepsilon. \end{cases} \nonumber \end{gather} The following limit relations hold: \begin{gather} \lim_{\varepsilon\to 0}H_{\varepsilon}(s)=H(s) \quad\forall s\in\mathbb{R}, \label{15} \\ \lim_{\varepsilon\to 0}H'_{\varepsilon}(s)=H'(s) \quad \forall s\in\mathbb{R}, \label{16} \\ \lim_{\varepsilon\to 0}H''_{\varepsilon}(s) = \begin{cases} H''(s) & \text{if } s\in \mathbb{R}\setminus \{0\}, \\ 0 & \text{if } s=0. \end{cases} \label{17} \end{gather} Using inclusions \eqref{18} and \eqref{13}, the equality $H_{\varepsilon}(0)=0$ and Lemma \ref{lem3.1}, we establish that $H_{\varepsilon}(u)\in{\mathaccent"7017 W}^{1,q}_{2,p}(\Omega)\cap L^{\infty}(\Omega)$, $D^{\alpha}H_{\varepsilon}(u)=H'_{\varepsilon}(u)D^{\alpha}u$ if $|\alpha|=1$, and $ D^{\alpha}H_{\varepsilon}(u)=H'_{\varepsilon}(u)D^{\alpha}u+H''_{\varepsilon}(u)D^{\beta}u D^{\gamma}u$ if $|\alpha|=2$ and $\alpha=\beta+\gamma$, $|\beta|=|\gamma|=1$. Hence, using \eqref{18}, \eqref{15}--\eqref{17} along with Dominated Convergence Theorem, we deduce that \begin{gather} \lim_{\varepsilon\to 0}\|H_{\varepsilon}(u)-H(u)\|_{L^q(\Omega)}=0, \label{14}\\ \lim_{\varepsilon\to0} \sum_{|\alpha|=1}\|D^{\alpha}H_{\varepsilon}(u) -w_\alpha\|_{L^q(\Omega)}=0, \quad \lim_{\varepsilon\to0} \sum_{|\alpha|=2}\|D^{\alpha}H_{\varepsilon}(u) -w_\alpha\|_{L^{p}(\Omega)}=0. \label{19} \end{gather} Using these limit relations, in the usual way we establish that for every $\alpha\in \Lambda$ there exists the generalized derivative $D^{\alpha}H(u)$, and $D^{\alpha}H(u)=w_\alpha$ a. e. on $\Omega$. Then, by \eqref{20}, \eqref{14} and \eqref{19}, the function $H(u)$ belong to ${\mathaccent"7017 W}^{1,q}_{2,p}(\Omega)\cap L^{\infty}(\Omega)$, and \eqref{21}--\eqref{23} imply that assertions (i) and (ii) hold. The proof is complete. \end{proof} The next result is similar to the corresponding part of Stampacchia's lemma \cite{Stam66}. \begin{lemma} \label{lem3.3} Let $\varphi$ be a nonincreasing nonnegative function on $[0, +\infty)$. Let $C>0$, $b_1\geq 0$, $b_2\geq 0$, $0\leq\tau_1<\tau_2$, $\gamma>1$ and $k_0\geq 0$. Let for every $k$ and $l$ such that $k_0 \max\{k_0,1\}$ and \begin{equation}\label{st2} d ^{(\tau_2-\tau_1)(k_0+d/2)+b_2-b_1} \geq 2^{2\tau_1d+b_1+ (2\gamma-1)(2\tau_2d+b_2)/(\gamma-1)} C [\varphi(k_0)]^{\gamma-1}. \end{equation} Then $ \varphi(k_0+d)=0$. \end{lemma} \begin{proof} We set $a=(2\tau_2d+b_2)/(\gamma-1)$, and let for every $j\in\mathbb{N}$, \begin{equation}\label{st3} k_j = k_0 + d - \frac{d}{2^j} . \end{equation} Then for every $j\in\mathbb{N}$ we have $$ k_01$, we establish that for every $j\in\mathbb{N}$, $$ \varphi(k_{j+1})\leq \frac{C 2^{2\tau_1d+b_1}\cdot2^{(j+1)(2\tau_2d+b_2)}} {d^{(\tau_2-\tau_1)(k_0+d/2)+b_2-b_1}} [\varphi(k_{j})]^{\gamma}. $$ By means of the latter inequality and \eqref{st2}, we establish by induction on $j$, that for every $j\in\mathbb{N}$, $$ \varphi(k_j) \leq 2^{-a(j-1)} \varphi(k_0). $$ Using this result and relation \eqref{st3} and taking into account that the function $\varphi$ is nonincreasing and nonnegative, we deduce that $\varphi(k_0+d)=0$. The proof is complete. \end{proof} \section{Proof of Theorem \ref{th2.4}}\label{proof2} Let the functions $g_2$, $g_3$, $g_4$ and $f$ belong to $L^{n/q}(\Omega)$, and let $u$ be a bounded generalized solution of problem \eqref{10}, \eqref{11}. We fix an arbitrary positive number $\lambda$ such that \begin{equation} \lambda>c_4/c_3. \label{24} \end{equation} By $c_i$, $i=6,7,\dots$, we shall denote positive constants depending only on $n$, $p$, $q$, $\operatorname{meas} \Omega$, $c$, $c_0$, $c_2$, $c_3$, $c_4$, $\lambda$ and the functions $g_2$, $g_3$, $g_4$ and $f$. We define the function $h:\mathbb{R}\to\mathbb{R}$ by $$ h(s) =(e^{\lambda |s|}-1)\operatorname{sign}s, \quad s\in\mathbb{R}. $$ We set $c_6=c_3\lambda-c_4$. By \eqref{24}, we have $c_6>0$. Elementary calculations show that \begin{equation} c_3h'-c_4|h|>c_6h' \quad \text{in } \mathbb{R}. \label{25} \end{equation} We set \begin{gather*} I' = \int_{\{u\neq 0\}} \big\{\sum_{|\alpha|=2} |A_\alpha (x,\nabla_2 u)|\big\} \big\{\sum_{|\beta|=1} |D^\beta u|^2\big\} |h''(u)| dx, \\ \Phi = \sum_{|\alpha|=1} |D^\alpha u|^q + \sum_{|\alpha|=2} |D^\alpha u|^p. \end{gather*} By Lemma \ref{lem3.2}, $h(u)\in\overset{\circ}{W}{}^{1,q}_{2,p}(\Omega)\cap L^{\infty}(\Omega)$. Then, by \eqref{12}, we have $$ \int_\Omega \big\{\sum_{\alpha\in\Lambda}A_\alpha(x,\nabla_2 u)D^\alpha h(u)+A_0(x,u)h(u)+B(x,u,\nabla_2 u)h(u)\big\} dx = \int_\Omega f h(u) dx. $$ From this equality and assertions (i) and (ii) of Lemma \ref{lem3.2} we deduce that \begin{align*} &\int_\Omega \big\{\sum_{\alpha\in\Lambda} A_\alpha (x,\nabla_2 u) D^\alpha u\big\} h'(u)dx+\int_\Omega A_0(x,u)h(u)dx\\ &\leq\int_\Omega |B(x,u,\nabla_2 u)||h(u)|dx +\int_\Omega |f|| h(u)| dx+I'. \end{align*} Hence, using \eqref{5}--\eqref{8} and the facts that $00$. It is obvious that $$ \frac{p-1}{p} + \frac{2}{q} + \frac{q-2p}{qp} = 1. $$ Using this equality and Young's inequality, we establish that if $\alpha\in \Lambda$, $|\alpha|=2$, and $\beta\in \Lambda$, $|\beta|=1$, then $$ |A_\alpha(x,\nabla_2 u)| |D^\beta u|^2 \leq \varepsilon^2 |A_\alpha(x,\nabla_2 u)|^{p/(p-1)} + \varepsilon^2 |D^\beta u|^q + \varepsilon^{2-2qp/(q-2p)} \ \ \ \text{on} \ \ \Omega. $$ From this and \eqref{4} we deduce that \begin{align*} I' & \leq n(c_2 + n)\varepsilon^2 \int_{\{u\neq0\}} \Phi |h''(u)| dx + n \varepsilon^2 \int_{\{u\neq0\}} g_2 |h''(u)|dx \\ &\quad + n^3 \varepsilon^{2-2qp/(q-2p)} \int_{\{u\neq0\}} |h''(u)| dx. \end{align*} Putting in this inequality $\varepsilon=(\frac{c_6}{2\lambda n(c_2+n)})^{1/2}$, and noting that $|h''|=\lambda h'$ and $|h''|=\lambda^{2}|h|+\lambda^{2}$ on $\mathbb{R}\setminus\{0\}$, we obtain $$ I' \leq\frac{c_6}{2}\int_\Omega h'(u)\Phi dx+c_7\int_\Omega(g_2+1)|h(u)|dx+c_8. $$ From this and \eqref{26} it follows that \begin{equation} \frac{c_6}{2}\int_\Omega h'(u)\Phi dx+c_0\int_\Omega |u|^{q-1}|h(u)|dx\leq c_9\int_\Omega F|h(u)|dx+c_{10} \label{27} \end{equation} where $F=g_2+\lambda g_3+g_4+|f|+1$. Now, we estimate the integral $\int_\Omega F|h(u)|dx$. We fix an arbitrary $K>0$. It is clear that \begin{gather} \begin{aligned} &\int_\Omega F|h(u)|dx\\ &=\int_{\{F>K, |u|\geq 1\}} F|h(u)|dx+\int_{\{FK, |u|< 1\}} F|h(u)|dx, \end{aligned} \label{28} \\ \int_{\{FK, |u|< 1\}} F|h(u)|dx<(e^{\lambda}-1)\int_\Omega F dx. \label{29} \end{gather} Before estimating the first integral in the right-hand side of equality \eqref{28}, we remark that there exists a positive constant $c_{q,\lambda}$ depending only on $q$ and $\lambda$ such that \begin{equation} |h(s)|\leq c_{q,\lambda}|h(s/q)|^q \quad \text{for every } s\geq 1 . \label{31} \end{equation} Note also that, by \eqref{2}, assertion (i) of Lemma \ref{lem3.2} and equality $(h'(s/q))^q=\lambda^{q-1}h'(s)$, $s\in\mathbb{R}$, we have \begin{equation} \Big(\int_\Omega |h(u/q)|^{q^{\ast}}dx\Big)^{q/q^{\ast}} \leq (c\lambda^{q-1}/q^q)\int_\Omega h'(u)\Phi dx. \label{30} \end{equation} Now, using Holder's inequality, \eqref{31} and \eqref{30}, we obtain \begin{align*} \int_{\{F>K, |u|\geq 1\}} F|h(u)|dx & \leq \Big(\int_{\{F>K\}} F^{n/q}dx\Big)^{q/n} \Big(\int_{\{|u|\geq 1\}} |h(u)|^{n/(n-q)}dx\Big)^{(n-q)/n} \\ & \leq c_{q,\lambda}\|F\|_{L^{n/q}(\{F>K\})}\Big(\int_{\Omega} |h(u/q)|^{q^{\ast}}dx\Big)^{q/q^{\ast}} \phantom{\int_{\Omega}^{\Omega}} \\ & \leq c_{q,\lambda}c\lambda^{q-1}q^{-q}\|F\|_{L^{n/q}(\{F>K\})}\int_\Omega h'(u)\Phi dx. \end{align*} From this along with \eqref{28}--\eqref{29} it follows that \begin{equation}\label{32'} \int_\Omega F|h(u)|dx \leq c_{11}\|F\|_{L^{n/q}(\{F>K\})}\int_\Omega h'(u)\Phi dx+\int_\Omega K|h(u)|dx+c_{12}. \end{equation} Now, choosing $K>0$ such that $c_9c_{11}\|F\|_{L^{n/q}(\{F>K\})}c_9K\}}|u|^{q-1}|h(u)|dx\\ &> c_9\int_{\{c_0|u|^{q-1}>c_9K\}} K |h(u)|dx, \end{aligned} \label{33} \\ \begin{aligned} &c_9\int_\Omega K|h(u)|dx\\ &=c_9\int_{\{ c_0|u|^{q-1}>c_9K\}}K |h(u)|dx+c_9\int_{\{ c_0|u|^{q-1}\leq c_9K\}}K |h(u)|dx \\ &\leq c_9\int_{\{ c_0|u|^{q-1}>c_9K\}}K |h(u)|dx+c_9K(e^{\lambda(c_9K/c_0)^{1/(q-1)}}-1) \operatorname{meas}\Omega . \end{aligned} \label{34} \end{gather} From \eqref{32}--\eqref{34} it follows that $$ \frac{c_6}{4}\int_\Omega h'(u)\Phi dx\leq c_{13}. $$ Hence, taking into account that for every $s\in\mathbb{R}$, $h'(s)=\lambda\exp(\lambda|s|)$, we deduce \eqref{West}. The proof is comlete. \section{Proof of Theorem \ref{th2.5}}\label{proof3} Let $r>n/q$, let the functions $g_2$, $g_3$, $g_4$ and $f$ belong to $L^{r}(\Omega)$, and let $M$ be a majorant for $\|g_2\|_{r}$, $\|g_3\|_{r}$, $\|g_4\|_{r}$ and $\|f\|_{r}$. Let $u$ be a generalized solution of problem \eqref{10}, \eqref{11} such that \begin{equation}\label{uboun} u\in L^{\infty}(\Omega). \end{equation} In view of the assumption $r>n/q$, we have \begin{equation}\label{num1} qr/(r-1)(r-1)/r,\\ \label{vartheta} \vartheta=q/n-1/r-2qp\delta/(q-2p),\\ \label{theta} \theta(q-1)-\vartheta q^{\ast}<0, \\ \label{t} t=2+1/\delta . \end{gather} We set \begin{equation}\label{lambda} \lambda=2c_4/c_3. \end{equation} Without loss of generality, we may assume that \begin{equation}\label{lambda1} \lambda>1. \end{equation} By \eqref{2} and \eqref{3.1}, for every $k>0$ we have \begin{equation}\label{3.2.00} \varphi(k) \leq C_2^{q^\ast/q}c^{q^\ast} k^{-q^\ast}. \end{equation} Therefore, there exists a positive number $k_{\ast}$ depending only on $n$, $p$, $q$, $t$, $\theta$, $\vartheta$, $c$, $c_2$, $c_3$, $c_4$, $C_2$, and $\|\tilde{F}\|_{r}$ such that \begin{gather}\label{3.2} \forall k \geq k_{\ast}, \quad 2(c_2+n)(\lambda t(t+1)nk)^{2}[\varphi(k)]^{1/(t-2)} < \min\{1/2, c_3/12\}, \\ \label{3.2.4} \forall k \geq k_{\ast} , \quad (c/q)^q(\lambda(t+1))^{q-1}\|\tilde{F}\|_{r}k^{\theta(q-1)} [\varphi(k)]^{\vartheta}2$. Hence, \begin{equation}\label{PsiC2} \Psi\in C^2(\mathbb{R}), \end{equation} and for every $s\in\mathbb{R}$ the following equalities hold: \begin{gather}\label{Psi'} \Psi'(s)=|s|^{\mu-1}\exp(\lambda|s|)(\mu+\lambda|s|) =\lambda|\Psi(s)|+\mu|s|^{\mu-1}\exp(\lambda|s|), \\ \label{Psi''} \Psi''(s)=|s|^{\mu-2}\exp(\lambda|s|)(\mu(\mu-1)+2\lambda\mu|s|+\lambda^{2} s^{2})\operatorname{sign} s. \hskip 35pt \end{gather} Let us prove the following assertion: \begin{itemize} \item[(A5)] if $s\in\mathbb{R}$, then \begin{equation}\label{3.8} c_3\Psi'(G_k(s))G'_k(s)-c_4|\Psi(G_k(s))|\geq \frac{c_3}{2} \Psi'(G_k(s))G'_k(s). \end{equation} \end{itemize} Indeed, if $s\in\mathbb{R}$ and $|s|\leq k$, then both sides of inequality \eqref{3.8} are equal zero and therefore, this inequality is true. Now, let $k<|s|<2k$ and $y=(|s|-k)/k$. By \eqref{G_k(s)}, \eqref{G'_k(s)} and the inequality $y<1$, we have \begin{equation} \label{G/G'} \begin{aligned} \frac{|G_k(s)|}{G'_k(s)} &=\frac{k}{t+1}\Big(y+\frac{t}{(t-1)(t-(t-1)y)}-\frac{1}{t-1}\Big)\\ &<\frac{k}{t+1}\Big(1+\frac{t}{t-1}-\frac{1}{t-1}\Big)=\frac{2k}{t+1} . \end{aligned} \end{equation} Using \eqref{mu}, \eqref{Psi}, \eqref{Psi'}, \eqref{G/G'} and the inequality $t>1$, we obtain \begin{align*} & c_3 \Psi'(G_k(s))G'_k(s)-c_4|\Psi(G_k(s))| \\ &=|G_k(s)|^{\mu-1}\exp(\lambda|G_k(s)|)G'_k(s)(c_3\mu+c_3\lambda|G_k(s)| -c_4|G_k(s)|/G'_k(s)) \\ &\geq|G_k(s)|^{\mu-1}\exp(\lambda|G_k(s)|)G'_k(s)(c_3\mu+c_3\lambda|G_k(s)| -c_4k) \\ &\geq \frac{c_3}{2}|G_k(s)|^{\mu-1}\exp(\lambda|G_k(s)|)G'_k(s) (2\mu-\frac{2c_4k}{c_3}+\lambda|G_k(s)|)=\frac{c_3}{2}\Psi'(G_k(s))G'_k(s). \end{align*} Thus, inequality \eqref{3.8} holds. Finally, let $|s|\geq 2k$. Using \eqref{lambda}, \eqref{Psi}, \eqref{Psi'} and the equality $G'_k(s)=1$, we obtain \begin{align*} &c_3\Psi'(G_k(s))G'_k(s)-c_4|\Psi(G_k(s))|\\ &=|G_k(s)|^{\mu-1}\exp(\lambda|G_k(s)|)(c_3\mu+(\lambda c_3-c_4)|G_k(s)|) \\ &\geq \frac{c_3}{2} |G_k(s)|^{\mu-1}\exp(\lambda|G_k(s)|) (\mu+\lambda|G_k(s)|) \\ &=\frac{c_3}{2} \Psi'(G_k(s))G'_k(s). \end{align*} Therefore, inequality \eqref{3.8} holds. Thus, inequality \eqref{3.8} holds for every $s\in\mathbb{R}$, and assertion (A5) is proved. \textit{Step 2.} Using inclusions \eqref{uboun}, \eqref{hC2}, \eqref{PsiC2}, the equalities $G_k(0)=\Psi(0)=0$ and Lemma \ref{lem3.1}, we establish that $\Psi(G_k(u))\in\overset{\circ}{W}{}^{1,q}_{2,p}(\Omega)\cap L^{\infty}(\Omega)$ and the following assertions hold: \begin{itemize} \item[(A6)] for every $n$-dimensional multi-index $\alpha, |\alpha|=1$, $$ D^\alpha \Psi(G_k(u)) = \Psi'(G_k(u))G'_k(u) D^\alpha u \ \ \ \ \text{a. e. in } \ \ \Omega ; $$ \item[(A7)] for every $n$-dimensional multi-index $\alpha, |\alpha|=2$, \begin{align*} D^\alpha \Psi(G_k(u)) &= \Psi'(G_k(u))G'_k(u) D^\alpha u +\big[\Psi''(G_k(u))(G'_k(u))^{2}\\ &\quad -\Psi'(G_k(u))h''_k(u)\big]D^{\beta}uD^{\gamma} u\quad \text{a. e. in } \Omega \end{align*} where $\alpha=\beta+\gamma$, $|\beta|=|\gamma|=1$. \end{itemize} We set \begin{gather*} I'_k = \int_\Omega \big\{\sum_{|\alpha|=2} |A_\alpha (x,\nabla_2 u)|\big\} \big\{\sum_{|\beta|=1} |D^\beta u|^2\big\}\Psi'(G_k(u)) |h''_k (u)|dx, \\ I''_k = \int_\Omega \big\{\sum_{|\alpha|=2} |A_\alpha (x,\nabla_2 u)|\big\} \big\{\sum_{|\beta|=1} |D^\beta u|^2\big\}\Psi''(G_k(u))(G'_k(u))^{2}dx. \end{gather*} Putting the function $\Psi(G_k(u))$ into \eqref{12} instead of $v$, we obtain \begin{align*} &\int_\Omega \big\{\sum_{\alpha\in\Lambda} A_\alpha (x,\nabla_2 u) D^\alpha \Psi(G_k(u))\big\} dx\\ &+\int_\Omega A_0(x,u)\Psi(G_k(u))dx +\int_\Omega B(x,u,\nabla_2 u)\Psi(G_k(u))dx\\ & = \int_\Omega f\Psi(G_k(u)) dx. \end{align*} From this equality and assertions (A6) and (A7) we deduce that \begin{align*} &\int_\Omega \big\{\sum_{\alpha\in\Lambda} A_\alpha (x,\nabla_2 u) D^\alpha u\big\}\Psi'(G_k(u))G'_k(u)dx+\int_\Omega A_0(x,u)\Psi(G_k(u))dx \\ &\leq I'_k + I''_k+\int_\Omega |B(x,u,\nabla_2 u)||\Psi(G_k(u))|dx + \int_\Omega |f||\Psi(G_k(u))| dx. \hskip 6pt \end{align*} Hence, using \eqref{5}--\eqref{8}, we obtain \begin{align*} &\int_\Omega \big\{c_3\Psi'(G_k(u))G'_k(u)-c_4|\Psi(G_k(u))|\big\} \Phi dx+c_0\int_\Omega |u|^{q-1}|\Psi(G_k(u))|dx \\ &\leq I'_k + I''_k + \int_\Omega g_3 \Psi'(G_k(u))G'_k(u)dx+\int_\Omega (g_4+|f|)|\Psi(G_k(u))| dx. \end{align*} In turn, from this and assertion (A5) it follows that \begin{equation} \label{3.9} \begin{aligned} &\frac{c_3}{2}\int_\Omega\Phi \Psi'(G_k(u))G'_k(u) dx\\ &\leq I'_k + I''_k + \int_\Omega g_3 \Psi'(G_k(u))G'_k(u)dx +\int_\Omega (g_4+|f|)|\Psi(G_k(u))| dx. \end{aligned} \end{equation} \textit{Step 3.} Let us obtain suitable estimates for the addends in the right-hand side of this inequality. First, assume that $\varphi(k)>0$. We set \begin{equation}\label{3.10} \varepsilon = [\varphi(k)]^{1/(t-2)}. \end{equation} Since $k\geq k_0$, by \eqref{3.2} and \eqref{3.4} we have $\varphi(k)<1$. Therefore, \begin{equation}\label{varepsilon} 0<\varepsilon<1. \end{equation} We shall prove the inequality \begin{equation} \label{3.20} \begin{aligned} I'_k&\leq \frac{c_3}{12}\int_\Omega\Phi\Psi'(G_k(u))G'_k(u) dx+\frac{1}{2}\int_\Omega g_2|\Psi(G_k(u))|dx \\ &\quad +\frac{1}{2} \varepsilon^{-2qp/(q-2p)} \int_\Omega|\Psi(G_k(u))|dx+c_{14}[\varphi(k)]^{1/r'}. \end{aligned} \end{equation} Obviously, $$ \frac{p-1}{p} + \frac{2}{q} + \frac{q-2p}{qp} = 1. $$ Using this equality and the Young's inequality, we establish that if $\alpha$ is an $n$-dimensional multi-index, $|\alpha|=2$, and $\beta$ is an $n$-dimensional multi-index, $|\beta|=1$, then \[ |A_\alpha(x,\nabla_2 u)| |D^\beta u|^2 \leq \varepsilon^2 |A_\alpha(x,\nabla_2 u)|^{p/(p-1)} + \varepsilon^2 |D^\beta u|^q + \varepsilon^{2-2qp/(q-2p)} \quad \text{on }\Omega. \] This and relation \eqref{4} yields \begin{equation} \label{3.11} \begin{aligned} I'_k &\leq n(c_2 + n)\varepsilon^2 \int_\Omega \Phi\Psi'(G_k(u)) |h''_k(u)| dx + n \varepsilon^2 \int_\Omega g_2 \Psi'(G_k(u)) |h''_k(u)|dx \\ &\quad + n^3 \varepsilon^{2-2qp/(q-2p)} \int_\Omega \Psi'(G_k(u))|h''_k(u)| dx. \end{aligned} \end{equation} Let us estimate the second integral in the right-hand side of \eqref{3.11}. By \eqref{3.7}, \eqref{mu}, \eqref{Psi'} and the inequality $k>1$, we have \begin{equation}\label{3.12} \int_\Omega g_2 \Psi'(G_k(u)) |h''_k(u)|dx\leq \lambda t^{2} \int_\Omega g_2(|\Psi(G_k(u))|+|G_k(u)|^{\mu-1}\exp(\lambda|G_k(u)|))dx. \end{equation} Also, it is clear that \begin{equation} \label{3.13} \begin{aligned} &\int_\Omega g_2|G_k(u)|^{\mu-1}\exp(\lambda|G_k(u)|)dx\\ &=\int_{\{|G_k(u)|<1\}} g_2|G_k(u)|^{\mu-1}\exp(\lambda|G_k(u)|)dx\\ &\quad +\int_{\{|G_k(u)|\geq1\}} g_2|G_k(u)|^{\mu-1}\exp(\lambda|G_k(u)|)dx \\ &\leq e^{\lambda}\int_{\{|u|\geq k\}} g_2dx+\int_\Omega g_2|\Psi(G_k(u))|dx \\ &\leq e^{\lambda}\|g_2\|_{r}[\varphi(k)]^{1/r'}+\int_\Omega g_2|\Psi(G_k(u))|dx. \end{aligned} \end{equation} From \eqref{3.12} and \eqref{3.13} it follows that \begin{equation}\label{3.14} \int_\Omega g_2 \Psi'(G_k(u)) |h''_k(u)|dx\leq 2\lambda t^{2} \int_\Omega g_2|\Psi(G_k(u))|dx+c_{15}\|g_2\|_{r}[\varphi(k)]^{1/r'}. \end{equation} Similar to \eqref{3.14} we obtain the following estimate of the third integral in the right-hand side of inequality \eqref{3.11}: \begin{equation}\label{3.15} \int_\Omega \Psi'(G_k(u)) |h''_k(u)|dx\leq 2\lambda t^{2} \int_\Omega |\Psi(G_k(u))|dx+c_{15}[\varphi(k)]. \end{equation} Before estimating the first integral in the right-hand side of inequality \eqref{3.11}, we remark that \begin{equation}\label{Psi'e} \Psi'(G_k(s))< 2\lambda e^{\lambda}k\varepsilon^{t(\mu-1)} \quad \text{if } k\leq |s|1$, we have \begin{align*} \Psi'(G_k(s)) &=|G_k(s)|^{\mu-1}(\lambda k+\lambda|G_k(s)|)\exp(\lambda|G_k(s)|)\\ & <\varepsilon^{t(\mu-1)}e^{\lambda\varepsilon^{t}}(\lambda k+\lambda)<2\lambda e^{\lambda} k \varepsilon^{t(\mu-1)}. \end{align*} Hence, assertion \eqref{Psi'e} is true. Next, it is clear that \begin{equation} \label{3.16} \begin{aligned} \int_\Omega \Phi\Psi'(G_k(u)) |h''_k (u)|dx &=\int_{\{k\leq |u|1$, we obtain \eqref{3.20}. \textit{Step 4.} Let us estimate the integral $I''_k$. We shall establish the inequality \begin{equation} \label{3.38} \begin{aligned} I''_k&\leq \frac{c_3}{12}\int_\Omega\Phi\Psi'(G_k(u))G'_k(u) dx+\frac{1}{2}\int_\Omega g_2|\Psi(G_k(u))|dx \\ &\quad +\frac{1}{2} \varepsilon^{-2qp/(q-2p)}\int_\Omega|\Psi(G_k(u))|dx +c_{16}[\varphi(k)]^{1/r'}. \end{aligned} \end{equation} Similar to \eqref{3.11}, we have \begin{equation} \label{3.21} \begin{aligned} I''_k &\leq n(c_2 + n)\varepsilon^2 \int_\Omega \Phi|\Psi''(G_k(u))| (G'_k(u))^{2} dx + n \varepsilon^2 \int_\Omega g_2 |\Psi''(G_k(u))| (G'_k(u))^{2}dx \\ &\quad + n^3 \varepsilon^{2-2qp/(q-2p)} \int_\Omega |\Psi''(G_k(u))| (G'_k(u))^{2} dx. \end{aligned} \end{equation} Let us estimate the first integral in the right-hand side of inequality \eqref{3.21}. By \eqref{Psi'}, \eqref{Psi''} and \eqref{3.6}, for every $s\in\mathbb{R}$, $$ |\Psi''(G_k(s))| (G'_k(s))^{2} \leq \mu^{2}|G_k(s)|^{\mu-2}\exp(\lambda|G_k(s)|)(G'_k(s))^{2}+2\lambda \Psi'(G_k(s))G'_k(s). $$ From this it follows that \begin{equation} \label{3.22} \begin{aligned} \int_\Omega \Phi|\Psi''(G_k(u))| (G'_k(u))^{2} dx & \leq \int_\Omega \Phi\mu^{2}|G_k(u)|^{\mu-2} \exp(\lambda|G_k(u)|)(G'_k(u))^{2} dx\\ &\quad + 2\lambda\int_\Omega \Phi\Psi'(G_k(u))G'_k(u)dx. \end{aligned} \end{equation} Clearly, \begin{equation} \label{3.23} \begin{aligned} &\int_\Omega \Phi\mu^{2}|G_k(u)|^{\mu-2}\exp(\lambda|G_k(u)|)(G'_k(u))^{2}dx \\ &= \int_{\{k\leq |u| < k(1+\varepsilon k^{-1/(t-2)})\}} \Phi\mu^{2}|G_k(u)|^{\mu-2}\exp(\lambda|G_k(u)|)(G'_k(u))^{2}dx \\ &\quad +\int_{\{|u|\geq k(1+\varepsilon k^{-1/(t-2)})\}} \Phi\mu^{2}|G_k(u)|^{\mu-2}\exp(\lambda|G_k(u)|)(G'_k(u))^{2}dx. \end{aligned} \end{equation} Now, observe that the following assertions hold: \begin{itemize} \item[(A8)] if $\varepsilon\in(0,1)$, $s\in\mathbb{R}$ and $k\leq |s| \leq k(1+\varepsilon k^{-1/(t-2)})$, then $$ \mu^{2}|G_k(s)|^{\mu-2}\exp(\lambda|G_k(s)|)(G'_k(s))^{2}\leq (\lambda t)^{2}e^{\lambda}\varepsilon^{\mu(t-2)}; $$ \item[(A9)] if $\varepsilon\in(0,1)$, $s\in\mathbb{R}$ and $|s|\geq k(1+\varepsilon k^{-1/(t-2)})$, then $$ \mu^{2}|G_k(s)|^{\mu-2}G'_k(s)\exp(\lambda|G_k(s)|)\leq \frac{\lambda t(t+1)k^{1/(t-2)}}{2\varepsilon} \Psi'(G_k(s)). $$ \end{itemize} Indeed, let $s\in\mathbb{R}$, $k\leq |s| \leq k(1+\varepsilon k^{-1/(t-2)})$ and $y=(|s|-k)/k$. Using assertions \eqref{G_k(s)}, \eqref{G'_k(s)} and (A4), equality \eqref{mu} and the inequalities $0\leq y \leq \varepsilon k^{-1/(t-2)}<1$, $k>1$ and $\mu>1$, we obtain \begin{align*} &\mu^{2}|G_k(s)|^{\mu-2}\exp(\lambda|G_k(s)|)(G'_k(s))^{2}\\ &= \lambda^{2}(t-(t-1)y)^{2}k^{\mu}y^{t\mu-2} \Big(1-\frac{t-1}{t+1}y\Big)^{\mu-2}\exp(\lambda|G_k(s)|)\\ &\leq (\lambda t)^{2} (ky^{t-2})^{\mu}e^{\lambda} \\ &\leq (\lambda t)^{2}e^{\lambda}\varepsilon^{\mu(t-2)}. \end{align*} Consequently, assertion (A8) is true. Now let $s\in\mathbb{R}$, $k(1+\varepsilon k^{-1/(t-2)})\leq |s| \leq 2k$ and $y=(|s|-k)/k$. Using assertions \eqref{G_k(s)} and \eqref{G'_k(s)}, equalities \eqref{mu} and \eqref{Psi'} and the inequality $\varepsilon k^{-1/(t-2)}\leq y \leq 1$, we obtain \begin{align*} &\mu^{2}|G_k(s)|^{\mu-2}G'_k(s)\exp(\lambda|G_k(s)|)\\ &=\lambda \mu k^{\mu-1}y^{t(\mu-1)}\Big(1-\frac{t-1}{t+1}y\Big)^{\mu-1} \frac{(t-(t-1)y)}{y(1-\frac{t-1}{t+1}y)}\exp(\lambda|G_k(s)|)\\ &\leq\frac{\lambda t(t+1)}{2y} \mu|G_k(s)|^{\mu-1}\exp(\lambda|G_k(s)|)\\ &\leq \frac{\lambda t(t+1)k^{1/(t-2)}}{2\varepsilon} \Psi'(G_k(s)). \end{align*} Finally, suppose that $s\in\mathbb{R}$ and $|s|\geq 2k$. Then, by the definitions of the functions $h_k$ and $G_k$, we have $$ |G_k(s)|=|s|-\frac{2kt}{t+1}\geq \frac{2k}{t+1} . $$ Therefore, \begin{equation}\label{s2k} k\leq (t+1)|G_k(s)|/2. \end{equation} Using \eqref{mu}, \eqref{Psi'}, \eqref{s2k}, the equality $G'_k(s)=1$ and taking into account the inequalities $t>2$, $k>1$ and \eqref{varepsilon}, we obtain \begin{align*} &\mu^{2}|G_k(s)|^{\mu-2}G'_k(s)\exp(\lambda|G_k(s)|)\\ &=\lambda\mu k|G_k(s)|^{\mu-2}\exp(\lambda|G_k(s)|) \\ &\leq\frac{\lambda(t+1)}{2} \mu|G_k(s)|^{\mu-1}\exp(\lambda|G_k(s)|)\\ &\leq\frac{\lambda t(t+1)k^{1/(t-2)}}{2\varepsilon} \Psi'(G_k(s)). \end{align*} Thus, assertion (A9) holds. From assertion (A8) and \eqref{3.1} it follows that \begin{equation}\label{3.24} \begin{aligned} &\int_{\{k\leq |u| < k(1+\varepsilon k^{-1/(t-2)})\}} \Phi\mu^{2}|G_k(u)|^{\mu-2}\exp(\lambda|G_k(u)|)(G'_k(u))^{2}dx\\ &\leq C_2(\lambda t)^{2}e^{\lambda}\varepsilon^{\mu(t-2)}, \end{aligned} \end{equation} and by assertion (A9), we have \begin{equation} \label{3.25} \begin{aligned} &\int_{\{|u|\geq k(1+\varepsilon k^{-1/(t-2)})\}} \Phi\mu^{2}|G_k(u)|^{\mu-2}\exp(\lambda|G_k(u)|)(G'_k(u))^{2}dx\\ &\leq \frac{\lambda t(t+1)k^{1/(t-2)}}{2\varepsilon} \int_{\Omega}\Phi\Psi'(G_k(u))G'_k(u)dx, \end{aligned} \end{equation} From \eqref{lambda}, \eqref{3.23}, \eqref{3.24} and \eqref{3.25} we deduce the inequality \begin{align*} &\int_\Omega \Phi\mu^{2}|G_k(u)|^{\mu-2}\exp(\lambda|G_k(u)|)(G'_k(u))^{2}dx \\ &\leq c_{17}\varepsilon^{\mu(t-2)}+\frac{\lambda t(t+1)k^{1/(t-2)}}{2\varepsilon}\int_{\Omega}\Phi\Psi'(G_k(u))G'_k(u)dx. \end{align*} In turn, from this inequality and \eqref{3.22} we obtain the following estimate for the first integral in the right-hand side of inequality \eqref{3.21}, \begin{equation}\label{3.27} \begin{aligned} &\int_\Omega \Phi|\Psi''(G_k(u))| (G'_k(u))^{2} dx\\ &\leq c_{17}\varepsilon^{\mu(t-2)}+\frac{\lambda t(t+1)k^{1/(t-2)}}{\varepsilon}\int_{\Omega}\Phi\Psi'(G_k(u))G'_k(u)dx. \end{aligned} \end{equation} Now, let us estimate the second integral in the right-hand side of inequality \eqref{3.21}. By \eqref{Psi} and \eqref{Psi''}, for every $s\in\mathbb{R}$, we have $$ |\Psi''(s)|\leq \mu^{2}|s|^{\mu-2}\exp(\lambda|s|)+2\lambda \mu |s|^{\mu-1}\exp(\lambda|s|)+\lambda^{2}|\Psi(s)| . $$ Hence, \begin{equation} \label{3.28} \begin{aligned} &\int_\Omega g_2 |\Psi''(G_k(u))| (G'_k(u))^{2}dx \\ &\leq\int_\Omega\mu^{2}|G_k(u)|^{\mu-2}(G'_k(u))^{2}\exp(\lambda|G_k(u)|)g_2dx\\ &\quad + 2\lambda\int_\Omega\mu|G_k(u)|^{\mu-1}(G'_k(u))^{2} \exp(\lambda|G_k(u)|)g_2dx\\ &\quad +\lambda^{2}\int_\Omega g_2|\Psi(G_k(u))|(G'_k(u))^{2}dx. \end{aligned} \end{equation} Clearly, \begin{equation} \label{3.29} \begin{aligned} &\int_\Omega\mu^{2}|G_k(u)|^{\mu-2}(G'_k(u))^{2}\exp(\lambda|G_k(u)|)g_2dx\\ &= \int_{\{k\leq |u|2$, $k>1$ and \eqref{varepsilon}, we have \begin{align*} \mu^{2}|G_k(s)|^{\mu-2}(G'_k(s))^{2}\exp(\lambda|G_k(s)|) &=\lambda^{2}k^{2}|G_k(s)|^{\mu-2}\exp(\lambda|G_k(s)|)\\ &\leq\frac{\lambda ^{2}(t+1)^{2}}{4}|G_k(s)|^{\mu}\exp(\lambda|G_k(s)|)\\ &=\frac{(\lambda t)^{2}(t+1)^{2}k^{2/(t-2)}}{4\varepsilon}|\Psi(G_k(s))| . \end{align*} Thus, assertion (A10) holds. From assertion (A10) it follows that \begin{equation} \label{3.31} \begin{aligned} &\int_{\{|u|\geq k(1+\varepsilon^{1/2}k^{-1/(t-2)})\}} \mu^{2}|G_k(u)|^{\mu-2}(G'_k(u))^{2}\exp(\lambda|G_k(u)|)g_2dx \\ &\leq \frac{(\lambda t)^{2}(t+1)^{2}k^{2/(t-2)}}{4\varepsilon} \int_{\Omega}g_2|\Psi(G_k(u))|dx. \end{aligned} \end{equation} Using \eqref{3.29}--\eqref{3.31}, we obtain \begin{equation} \label{3.33} \begin{aligned} &\int_{\Omega}\mu^{2}|G_k(u)|^{\mu-2}(G'_k(u))^{2}\exp(\lambda|G_k(u)|)g_2dx\\ &\leq c_{18}[\varphi(k)]^{\mu/2+1/r'} + \frac{(\lambda t)^{2} (t+1)^{2}k^{2/(t-2)}}{4\varepsilon}\int_{\Omega}g_2|\Psi(G_k(u))|dx. \end{aligned} \end{equation} Before estimating the second integral in the right-hand side of inequality \eqref{3.28}, we note that for every $s\in\mathbb{R}$ the following inequality holds: \begin{equation}\label{3.34} \mu|G_k(s)|^{\mu-1}(G'_k(s))^{2}\exp(\lambda|G_k(s)|)\leq \frac{\lambda t^{2}(t+1)}{2} |\Psi(G_k(s))| . \end{equation} Indeed, if $s\in\mathbb{R}$ and $|s|\leq k$, then both sides of inequality \eqref{3.34} are equal zero and therefore, this inequality is true. Now, let $k<|s|<2k$ and $y=(|s|-k)/k$. Using \eqref{G_k(s)}, \eqref{G'_k(s)}, \eqref{mu} and the inequalities $02$, we obtain \begin{align*} \mu|G_k(s)|^{\mu-1}(G'_k(s))^{2} &=\lambda k^{\mu}y^{t\mu+t-2}\Big(1-\frac{t-1}{t+1}y\Big)^{\mu} \frac{(t-(t-1)y)^{2}}{(1-\frac{t-1}{t+1}y)} \\ &< \lambda k^{\mu}y^{t\mu}\Big(1-\frac{t-1}{t+1}y\Big)^{\mu} \frac{t^{2}(t+1)}{2}\\ &=\frac{\lambda t^{2}(t+1)}{2} |G_k(s)|^{\mu}. \end{align*} These relations and \eqref{Psi} imply that inequality \eqref{3.34} holds. Finally, let $|s|\geq 2k$. Then, by \eqref{mu}, \eqref{Psi}, \eqref{s2k}, the equality $G'_k(s)=1$ and the inequality $t>2$, we obtain \begin{align*} \mu|G_k(s)|^{\mu-1}(G'_k(s))^{2}\exp(\lambda|G_k(s)|) &=\lambda k |G_k(s)|^{\mu-1}\exp(\lambda|G_k(s)|)\\ &\leq \frac{\lambda(t+1)}{2} |G_k(s)|^{\mu}\exp(\lambda|G_k(s)|)\\ &=\frac{\lambda t^2(t+1)}{2} |\Psi(G_k(s))|. \end{align*} Therefore, inequality \eqref{3.34} holds. Thus, inequality \eqref{3.34} holds for every $s\in\mathbb{R}$. From \eqref{3.34} it follows that \begin{equation}\label{3.35} \int_\Omega\mu|G_k(u)|^{\mu-1}(G'_k(u))^{2}\exp(\lambda|G_k(u)|)g_2dx \leq \frac{\lambda t^{2}(t+1)}{2}\int_\Omega g_2|\Psi(G_k(u))|dx. \end{equation} In turn, using \eqref{3.28}, \eqref{3.33}, \eqref{3.35}, \eqref{3.6}, \eqref{lambda1} and \eqref{varepsilon} along with the inequalities $t>2$ and $k>1$, we deduce that \begin{equation} \label{3.36} \begin{aligned} &\int_\Omega g_2|\Psi''(G_k(u))|(G'_k(u))^{2}dx \\ &\leq c_{18}[\varphi(k)]^{\mu/2+1/r'} + \frac{3(\lambda t)^{2}(t+1)^{2}k^{2/(t-2)}}{4\varepsilon}\int_\Omega g_2|\Psi(G_k(u))|dx. \end{aligned} \end{equation} Similar to \eqref{3.36} we have \begin{equation} \label{3.37} \begin{aligned} &\int_\Omega |\Psi''(G_k(u))|(G'_k(u))^{2}dx \\ &\leq c_{19}[\varphi(k)]^{1+\mu/2}+\frac{3(\lambda t)^{2}(t+1)^{2}k^{2/(t-2)}}{4\varepsilon}\int_\Omega |\Psi(G_k(u))|dx. \end{aligned} \end{equation} Now, using \eqref{3.21}, \eqref{3.27}, \eqref{3.36} and \eqref{3.37} and taking into account \eqref{delta}, \eqref{t}, \eqref{3.2}, \eqref{3.10} and \eqref{varepsilon}, we obtain \eqref{3.38}. \textit{Step 5.} Let us prove that for the third integral in the right-hand side of inequality \eqref{3.9} the following inequality holds: \begin{equation}\label{3.41} \int_\Omega g_3 \Psi'(G_k(u))G'_k(u)dx\leq c_{20}\varphi(k)+\frac{\lambda t(t+1)k^{1/(t-1)}}{\varepsilon}\int_\Omega g_3|\Psi(G_k(u))|dx . \end{equation} In fact, by \eqref{3.6} and \eqref{Psi'}, we have \begin{equation} \label{3.39} \begin{aligned} &\int_\Omega g_3 \Psi'(G_k(u))G'_k(u)dx\\ &\leq\int_\Omega \mu |G_k(u)|^{\mu-1}G'_k(u)\exp(\lambda|G_k(u)|)g_3dx +\lambda\int_\Omega g_3 |\Psi(G_k(u))|dx . \end{aligned} \end{equation} It is clear that \begin{equation} \label{3.40} \begin{aligned} &\int_\Omega \mu |G_k(u)|^{\mu-1}G'_k(u)\exp(\lambda|G_k(u)|)g_3dx\\ &=\int_{\{k\leq |u|1$, $k>1$ and $\mu>1$, from \eqref{3.39}, \eqref{3.40} and assertions (A11) and (A12) we deduce \eqref{3.41}. \textit{Step 6.} Using \eqref{tildeF}, \eqref{3.2}, \eqref{3.9}--\eqref{3.20}, \eqref{3.38} and \eqref{3.41}, and taking into account that $k\geq k_{\ast}$ and $t>2$, we obtain that \begin{equation}\label{3.42} \frac{c_3}{3}\int_\Omega \Phi\Psi'(G_k(u))G'_k(u)dx\leq \varepsilon^{-2qp/(q-2p)}\int_\Omega \tilde{F}|\Psi(G_k(u))|dx+c_{21}[\varphi(k)]^{1/r'}. \end{equation} For the integral in the right-hand side of this inequality we shall establish the estimate \begin{equation} \label{3.47} \begin{aligned} &\int_\Omega \tilde{F}|\Psi(G_k(u))|dx\\ &\leq c_{22}k^{(1-\theta)\mu}e^{\lambda k}[\varphi(k)]^{1/r'}\\ &\quad +\frac{c^q(\lambda(t+1))^{q-1}}{q^q}\|\tilde{F}\|_{r}k^{\theta(q-1)} [\varphi(k)]^{q/n-1/r}\int_{\Omega}\Phi\Psi'(G_k(u))G'_k(u)dx. \end{aligned} \end{equation} Using H\"{o}lder's inequality and the definition of the function $\Psi$, we obtain \begin{equation} \label{3.43} \begin{aligned} &\int_\Omega \tilde{F}|\Psi(G_k(u))|dx \\ &= \int_{\{|G_k(u)|k^{1-\theta}, \\ 0 & \text{if } |s|\leq k^{1-\theta}. \end{cases} $$ Using the definitions of the functions $\Psi$ and $w$ and H\"{o}lder's inequality, we establish that \begin{equation}\label{3.44} \begin{aligned} &\int_{\{|G_k(u)|\geq k^{1-\theta}\}} \tilde{F}|\Psi(G_k(u))|dx\\ &\leq 2^{q-1}\|\tilde{F}\|_{r}k^{(1-\theta)\mu}e^{\lambda k^{1-\theta}}[\varphi(k)]^{1/r'}+\int_{\Omega}\tilde{F}|w(G_k(u))|^qdx. \end{aligned} \end{equation} Taking into account \eqref{num1}, \eqref{2} and \eqref{3.6} and using H\"{o}lder's inequality, we obtain \begin{equation} \label{3.45} \begin{aligned} &\int_{\Omega}\tilde{F}|w(G_k(u))|^qdx\\ &\leq \|\tilde{F}\|_{r}\|w(G_k(u))\|_{qr'}^q \\ &\leq \|\tilde{F}\|_{r}\|w(G_k(u))\|_{q^{\ast}}^q[\varphi(k)]^{q/n-1/r}\\ &\leq \frac{c^q2^{q-1}}{q^q}\|\tilde{F}\|_{r}[\varphi(k)]^{q/n-1/r}\\ &\quad \times \int_{\{|G_k(u)|\geq k^{1-\theta}\}} (\mu^q|G_k(u)|^{\mu-q}+\lambda^q|G_k(u)|^{\mu}) \exp(\lambda|G_k(u)|)G'_k(u)\Phi dx. \end{aligned} \end{equation} To proceed estimating the integral in the left-hand side of \eqref{3.45}, we observe that the following assertion holds: If $s\in\mathbb{R}$ and $|G_k(s)|\geq k^{1-\theta}$, then \begin{equation}\label{Gk(s)theta} \mu^q|G_k(s)|^{\mu-q}\leq(\lambda(t+1)/2)^{q-1}k^{\theta(q-1)} \mu|G_k(s)|^{\mu-1}. \end{equation} Indeed, let $s\in\mathbb{R}$, $|G_k(s)|\geq k^{1-\theta}$ and $k<|s|<2k$. Then, setting $y=(|s|-k)/k$ and taking into account the inequality $0k^{-\theta}$. Using this inequality and assertion \eqref{G_k(s)}, we obtain $$ \mu^q|G_k(s)|^{\mu-q}=\frac{\lambda^{q-1}\mu|G_k(s)|^{\mu-1}}{y^{t(q-1)}(1-\frac{t-1}{t+1}y)^{q-1}} \leq(\lambda(t+1)/2)^{q-1}k^{\theta(q-1)}\mu|G_k(s)|^{\mu-1}. $$ Now, let $|s|\geq 2k$. Then, by \eqref{mu}, \eqref{s2k} and the inequality $k^{\theta(q-1)}\geq 1$, we have $$ \mu^q|G_k(s)|^{\mu-q}=\lambda^{q-1}\mu k^{q-1}|G_k(s)|^{\mu-q}\leq (\lambda(t+1)/2)^{q-1}k^{\theta(q-1)}\mu|G_k(s)|^{\mu-1}. $$ Thus, assertion \eqref{Gk(s)theta} holds. From \eqref{3.45} and assertion \eqref{Gk(s)theta}, taking into account the definition of the function $\Psi$ and the inequalities $t>1$ and $k\geq 1$, we deduce that \begin{equation} \label{3.46} \begin{aligned} &\int_{\Omega}\tilde{F}|w(G_k(u))|^qdx \\ &\leq \frac{c^q(\lambda(t+1))^{q-1}}{q^q}\|\tilde{F}\|_{r} k^{\theta(q-1)}[\varphi(k)]^{q/n-1/r}\int_{\Omega}\Phi\Psi'(G_k(u))G'_k(u)dx. \end{aligned} \end{equation} In turn, using \eqref{3.43}, \eqref{3.44} and \eqref{3.46} along with the inequalities $k>1$ and $\theta>0$, we obtain \eqref{3.47}. Inequalities \eqref{3.42} and \eqref{3.47} along with the inequalities $0<\varphi(k)<1$, $k>1$, $\theta<1$, \eqref{vartheta}, \eqref{t}, \eqref{3.2.4} and \eqref{3.10} imply that \begin{equation}\label{3.48} \frac{c_3}{6}\int_\Omega \Phi\Psi'(G_k(u))G'_k(u)dx\leq(c_{21}+c_{22})k^{(1-\theta)\mu}e^{\lambda k}[\varphi(k)]^{\vartheta+q/q^{\ast}}. \end{equation} \textit{Step 7.} Let us estimate from below the integral in the left-hand side of inequality \eqref{3.48}. This will allow us to apply Lemma \ref{lem3.3} and to obtain the conclusion of the theorem. We fix $l\in (k,2k]$. Using \eqref{2}, \eqref{lambda1}, \eqref{3.6}, \eqref{mu} and \eqref{Psi'} and the inequality $\mu\geq q$, we obtain \begin{equation} \label{3.49} \begin{aligned} &\int_\Omega \Phi\Psi'(G_k(u))G'_k(u)dx\\ &\geq \frac{(q/2\lambda)^q}{k^q}\sum_{|\alpha|=1} \int_\Omega|D^{\alpha}(|G_k(u)|^{\mu/q+1}\operatorname{sign}G_k(u))|^qdx \\ &\geq \frac{(q/2\lambda c)^q}{k^q} \Big(\int_{\{|u|\geq l\}}|G_k(u)|^{(\mu/q+1)q^{\ast}}dx\Big)^{q/q^{\ast}}. \end{aligned} \end{equation} From assertion (A3) it follows that \begin{equation}\label{3.50} \int_{\{|u|\geq l\}}|G_k(u)|^{(\mu/q+1)q^{\ast}}dx\geq \Big(\frac{2}{t+1}\Big)^{(\mu/q+1)q^{\ast}} \frac{(l-k)^{t(\mu/q+1)q^{\ast}}}{k^{(t-1)(\mu/q+1)q^{\ast}}} \varphi(l). \end{equation} From \eqref{3.48}--\eqref{3.50}, taking into account the equality $\mu=\lambda k$, we deduce that $$ \varphi(l)\leq c_{23}\Big[\frac{e(t+1)}{2}\Big]^{\lambda q^{\ast}k/q} \frac{k^{\lambda q^{\ast}(t-\theta) k/q+tq^{\ast}}}{(l-k)^{\lambda q^{\ast}tk/q+tq^{\ast}}} [\varphi(k)]^{1+\vartheta q^{\ast}/q}. $$ This and the inequality $(e(t+1)k^{-\theta/2}/2)^{\lambda q^{\ast}k/q} \leq c_{24}$ allow us to conclude that the following assertion holds: If $k_0\leq kn/q$, let the functions $g_2$, $g_3$, $g_4$ and $f$ belong to $L^{r}(\Omega)$. Let for every $i\in \mathbb{N}$, $T_i: \mathbb{R}\to \mathbb{R}$ be the function such that $$ T_i(s) = \begin{cases} s & \text{if } |s|\leq i, \\ i\operatorname{sign} s &\text{if } |s|> i. \end{cases} $$ Now, for every $i\in \mathbb{N}$ we define the function $B_i:\Omega\times \mathbb{R}\times \mathbb{R}^{n,2} \to \mathbb{R}$ by $$ B_i(x,s,\xi)=T_i(B(x,s,\xi)), \quad (x,s,\xi)\in\Omega\times \mathbb{R}\times \mathbb{R}^{n,2}. $$ Obviously, for every $i\in\mathbb{N}$ and for every $(x,s,\xi)\in\Omega\times \mathbb{R}\times \mathbb{R}^{n,2}$, \begin{gather}\label{5.1} |B_i(x,s,\xi)|\leq i, \\ \label{5.2} |B_i(x,s,\xi)|\leq c_4\big\{\sum_{|\alpha|=1}|\xi_\alpha|^q + \sum_{|\alpha|=2}|\xi_\alpha|^{p} \big\}+ g_4(x). \end{gather} From \eqref{2}--\eqref{5}, \eqref{grA}, \eqref{9}, \eqref{6} and \eqref{5.1} and the results of \cite{Lions} on solvability of equations with pseudomonotone operators it follows that if $i\in \mathbb{N}$, then there exists a function $u_i\in {\mathaccent"7017 W} ^{1,q}_{2,p}(\Omega)$ such that for every function $v\in {\mathaccent"7017 W}^{1,q}_{2,p}(\Omega)$, \begin{equation}\label{5.3} \int_\Omega \big\{\sum_{\alpha\in\Lambda}A_\alpha(x,\nabla_2 u_i)D^\alpha v+A_0(x,u_i)v+B_i(x,u_i,\nabla_2 u_i)v\big\} dx = \int_\Omega f v dx. \end{equation} Hence, on the basis of the inclusions $g_2$, $g_3$, $f\in L^{r}(\Omega)$ and $B_i(x,u_i,\nabla_2 u_i)\in L^{\infty}(\Omega)$ and a slight modification (due to the presence in \eqref{5.3} of the term $A_0$ satisfying conditions \eqref{8} and \eqref{grA}) of the proof of assertion (iii) of \cite[Theorem 1]{KovVoi} we establish that for every $i\in\mathbb{N}$, $$ u_i\in L^{\infty}(\Omega). $$ Using this inclusion, inequality \eqref{5.2} and Theorems \ref{th2.4} and \ref{th2.5}, we obtain that for every $i\in\mathbb{N}$, \begin{gather}\label{5.4} \int_\Omega \Big(\sum_{|\alpha|=1} |D^\alpha u_i|^q + \sum_{|\alpha|=2} |D^\alpha u_i|^p\Big) dx \leq C_2, \\ \label{5.5} \|u_i\|_{\infty} \leq C_1. \end{gather} By \eqref{5.4}, \eqref{2} and in view of the compactness of the embedding ${\mathaccent"7017 W}^{1,q}(\Omega)$ into $L^{\lambda}(\Omega)$ with $\lambda