\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 27, pp. 1--18.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/27\hfil Fourth-order critical growth problem] {Multiple solutions of a fourth-order nonhomogeneous equation with \\ critical growth in $\mathbb{R}^4$} \author[A. Sarkar \hfil EJDE-2017/27\hfilneg] {Abhishek Sarkar} \address{Abhishek Sarkar \newline NTIS, University of West Bohemia, Technick\'a 8, 306 14 Plze\v{n}, Czech Republic} \email{sarkara@ntis.zcu.cz} \dedicatory{Communicated by Mitsuharu Otani} \thanks{Submitted September 12, 2016. Published January 24, 2017.} \subjclass[2010]{35J30, 35J40, 35J60} \keywords{Biharmonic; critical exponent; multiple solutions} \begin{abstract} In this article we study the existence of at least two positive weak solutions of an nonhomogeneous fourth-order Navier boundary-value problem involving critical exponential growth on a bounded domain in $\mathbb{R}^4$, with a parameter $\lambda >0$. We establish upper and lower bounds for $\lambda$, which determine multiplicity and non-existence of solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} Let $\Omega \subset \mathbb{R}^{4}$ be a bounded domain with the boundary $\partial \Omega \in C^{2,\sigma}$ for some $0 < \sigma <1$. In this context, we study the existence of multiple solutions in $W^{2,2}_{\mathcal{N}}(\Omega) = \{u \in W^{2,2}(\Omega): u = 0 \text{ on } \partial \Omega\}$, for the following fourth-order Navier boundary value problem \begin{equation} \label{eP} \begin{gathered} \Delta^2 u = \mu u |u|^pe^{u^2} + \lambda h(x) \quad \text{in } \Omega,\\ u,-\Delta u > 0 \quad \text{in } \Omega, \\ u=\Delta u=0 \quad \text{on } \partial \Omega, \end{gathered} \end{equation} where $h \geq 0$ in $\Omega$, $\|h\|_{L^2(\Omega)} = 1$, $ \lambda >0$, $\mu =1$ if $p>0$ and $\mu \in (0,\lambda_1(\Omega))$ if $p=0$. Where $\lambda_1(\Omega)$ and $\phi_1$ denote the first eigenvalue and the corresponding eigenfunction of $\Delta^2 $ on $W^{2,2}_{\mathcal{N}}(\Omega)$ respectively with respect to the Navier boundary condition. We note that $\lambda_1 >0$ and $\phi_1$ is strictly positive (see \cite{Drabek}). The existence of multiple solutions for analogous problems in higher dimensions with critical exponent, have been studied in \cite{Peral,Lu} for the Dirichlet boundary condition, and in \cite{Zhang} for Navier boundary condition. The existence of multiple solutions for the fourth-order nonhomogeneous quasilinear equation has been studied in \cite{Santos}. The corresponding problem for second order elliptic equations have been studied in \cite{Prashanth} for dimension two and in \cite{Tarantello} for higher dimensions. We note that, the critical growth for the fourth-order equations is $u \mapsto |u|^{8/(N-4)}u$ for $N \geq 5$, from the point of view of the Sobolev imbedding theorem in $\mathbb{R}^n$. In 1971, Moser \cite{Moser} proved the following theorem. \begin{theorem} \label{Moser_Thm} Let $\Omega \subset \mathbb{R}^N$, $N \geq 2$ be a bounded domain. There exists a constant $C_N >0$ such that for any $u \in W^{1,N}_0(\Omega)$, $N\geq 2$ with $\|\nabla u\|_{L^N(\Omega)} \leq 1$, then \begin{equation} \int_{\Omega} e^{\alpha |u|^p}\,\mathrm{d} x \leq C_N|\Omega|, \quad \forall\alpha \leq \alpha_N, \label{Moser_Sharp} \end{equation} where $$ p = \frac{N}{N-1}, \quad \alpha_N:= N w^{\frac{1}{N-1}}_{N-1}, $$ and $w_{N-1}$ is the surface measure of the unit sphere $\mathbb{S}^{N-1} \subset \mathbb{R}^N$. Furthermore, the integral on the left hand side can be made arbitrarily large if $\alpha > \alpha_N$ by appropriate choice of $u$ with $\|\nabla u\|_{L^N(\Omega)} \leq 1$. The embedding $$ W^{1,N}_0(\Omega) \ni u \mapsto e^{\alpha |u|^{\frac{N}{N-1}}} \in L^1(\Omega), $$ is compact for $\alpha < \alpha_N$ and it is not compact for $\alpha = \alpha_N$. \end{theorem} In 1988, Adams \cite{Adams} extended the above result of Moser to higher order Sobolev spaces. To state the main theorem by Adams, we denote the $m$-th order derivatives of $u \in C^m(\Omega)$, by \[ \nabla^m u = \begin{cases} \Delta ^{m/2} u , &\text{for $m$ even},\\ \nabla \Delta^{(m-1)/2} u, &\text{for $m$ odd}. \end{cases} \] Now denote by $W^{m,\frac{N}{m}}_0(\Omega)$ the completion of $C^{\infty}_0(\Omega)$, under the Sobolev norm \begin{equation} \|u\|_{W^{m,\frac{N}{m}}(\Omega)} = \Big( \|u\|^{N/m}_{N/m} + \sum_{|\alpha|=1}^m \|D^{\alpha} u\|^{N/m}_{N/m}\Big)^{m/N}. \label{full-norm} \end{equation} Adams proved the following embedding. \begin{theorem}\label{Adams_Thm} Let $\Omega \subset \mathbb{R}^N$ be a bounded domain. If $m$ is a positive integer and $m \leq N$, then there exists a constant $C_0=C_0(N,m)>0$, such that for any $u \in W^{m,\frac{N}{m}}_0(\Omega)$ with $\| \nabla^m u\|_{L^{N/m}(\Omega)} \leq 1$, then \begin{equation} \frac{1}{|\Omega|} \int_{\Omega} \exp\Big(\beta |u(x)|^{N/(N-m)}\Big) \,\mathrm{d} x \leq C_0, \label{Adams_Sharp} \end{equation} for all $\beta \leq \beta_{N,m}$, where \[ \beta_{N,m}=\begin{cases} \frac{N}{w_{N-1}} \big[\frac{\pi ^{\frac{N}{2}}2^m \Gamma(\frac{m+1}{2})} {\Gamma(\frac{N-m+1}{2})}\big]^{N/(N-m)}, & \text{when $m$ is odd,}\\[4pt] \frac{N}{w_{N-1}}\big[\frac{\pi ^{\frac{N}{2}}2^m \Gamma(\frac{m}{2})} {\Gamma(\frac{N-m}{2})}\big]^{N/(N-m)} , & \text{when $m$ is even}. \end{cases} \] Furthermore, for any $\beta > \beta_{N,m}$, the integral can be made as large as possible by appropriate choice of $u$ with $\|\nabla ^m u\|_{L^{N/m}(\Omega)} \leq 1$. \end{theorem} Now we define a subspace of $W^{m,\frac{N}{m}}(\Omega)$, by $$ W^{m,\frac{N}{m}}_{\mathcal{N}}(\Omega) := \big\{u \in W^{m,\frac{N}{m}}(\Omega): \Delta^j u|_{\partial \Omega} =0 \text{ for } 0 \leq j\leq \big[\frac{m-1}{2}\big]\big\}. $$ Note that, $W_0^{m,\frac{N}{m}}(\Omega)$ is strictly contained in $W^{m,\frac{N}{m}}_{\mathcal{N}}(\Omega)$. Therefore, one has \begin{align*} &\sup_{u \in W^{m,\frac{N}{m}}_0(\Omega), \|\nabla^m u\|_{L^{N/m}(\Omega)} \leq 1} \int_{\Omega} e^{\beta_{N,m} |u|^{N/(N-m)}}\,\mathrm{d} x \\ &\leq \sup_{u \in W^{m,\frac{N}{m}}_{\mathcal{N}}(\Omega), \|\nabla^m u\|_{L^{N/m}(\Omega)} \leq 1} \int_{\Omega} e^{\beta_{N,m} |u|^{N/(N-m)}}\,\mathrm{d} x. \end{align*} In 2012, Tarsi \cite{Tarsi} established that the Adams' inequality is also valid for the larger space $W^{m,\frac{N}{m}}_{\mathcal{N}}(\Omega)$. The key idea was to embed $W^{m,\frac{N}{m}}_{\mathcal{N}}(\Omega)$ into a Zygmund space. We state her embedding theorem below. \begin{theorem}\label{Tarsi_Thm} Let $N >2$, $m 0$, such that for all $u \in W^{m,\frac{N}{m}}_{\mathcal{N}}(\Omega)$ with $\|\nabla^m u \|_{L^{N/m}(\Omega)} \leq 1$, we have \begin{equation} \int_{\Omega} e^{\beta |u|^{N/(N-m)}}\,\mathrm{d} x \lambda^*$. \end{theorem} Though the Palais-Smale condition fails due to the presence of critical exponent, first we adapt the method of Tarantello (cf. \cite{Tarantello}) to prove the existence of a first solution by a decomposition of Nehari manifold into three parts. Then, for the existence of second solution, we rely on a refined version of the Mountain-Pass Lemma, which was introduced by Ghoussoub and Preiss in \cite{Preiss}. \section{Decomposition of the Nehari Manifold} Let $f(u) = \mu |u|^pue^{u^2}$. The corresponding energy functional associated with problem \eqref{eP}, is \begin{equation} J(u)= \frac{1}{2} \int_{\Omega}|\Delta u|^2- \int_{\Omega} F(u) - \lambda \int_{\Omega} hu , \label{energy} \end{equation} where $F(u) = \int_0^u f(s)ds$. As the energy functional is not bounded from below on $W^{2,2}_{\mathcal{N}}(\Omega)$, we need to study $J$ on the Nehari manifold \begin{equation} \mathcal{M}:= \{u \in W^{2,2}_{\mathcal{N}}(\Omega): \langle J'(u),u \rangle =0\}, \end{equation} where $J'(u)$ denotes the Fr\'{e}chet derivative of $J$ at $u$ and $\langle \cdot,\cdot \rangle$ is the inner product. Here we note that, $\mathcal{M}$ contains every solution of the problem \eqref{eP}. For any $u \in W^{2,2}_{\mathcal{N}}(\Omega)$, we note that \begin{gather*} \langle J'(u),u \rangle = \int_{\Omega} |\Delta u|^2 - \int_{\Omega} f(u)u - \lambda \int_{\Omega} hu, \\ \langle J''(u)u,u \rangle = \int_{\Omega}|\Delta u|^2 - \int_{\Omega} f'(u)u^2. \end{gather*} Similar to the method used by Tarantello \cite{Tarantello}, we split $\mathcal{M}$ into three parts \begin{gather*} \mathcal{M}^0 = \{u \in \mathcal{M}: \langle J''(u)u,u \rangle = 0\},\\ \mathcal{M}^+ = \{u \in \mathcal{M}: \langle J''(u)u,u \rangle > 0\},\\ \mathcal{M}^- = \{u \in \mathcal{M}: \langle J''(u)u,u \rangle < 0\}. \end{gather*} \section{Topological Properties of $\mathcal{M}^0,\mathcal{M}^+, \mathcal{M}^-$} Our first aim is to show, $\mathcal{M}^0 = \{0\}$ for some small $\lambda$. For this, let $\zeta >0$ if $p>0$ and $\zeta < \frac{\lambda_1-\mu}{\mu}$ if $p=0$. Define, $ \Lambda := \{u \in W^{2,2}_{\mathcal{N}}(\Omega): \int_{\Omega} |\Delta u|^2 \leq (1+\zeta)\int_{\Omega} f'(u)u^2\}$. Then, Lemma \ref{nonempty} implies that $\Lambda \neq \{0\}$. We now assume the following important hypotheses \begin{gather} \lambda >0, \quad \|h\|_{L^2(\Omega)} = 1 , \nonumber \\ \inf_{u \in \Lambda \setminus \{0\}}\Big(\mu \int_{\Omega}(p+2u^2)|u|^{p+2}e^{u^2} - \lambda \int_{\Omega}hu\Big)>0. \label{Nehari1} \end{gather} Condition \eqref{Nehari1} forces $\lambda$ to be suitably small. Indeed, we can prove the following result. \begin{proposition} \label{lambdastar} Let \begin{equation} \lambda < \mu C_0^{\frac{p+3}{p+4}}|\Omega|^{-(\frac{p+2}{2p+8})}, \label{lambda} \end{equation} where $C_0 = \inf_{u \in \Lambda \setminus \{0\}} \int_{\Omega}(p+2u^2)|u|^{p+2}e^{u^2} >0$. Then \eqref{Nehari1} holds. \end{proposition} \begin{proof} \textbf{Step 1:} $\inf_{ u \in \Lambda \setminus \{0\}} \|u\|_{W^{2,2}_{\mathcal{N}}(\Omega)} >0$. Toward a contradiction, suppose that, there exists a sequence $\{u_n\} \subset \Lambda \setminus \{0\}$, with $\|u_n\|_{W^{2,2}_{\mathcal{N}}(\Omega)} \to 0$ as $n \to \infty$. Let $v_n = \frac{u_n}{\|u_n\|_{W^{2,2}_{\mathcal{N}}(\Omega)}}$. Then $\|v_n\|_{W^{2,2}_{\mathcal{N}}(\Omega)} =1$ and $v_n$ satisfies \begin{equation} 1 \leq (1+\zeta) \int_{\Omega} f'(u_n) v_n^2, \quad \forall n. \label{lambda1} \end{equation} Since $u_n \to 0$ in $W^{2,2}_{\mathcal{N}}(\Omega)$, by Adams' inequality for the higher order derivative in Theorem \ref{Tarsi_Thm}, we obtain $f'(u_n) \to f'(0)$ in $L^r(\Omega)$ for all $r \geq 1$. Since $v_n$ is bounded in $W^{2,2}_{\mathcal{N}}(\Omega)$, $v_n$ has a weak limit say $v$ in $W^{2,2}_{\mathcal{N}}(\Omega)$. Certainly $\|v\|_{W^{2,2}_{\mathcal{N}}(\Omega)} \leq 1$ and up to a subsequence which we still denote by $v_n$ which converges strongly to $v$ in $L^r(\Omega)$ for all $r \geq 1$. Hence from \eqref{lambda1}, we obtain \begin{equation} \int_{\Omega} |\Delta v|^2 \leq 1 \leq (1+\zeta) f'(0) \int_{\Omega} v^2. \label{lambda2} \end{equation} This gives a contradiction for the case $p>0$ since $f'(0)= 0$ in this case. For the case $p=0$, by the assumption $$ \int_{\Omega} |\Delta v|^2 \geq \lambda_1 \int_{\Omega} v^2 > (1+\zeta) \mu \int_{\Omega}v^2, $$ which gives a contradiction to \eqref{lambda1} since $f'(0)= \mu$. This completes Step 1. Since $\inf_{ u \in \Lambda \setminus \{0\}} \|u\|_{W^{2,2}_{\mathcal{N}}(\Omega)} >0$, from the definition of $\Lambda$, we obtain \begin{equation} 0< \inf_{u \in \Lambda \setminus \{0\}} \int_{\Omega}f'(u)u^2 =\inf_{u \in \Lambda \setminus \{0\}}\mu \int_{\Omega}(p+1+2u^2)e^{u^2}|u|^{p+2}. \label{Nehari123} \end{equation} Using \eqref{Nehari123}, we can easily check that \begin{equation} C_0=\inf_{u \in \Lambda \setminus \{0\}}\int_{\Omega}(p+2u^2)|u|^{p+2}e^{u^2} >0. \label{lambda-inf} \end{equation} \smallskip \noindent\textbf{Step 2:} Finally, we have \begin{align*} \lambda \big|\int_{\Omega} hu\big| &\leq \lambda \|u\|_{L^2(\Omega)} \leq \lambda |\Omega|^{\frac{p+2}{2p+8}} \Big(\int_{\Omega} |u|^{p+4} \Big)^{1/(p+4)}\\ &\leq \frac{\lambda |\Omega|^{\frac{p+2}{2p+8}}} {(\mu \int_{\Omega}(p+2u^2)|u|^{p+2}e^{u^2})^{\frac{p+3}{p+4}}} \Big(\mu \int_{\Omega} (p+2u^2)|u|^{p+2}e^{u^2}\Big)\\ &\leq \Big(\frac{\lambda |\Omega|^{\frac{p+2}{2p+8}}}{\mu C_0^{\frac{p+3}{p+4}}}\Big) \Big(\mu \int_{\Omega}(p+2u^2)|u|^{p+2}e^{u^2}\Big). \end{align*} Hence, from the above inequality together with \eqref{lambda} and \eqref{lambda-inf}, we can complete the proof. \end{proof} \begin{lemma} \label{Null Set} Suppose $\lambda >0$ be such that \eqref{Nehari1} holds. Then $\mathcal{M}^0 = \{0\}$. \end{lemma} \begin{proof} For the sake of contradiction, suppose there exists $u \in \mathcal{M}^0$ and $u \not\equiv 0$. Then, we have \begin{gather} \int_{\Omega}|\Delta u|^2 = \int_{\Omega} f(u) u + \lambda \int_{\Omega}hu, \label{Nehari2}\\ \int_{\Omega}|\Delta u|^2 = \int_{\Omega} f'(u)u^2. \label{Nehari3} \end{gather} It follows from \eqref{Nehari3}, that $$ \int_{\Omega}|\Delta u|^2 = \int_{\Omega} f'(u)u^2 < (1+\zeta) \int_{\Omega} f'(u)u^2, $$ therefore, $ u \in \Lambda \setminus \{0\}$. From \eqref{Nehari2} and \eqref{Nehari3}, we obtain $$ \lambda \int_{\Omega} hu = \int_{\Omega}(f'(u)u-f(u))u = \mu \int_{\Omega}(p+2u^2)|u|^{p+2}e^{u^2}, $$ which violates the condition \eqref{Nehari1}. Therefore, $\mathcal{M}^0 = \{0\}$. \end{proof} Now we discuss the topological properties of $\mathcal{M}^+$ and $\mathcal{M}^-$. The study of Nehari manifold is closely related to the behaviour of the map $s \mapsto J(su)$. This technique was first introduced in \cite{Dra-Poh} by Dr\'{a}bek and Pohozaev. Given $u \in {W^{2,2}_{\mathcal{N}}(\Omega)} \setminus \{0\}$, we define a map, $\xi_u: \mathbb{R}^+ \to \mathbb{R}$ by \begin{equation} \xi_u(s) = s \int_{\Omega} |\Delta u|^2 - \int_{\Omega} f(su)u. \label{xi} \end{equation} The choice of the above function is a consequence of the following expression \begin{equation*} \langle J'(su), su \rangle = s \Big(s \int_{\Omega} |\Delta u|^2 - \int_{\Omega} f(su)u - \lambda \int_{\Omega}hu \Big). \end{equation*} So, $\xi_u(s)=\lambda \int_{\Omega}hu$ if and only if $su \in \mathcal{M}$ for $s >0$. Now we are ready to prove the following lemma. \begin{lemma} \label{nonempty} For every $u \in W^{2,2}_{\mathcal{N}}(\Omega) \setminus \{0\}$, there exists unique $s_* =s_*(u) >0$, such that $\xi_u(\cdot)$ has its maximum at $s_*$ with $\xi_u(s_*)>0$. Also there holds, $s_*u \in \Lambda \setminus \{0\}$. \end{lemma} \begin{proof} Differentiating \eqref{xi}, we have \begin{equation} \xi'_u(s) = \int_{\Omega} |\Delta u|^2- \int_{\Omega} f'(su)u^2. \label{xi1} \end{equation} Therefore, \begin{equation} s^2 \xi'_u(s) = \int_{\Omega}|\Delta (su)|^2 - \int_{\Omega}f'(su)(su)^2 = \langle J''(su)su,su \rangle. \label{xi2} \end{equation} Now we observe that, $\xi_u(\cdot)$ is a concave function on $\mathbb{R}^+$ since \begin{equation} \xi''_u(s)=-\int_{\Omega}f''(su)u^3 < 0. \label{xi3} \end{equation} Also from the range of $\mu$, we obtain \[ \lim_{s \to 0+} \xi'_u(s) >0 , \quad \lim_{s \to \infty} \xi_u(s) = -\infty. \] Hence there exists a unique $s_*=s_*(u)>0$, such that $\xi_u$ is increasing on $(0,s_*)$, decreasing on $(s_*,\infty)$ and $\xi'_u(s_*)=0$. Now using \eqref{xi1} and $\xi'_u(s_*)=0$, we deduce \begin{equation} \begin{aligned} \xi_u(s_*) &= s_* \int_{\Omega} f'(s_*u) u^2 - \int_{\Omega} f(s_*u)u \\ &= \frac{1}{s_*} \int_{\Omega} (f'(s_*u)s_*u - f(s_*u))s_*u \\ &=\frac{\mu}{s_*} \int_{\Omega} (p+2(s_*u)^2)|s_*u|^{p+2}e^{(s_*u)^2} > 0. \end{aligned} \label{xi4} \end{equation} Here we note that, $f'(s)s-f(s) =\mu(p+2s^2)|s|^p s e^{s^2}$. Finally $$ s_* \xi'_u(s_*) = \int_{\Omega} |\Delta(s_*u)|^2 - \int_{\Omega}f'(s_*u)(s_*u)^2 =0, $$ which implies, $s_*u \in \Lambda \setminus \{0\}$. \end{proof} \begin{lemma} \label{min-thm} Let $\lambda$ be such that \eqref{Nehari1} holds. Then, for every $u \in W^{2,2}_{\mathcal{N}}(\Omega) \setminus \{0\}$, there exists a unique $s_- = s_-(u)>0$ such that $s_-u \in \mathcal{M}^-$, $s_- > s_*$ and $J(s_-u) = \max_{s \geq s_*} J(su) \hspace{2mm}\forall s \in [s_*, \infty)$. Furthermore, if $\int_{\Omega} hu >0$, then there exists a unique $s_+=s_+(u) >0$ such that $s_+u \in \mathcal{M}^+$. In particular, $ s_+ 0. $$ Next from \eqref{Nehari1} and \eqref{xi4}, we obtain \[ \xi_u(s_*) - \lambda \int_{\Omega} hu = \frac{1}{s_*}\biggl\{ \mu \int_{\Omega} (p+2(s_*u)^2)|s_*u|^{p+2}e^{(s_*u)^2} - \lambda \int _{\Omega} h(s_*u)\biggr\} >0. \] Since $\xi_u(\cdot)$ is strictly decreasing in $(s_*,\infty)$ and $\lim_{s \to \infty} \xi_u(s) = -\infty$, there exists a unique $s_- = s_-(u) > s_*$, such that $\xi_u(s_-) = \lambda \int_{\Omega} hu$. That is $s_-u \in \mathcal{M}$. One has $s_- > s_*$ and $\rho_u'(s_) <0$, we obtain $s_-u \in \mathcal{M^-}$. On the other hand, when $\int_{\Omega} hu >0$, we have $\lim_{s \to 0+} \xi_u(s) <0$, which implies, $\xi_u(s)- \lambda \int_{\Omega} hu <0$ for $s$ near $0$. Hence there exists a unique $s_+$, such that $\xi_u(s_+) = \lambda \int_{\Omega} hu$ which implies $s_+u \in \mathcal{M}$. From the graph, we see that $\xi_u(\cdot)$ is strictly increasing in $(0,s_*)$. Hence we have, $s_+u \in \mathcal{M}^+$. And the remaining properties of $s_-, s_+$ can be proved by analyzing the identity $\rho_u(s) = \xi_u(s) - \lambda \int_{\Omega}hu$. \end{proof} \begin{remark} \rm If we define the positive cone $\mathcal{P}=\{ u \in W^{2,2}_{\mathcal{N}}(\Omega): \int_{\Omega}hu >0\}$ in $W^{2,2}_{\mathcal{N}}(\Omega)$. Then, we obtain $\mathcal{M^+} \subset \mathcal{P}$. \end{remark} The next corollary shows some topological properties of $\mathcal{M}^+, \mathcal{M}^-$. \begin{corollary} \label{coro3.5} Let $S_{W^{2,2}_{\mathcal{N}}(\Omega)} = \{u \in W^{2,2}_{\mathcal{N}}(\Omega): \|u\|_{W^{m,2}_{\mathcal{N}}(\Omega)} = 1\}$. Then there exists a homeomorphism $S_-: S_{W^{2,2}_{\mathcal{N}}(\Omega)} \to \mathcal{M}^-$ defined by $S_-(u) = s_-(u)u$. Also $\mathcal{M}^+$ is homeomorphic to $S_{W^{2,2}_{\mathcal{N}}(\Omega)} \cap \mathcal{P}$. \end{corollary} \begin{proof} The function $S_-$ is continuous, because $s_-$ is continuous as an application of implicit function theorem applied to the map, $(s,u) \mapsto \xi_u(s) - \lambda \int_{\Omega}hu$. Also we deduce the continuity of $(S_-)^{-1}$ by the fact that $(S_-)^{-1}(w) = w/\|w\|$. In a similar manner, we can prove that $ \mathcal{M}^+$ is homeomorphic to $S_{W^{m,2}_{\mathcal{N}}(\Omega)} \cap \mathcal{P}$. \end{proof} We set, $\theta_0 = \inf\{J(u) : u \in \mathcal{M}\}$. Relying on the embedding of $W^{2,2}_{\mathcal{N}} (\Omega) \hookrightarrow L^q(\Omega)$ for all $1 \leq q <\infty$ and using the estimate, $F(s) \leq \frac{\mu|s|^p}{2}(e^{s^2}-1)$ for all $s \in \mathbb{R}$, we have the following two lemmas on the lower bound and upper bound of $\theta_0$ in terms of $\lambda$, $\mu$. \begin{lemma} \label{upper bound} There exists $C_1=C_1(p)>0$, such that $$ \theta_0 \geq -C_1 \lambda^{\frac{p+4}{p+3}}. $$ \end{lemma} \begin{proof} Let $u \in \mathcal{M}$. Then \begin{align*} J(u)&=\frac{1}{2} \int_{\Omega} |\Delta u|^2 -\int_{\Omega}F(u) -\lambda \int_{\Omega}hu\\ &=\int_{\Omega} \big[ \frac{1}{2}f(u)u-F(u)\big] -\frac{\lambda}{2} \int_{\Omega}hu. \end{align*} We note that a simple calculation gives \begin{equation} F(s) \leq \frac{\mu |s|^p}{2}(e^{s^2}-1), \quad \forall s \in \mathbb{R}. \label{F-estimate} \end{equation} Using \eqref{F-estimate}, we deduce \begin{equation} \begin{aligned} J(u) &\geq \frac{\mu}{2} \int_{\Omega} ((u^2-1)e^{u^2}+1)|u|^p - \frac{\lambda}{2} \int_{\Omega}hu \\ &\geq \frac{c\mu}{2} \int_{\Omega} |u|^{p+4} -\frac{\lambda}{2}\int_{\Omega} hu, \end{aligned} \label{lower-bound-lemma} \end{equation} since $(s^2-1)e^{s^2}+1 \geq cs^4$ for some $c>0$ and for all $s \in \mathbb{R}$. By applying H\"{o}lder inequality, we obtain \begin{equation} \int_{\Omega} hu \leq |\Omega|^{\frac{p+2}{2(p+4)}} \|u\|_{L^{p+4}(\Omega)}. \label{lower-bound-holder} \end{equation} From \eqref{lower-bound-lemma} and \eqref{lower-bound-holder}, we obtain \begin{equation} J(u) \geq \frac{c \mu}{2} \|u\|^{p+4}_{L^{p+4}(\Omega)} - \Big(\frac{\lambda |\Omega|^{\frac{p+2}{2(p+4)}}}{2}\Big) \|u\|_{L^{p+4}(\Omega)}. \label{lower-bound} \end{equation} By considering the global minimum of the function \[ \omega(x)= (\frac{c \mu}{2}) x^{p+4} - \Big( \frac{\lambda |\Omega|^{\frac{p+2}{2(p+4)}}}{2}\Big) x, \] we deduce $\theta_0 \geq -C_1 \lambda^{\frac{p+4}{p+3}}$. \end{proof} \begin{lemma}\label{uppbdd} There exists $C_2 >0$, such that $$ \theta_0 \leq - \frac{\mu (p+1)p}{2(p+2)}C_2. $$ \end{lemma} \begin{proof} Choose $v \in W^{2,2}_{\mathcal{N}}(\Omega) \setminus \{0\}$, such that $\int_{\Omega} hv >0$. Therefore, by Lemma \ref{min-thm}, there exists $s_+=s_+(v)>0$, such that $s_+v \in \mathcal{M}^+$. Hence \begin{equation} \begin{aligned} J(s_+v) &= -\frac{s_+^2}{2} \int_{\Omega} |\Delta u|^2 + \int_{\Omega}[ f(s_+v)s_+v - F(s_+v)] \\ & \leq \int_{\Omega}\Big[ f(s_+v)s_+v -F(s_+v) - \frac{1}{2}f'(s_+v)(s_+v)^2\Big], \end{aligned}\label{upp} \end{equation} since $s_+v \in \mathcal{M}$. Now we consider the function $$ \gamma(s) = f(s)s - F(s) - \frac{1}{2} f'(s) s^2. $$ We note that $\gamma'(s)= -\frac{1}{2} f''(s)s^2$. Since $\gamma(0)=0$, it follows that $\gamma(s) \leq 0$ for all $s \in \mathbb{R}$. Also we can verify the following limits \begin{gather*} %(i) \lim_{s \to 0} \frac{\gamma(s)}{|s|^{p+2}} = -\frac{\mu p (p+1)}{2(p+2)} \quad \text{if }p >0, \\ \lim_{s \to 0} \frac{\gamma(s)}{s^4} = -\frac{3}{4}\mu \quad \text{if } p=0, \\ \lim_{s \to \infty} \frac{\gamma(s)}{|s|^{p+4}e^{s^2}} = -\mu \quad \forall p \geq 0. \end{gather*} From these two estimates, we obtain \begin{equation} \gamma(s) \leq -\frac{\mu p(p+1)}{(p+2)}(p+2s^2)|s|^{2(p+2)}e^{s^2} ,\quad \forall s \in \mathbb{R}. \label{upp1} \end{equation} Therefore, using \eqref{upp} and \eqref{upp1} we obtain \begin{equation} \begin{aligned} J(s_+v) &\leq -\frac{\mu p(p+1)}{2(p+2)} \int_{\Omega} (p+2|s_+v|^2) |s_+v|^{p+2} e^{|s_+v|^2} \\ &\leq -\frac{\mu p(p+1)}{2(p+2)} \int_{\Omega} |s_+v|^{p+4}. \end{aligned} \end{equation} Hence \[ \theta_0 \leq -\frac{\mu p(p+1)}{2(p+2)} C_2,\quad \text{where } C_2 = \int_{\Omega} |s_+v|^{p+4}. \] \end{proof} As a consequence of Lemma \ref{Null Set}, we have the following lemma. \begin{lemma} \label{Implicit-Inner} Let $\lambda$ and $h$ satisfy \eqref{Nehari1}. Given $u \in \mathcal{M} \setminus \{0\}$, there exists $\delta >0$ and a differentiable function $s:\{w \in W^{2,2}_{\mathcal{N}}(\Omega): \|w\|_{W^{2,2}_{\mathcal{N}}(\Omega)} < \delta \}\to \mathbb{R}$, with $$ s(0)=1,\quad s(w)(u-w) \in \mathcal{M}, \quad \forall \|w\|_{W^{2,2}_{\mathcal{N}}(\Omega)} < \delta $$ and \begin{equation} \langle s'(0), v \rangle= \frac{2 \int_{\Omega} \Delta u \Delta v - \int_{\Omega} (f'(u)u+f(u))v - \lambda \int_{\Omega}hv} {\int_{\Omega} |\Delta u |^2 - \int_{\Omega}f'(u)u^2} . \label{Nehari4} \end{equation} \end{lemma} \begin{proof} Define a function $G: \mathbb{R} \times W^{2,2}_{\mathcal{N}}(\Omega) \to \mathbb{R}$ by $$ G(s,w) = s \int_{\Omega} |\Delta (u-w)|^2 - \int_{\Omega} f(s(u-w))(u-w) - \lambda \int_{\Omega} h(u-w). $$ Then $ G \in C^1( \mathbb{R} \times W^{2,2}_{\mathcal{N}}(\Omega); \mathbb{R})$ and since $u \in \mathcal{M}$ it implies $$ G(1,0)= \int_{\Omega} |\Delta u|^2 - \int_{\Omega}f(u)u - \lambda \int_{\Omega} hu = 0. $$ Also $G_s(1,0) \neq 0$, indeed $$ G_s(1,0)= \int_{\Omega} |\Delta u|^2 - \int_{\Omega} f'(u)u^2 \neq 0, $$ thanks to Lemma \ref{Null Set}. Then by the Implicit Function Theorem, there exists $\delta >0$ and a map $s:\{w \in W^{2,2}_{\mathcal{N}}(\Omega): \|w\| < \delta\} \to \mathbb{R}$ of class $C^1$ that satisfies \begin{gather*} G(s(w),w)= 0, \quad \text{if } \|w\|_{W^{2,2}_{\mathcal{N}}(\Omega)} < \delta,\\ s(0) = 1. \end{gather*} Then \begin{align*} 0 &= s(w)G(s(w),w)\\ &= \int_{\Omega} (s(w) |\Delta (u-w)|)^2 - \int_{\Omega}f(s(w)(u-w))s(w)(u-w) \\ &\quad -\lambda \int_{\Omega}hs(w)(u-w), \end{align*} that is $s(w)(u-w) \in \mathcal{M}$ for all $w \in W^{2,2}_{\mathcal{N}}(\Omega)$ with $\|w\| < \delta$. Now if we differentiate the identity $G(s(w),w)=0$ with respect to $w$, we obtain \[ 0 = \langle G_s(s(w),w)s'(w) + G_w(s(w),w), v \rangle, \quad \forall v \in W^{2,2}_{\mathcal{N}}(\Omega). \] Put $w=0$ in the above identity and we obtain \[ 0 = \langle G_s(1,0) s'(0) + G_w(1,0), v \rangle = G_s(1,0) \langle s'(0), v \rangle + \langle G_w(1,0), v \rangle. \] Therefore, \[ \langle s'(0), v \rangle = -\frac{\langle G_w(1,0),v \rangle}{G_s(1,0)} = \frac{2 \int_{\Omega} \Delta u \Delta v - \int_{\Omega} (f'(u)u+f(u))v - \lambda \int_{\Omega}hv}{\int_{\Omega} |\Delta u|^2 - \int_{\Omega}f'(u)u^2}. \] \end{proof} \section{Local minimum of $J$ in $W^{2,2}_{\mathcal{N}}(\Omega)$} We are now in a position to prove the existence of a local minimizer for $J$, which ensures the existence of a first solution. Note that, $\mathcal{M} \subset W^{2,2}_{\mathcal{N}}(\Omega)$ is closed, hence a complete metric space. Now $J$ is bounded below on $\mathcal{M}$. By the Ekeland's Variational Principle, there exists a sequence $\{u_n\} \subset \mathcal{M} \setminus \{0\}$, satisfying \begin{equation} J(u_n) < \theta_0 + \frac{1}{n}, \quad J(v) \geq J(u_n)- \frac{1}{n}\|v-u_n\|_{W^{2,2}_{\mathcal{N}}(\Omega)}, \quad \forall v \in \mathcal{M} . \label{Ekeland} \end{equation} \begin{proposition} \label{norm-dual} Let $\lambda$ and $h$ satisfy \eqref{Nehari1}. Then, we have $$ \lim _{n \to \infty}\|J'(u_n)\|_{(W^{2,2}_{\mathcal{N}}(\Omega))^{-1}} = 0. $$ \end{proposition} \begin{proof} We proceed in three steps. \smallskip \noindent\textbf{Claim 1:} $\liminf_{n \to \infty}\|u_n\|_{W^{2,2}_{\mathcal{N}}(\Omega)}>0$. Suppose this claim is false. Then, there exists a subsequence of $\{u_n\}$, which we still denote by $\{u_n\}$, such that $\|u_n\|_{W^{2,2}_{\mathcal{N}}(\Omega)} \to 0$ as $n \to \infty$. Therefore, $J(u_n) \to 0$ as $n \to \infty$ by continuity of the functional $J$. Which contradicts Lemma \ref{uppbdd}. \smallskip \noindent\textbf{Claim 2:} $\liminf_{n \to \infty} \int_{\Omega} (p+2u_n^2)|u_n|^{p+2}e^{u_n^2} >0$. We argue by contradiction. Assume there exists a subsequence of $\{u_n\}$, which is still denoted by $\{u_n\}$, satisfying \begin{equation} \lim_{n \to \infty} \int_{\Omega}(p+2u_n^2)|u_n|^{p+2}e^{u_n^2} \to 0 \quad \text{as } n \to \infty . \label{lim-dual} \end{equation} Here we note that, $u_n \to 0$ in $L^q(\Omega)$ for all $q \in [1,\infty)$, by using \eqref{lim-dual}, and for the case $p>0$, we obtain \[ \int_{\Omega} f(u_n)u_n = \mu \int_{\Omega} |u_n|^{p+2} e^{u_n^2} \to 0 \quad \text{ as } n \to \infty. \] Therefore, we have $\int_{\Omega}f(u_n)u_n \to 0$ and $\int_{\Omega} hu_n \to 0$ as $n \to \infty$. Since $\{u_n\} \subset \mathcal{M}$, we deduce $\|u_n\|_{W^{2,2}_{\mathcal{N}}} \to 0$ as $n \to \infty$, hence a contradiction to Claim 1. Similar argument also holds for $p=0$. \smallskip \noindent\textbf{Claim 3:} $\liminf_{n \to \infty} | \int_{\Omega}|\Delta u_n|^2 - \int_{\Omega} f'(u_n)u_n^2| >0$. Suppose the claim does not hold. Then for a subsequence $\{u_n\}$, we have \begin{equation} \int_{\Omega} |\Delta u_n|^2 - \int_{\Omega} f'(u_n)u_n^2 = o_n(1). \label{Nehari1234} \end{equation} From \eqref{Nehari1234} and using Claim 1, we infer that \[ \liminf_{n \to \infty}f'(u_n)u_n^2 >0. \] Therefore, we have $u_n \in \Lambda \setminus \{0\}$ for large $n$. Since $\{u_n\} \subset \mathcal{M}$, we obtain \begin{align*} o_n(1) &= \lambda \int_{\Omega} hu_n + \int_{\Omega} (f(u_n)-f'(u_n)u_n)u_n\\ &= -\mu \int_{\Omega} (p+2u_n^2)|u_n|^{p+2}e^{u_n^2} + \lambda \int_{\Omega} hu_n, \end{align*} which contradicts \eqref{Nehari1}. This completes the proof of the claim. Now we complete the proof of the proposition. Suppose the statement of the proposition is false, i.e., $\|J'(u_n)\|_{(W^{2,2}_{\mathcal{N}}(\Omega))^{-1}} >0$, for all large $n$ (otherwise obvious). Now we set, $u=u_n \in \mathcal{M}$ and $w=\delta \frac{J'(u_n)}{\|J'(u_n)\|}$ for $\delta >0$ small (by Riesz representation theorem, we identify $J'(u_n)$ as an element in $W^{2,2}_{\mathcal{N}}(\Omega)$). Applying Lemma \ref{Implicit-Inner}, we obtain $s_n(\delta):= s[ \delta \frac{J'(u_n)}{\|J'(u_n)\|}] >0$, such that \[ w_{\delta}= s_n(\delta) \big[ u_n - \delta \frac{J'(u_n)}{\|J'(u_n)\|}\big] \in \mathcal{M}. \] Now from \eqref{Ekeland} and with the help of Taylor expansion, we have \begin{align*} \frac{1}{n} \|w_{\delta} - u_n\| &\geq J(u_n) - J(w_{\delta})\\ &=(1-s_n(\delta)) \langle J'(w_{\delta}),u_n \rangle + \delta s_n(\delta) \big\langle J'(w_{\delta}), \frac{J'(u_n)}{\|J'(u_n)\|} \big\rangle + o(\delta). \end{align*} Dividing by $\delta>0$ and taking limit as $\delta \to 0$, we obtain \[ \frac{1}{n}(1+|s_n'(0)| \|u_n\|) \geq -s'_n(0) \langle J'(u_n),u_n\rangle +\|J'(u_n)\| =\|J'(u_n)\|. \] Hence \[ \|J'(u_n)\| \leq \frac{1}{n}(1+|s_n'(0)|\|u_n\|). \] We complete the proof by noticing that, $|s_n'(0)|$ is uniformly bounded on $n$ by \eqref{Nehari4} and using the Claim 2. \end{proof} \begin{theorem} Let $\lambda$ and $h$ satisfy \eqref{Nehari1}. Then there exists a nonnegative function $ u_0 \in \mathcal{M}^+$, such that $J(u_0)= \inf_{u \in \mathcal{M}\setminus \{0\}} J(u)$. Moreover, $u_0$ is a local minimum for $J$ in $W^{2,2}_{\mathcal{N}}(\Omega)$. \label{1st solution} \end{theorem} \begin{proof} Let $\{u_n\}$ be a sequence which minimizes $J$ on $\mathcal{M} \setminus \{0\}$ as in \eqref{Ekeland}. \smallskip \noindent\textbf{Step 1:} $\liminf_{n \to \infty} \int_{\Omega} hu_n >0$ and hence $u_n \in \mathcal{M}^+$. Indeed, $u_n \in \mathcal{M}$ and by using Lemma \ref{uppbdd}, there exists $C>0$, such that \begin{equation} \begin{aligned} J(u_n) &= \frac{p}{2(p+2)} \int_{\Omega} |\Delta u_n|^2 + \int_{\Omega} ( \frac{1}{p+2} f(u_n)u_n - F(u_n)) \\ & \quad -\lambda \frac{p+1}{p+2} \int_{\Omega} hu_n < -C . \end{aligned} \label{liminf} \end{equation} Now we note that, $F(s) < \frac{1}{p+2}f(s)s$ for all $ s \in \mathbb{R}$. Using \eqref{liminf}, we conclude \[ \liminf _{n \to \infty} \int_{\Omega}hu_n >0. \] \noindent\textbf{Step 2:} $ \limsup _{n \to \infty} \|u\|_{W^{2,2}_{\mathcal{N}}(\Omega)} < \infty$. \\ \textit{Case 1.} If $p>0$, then by using \eqref{liminf}, we obtain directly $$ \int_{\Omega}|\Delta u_n|^2 \leq \lambda \int_{\Omega} hu_n, $$ and then with the help of Sobolev embedding we derive $\{u_n\}$ is bounded in $W^{2,2}_{\mathcal{N}}(\Omega)$. \\ \textit{Case 2.} If $p=0$, by using the fact that $\frac{1}{2}f(s)s - F(s) \geq Cs^4$ for all $s \in \mathbb{R}$ and for some $C>0$, we deduce that $\{u_n\}$ is a bounded sequence in $L^2(\Omega)$. It implies that $\{F(u_n)\}$ is a bounded sequence in $L^1(\Omega)$ using \eqref{liminf} and hence $\{u_n\}$ is a bounded sequence in $W^{2,2}_{\mathcal{N}}(\Omega)$. \smallskip \noindent\textbf{Step 3:} Existence of $u_0 \in \mathcal{M}^+$. From the previous step up to a subsequence, $u_n \rightharpoonup u_0$ weakly in $W^{2,2}_{\mathcal{N}}(\Omega)$. Now from the Proposition \ref{norm-dual}, we note that $\{f(u_n)u_n\}$ is a bounded sequence in $L^1(\Omega)$. Therefore, by recalling Vitali convergence theorem (for details see Lemma \ref{mountain}), we obtain $$ \int_{\Omega}f(u_n) \phi \to \int_{\Omega}f(u_0)\phi, \quad \text{for all } \phi \in W^{2,2}_{\mathcal{N}}(\Omega). $$ Hence, $u_0$ will solve \eqref{eP}. It is obvious that $u_0 \neq 0$ as $h \neq 0$, that is $u_0 \in \mathcal{M}$. We see that $ \theta_0 \leq J(u_0)$. From \eqref{Ekeland} we obtain by using Fatou's Lemma that $\theta_0 = \liminf_{n \to \infty} J(u_n) \geq J(u_0)$. Therefore, $u_0$ minimizes $J$ on $\mathcal{M} \setminus \{0\}$. Now we have to show $u_0 \in \mathcal{M}^+$. Since $u_0$ satisfies $\int_{\Omega} hu_0>0$, by Lemma \ref{maxima-minima}, there exists $s_+(u_0)$ such that $s_+(u_0) u_0 \in \mathcal{M}^+$. We claim $s_+(u_0)=1$. Suppose $s_+(u_0) <1$, then $s_-(u_0)=1$ and hence $u_0 \in \mathcal{M}^-$. By using Lemma \ref{maxima-minima}, we obtain $$ J(s_+(u_0)u_0) < J(u_0) = \theta_0, $$ which is impossible since $s_+(u_0)u_0 \in \mathcal{M}\setminus\{0\}$. \smallskip \noindent\textbf{Step 4:} $u_0$ is a local minimum for for $J$ in $W^{2,2}_{\mathcal{N}}(\Omega)$. We see that $s_+(u_0) =1$, since $u_0 \in \mathcal{M}^+$ (from Step 3). Also from \eqref{maxima-minima}, we have \begin{equation*} s_+(u_0)=1 0$, we have \begin{equation} 1 < s_*(u_0-w), \quad \forall \|w\|_{W^{2,2}_{\mathcal{N}}(\Omega)} < \delta. \label{minimizer} \end{equation} By Lemma \ref{Implicit-Inner}, for $\delta>0$ small enough if necessary, there exists $s: \{w \in W^{2,2}_{\mathcal{N}}(\Omega): \|w\| < \delta\} \to \mathbb{R}$, such that $s(w)(u_0-w) \in \mathcal{M}$ and $s(0)=1$. Whenever $s(w) \to 1 $ as $\|w\| \to 0$, we have \[ s(w) < s_*(u_0-w), \hspace{1.5mm} \forall w \in W^{2,2}_{\mathcal{N}}(\Omega) \quad \text{with } \|w\| < \delta. \] Hence, we obtain $s(w)(u_0-w) \in \mathcal{M}^+$, using the above inequality and Lemma \ref{maxima-minima}. Again by using the Lemma \ref{maxima-minima}, we see that \[ J(u_0-w) \geq J(s(w)(u_0-w)) \geq J(u_0), \quad \forall s \in [0,s_*(u_0-w)]. \] Therefore, from \eqref{minimizer}, we observe that $J(u_0-w) \geq J(u_0)$ for every $\|w\|_{W^{2,2}_{\mathcal{N}}(\Omega)} <\delta$. Consequently, $u_0$ is a local minimizer. \smallskip \noindent\textbf{Step 5:} A positive local minimum for $J$. When $u_0, -\Delta u_0 > 0$, we are done. Otherwise, we obtain positive solution by the following procedure (cf. \cite{Santos}). Since $-\Delta u_0 \in L^2(\Omega)$ (also we note that $-\Delta u_0 \not\equiv 0$, as $u_0 \in \mathcal{M}^+$), by standard elliptic PDE theory (see e.g \cite[Theorem 9.1.4]{Wu}), the boundary-value problem \[ \begin{gathered} -\Delta v = |-\Delta u_0| \quad \text{in } \Omega,\\ v =0 \quad \text{on } \partial \Omega, \end{gathered} \] has a strong solution in $W^{2,2}_{\mathcal{N}}(\Omega)$. Note that, $\|v\|_{W^{2,2}_{\mathcal{N}}} = \|u_0\|_{W^{2,2}_{\mathcal{N}}}$ and also by maximum principle, we obtain $ v > |u_0|$ in $\Omega$. Hence, we obtain \begin{equation} \|v\| = \|u_0\|,\quad |v|_{p+1} \geq |u_0|_{p+1}, \quad \int_{\Omega} h v >0. \label{hv} \end{equation} From \eqref{hv}, we have \begin{equation*} 1 = s_+(u_0) \leq s_+(v) \leq s_-(v) \leq s_-(u_0). \end{equation*} Then Lemma \ref{min-thm} implies \begin{equation} J(s_+(v)v) =\min_{s \in [0,s_-(v)]} J(sv) \leq J(v) \leq J(u_0), \label{min1} \end{equation} where the first inequality is not strict if and only if $s_+(v)=1$. Since $J(s_+(v)v) \geq \theta_0$ and $J(u_0) =\theta_0$, we obtain $s_+(v)=1$ from \eqref{min1}, which implies $v \in \mathcal{M}^+$ and $J(v)=J(u_0)=\theta_0$. Then $v$ is also a local minimum for $J$. When $u_0$ does not satisfy $u_0, -\Delta u_0 > 0$ a.e in $\Omega$, we replace $u_0$ by $v$. \end{proof} \section{Existence of a second solution} The existence of a second solution for \eqref{eP}, depends on whether we can apply some version of Mountain Pass Lemma. We wish to look for a solution of the form $u_1 = v + u_0$, where $u_0$ is the local minimum for the functional \eqref{energy}. Then, we see that $u_1$ will solve \eqref{eP}, whenever $v$ solves the equation \begin{equation} \label{eP1} \begin{gathered} \Delta^2 v = f(v+u_0)-f(u_0)\quad \text{in }\Omega,\\ v, -\Delta v > 0 \quad \text{in }\Omega, \\ v = \Delta v = 0\quad \text{on } \partial \Omega. \end{gathered} \end{equation} We can write the above problem as \begin{equation} \label{ePtilde} \begin{gathered} \Delta^2 v = \tilde{f}(x,v) \quad \text{in }\Omega,\\ v, -\Delta v > 0 \quad \text{in }\Omega, \\ v = \Delta v = 0 \quad \text{on } \partial \Omega, \end{gathered} \end{equation} when we define the map $\tilde{f}: \Omega \times \mathbb{R} \to \mathbb{R}$ by \[ \tilde{f}(x,s) = \begin{cases} f(s+u_0(x))-f(u_0(x)) &\text{if } s \geq 0, \\ 0 &\text{if }s<0. \end{cases} \] The energy functional corresponding to \eqref{ePtilde} is $J_{u_0}: {W^{2,2}_{\mathcal{N}}(\Omega)} \to \mathbb{R}$, defined by \[ J_ {u_0}(v) = \frac{1}{2}\int_{\Omega}|\Delta v|^2 - \int_{\Omega} \tilde{F}(x,v), \] where $\tilde{F}(x,s) = \int_0^s \tilde{f}(x,t)dt$. Now onwards, we denote $J_{u_0}$ by $J_0$. These type of functionals were studied in many articles, for example see \cite{Zhao-Chang,Peral}. We now state the Generalized Mountain Pass Lemma that was introduced by Ghoussoub and Preiss \cite{Preiss}. \begin{definition} \rm Let $H$ be a closed subset of the Banach Space $W^{2,2}_{\mathcal{N}}(\Omega)$. We say that a sequence $\{v_n\} \subset W^{2,2}_{\mathcal{N}}(\Omega)$ is a Palais-Smale sequence for $J_0$ at the level $c$ around $H$, if \begin{itemize} \item[(i)] $\lim_{n \to \infty} \operatorname{dist}(v_n,H) =0$, \item[(ii)] $\lim_{n \to \infty} J_0(v_n) = c$, \item[(iii)] $\lim_{n \to \infty} \|J_0'(v_n)\|_{(W^{2,2}_{\mathcal{N}}(\Omega))^{-1}} = 0$. \end{itemize} In short, we say such a sequence is a $(PS)_{H,c}$ sequence. \end{definition} \begin{remark} \rm In case $H =W^{2,2}_{\mathcal{N}}(\Omega)$, the above definition coincides with the usual Palais-Smale sequence at the level $c$. \end{remark} \begin{lemma} \label{mountain} Let $H \subset W^{2,2}_{\mathcal{N}}(\Omega)$ be a closed set, $c \in \mathbb{R}$. Assume, $\{v_n\} \subset W^{2,2}_{\mathcal{N}}(\Omega)$ be a $(PS)_{H,c}$ sequence. Then (up to a subsequence), $v_n \rightharpoonup v_0$ weakly in $W^{2,2}_{\mathcal{N}}(\Omega)$, and \begin{equation} \lim _{n \to \infty} \int_{\Omega} \tilde{f}(x,v_n) = \int_{\Omega} \tilde{f}(x,v_0), \quad \lim_{n \to \infty} \int_{\Omega} \tilde{F}(x,v_n) = \int_{\Omega} \tilde{F}(x,v_0). \label{Vitali} \end{equation} \end{lemma} \begin{proof} From the fact that $\{v_n\}$ is a $(PS)_{H,c}$ sequence, we have \begin{gather} \frac{1}{2}\int_{\Omega} |\Delta v_n|^2 - \int_{\Omega} \tilde{F}(x,v_n) = c+ o_n(1), \label{GPS1}\\ \big|\int_{\Omega} \Delta v_n \Delta \phi - \int_{\Omega}\tilde{f}(x,v_n) \phi \big| \leq o_n(1) \|\phi\|_{W^{2,2}_{\mathcal{N}}(\Omega)}, \quad \forall \phi \in W^{2,2}_{\mathcal{N}}(\Omega). \label{GPS2} \end{gather} Now we make the following claim. \smallskip \noindent\textbf{Claim:} $\sup_n \|v_n\|_{W^{2,2}_{\mathcal{N}}(\Omega)} < \infty$ and $\sup_n \int_{\Omega} \tilde{f}(x,v_n) < \infty$. Given any $\epsilon >0$, there exists $s_{\epsilon} >0$, such that \begin{equation} \int_{\Omega} \tilde{F}(x,s) \leq \epsilon \int_{\Omega}s \tilde{f}(x,s), \quad \forall |s| \geq s_{\epsilon}. \label{GPS3} \end{equation} Using \eqref{GPS1} and \eqref{GPS3}, we have \begin{equation} \begin{aligned} \frac{1}{2}\int_{\Omega} |\Delta v_n|^2 &\leq \int_{\Omega \cap \{|v_n| \leq s_{\epsilon}\}} \tilde{F}(x,v_n) + \int_{\Omega \cap \{|v_n| \geq s_{\epsilon}\}} \tilde{F}(x,v_n) +c+o_n(1) \\ &\leq \int_{\Omega \cap \{|v_n| \leq s_{\epsilon}\}}\tilde{F}(x,v_n) + \epsilon \int_{\Omega} \tilde{f}(x,v_n)v_n +c+o_n(1) \\ &\leq C_{\epsilon} + \epsilon \int_{\Omega} \tilde{f}(x,v_n)v_n. \end{aligned} \label{GPS4} \end{equation} Now from \eqref{GPS4} and by substituting $\phi = v_n$ in \eqref{GPS2}, we obtain \begin{align*} \int_{\Omega}\tilde{f}(x,v_n)v_n &\leq \int_{\Omega}|\Delta v_n|^2 + o_n(1) \|v_n\|_{W^{2,2}_{\mathcal{N}}(\Omega)}\\ &\leq 2 C_{\epsilon} + 2\epsilon \int_{\Omega} \tilde{f}(x,v_n)v_n +o_n(1)\|v_n\|_{W^{2,2}_{\mathcal{N}}(\Omega)}, \end{align*} Hence, by choosing $\epsilon$ small enough if needed, we obtain \begin{equation} \int_{\Omega} \tilde{f}(x,v_n)v_n \leq \frac{2C_{\epsilon}}{1-2\epsilon} + o_n(1)\|v_n\|_{W^{2,2}_{\mathcal{N}}(\Omega)}. \label{GPS5} \end{equation} We conclude the claim by using \eqref{GPS2} and \eqref{GPS5}. Also note that, $\sup_n\int_{\Omega}\tilde{f}(x,v_n)v_n <\infty$. Since $\{v_n\} \subset W^{2,2}_{\mathcal{N}}(\Omega)$ is bounded, up to a subsequence $v_n \rightharpoonup v_0$ weakly in $W^{2,2}_{\mathcal{N}}(\Omega)$, for some $v_0 \in W^{2,2}_{\mathcal{N}}(\Omega)$. Let $|A|$ denote the Lebesgue measure of $A \subset \mathbb{R}^{4}$. Now we set \[ C:=\sup_n \int_{\Omega}|\tilde{f}(x,v_n)v_n|, \] and notice that $C<\infty$, from the above claim. Given $\epsilon >0$, we define \[ \mu_{\epsilon}= \max_{x \in \bar{\Omega}, |s|\leq \frac{2C}{\epsilon}} |\tilde{f}(x,s)|. \] Then, for any $A \subset \Omega$ with $|A| \leq \frac{\epsilon}{2\mu_{\epsilon}}$, we have \begin{align*} \int_{A} |\tilde{f}(x,v_n)| &\leq \int_{A \cap \{|v_n| \geq \frac{2C}{\epsilon}\}} \frac{|\tilde{f}(x,v_n)v_n|}{|v_n|} + \int_{A \cap \{|v_n| \leq \frac{2C}{\epsilon}\}} |\tilde{f}(x,v_n)|\\ &\leq \frac{\epsilon}{2} + \mu_{\epsilon}|A| \leq \epsilon. \end{align*} Therefore, $\{\tilde{f}(x,v_n)\}$ is an equi-integrable family in $L^1(\Omega)$ and so is $\{\tilde{F}(x,v_n)\}$ (we note that, $|\tilde{F}(x,t)| \leq C_1 |t||\tilde{f}(x,t)|$ for all $(x,t) \in \bar{\Omega} \times \mathbb{R}$ and for some $C_1>0$). By applying the Vitali convergence theorem, we complete the proof. \end{proof} Certainly, $J_0(0)=0$ and $v=0$ is a local minimum for $J_0$. Also, we have $$ \lim_{s \to \infty}J_0(sv) = -\infty, \quad \forall v \in W^{2,2}_{\mathcal{N}}(\Omega)\setminus \{0\}. $$ Hence, we can fix $e \in W^{2,2}_{\mathcal{N}}(\Omega)\setminus \{0\}$, such that $J_0(e) <0$. Now, we define the mountain pass level $$ c_0 = \inf_{\gamma \in \Gamma} \sup_{s \in [0,1]} J_0(\gamma(s)), $$ where $\Gamma = \{ \gamma \in C([0,1],W^{2,2}_{\mathcal{N}}(\Omega)): \gamma(0)=0, \gamma(1) = e\}$. Note that, from the definition, we have $c_0 \geq 0$. Define, $R_0 =\|e\|_{W^{2,2}_{\mathcal{N}}(\Omega)}$. We note that, $\inf\{J_0(v):\|v\|_{W^{2,2}_{\mathcal{N}}(\Omega)}=R\} = 0$ for all $R \in (0,R_0)$. We consider, $H = W^{2,2}_{\mathcal{N}}(\Omega)$ if $c_0 >0$ and $H=\{\|v\|_{W^{2,2}_{\mathcal{N}}(\Omega)} =\frac{R_0}{2}\}$ if $c_0=0$. We now present an upper bound for $c_0$. \begin{lemma}\label{MPLev} The upper bound $c_0$ for the Mountain Pass level satisfies \begin{equation} c_0 < 16 \pi^2. \label{MPL1} \end{equation} \end{lemma} \begin{proof} Without loss of generality, we can assume that the unit ball $B_0(1) \subset \Omega$. For any $\epsilon >0$, we define \begin{equation} \tilde{\tau}_n(x) := \begin{cases} \sqrt{\frac{1}{16 \pi^2}\log n}+\frac{1}{\sqrt{16 \pi^2 \log n}} (1-n|x|^2), &\text{if } |x| \in [0, \frac{1}{\sqrt{n}}),\\[4pt] -\sqrt{\frac{1}{4\pi^2 \log n}} \log |x|, &\text{if } |x| \in [\frac{1}{\sqrt{n}},1),\\[4pt] \chi_n(x), & \text{if } |x| \in [1,\infty), \end{cases} \end{equation} where \[ \chi_n \in \mathcal{C}^{\infty}_0(\Omega), \quad \chi_n\big|_{\partial B_1(0)}= \chi_n|_{\partial \Omega} =0. \] Furthermore, $\Delta \chi_n|_{\partial \Omega} = 0$ and $ \chi_n, |\nabla \chi_n|, \Delta \chi_n$ are of $O(\frac{1}{\sqrt{2 \log n}})$ as $n \to \infty$. Then, $\tilde{\tau}_n \in W^{2,2}_{\mathcal{N}}(\Omega)$. Now we normalize $\tilde{\tau}_n$, by setting $$ {\tau}_n := \frac{\tilde{\tau}_n}{\|\tilde{\tau}_n\|_{W^{2,2}_{\mathcal{N}} (\Omega)}} \in W^{2,2}_{\mathcal{N}}(\Omega). $$ Suppose \eqref{MPL1} is not true. This implies for all $n$, there is $s_n>0$ (see \cite{Lam-Lu}), such that \[ J_0(s_n \tau_n)= \sup_{s>0}J_0(s \tau_n) \geq 16 \pi^2, \quad \forall n. \] Hence \begin{equation} \frac{s_n^2}{2} - \int_{\Omega} \tilde{F}(x,s_n \tau_n) \geq 16 \pi^2, \quad \forall n. \label{MPL201} \end{equation} In particular \begin{equation} s_n^2 \geq 32 \pi^2, \quad \forall n. \label{MPL2} \end{equation} It follows that, $\frac{d}{ds}J_0(s \tau_n)=0$ at the point of maximum $s=s_n$ for $J_0$, we obtain \begin{equation} s_n^2 = \int_{\Omega} \tilde{f}(x,s_n \tau_n)(s_n \tau_n). \label{MPL3} \end{equation} Now, from the definition of $\tilde{f}$, we have $\inf_{x \in \bar{\Omega}} \tilde{f}(x,s) \geq e^{s^2}$ for $|s|$ large. Then, from \eqref{MPL2} for sufficiently large $n$, we obtain \begin{equation} \begin{aligned} s_n^2 &\geq \int_{\{|x| \leq \frac{1}{\sqrt{n}}\}} \tilde{f}(x,s_n \tau_n)(s_n \tau_n) \\ &\geq \int_{\{|x| \leq \frac{1}{\sqrt{n}}\}} e^{s_n^2 \tau_n ^2}(s_n \tau_n) \\ &\geq e^{{s_n^2} \frac{\log n}{16 \pi^2}} \frac{s_n}{\sqrt{16 \pi^2}} \sqrt{\log n} \frac{\pi^2}{2} \frac{1}{n^2} \\ &=\frac{\pi}{8} e^{(\frac{s_n^2}{16\pi^2}-2)\log n} s_n (\log n)^{\frac{1}{2}}. \end{aligned} \label{MPL4} \end{equation} Using \eqref{MPL2} and \eqref{MPL4}, it follows that $s_n$ is bounded and also $s_n^2 \to 32 \pi^2$. Also from \eqref{MPL4}, we have $s_n \geq \frac{\pi}{8} (\log n)^{\frac{1}{2}}$ for all large $n$, which gives a contradiction. \end{proof} We now prove the theorem regarding the existence of a second solution. \begin{theorem} \label{2nd solution} Given a local minimum $u_0$ of $J$ in $W^{2,2}_{\mathcal{N}}(\Omega)$, there exists an element $v_0 \in W^{2,2}_{\mathcal{N}}(\Omega)$ with $v_0 > 0$ in $\Omega$, such that $J_0'(v_0)=0$. \end{theorem} \begin{proof} From Lemma \ref{MPLev}, we have $c_0 \in [0, 16 \pi^2)$. Consider $\{v_n\}$ be a Palais-Smale sequence for $J_0$ at the level $c_0$ around $H$ (such a $(PS)_{H,c_0}$ sequence exists \cite{Preiss}). Then, up to a subsequence, $v_n \rightharpoonup v_0$ weakly in $W^{2,2}_{\mathcal{N}}(\Omega)$ for some $v_0 \in W^{2,2}_{\mathcal{N}}(\Omega)$ by Lemma \ref{mountain} and \eqref{Vitali} holds. We can easily check that, $v_0$ is a solution of \eqref{ePtilde} and therefore a critical point of $J_0$. It remains to show that $v_0$ is not a trivial solution. We prove this by contradiction. \smallskip \noindent\textbf{Case I.} $c_0 = 0$, $ v_0 = 0$. We note that, $H= \{\|v\|_{W^{2,2}_{\mathcal{N}}(\Omega)} = \frac{R_0}{2}\}$ in this case. Also, \[ o_n(1)=J_0(v_n)=\frac{1}{2} \int_{\Omega}|\Delta v_n|^2 - \int_{\Omega}\tilde{F}(x,v_n) =\frac{1}{2} \int_{\Omega}|\Delta v_n|^2 +o_n(1), \] which contradicts that $\operatorname{dist}(v_n,H) \to 0$ as $n \to \infty$. \smallskip \noindent\textbf{Case II.} $c_0 \in (0,16 \pi^2)$, $v_0=0$. Using the fact that $J_0(v_n) \to c_0$, we see that for given any $\epsilon > 0$, $\|v_n\|^2_{W^{2,2}_{\mathcal{N}}(\Omega)} \leq 32 \pi^2 -\epsilon$ for all large $n$. Let, $0 < \delta < \frac{\epsilon}{32 \pi^2}$ and $q = \frac{32 \pi^2}{(1+\delta)(32 \pi^2 -\epsilon)} >1$. We have $$ \int_{\Omega} |\tilde{f}(x,v_n)v_n|^q \leq C \int_{\Omega} e^{((1+\delta)q\|v_n\|^2)(\frac{v_n^2}{\|v_n\|^2})^2}, $$ since, $\sup_{x \in \bar{\Omega}}|\tilde{f}(x,s)s| \leq C e^{(1+\delta)s^2}$ for all $ s \in \mathbb{R}$ and for some $C>0$. Now using Tarsi's embedding \eqref{Tarsi}, we obtain $\sup_{x \in \bar{\Omega}} \int_{\Omega} |\tilde{f}(x,v_n)v_n|^q < \infty$ since $(1+\delta)q\|v_n\|^2 \leq 32 \pi^2$. Since $v_n \to 0$ pointwise almost everywhere in $\Omega$, by recalling Vitali convergence theorem, one obtains $\int_{\Omega} \tilde{f}(x,v_n)v_n \to 0$ as $n \to \infty$. Therefore, \begin{align*} o_n(1) \|v_n\|_{W^{2,2}_{\mathcal{N}}(\Omega)} &= \langle J_0'(v_n),v_n \rangle = \frac{1}{2} \int_{\Omega}|\Delta v_n|^2 - \int_{\Omega} \tilde{f}(x,v_n)v_n\\ &= \frac{1}{2} \int_{\Omega} |\Delta v_n|^2+ o_n(1), \end{align*} which contradicts, $\frac{1}{2} \int_{\Omega} |\Delta v_n|^2 \to c_0$ as $n \to \infty$. Therefore, $v_0 \not\equiv 0$ in $\Omega$ and the positivity of $v_0$ and $-\Delta v_0$ follows from the fact that, $\tilde{f}(x,s) \geq 0$ for all $(x,s) \in \Omega \times \mathbb{R}$ and using the maximum principle. \end{proof} \section{Proof of Theorem \ref{Multiplicity}} Define, $\lambda_* := \mu C_0^{\frac{p+3}{p+4}}|\Omega|^{-\frac{p+2}{2p+8}}$ where $C_0$ is same as in the Proposition \eqref{lambdastar}. Then, condition \eqref{Nehari1} is true whenever $0<\lambda < \lambda_*$. From the Theorem \ref{1st solution} and \ref{2nd solution}, we show the existence of at least two positive solutions for \eqref{eP}. Also, we define $$ \lambda^* := p \mu^{-1/p} \Big( \frac{\lambda_1}{p+1}\Big)^{\frac{p+1}{p}} \Big( \frac{\int_{\Omega}\phi_1}{\int_{\Omega}h \phi_1}\Big). $$ We prove, there is no solution of \eqref{eP} when $\lambda > \lambda^*$. Assume, $u_{\lambda}$ be a solution of \eqref{eP}. Now multiply \eqref{eP} by $\phi_1$ and then integrating by parts over $\Omega$, we obtain \begin{align*} \int_{\Omega} \phi_1( \Delta^2 u_{\lambda} ) = \int_{\Omega}f(u_{\lambda})\phi_1 + \lambda \int_{\Omega}h \phi_1, \end{align*} which implies \begin{equation} \lambda \int_{\Omega} h \phi_1 = \int_{\Omega}(\lambda_1u_{\lambda}-f(u_{\lambda}))\phi_1 . \label{lambdaupper} \end{equation} We see that, $\lambda_1t-f(t) \leq \lambda_1t-\mu t^{p+1}=\Theta(t)$ for all $t >0$. The global maximum for the function $\Theta$ is $p \mu^{-1/p} (\frac{\lambda_1}{p+1} )^{\frac{p+1}{p}}$ on $(0,\infty)$. Then, from \eqref{lambdaupper} and the definition of $\lambda^*$, we obtain $\lambda \leq \lambda^*$. This completes the proof of Theorem \ref{Multiplicity}. \subsection*{Acknowledgements} This research was supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports. The author wishes to express his sincere thanks to Prof. Prashanth K. 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