\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2003(2003), No. 42, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \thanks{\copyright 2003 Southwest Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE--2003/42\hfil A discontinuous problem involving the p-Laplacian] {A discontinuous problem involving the p-Laplacian operator and critical exponent in $\mathbb{R}^N$} \author[C. O. Alves \& A. M. Bertone\hfil EJDE--2003/42\hfilneg] {Claudianor Oliveira Alves \& Ana Maria Bertone} \address{Claudianor Oliveira Alves\hfill\break Universidade Federal de Campina Grande, Departamento de Matem\'atica \\ 58109-970 Campina Grande-PB, Brazil} \email{coalves@dme.ufpb.br} \address{Ana Maria Bertone\hfill\break Universidade Federal da Para\'\i ba, Departamento de Matem\'atica\\ 58059-900 Jo\~ao Pessoa-PB, Brazil} \email{anita@mat.ufpb.br} \date{} \thanks{Submitted September 23, 2002. Published April 16, 2003.} \thanks{Partially supported by PRONEX-MCT/Brazil and Millennium Institute for the Global\hfill\break\indent Advancement of Brazilian Mathematics - IM-AGIMB} \subjclass[2000]{35A15, 35J60, 35H30} \keywords{Variational methods, discontinuous nonlinearities, critical exponents} \begin{abstract} Using convex analysis, we establish the existence of at least two nonnegative solutions for the quasilinear problem $$ -\Delta_{p}u=H(u-a)u^{p^*-1} +\lambda h(x)\quad\mbox{in }\mathbb{R}^N $$ where $\Delta_{p}u$ is the $p$-Laplacian operator, $H$ is the Heaviside function, $p^*$ is the Sobolev critical exponent, and $h$ is a positive function. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{thm}{Theorem}[section] \newtheorem{prop}[thm]{Proposition} \newtheorem{lem}[thm]{Lemma} \section{Introduction} The interest in the study of nonlinear partial differential equations with discontinuous nonlinearities has increased because many free boundary problems arising in mathematical physics may be stated in this from. Among these problems, we have the obstacle problem, the seepage surface problem, and the Elenbaas equation; see for example \cite{chang1, chang2, chang3}. Among the typical examples, we have chosen the model for the heat conductivity in electrical media. This model has a discontinuity in its constitutive laws. In fact, considering a domain $\Omega\subset\mathbb{R}^3$ (which in particular could be taken as the whole space $\mathbb{R}^3$ \cite{AmbCalaDobarro}) with electrical media, the thermal and electrical conductivity are denoted by $K(x,t)$ and $\sigma(x,t)$, respectively. Here $x$ is in $\Omega$ and $t$ represents the temperature. Since we are considering an electrical media, the function $\sigma$ may have discontinuities in $t$, and the distribution of the temperature is unknown. The differential equation describing this distribution is \begin{equation}\label{heat} -\sum_{i=1}^n\frac{\partial}{\partial x_i}\big(K(x,u(x))\frac{\partial u(x)}{\partial x_i}\big)=\sigma(x,u(x)). \end{equation} Note that this equation is related to a free boundary problem in which the jump surface of the electrical conductivity is unknown. We describe this surface as being the set \begin{equation}\label{gamma} \Gamma_\alpha(u)=\{x\in\Omega,\ u(x)=\alpha, \mbox{ $\sigma$ is discontinuous at $\alpha$} \}. \end{equation} When the thermal conductivity $K$ is constant and the electrical conductivity $\sigma$ has a single jump and a critical growth, the model becomes \begin{equation}\label{special} -\Delta u=H(u-a)u^{2^*-1}+\lambda h(x)\quad\mbox{in }\Omega\,. \end{equation} Here $H$ is the Heaviside function (i.e. $H(t)=0$ if $t\le0$ and $H(t)=1$ if $t>0$), $2^*\equiv 2N/(N-2)$ is the well known Sobolev critical exponent for $N>2$, $\lambda$ is a positive parameter, and $h$ is a measurable function defined in $\Omega$. Note that in this model the jump surface of the solution (\ref{gamma}) is represented by the set \begin{equation}\label{gammaa} \Gamma_a(u)=\{x\in\mathbb{R}^N,\ u(x)=a\}. \end{equation} Related to problem (\ref{special}) for the special case of $a=0$, i.e., without jump discontinuities, we cite the works of Tarantello \cite{Tarantello} when p=2, and Alves \cite{Alves2}, Cao, Li \& Zhou \cite{CaoZhou} and Gon\c calves \& Alves \cite{GAlves} for the case $p \geq 2$. In the case $a>0$, we cite the work of Alves, Bertone \& Gon\c calves \cite{AlvesBertoneGoncalves}. In this paper we employ variational techniques to study existence and multiplicity of nonnegative solutions of a family of elliptic equations of type (\ref{special}) in the whole space $\mathbb{R}^N$. More precisely, we shall study the quasilinear problem \begin{equation}\label{main} -\Delta_pu=H(u-a)u^{p^*-1}+\lambda h \mbox{ in $\mathbb{R}^N$,} \end{equation} where here $p^*$ is the critical Sobolev exponent defined by $\frac{p N}{N-p}$ with $N>p$. We consider $a>0$ and $\lambda>0$ real parameters, $h:\mathbb{R}^N \to (0, \infty )$ a positive measurable function with \begin{equation} h\in L^\theta(\mathbb{R}^N)\cap L^1(\mathbb{R}^N),\quad \frac{1}{\theta}+\frac{1}{p^*}=1. \label{H} \end{equation} As a solution of (\ref{main}) we understand a function $u\in \mathcal{D}^{1,p}$ verifying \begin{equation}\label{sol} -\Delta_pu(x)-\lambda h(x)\in\widehat f(u(x))\quad \mbox{ a.e in }\mathbb{R}^N, \end{equation} where $\widehat f$ is the multi-valued function $$ \widehat f(s)=\begin{cases} \{f(s)\},&\mbox{if }s\neq a\\ [f(a--),f(a+)], &\mbox{if }s=a, \end{cases} $$ with $f(t)=H(t-a)t^{p^*-1}$, $f(t+0)=\lim_{\delta\to 0^+} f(t+\delta)$, $f(t-0)=\lim_{\delta\to0^+}f(t-\delta)$. We recall that the solutions of (\ref{main}) are exactly the critical points of the functional $I_{\lambda, a}:\mathcal{D}^{1,p} \to \mathbb{R}$ given by \begin{equation}\label{energy} I_{\lambda,a}(u)=\frac{1}{p}\|u\|^p-\int_{\mathbb{R}^N}F(u)dx -\lambda\int_{\mathbb{R}^N}h(x)udx. \end{equation} where $F(u)=\int_0^u f(t)dt$ and $\mathcal{D}^{1,p}$ is the closure of $C_0^\infty(\mathbb{R}^N)$ with respect to the norm $$ \|\phi\|^p=\int_{\mathbb{R}^N}|\nabla\phi(x)|^pdx. $$ The set $\Gamma_a(u)$ has a relevant role when its Lebesgue measure is zero, since the solutions would satisfy (\ref{main}) in the ``strong" sense, i.e., \begin{equation}\label{strong} -\Delta_p u(x)=H(u(x)-a)u(x)^{p^*-1} + \lambda h(u(x)), \end{equation} almost everywhere (a.e. for short) in $\mathbb{R}^N$. Another important remark is that we are considering only {\sl nontrivial} solutions which means that the functions $u\not\equiv0$ and verify $\mathop{\rm meas}\{x\in\mathbb{R}^N,\ u(x)>a\}>0$. We observe that there exists a function $w_{\lambda}$ which satisfies \begin{equation}\label{linear} -\Delta_p u=\lambda h(x),\ u(x)>0\mbox{ in $\mathbb{R}^N$ } \end{equation} and $|w_{\lambda}|_\infty\le a$, then it is a solution of (\ref{main}) when $\lambda$ is small. Furthermore, we will denote by $w=w_\lambda$ the unique solution of (\ref{linear}). Our main result is the following. \begin{thm}\label{mainthm} Assume that $h$ satisfies \eqref{H}. Then, there exists $\lambda_*>0$ and $a_*>0$ such that if $\lambda\in (0,\lambda_*)$ and $a\in(0,a_*)$, problem \eqref{main} has two nonnegative solutions $u_i$, $i=1,2$ with the following properties: \begin{itemize} \item[(i)] $\Delta_p{u_i} \in L^{\theta}(\mathbb{R}^N)$; \item[(ii)] $\mathop{\rm meas}\{x\in\mathbb{R}^N, u_i(x)>a \}>0$, $i=1,2$; \item[(iii)] $\mathop{\rm meas}\Gamma_a(u_i)=0$; \item[(iv)] $I_{\lambda,a}(u_2)<00$, $e\in X$ with $\|e\|>r_1$ such that \begin{equation}\label{geom} \Phi(u)\ge\eta\mbox{ if $\|u\|=r_1$, $\Phi(e)\le0$.} \end{equation} If $c\equiv\inf_{\gamma\in\Gamma}\max_{0\le t\le 1}\Phi(\gamma(t))$ and $$ \Gamma\equiv\{\gamma\in{\mathcal C}([0,1], \ X)\ |\ \gamma(0)=0,\ \gamma(1)=e\}, $$ then $c>0$ and there exists a sequence $\{u_n\}\subset X$ satisfying (\ref{ps}). \end{thm} \begin{prop}\label{grad} Let $\Phi(u)=\int_{\mathbb{R}^N}F(u)dx$ be the functional defined in (\ref{energy}). Then, $\Phi\in Lip_{loc}(L^{p^*}(\mathbb{R}^N);\mathbb{R})$, $ \partial \Phi(u) \subset (L^{p^{*}}(\mathbb{R}^N))^{\prime}$ and if $\omega\in\partial\Phi(u)$, it satisfies \begin{equation}\label{omega} \omega(x)\in\widehat f(u(x)), \mbox{ a.e. $x\in\mathbb{R}^N$}. \end{equation} \end{prop} \section{Preliminary Results} Hereafter we shall use $L^s,\ s>1$ to represent the Lebesgue space $L^s(\mathbb{R}^N)$ and $|\cdot|_s$ its usual norm. Besides, if $g$ is a Lebesgue integrable function, we shall write $\int g$ for $\int_{\mathbb{R}^N}gdx$ and $S$ denotes the best Sobolev constant of the imbedding $\mathcal D^{1,p}\hookrightarrow L^{p^*}$, that is, $$ S=\min_{u\in\mathcal D^{1,p},\,u\neq 0} \frac{\int|\nabla{u}|^p}{\left(\int|u|^{p^*} \right)^{\frac{p}{p^*}}} $$ Our first Lemma is a version for vectorial functions in $\mathbb{R}^{N}$ of a result due to Brezis \& Lieb ( see \cite{BL} ). Its proof can be found in \cite{Alves}. \begin{lem} \label{lm3.1} Let $\eta _n:\mathbb{R}^N\to \mathbb{R}^K\;(K\geq 1)\;$ with $\eta _n\in L^p(\mathbb{R}^N)\times\ldots\times L^p(\mathbb{R}^N)$\ $(p\geq 2)$, $\eta _n(x)\to 0\;$ a.e in $\mathbb{R}^N\;$ and $A(y)=\left| y\right| ^{p-2}y$, for all $y\in\mathbb{R}^K.$ Then, if \linebreak $|\eta _n| _{L^p(\mathbb{R}^N)}\leq C$, for all $n\in\mathbb{N}$ we have \[ \int_{\mathbb{R}^N}\left| A(\eta _n+w)-A(\eta _n)-A(w)\right|^{\frac p{p-1}}=o_n(1), \] for each $w\in L^p(\mathbb{R}^N)\times\ldots\times L^p(\mathbb{R}^N)\;$ fixed. \label{alves1} \end{lem} The next lemma is standard and its proof use similar arguments to those in \cite{AlvesBertoneGoncalves}. It shows that the functional $I_{\lambda,a}$ verifies the mountain pass geometry. \begin{lem}\label{prel1} There is $\lambda_0>0$ such that for $\lambda\in(0,\lambda_0)$ the functional $I_{\lambda,a}$ verifies the mountain pass geometry (\ref{geom}), for all $a>0$. \end{lem} Using the lemma above, we conclude by Theorem \ref{mountain} that there exists $\{u_{n} \}$ in $\mathcal{D}^{1,p}$ such that $$ I_{\lambda,a}(u_n)\to c\mbox{ and } \|w_n\|=\min_{w_{n}\in\partial I_{\lambda,a}(u_n)}\|w\| \to 0 $$ \begin{lem}\label{prel2} The functional $I_{\lambda,a}$ satisfies the condition $(PS)_c$, for $$ c\in(-\infty,\frac{1}{N}S^{\frac{N}{p}}-c_1\lambda^\frac{p}{p-1}), $$ where $c_1=c_1(N,S,\theta,|h|_\theta)$ is a positive constant that verifies the following inequality $$ \frac{1}{N}t^p-\frac{\lambda}{\theta}|h|_\theta t\ge-c_1 \lambda^{\frac{p-1}{p}},\quad \mbox{for all }t\ge0. $$ \end{lem} \begin{proof} Suppose $u_n$ satisfies (\ref{ps}). One has $u_n$ bounded and there exists $u_0\in \mathcal{D}^{1,p}$ such that $u_n$ converges weakly in $\mathcal D^{1,p}$ and a.e. in $\mathbb{R}^N$ to $u_0$. Let $v_n=u_n-u_0$ and suppose that $\|v_n\|^{p} \to l>0$. Thus, $$ \langle w_n,v_n\rangle =\int|\nabla u_n|^{p-2}\nabla u_n\nabla v_n-\lambda\int h(x)v_n-\langle \rho_n,v_n\rangle $$ where $\rho_{n} \in \partial \Phi(u_{n})$. Using Proposition 2.1, we have \[ 0 \leq \rho_{n}(x) \leq u_{n}^{p^{*}-1}(x) \quad \mbox{a.e in } \mathbb{R}^{N} \] and repeating similar arguments explored in \cite{GAlves}, it is possible to show the existence of a set $\Gamma \subset \mathbb{R}^{N}$ empty or finite such that $\{u_{n} \}$ is strongly convergent in $L^{p^{*}}(K)$ for all $K \subset (\mathbb{R}^{N} \setminus \Gamma )$ compact set. The above information imply that, up to subsequence, we can assume \[ \rho_{n}(x) \to \rho_{0}(x) \quad \mbox{a.e in } \mathbb{R}^{N}. \] The above properties involving the sequences $\{u_{n}\}$ and $\{ \rho_{n} \}$ together with the arguments explored in \cite{GAlves} are sufficient to show \[ \nabla{u_{n}}(x) \to \nabla{u_{0}}(x) \quad \mbox{a.e in } \mathbb{R}^{N}. \] From Lemma \ref{alves1} we have $$ \langle w_n,v_n\rangle =\int|\nabla v_n|^p+\int |\nabla u_0|^{p-2} \nabla u_0 \nabla v_n-\langle \rho_n,v_n\rangle +o_n(1). $$ Hence, $\langle w_n,v_n\rangle =l-\langle \rho_n,v_n\rangle +o_n(1)$, which implies \begin{equation}\label{estimate5} \lim_{n\to \infty}\langle \rho_n,v_n\rangle =l. \end{equation} Moreover, by recalling that $$ \langle \rho_n,v_n\rangle \le\int f(u_n+0){v_n}_+ + \int f(u_n-0)(-{v_n}_-), $$ we get $$ \langle \rho_n,v_n\rangle \le \int u_n^{p^*-1}{v_n}_+= \int_{u_n>u_0}u_n^{p^*-1}(u_n-u_0). $$ Consequently, $$ \langle \rho_n,v_n\rangle \le \int u_0^{p^*}+\int|v_n|^{p^*}- \int_{u_n\le u_0}u_n^{p^*}-\int u_n^{p^*-1}u_0+\int_{u_n\le u_0}u_n^{p^{*}-1}u_0+o_n(1). $$ Therefore, $$ \langle \rho_n,v_n\rangle \le\int|v_n|^{p^*}+o_n(1). $$ The last inequality implies that \begin{equation}\label{estimate6} \lim_{n\to \infty}\langle \rho_n,v_n\rangle \le\lim_{n\to \infty}\int|v_n|^{p^*}. \end{equation} Now, from (\ref{estimate5}) and (\ref{estimate6}), we obtain that $Sl^{\frac{p}{p^*}}\le l$, which infers \begin{equation}\label{estimate7} S^{\frac{N}{p}}\le l. \end{equation} On the other hand, we have $$ I_{\lambda,a}(u_n)+o_n(1)=I_{\lambda,a}(u_n)- \frac{1}{p^*}\langle w_n,u_n\rangle $$ that is, $$ I_{\lambda,a}(u_n)+o_n(1) \geq \frac{1}{N}\|u_n\|^p-\frac{\lambda}{\theta} \int h(x) u_n. $$ Thus, $$ I_{\lambda,a}(u_n)+o_n(1)\geq \frac{1}{N}\|v_n\|^p-\lambda^{\frac{p}{p-1}} c_1+o_n(1), $$ where $ c_1=c_1(N,S,\theta,|h|_\theta)$ is the constant stated in the Lemma. \\ From the last inequality and (\ref{estimate7}), we get $$ c\ge \frac{S^{\frac{N}{p}}}{N}-\lambda^{\frac{p}{p-1}} c_1, $$ which contradicts that $c\in(-\infty, \frac{S^{\frac{N}{p}}}{N}-\lambda^{\frac{p}{p-1}} c_1)$. Therefore, we should have $l=0$ and consequently $u_n\to u_0$ in $\mathcal D^{1,p}$. This finished the proof of the lemma. \end{proof} \begin{lem}\label{prel3} There exists $\lambda_1>0$, $a_*>0$, and $e\in\mathcal D^{1,p}$ such that, for $\lambda\in(0,\lambda_1)$ and $a\in(0,a_*)$, we have $e\in B^c_\rho(0)$ with $I_{\lambda,a}(e)<0$, and \begin{equation}\label{levels} 00$ such that $\frac{S^{\frac{N}{p}}}{N}-\lambda_2^{\frac{p}{p-1}} c_1>0 \,\, \forall \lambda \in (0, \lambda_{2}).$ % It is known by Talenti in \cite{Talenti} that the family of functions $$ w_\varepsilon (x)=\frac{\Big[N\varepsilon \big( \frac{N-p}{p-1} \big)^{p-1} \Big]^ {\frac{N-p}{p^{2}}}}{(\varepsilon+|x|^{\frac{p}{p-1}})^{\frac{N-p}{p}}} \quad \varepsilon >0 $$ satisfies $$ \|w_\varepsilon\|^p=|w_\varepsilon|_{p^*}^{p^*}=S^{\frac{N}{p}}. $$ Note that, there is $t_{0}>0$ such that for $t\le t_0$, we have $$ I_{\lambda,a}(t w_\varepsilon)\le\frac{1}{N}S^{\frac{N}{p}}- c_1\lambda^{\frac{p}{p-1}} \,\,\, \forall \lambda \in (0, \lambda_{2}). $$ Moreover, if $t\ge t_0$ $$ \Omega_a=\{t_0w_\varepsilon>a\} \subset\{tw_\varepsilon>a\}, $$ thus, \begin{align*} I_{\lambda,a}(t w_\varepsilon) &\le \frac{t^p}{p}S^{\frac{N}{p}}- \lambda t\int h(x)w_\varepsilon -\int_{\Omega_a}F(tw_\varepsilon)\\ &=\frac{t^p}{p}S^{\frac{N}{p}}-\lambda t\int h(x)w_\varepsilon- \frac{t^{p^*}}{p^*}\int_{\Omega_a} w_\varepsilon^{p^*}+\frac{a^{p^*}}{p^*}|\Omega_a|. \end{align*} Therefore, the function $$ P(t)=\frac{t^p}{p}S^{\frac{N}{p}}-\lambda t\int h(x)w_\varepsilon- \frac{t^{p^*}}{p^*}\int_{\Omega_a} w_\varepsilon^{p^*}+\frac{a^{p^*}}{p^*}|\Omega_a|, $$ has a maximum at $\gamma_1>0$ and the function $g(t)=\frac{t^p}{p}- \frac{t^{p^*}}{p^*}$ attains its maximum in $t=1$. As a consequence we get $$ I_{\lambda,a}(t w_\varepsilon)\le\frac{1}{N}S^{\frac{N}{p}}- \lambda t_{0} \int h(x)w_\varepsilon +\frac{\gamma_1^{p^*}}{p^*}\int_{\Omega_a^c}w_\varepsilon^{p^*}+ \frac{a^{p^*}}{p^*}|\Omega_a|. $$ Now, noticing that $$ |\Omega_a| \le \frac{\omega_NK_\varepsilon t_0^{\frac{N}{N-p}}} {a^{\frac{N}{N-p}}}, $$ where $K_\varepsilon$ is a constant that dependents of $\varepsilon$, one obtains $$ a^{p^*}|\Omega_a|\to 0\quad \mbox{as } a\to 0. $$ Then, by taking $\lambda_3>0$ such that $$ \lambda t_{0} \int h(x)w_\varepsilon>\lambda^{\frac{p}{p-1}}c_1, $$ for all $\lambda\in(0,\lambda_3)$, we choose $a^{*}=a(\lambda_3)$ satisfying $$ -\lambda t_{0} \int h(x)w_\varepsilon +\frac{\gamma_1^{p^*}}{p^*}\int_{\Omega_a^c}w_\varepsilon^{p^*} +\frac{a^{p^*}}{p^*}|\Omega_a|<-\lambda^{\frac{p}{p-1}}c_1 \quad \forall a \in (0,a^{*}). $$ Finally, for $a\in(0,a_*)$ and $\lambda\in(0,\lambda_1)$, with $\lambda_1=\min\{\lambda_2,\lambda_3\}$, we have $$ I_{\lambda,a}(t w_\varepsilon)\le\frac{1}{N}S^{\frac{N}{p}}- c_1\lambda^{\frac{p}{p-1}} \quad \forall \lambda \in (0, \lambda_{2}) \quad \forall t \geq t_{0}, $$ and the proof is complete \end{proof} \section{Proof of Theorem \ref{mainthm}} \subsection{First solution (Mountain Pass)} Let $\lambda_*=\min\{\lambda_0,\lambda_1\}$, where $\lambda_0$ and $\lambda_1$ were given by Lemmas \ref{prel1} and \ref{prel3}. By Theorem \ref{mountain} there exists a sequence $(PS)_c$, for $c$ defined in (\ref{levels}). Therefore we obtain $\rho_n\in\partial\Phi(u_n)$ such that \begin{equation}\label{wn} w_n=Q'(u_n)-\Psi'(u_n)-\rho_n, \end{equation} where here $w_n$ was defined in (\ref{ps}), and $$ Q(u)=\frac{1}{p}\int|\nabla u|^pdx,\quad \Psi(u)=\lambda\int h(x)u(x)dx. $$ Using straightforward arguments, we find that $\{u_n\}$ is bounded in $\mathcal{D}^{1,p}$. Moreover, using the fact that \[ \langle w_{n},u_{n-}\rangle =o_{n}(1) \] we have $\|{u_n}_-\| \to 0$, where $u_{n-}$ is the negative part of $u_{n}$. Then there exists a nonnegative $u_{1} \in\mathcal D^{1,p}$ such that $u_n \rightharpoonup u_{1}$, $u_n(x)\to u_1(x)$, a.e. $x\in\mathbb{R}^N$. Besides, there exists $\rho_0\in L^{\theta}$ such that $\rho_n\rightharpoonup \rho_0$ in $L^{\theta}$. Now, since $\rho_n\in\partial\Phi(u_n)$, repeating the same arguments explored in the proof of Lemma 3.3 we have \begin{gather*} \rho_n(x)\in\widehat{f}(u_n(x)),\quad\mbox{a.e. }x\in\mathbb{R}^N,\\ \rho_0(x) \in \widehat{f}(u_{1}(x)),\quad \mbox{a.e. }x\in\mathbb{R}^N, \end{gather*} and for $\varphi\in\mathcal D^{1,p}$, \begin{equation} \int|\nabla u_1|^{p-2}\nabla u_1\nabla\varphi-\lambda\int h(x) \varphi -\int \rho_0 \varphi =0. \label{dual} \end{equation} \subsection*{Proof of i):} $\Delta_{p}u_{1} \in L^{\theta}$. In this subsection, we shall adapt for our problem some arguments that could be found in \cite{AC}. From (\ref{dual}), we have $$ -\Delta_{p}u_1=J_1+J_2\quad \mbox{in }(\mathcal{D}^{1,p})', $$ where $J_1,J_2:\mathcal D^{1,p}\to \mathbb{R}$ are linear functionals: $$ J_1(v)=\lambda\int h(x)v \quad\mbox{and}\quad J_2(v)=\int\rho_0(x)v. $$ Note that $J_1, J_2\in(L^{p^*})'\subset(\mathcal{D}^{1,p})'$. Thus, by Riesz's Theorem, $J_1,\ J_2\in L^\theta$ and so $ \Delta_p{u_1} \in L^\theta$. Since (\ref{dual}) holds, then $$ -\Delta_{p}u_1=\lambda h+\rho_0\quad \mbox{a.e }\mathbb{R}^N $$ and, from this equality, we get $$ -\Delta_pu_1(x)-\lambda h(x)\in\widehat f(u(x)),\quad \mbox{a.e. } x\in\mathbb{R}^N. $$ This has proved that $u_1$ is a solution of (\ref{main}). \subsection*{Proof of ii):} $ \mathop{\rm meas} \{x\in\mathbb{R}^{N}; u_{1}>a \}>0$. Now, we shall prove that $u_1$ is a nontrivial solution. By Lemmas \ref{prel2} and \ref{prel3} we get $u_n\to u_1$ and $I(u_1)>0$, so that $u_1\not\equiv0$. Suppose, by contradiction, that $u_1\le a$ in $\mathbb{R}^N$. Then, $u_1$ would verify $$ \|u_1\|^p=\lambda\int h(x)u_1,\quad \mbox{in }\mathbb{R}^N, $$ and as a consequence $$ I(u_1)=\frac{-\lambda(p-1)}{p}\int h(x)u_1<0. $$ This contradicts the fact that $I(u_1)>0$. \subsection{Proof of iii):} $ \mathop{\rm meas} (\Gamma_a(u_1))=0$. Assume by contradiction that $ \mathop{\rm meas} (\Gamma_a(u_i))>0$. By using the Morrey-Stampacchia's Theorem (see \cite{Morrey} and \cite{Stampacchia}), we have that $-\Delta_{p}u(x)=0$ a.e. $x\in\Gamma_a(u)$. Hence, $$ -\lambda h(x)\in [0,a^{p^*}], $$ which is a contradiction. Thus $\mathop{\rm meas}(\Gamma_a(u_i))=0$, $i=1,2$. \subsection{Second solution (Local Minimization)} To prove the existence of $u_2$, we observe that, fixed a positive function $\psi \in C_{0}^{\infty}(\mathbb{R}^{N})$, we have \[ \lim_{t \to 0}I_{\lambda,a}(t \psi)<0. \] Consequently $$ \widetilde c=\inf_{\overline{B_\rho}}I_{\lambda,a}<0, \quad \mbox{for }a\in(0,a_*), $$ and $-\infty<\widetilde c<0$. Now, considering $I_{\lambda,a}|_{\overline B_\rho}$, we apply the Ekeland variational principle (see \cite{Ekeland}) to obtain $u_\varepsilon\in\overline{B_\rho}$ such that \begin{equation}\label{estimate2} I_{\lambda,a}(u_\varepsilon)<\inf_{\overline{B_\rho}} I_{\lambda,a}+\varepsilon, \end{equation} and \begin{equation}\label{estimate3} I_a(u_\varepsilon)0$ be small enough that $u_\gamma=u_\varepsilon+\gamma v\in B_\rho$, and $v\in \mathcal{D}^{1,p}$. From (\ref{estimate3}) we get $$ I_{\lambda,a}(u_\varepsilon+\gamma v)-I_{\lambda,a}(u_\varepsilon) +\gamma \varepsilon\|v\|\ge0. $$ Thus we have $$ -\varepsilon\|v\|\le\limsup_{\gamma \downarrow0} \frac{I_{\lambda,a}(u_\varepsilon+\gamma v)-I_{\lambda,a} (u_\varepsilon)}{\gamma} \le I_{\lambda,a}^0(u_\varepsilon;v). $$ Now, since the equality below $$ I_{\lambda,a}^0(u;v)=\max_{\mu\in\partial I_{\lambda,a}(u)}\langle \mu,v\rangle,\ u,\ v\in \mathcal{D}^{1,p}, $$ holds, it follows that $$ -\varepsilon\|v\|\le I_{\lambda,a}^0(u_\varepsilon,v) =\max_{\omega\in\partial I_{\lambda,a}(u_\varepsilon)} \langle \omega,v\rangle , \quad\mbox{for all } v\in \mathcal{D}^{1,p}. $$ Interchanging $v$ and $-v$ we obtain $$ -\varepsilon\|v\|\le\max_{\omega\in\partial I_{\lambda,a}(u_\varepsilon)} \langle \omega,-v\rangle =-\min_{\omega\in\partial I_{\lambda,a}(u_\varepsilon)}\langle \omega,v\rangle ,\quad v\in \mathcal{D}^{1,p} $$ Therefore, $$ \min_{\omega\in\partial I_{\lambda,a}(u_\varepsilon)} \langle \omega,v\rangle \le\varepsilon\|v\|,\quad v\in \mathcal{D}^{1,p}, $$ concluding that $$ \sup_{\|v\|=1}\min_{\omega\in\partial I_{\lambda,a}(u_\varepsilon)} \langle \omega,v\rangle \le\varepsilon. $$ Finally, by Ky Fan's Min-max theorem (\cite[Proposition 1.8]{BrezisNirStam}), we get $$ \min_{\omega\in\partial I_{\lambda,a}(u_\varepsilon)} \sup_{v\in B_1}\langle\omega,v\rangle\le\varepsilon, $$ which along with (\ref{estimate2}) yields the existence of $u_n\in B_\rho$ such that $I_{\lambda,a}(u_n)\to c$, and ${\min_{\omega\in\partial I_{\lambda,a}(u_n)}}\|\omega\|\to 0$. Therefore, by Lemma \ref{prel2}, there exists $u_2\in\mathcal D^{1,p}$ and a subsequence $u_{n_i}$ of $u_n$ such that $u_{n_i}\to u_2$ in $\mathcal{D}^{1,p}$ and $I_{\lambda,a}(u_2)=c=\inf_{\overline B_\rho}I<0$. Moreover, we have that $u_2>a$ in a open $\omega\in\mathbb{R}^N$ because, otherwise, we would have $u(x)\le a$ in $\mathbb{R}^N$. This implies that $u_2$ is a solution of (\ref{linear}) and by uniqueness we would have $u_2=u_2(a)$, for all $a\in(0, a^*).$ On the other hand, $$ S|u_2|_{p^*}^p\le\|u_2\|^p<\lambda\int h(x)u_2(x)\le\lambda|h|_1a, $$ which implies that $u_2$ goes to zero in $\mathcal{D}^{1,p}$ as $a$ goes to zero, hence $ii)$ and $iv)$ hold for $u_{2}$. The arguments to proof that $u_{2}$ also verifies $i)$ and $iii)$ are the same explored in the section 4.1. 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