\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 58, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/58\hfil Multiplicity theorems] {Multiplicity theorems for semipositone $p$-Laplacian problems} \author[X. Shang\hfil EJDE-2011/58\hfilneg] {Xudong Shang} \address{Xudong Shang \newline School of Mathematics, Nanjing Normal University, Taizhou College, 225300, Jiangsu, China} \email{xudong-shang@163.com} \thanks{Submitted December 17, 2010. Published May 4, 2011.} \subjclass[2000]{35J92, 35B05, 46T20} \keywords{Weak solutions; operator of type $(S)_{+}$; Semipositone; \hfill\break\indent $p$-Laplacian problems} \begin{abstract} In this article, we study the existence of solutions for the semipositone $p$-Laplacian problems. Under a subliner behavior at infinity, using degree theoretic arguments based on the degree map for operators of type $(S)_{+}$, we prove the existence of at least two nontrivial solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \section{Introduction} In this article, we study the existence of multiple solutions for the following nonlinear elliptic boundary-value problem $$\label{e1.1} \begin{gathered} -\Delta_{p}u = \lambda f(u) \quad x \in \Omega , \\u = 0 \quad x \in \partial\Omega, \end{gathered}$$ where $-\Delta_{p}u = - \operatorname{div}(|\nabla u|^{p-2}\nabla u)$ is the p-Laplacian operator, $p > 1$, $\lambda > 0$, $\Omega \subseteq \mathbb{R}^n(n\geq 1)$ is a bounded open set with smooth boundary, $f: [0, +\infty)\to \mathbb{R}$ is a continuous function satisfying the condition $f(0)< 0$. Such problems are usually referred in the literature as semipositone problems comparing with the positone case of $f(0)\geq 0$. Such semipositone problems arise in buckling of mechanical systems, design of suspension bridges, chemical reactions, astrophysics. As pointed out by Lions in \cite{L}, semipositone problems are mathematically very challenging. The semilinear semipositone problems have been studied for more than a decade. The usual approaches to such semipositone problems are through quadrature methods \cite{CHS,CS}, the method of sub-super-solution \cite{CGS}, bifurcation theory \cite{AAB,SS}. We refer the reader to the survey paper \cite{CMS} and references therein. See \cite{CCS,PS} for related results for multiparameter semipositone problems. Costa, Tehrani and Yang \cite{CTJ} studied the semipositone problems $$\label{e1.2} \begin{gathered} -\Delta u = \lambda f(u) \quad x \in \Omega , \\u = 0 \quad x \in \partial\Omega. \end{gathered}$$ They applied variational methods for locally Lipschitz functional and obtained positive solutions for sublinear and superlinear cases. Let us consider the sublinear case. It is well know that the main difficulty in proving the existence of a positive solution for \eqref{e1.1} consists in finding a positive sub-solution. As a matter of fact, it can be easily seen, no positive sub-solution can exist if $f(u)$ does not assume positive values. Our main objective in this article is to using degree theoretic arguments based on the degree map for operators of type $(S)_{+}$, improve the problem \eqref{e1.2} to quasilinear case, we obtain two nontrivial solutions for problem \eqref{e1.1} in the sublinear case. The hypotheses on the nonlinearity $f$ in problem \eqref{e1.1} are as follows: \begin{itemize} \item[(F1)] $f(0)< 0$, \item[(F2)] $\lim_{s\to +\infty}\frac{f(s)}{s^{p-1}} = 0$, \item[(F3)] $F(\beta) > 0$ for some $\beta>0$, where $F(s)=\int^{s}_0f(t)dt$. \end{itemize} Under the above assumptions, we state our main result for problem \eqref{e1.1}. \begin{theorem} \label{thm1.1} Suppose that {\rm (F1)--(F3)} hold. Then, there exists $\Lambda_0> 0$ such that \eqref{e1.1} has at least two nontrivial solutions for all $\lambda >\Lambda_0$. \end{theorem} The rest of this article is organized as follows. In section 2, we shall present some mathematical background needed in the sequel. Section 3 contains the proof of our main result. \section{Preliminaries} First, we recall some basic facts about the spectrum of $(-\Delta_{p},W_0^{1,p}(\Omega))$ with weights. Let $v\in L^{\infty}_{+}(\Omega)$, $v\neq 0$, $L^{\infty}_{+}(\Omega) =\{u\in L^{\infty}(\Omega):u\geq 0,\, x\in\Omega \}$. Consider the nonlinear weighted eigenvalue problem $$\label{e2.1} \begin{gathered} -\Delta_{p}u = \lambda v(x)|u|^{p-2}u \quad x \in \Omega ,\\ u = 0 \quad x \in \partial\Omega. \end{gathered}$$ This problem has a smallest eigenvalue denoted by $\lambda_{1}(v)$ which is positive, isolated, simple and admits the variational characterization $$\label{e2.2} \lambda_{1}(v) = \inf \big\{\frac{\int_{\Omega}|\nabla u|^{p}dx}{\int_{\Omega}v(x)|u|^{p}dx}: u\in W_0^{1,p}(\Omega), u\not=0\big\}.$$ In \eqref{e2.2} the infimum is attained at a corresponding eigenfunction $\phi_{1}$ taken to satisfy $\|\phi_{1}\|_{p}=1$. If $v_{1},v_{2}\in L^{\infty}_{+}(\Omega)\setminus \{0\}$ are two weight functions such that $v_{1}\leq v_{2}$ a.e. on $\Omega$ with strict inequality on a set of positive measure, then $\lambda_{1}(v_{2})<\lambda_{1}(v_{1})$. As usually we denote $\lambda_{1}=\lambda_{1}(1)$. If the $u\in W_0^{1,p}(\Omega)$ is an eigenfunction corresponding to an eigenvalue $\lambda \neq \lambda_{1}(v)$, then $u$ must change sign. We extend $f$ as $f(s)=f(0)$ for all $s<0$. It's well know that $u$ is a weak solution to \eqref{e1.1} if $u\in W_0^{1,p}(\Omega)$ and $\int_{\Omega} |\nabla u|^{p-2}\nabla u\nabla\varphi dx - \lambda\int_{\Omega} f(u)\varphi dx = 0$ for every $\varphi \in W_0^{1,p}(\Omega)$. For each $u\in W_0^{1,p}(\Omega)$, we define $I,K: W_0^{1,p}(\Omega)\to W_0^{-1,p'}(\Omega)$ by \begin{gather*} \langle I(u),\varphi \rangle = \int_{\Omega} |\nabla u|^{p-2}\nabla u\nabla\varphi dx, \quad \forall \varphi \in W_0^{1,p}(\Omega), \\ \langle K(u),\varphi \rangle = \int_{\Omega} f(u)\varphi dx , \quad \forall \varphi \in W_0^{1,p}(\Omega). \end{gather*} Hence, the weak solution of \eqref{e1.1} are exactly the solutions of the equation $I - \lambda F = 0$. \begin{definition}[\cite{Z}] \label{def2.1}\rm Let $X$ be a reflexive Banach space and $X^{*}$ its topological dual. We recall the mapping $A: X\to X^{*}$ is of type $(S)_{+}$, if any sequence ${u_{n}}$ in $X$ satisfying $u_{n}\rightharpoonup u_0$ in $X$ and $$\limsup_{n\to +\infty} \langle A(u_{n}), u_{n}-u_0 \rangle \leq 0$$ contains a convergent subsequence. \end{definition} Now consider triples $(A,\Omega,x_0)$ such that $\Omega$ is a nonempty, bounded, open set in $X$, $A: \overline{\Omega}\to X^{*}$ is a demicontinuous mapping of type $(S)_{+}$ and $x_0\not\in A(\partial \Omega)$. On such triples Browder \cite{B} defined a degree denoted by $\deg (A,\Omega,x_0)$, which has the following three basic properties: \begin{itemize} \item[(i)] (Normality) If $x_0\in A(\Omega)$ then $\deg (A,\Omega,x_0) = 1$; \item[(ii)] (Domain additivity) If $\Omega_{1},\Omega_{2}$ are disjoint open subsets of $\Omega$ and $x_0\not\in A(\overline{\Omega}\backslash(\Omega_{1}\cup\Omega_{2}))$ then $\deg (A,\Omega,x_0) = \deg (A,\Omega_{1},x_0) + \deg (A,\Omega_{2},x_0)$; \item[(iii)] (Homotopy invariance) If $\{A_{t}\}_{t\in [0,1]}$ is a homotopy of type $(S)_{+}$ such that ${A_{t}}$ is bounded for every $t\in [0,1]$ and $x_0:[0,1]\to X^{*}$ is a continuous map such that $x_0(t)\not\in A_{t}(\partial\Omega)$ for all $t\in [0,1]$, then $\deg (A_{t},\Omega,x_0(t))$ is independent of $t\in [0,1]$. \end{itemize} \begin{remark} \label{rmk2.2}\rm The operator $A$ is of type $(S)_{+}$ and $B$ is compact implies that $A+B$ is of type $(S)_{+}$. \end{remark} \begin{lemma}[\cite{MMP}] \label{lem2.3} If $X$ is a reflexive Banach space, $U\subset X$ is open, $\psi\in C^{1}(U)$, $\psi'$ is of type $(S)_{+}$, and there exist $x_0\in X$ and numbers $\gamma<\mu$ and $r>0$ such that \begin{itemize} \item[(i)] $V=\{\psi<\mu\}$ is bounded and $\overline{V}\subset U$; \item[(ii)] $\{\psi\leq\gamma\}\subseteq\overline{B_{r}(x_0)} \subset V$; \item[(iii)] $\psi'(x)\neq 0$ for all $x\in\{\gamma\leq\psi\leq\mu\}$, \end{itemize} then $\deg (\psi',V,0)=1$. \end{lemma} \section{Proof of main results} In this section, first several technical results will be established. \begin{lemma} \label{lem3.1} The mapping $I: W_0^{1,p}(\Omega)\to W_0^{-1,p'}(\Omega)$ is of type $(S)_{+}$. \end{lemma} \begin{proof} Assume that $u_{n}\rightharpoonup u$ in $W_0^{1,p}(\Omega)$ and $$\limsup_{n\to +\infty} \langle I(u_{n}), u_{n}-u\rangle \leq0.$$ Then we obtain $$\limsup_{n\to +\infty} \langle I(u_{n})-I(u), u_{n}-u\rangle \leq0.$$ By the monotonicity property of $I$ we have $$\lim_{n\to +\infty} \langle I(u_{n})-I(u), u_{n}-u\rangle = 0;$$ i.e., $$\label{e3.1} \lim_{n\to +\infty} \int_{\Omega}(|\nabla u_{n}|^{p-2}\nabla u_{n}-|\nabla u|^{p-2}\nabla u)(\nabla u_{n}-\nabla u)dx = 0.$$ Observe that for all $x,y\in \mathbb{R}^n$, $|x-y|^{p}\leq \begin{cases} (|x|^{p-2}x-|y|^{p-2}y)(x-y) & \text{if } p\geq 2,\\ [(|x|^{p-2}x-|y|^{p-2}y)(x-y)]^{p/2}(|x|+|y|)^{(2-p)p/2} &\text{if } 10, we now calculate the \deg (J,B_{R},0). \begin{lemma} \label{lem3.4} Under hypotheses {\rm (F2)}, there exists R_0>0 such that $$\label{e3.3} \deg (J,B_{R},0) = 0 \quad \text{for all } R\geq R_0.$$ \end{lemma} \begin{proof} Let $$\label{e3.4} \langle T(u),\varphi\rangle = \int_{\Omega} (u^{+})^{p-1}\varphi dx, \quad \forall \varphi \in W_0^{1,p}(\Omega).$$ where the u^{+}= \max \{u,0\}, u^{-}= \max \{-u,0\}. Since T is a completely continuous operator, the homotopy H_{1}(t, u):[0,1]\times W_0^{1,p}(\Omega)\to W^{-1,p'}(\Omega) defined by $$\label{e3.5} \langle H_{1}(t, u),\varphi \rangle = \int_{\Omega} |\nabla u|^{p-2}\nabla u \nabla\varphi dx - (1-t)\int_{\Omega} \lambda f(u)\varphi dx - t \int_{\Omega} k(x)(u^{+})^{p-1}\varphi dx$$ for all u,\varphi \in W_0^{1,p}(\Omega), t\in[0,1], k(x)\in L^{\infty}_{+}(\Omega)\setminus \{0\} and k(x)< \lambda_{1}. Clearly H_{1}(t, u) is of type (S)_{+}. We claim that there exists R_0 > 0 such that $$\label{e3.6} H_{1}(t, u)\not= 0 \quad \text{for all } t\in[0,1],\; u\in\partial B_{R},\; R\geq R_0.$$ Suppose that is not true. Then we can find sequences \{t_{n}\}\subset [0,1] and \{u_{n}\}\subset W_0^{1,p}(\Omega) such that t_{n}\to t\in [0,1], \|u_{n}\|\to \infty and $$\label{e3.7} \int_{\Omega} |\nabla u_{n}|^{p-2}\nabla u_{n} \nabla\varphi dx = (1-t_{n})\int_{\Omega} \lambda f(u_{n})\varphi dx + t_{n} \int_{\Omega} k(u_{n}^{+})^{p-1}\varphi dx$$ for all \varphi \in W_0^{1,p}(\Omega). Let h_{n} = \frac{u_{n}}{\|u_{n}\|}, we may assume that there exists h\in W_0^{1,p}(\Omega) satisfying \[ h_{n}\rightharpoonup h \quad \text{in } W_0^{1,p}(\Omega),\quad h_{n}\to h \quad \text{ in } L^{p}(\Omega),\quad h_{n}(x)\to h(x)\quad \text{a.e. on } \Omega.$ Acting with the test function $h_{n} -h \in W_0^{1,p}(\Omega)$ in \eqref{e3.7} we find $$\label{e3.8} \begin{split} &\int_{\Omega} |\nabla h_{n}|^{p-2}\nabla h_{n} \nabla(h_{n} - h) dx \\ &= (1-t_{n})\lambda \int_{\Omega}\frac{f(u_{n})}{\|u_{n}\|^{p-1}}(h_{n} - h) dx + t_{n} \int_{\Omega} k(h_{n}^{+})^{p-1}(h_{n} - h) dx. \end{split}$$ We are already show that \begin{gather*} (1-t_{n})\lambda \int_{\Omega}\frac{f(u_{n})}{\|u_{n}\|^{p-1}}(h_{n} - h) dx \to 0 \quad n\to\infty, \\ t_{n} \int_{\Omega} k(h_{n}^{+})^{p-1}(h_{n} - h) dx\to 0\quad n\to\infty. \end{gather*} Using this and \eqref{e3.8}, we obtain $\lim_{n\to +\infty} \int_{\Omega} |\nabla h_{n}|^{p-2}\nabla h_{n} \nabla(h_{n} - h) dx = 0,$ i.e., $\lim_{n\to +\infty} \langle I(h_{n}),h_{n}-h\rangle = 0$. By Lemma \ref{lem3.1} we obtain $h_{n}\to h$ in $W_0^{1,p}(\Omega)$ as $n\to\infty$ and $\|h\|=1$. This shows that $h\not=0$. Acting with the test function $h\in W_0^{1,p}(\Omega)$ in \eqref{e3.8}, we have $$\label{e3.9} \int_{\Omega} |\nabla h_{n}|^{p-2}\nabla h_{n} \nabla h dx = (1-t_{n})\lambda \int_{\Omega}\frac{f(u_{n})}{\|u_{n}\|^{p-1}}h dx + t_{n} \int_{\Omega} k(h_{n}^{+})^{p-1}h dx.$$ Passing to the limit in \eqref{e3.9} as $n\to\infty$, using hypothesis (F2) we find $$\label{e3.10} \int_{\Omega} |\nabla h|^{p}dx = \int_{\Omega} tk(h^{+})^{p} dx.$$ Acting with the test function $h^{-}\in W_0^{1,p}(\Omega)$ we obtain $h\geq 0$. Hence $$\label{e3.11} \int_{\Omega} |\nabla h|^{p}dx = \int_{\Omega} tkh^{p} dx.$$ If $t=0$, then $h=0$, a contradiction. So assume $t\in (0,1]$, exploiting the monotonicity of the principal eigenvalue on the weight function, we obtain $$\label{e3.12} 1=\lambda_{1}(\lambda_{1})< \lambda_{1}(k)\leq\lambda_{1}(tk).$$ We infer that $h=0$, which contradicts to the fact that $h\neq 0$. This contradiction shows the claim stated in \eqref{e3.6}. Due to \eqref{e3.6} we are allowed to use the homotopy invariance of the degree map, which through the homotopy $H_{1}(t,u)$ yields $$\label{e3.13} \deg (J,K_{R},0) = \deg (H_{1}(1,u),B_{R},0) \quad \text{for all } R\geq R_0.$$ Due to \eqref{e3.13}, the problem reduces to computing $\deg (H_{1}(1,u),B_{R},0)$. To this end let the homotopy $H_{2}(t,u): [0,1]\times W_0^{1,p}(\Omega)\to W^{-1,p'}(\Omega)$ be defined by $\langle H_{2}(t,u),\varphi\rangle = \int_{\Omega} |\nabla u|^{p-2}\nabla u \nabla\varphi dx + t\int_{\Omega}m(x)(u^{+})^{p-1}\varphi dx -\int_{\Omega}k(x)(u^{+})^{p-1}\varphi dx$ for all $u,\varphi \in W_0^{1,p}(\Omega)$, $t\in [0,1]$, $m(x)\in L_{+}^{\infty}(\Omega)$ and $m(x)> \lambda_{1}$. Clearly, $H_{2}(t,u)$ it is a homotopy of type $(S)_{+}$. Let us check that $H_{2}(t,u)\neq0$ for all $t\in[0,1]$ and $u\in\partial B_{R}$. Arguing by contradiction, assume that there exist $u \in W_0^{1,p}(\Omega)$ with $\|u\|=R$ and $t\in[0,1]$ such that $$\label{e3.14} \int_{\Omega} |\nabla u|^{p-2}\nabla u \nabla\varphi dx = -t\int_{\Omega} m(x)(u^{+})^{p-1}\varphi dx +\int_{\Omega}k(x)(u^{+})^{p-1}\varphi dx$$ for all $\varphi \in W_0^{1,p}(\Omega)$. Acting with the test function $u^{-}\in W_0^{1,p}(\Omega)$, we obtain $u\geq 0$. So $$\label{e3.15} \int_{\Omega} |\nabla u|^{p-2}\nabla u \nabla\varphi dx = -t\int_{\Omega} m(x)u^{p-1}\varphi dx +\int_{\Omega}k(x)u^{p-1}\varphi dx$$ Acting with the test function $u$ in \eqref{e3.15}, we have $$\label{e3.16} \int_{\Omega} |\nabla u|^{p}dx = \int_{\Omega}(k(x)-tm(x))u^{p}dx <(1-t)\lambda_{1}\int_{\Omega}u^{p}dx.$$ From this inequality, we conclude that $$\label{e3.17} \lambda_{1} \leq \frac{\int_{\Omega}|\nabla u|^{p}dx}{\int_{\Omega}|u|^{p}dx}<(1-t)\lambda_{1}.$$ The contradiction obtained justifies the desired conclusion. By the homotopy invariance of the degree map, we have $$\label{e3.18} \deg (H_{1}(1,u),B_{R},0) = \deg (H_{2}(1,u),B_{R},0) \quad \text{for all } R\geq R_0.$$ We choose $\|m(x)\|_{L^{\infty}}$ sufficiently large such that $$\label{e3.19} \int_{\Omega} |\nabla u|^{p-2}\nabla u \nabla\varphi dx -\int_{\Omega}k(x)u^{p-1}\varphi dx\neq -\int_{\Omega} m(x)u^{p-1}\varphi dx$$ for all $u \in B_{R}$. Then we obtain $\deg (H_{2}(1,u),B_{R},0) = 0 \quad \text{for all } R\geq R_0.$ Hence $\deg (J,B_{R},0) = 0 \quad \text{for all } R\geq R_0.$ The proof is complete. \end{proof} Now we can give the proof of our main result. \begin{proof}[Proof of Theorem \ref{thm1.1}] From the assumption of $f$, we see that for all $\epsilon> 0$, there exists $\theta > 0$ such that $$\label{e3.20} |f(s)|\leq\epsilon|s|^{p-1}+\theta, \quad \text{for all }x\in\Omega,\; s\in \mathbb{R}$$ Define $\phi:W_0^{1,p}(\Omega)\to \mathbb{R}$ as $\phi(u)=\frac{1}{p}\int_{\Omega}|\nabla u|^{p}dx - \lambda\int_{\Omega}F(u)dx.$ It is well know that under \eqref{e3.20}, $\phi$ is well defined on $W_0^{1,p}(\Omega)$, weakly lower semi-continuous and coercive. So, we can find $u_{1}\in W_0^{1,p}(\Omega)$ such that $$\label{e3.21} \phi(u_{1})=\inf_{W_0^{1,p}(\Omega)}\phi(u).$$ By the assumption (F3), we letting $\Omega_{\varepsilon}=\{x\in\Omega: dist(x,\partial\Omega)>\varepsilon \}$, \begin{gather*} u_0(x)=\beta \quad \text{for all }x\in\Omega_{\varepsilon},\\ 0\leq u_0(x)\leq \beta \quad \text{for all } x\in\Omega\setminus\Omega_{\varepsilon}. \end{gather*} Then \begin{align*} \phi(u_0) &=\frac{1}{p}\int_{\Omega}|\nabla u_0|^{p}dx - \lambda(\int_{\Omega_{\varepsilon}}F(u_0)dx +\int_{\Omega\setminus\Omega_{\varepsilon}}F(u_0)dx)\\ &\leq\frac{1}{p}\| u_0\|^{p} - \lambda(F(\beta) |\Omega_{\varepsilon}|-c(1+\beta^{p})|\Omega \setminus\Omega_{\varepsilon}|), \end{align*} when $\epsilon > 0$ sufficiently small, there exists $\Lambda_0>0$ such that $\phi(u_0)<0$ for all $\lambda>\Lambda_0$. So, $\phi(u_{1})<\phi(u_0)<0$, which shows $u_{1}\neq 0$. \eqref{e3.21} implies $$\label{e3.22} \int_{\Omega} |\nabla u_{1}|^{p-2}\nabla u_{1}\nabla\varphi dx = \lambda\int_{\Omega} f(u_{1})\varphi dx$$ for all $\varphi \in W_0^{1,p}(\Omega)$. So $u_{1}$ is a nontrivial solution of \eqref{e1.1}. Since $u_{1}$ is a global minimizer of $\phi$, without loss of generality, we can choose $r_{1}>0$ such that $$\label{e3.23} \phi(u_{1})<\phi(u),\quad \phi'(u)\neq 0\quad\text{for all } u\in\overline{B_{r_{1}}(u_{1})}\backslash\{u_{1}\},$$ and for all $r\in (0,r_{1})$ there holds $\mu = \inf\{\phi(u): u\in\overline{B_{r_{1}}(u_{1})} \setminus B_{r}(u_{1})\} - \phi(u_{1}) > 0.$ Define the set $V=\{u\in B_{\frac{r}{2}}(u_{1}): \phi(u)- \phi(u_{1}) < \mu\}$ which is an open and bounded neighborhood of $u_{1}$. Furthermore, find a number $r_0\in (0,\frac{r}{2})$ with $\overline{B_{r_0}(u_{1})}\subset V$ and $\gamma$ such that $0 < \gamma < \inf\{\phi(u): u\in\overline{B_{r_{1}}(u_{1})}\setminus B_{r_0}(u_{1})\} - \phi(u_{1}).$ Let $U = B_{r_{1}}(u_{1})$ and $\psi = \phi|_{B_{r_{1}}(u_{1})}- \phi(u_{1})$. By the \eqref{e3.23} we know that $0 \not \in \phi'(\overline{V}\backslash B_{r}(u_{1}) )$, using Lemma \ref{lem2.3}, we conclude that $$\label{e3.24} \deg (J,B_{r}(u_{1}),0) = \deg (J,V,0) = 1.$$ From Lemma \ref{lem3.4}, we can find number $R_0$ such that $$\label{e3.25} \deg (J,B_{R},0) = 0 \quad \text{if } R\geq R_0.$$ Now fix $R_0$ in \eqref{e3.25} sufficiently large such that $B_{r}(u_{1})\subset B_{R}$. Since the domain additivity of type $(S)_{+}$ $\deg (J,B_{R},0) = \deg (J,B_{r}(u_{1}),0) + \deg (J,B_{R}\backslash B_{r}(u_{1}),0) .$ we obtain $\deg (J,B_{R}\backslash B_{r}(u_{1}),0)= -1 .$ Hence, there exists $u_{2}\in B_{R}\backslash B_{r}(u_{1})$ solving the problem \eqref{e1.1}. 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