\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 142, 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 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/142\hfil Irregular oblique derivative problems] {Irregular oblique derivative problems for second-order nonlinear elliptic equations on infinite domains} \author[G. C. Wen\hfil EJDE-2012/142\hfilneg] {Guo Chun Wen} % in alphabetical order \address{Guo Chun Wen \newline LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China} \email{Wengc@math.pku.edu.cn} \thanks{Submitted July 20, 2012. Published August 20, 2012.} \subjclass[2000]{35J65, 35J25, 35J15} \keywords{Irregular oblique derivative problem; nonlinear elliptic equations; \hfill\break\indent infinite domains} \begin{abstract} In this article, we study irregular oblique derivative boundary-value problems for nonlinear elliptic equations of second order in an infinite domain. We first provide the formulation of the above boundary-value problem and obtain a representation theorem. Then we give a priori estimates of solutions by using the reduction to absurdity and the uniqueness of solutions. Finally by the above estimates and the Leray-Schauder theorem, the existence of solutions is proved. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Formulation of the problem} Let $D$ be an $(N+1)$-connected domain including the infinite point with the boundary $\Gamma=\cup^{N}_{j=0} \Gamma_{j}$ in $\mathbb{C}$, where $\Gamma\in C_{\mu}^{2}\,(0<\mu<1)$. Without loss of generality, we assume that $D$ is a circular domain in $|z|>1$, where the boundary consists of $N+1$ circles $\Gamma_0=\Gamma_{n+1}=\{|z|=1\}$, $\Gamma_j=\{|z-z_j|=r_j\}$, $j=1,\dots,N$ and $z=\infty\in D$. In this article, the notation is as the same in References \cite{b1,v1,w1,w2,w3,w4}. We consider the second-order nonlinear elliptic equation in the complex form $$\begin{gathered} u_{z\bar z}=F(z,u,u_{z},u_{zz}),\quad F=\operatorname{Re} [Qu_{zz}+A_{1}u_{z}] +\hat{A}_2u+A_3, \\ Q=Q(z,u,u_{z},u_{zz}),A_{j}=A_{j}(z,u,u_{z}),\quad j=1,2,3,\; \hat{A}_2=A_2+|u|^\sigma, \end{gathered} \label{e1.1}$$ satisfying the following conditions. \subsection*{Condition (C)} (1) $Q(z,u,w,U),A_{j}(z,u,w)(j=1,2,3)$ are continuous in $u\in\mathbb{R}$, $w\in\mathbb{C}$ for almost every $z\in D,\,U\in\mathbb{C}$, and $Q=0$, $A_{j}=0$ $(j=1,2,3)$ for $z\not\in D$, $\sigma$ is a positive number. (2) The above functions are measurable in $D$ for all continuous functions $u(z)$, $w(z)$ in $\overline{D}$, and satisfy $$L_{p,2}[A_j(z,u,w),\overline{D}]\le k_0,\quad j=1,2,\; L_{p,2}[A_3(z,u,w),\bar D]\le k_1, \label{e1.2}$$ in which $p_0,p\,(20$ in $D$, then there exists a point $z^*\in\Gamma$ such that $M=U(z^*)=\max_{\overline D}U(z)>0$. When $z^*\in\Gamma'$, noting that $\cos(\nu,n)\equiv0$, $c_1(z) \equiv0$, $\partial\Psi(z)/{\partial\nu}\equiv0$ on $\Gamma'$, we have $\partial U/\partial\nu\equiv0$, $U(z)\equiv M$ on $\Gamma_j(1\le j\le J')$, this contradicts the point conditions in \eqref{e1.10}. When $z^*\in\Gamma''$, if $\cos(\nu,n)>0$ at $z^*$, from \cite[Corollary 2.11, Chapter III]{w1}, we have $\partial U/\partial\nu>0$ at $z^*$, this contradicts \eqref{e1.10} on $\Gamma''$. If $\cos(\nu,n)=0$ and $c^*_1(z^*)\neq0$ at $z^*$, then $\partial U/\partial\nu+c_1^*(z)U\ne0$ at $z^*$, it is also impossible. Denote by $L$ the longest curve of $\Gamma$ including the point $z^*$, such that $\cos(\nu,n)=0$ and $c^*(z)=0$, thus $u(z)=M$ on $L$, from the point conditions in \eqref{e1.10}, any point of $\tilde T=\{z_0,z_1,\dots,z_{K'}\}$ cannot be an end point of $L$, then there exists a point $z'\in\Gamma''$, such that at $z'$, $\cos(\nu,n)>0$ $(<0)$, ${\partial U}/{\partial n}>0$, $\cos(\nu,s)>0$ $(<0)$, ${\partial U}/{\partial s}\ge0$, or $\cos(\nu,n)<0$ $(>0)$, ${\partial U}/{\partial n}>0$, $\cos(\nu,s)>0\,(<0)$, ${\partial U}/{\partial s}\le0$, hence $$\frac{\partial U}{\partial\nu}=\cos(\nu,n)\frac{\partial U} {\partial n}+\cos(\nu,s)\frac{\partial U}{\partial s}>0,\quad\text{or } <0\text{ at }z'$$ holds, where $s$ is the tangent vector at $z'\in\Gamma''$, and then $$\frac{\partial U}{\partial\nu}+c^*_1U>0,\quad\text{or } \frac{\partial U}{\partial\nu}+c^*_1U<0\text{ at }z',$$ it is also impossible. This shows that $u(z)$ cannot attain its maximum $M$ at a point $z^*\in\Gamma$. Similarly we can prove that $u(z)$ cannot attain its minimum at a point $z_*\in\Gamma$, hence $u(z)=0$ on $\Gamma$, thus $u(z)=0$ in $\overline D$. \end{proof} By a similar way as stated before, we can prove the uniqueness theorem of solutions of Problem (P) for equation \eqref{e1.1} with $\sigma=0$ as follows. \begin{corollary} \label{coro1.2} Suppose that\eqref{e1.1} with $\sigma=0$ satisfies Condition {\rm (C)} and the following condition, for any real functions $u_j(z)\in C^1(\overline{D}),V_j(z)\in L_{p_0,2}(\overline{D})(j=1,2)$, the following equality holds: \begin{align*} & F(z,u_1,u_{1z},V_1)-F(z,u_2,u_{2z},V_2)\\ &=\operatorname{Re}[\tilde Q(V_1-V_2)+\tilde A_1(u_1-u_2)_z]+\tilde A_2(u_1-u_2)\quad \text{in }D, \end{align*} where $|\tilde Q|\le q_0$ in $D$, $A_1,\tilde A_2\in L_{p_0,2}(\overline D)$. Then Problem {\rm (P)} for equation \eqref{e1.1} has at most one solution. \end{corollary} \section{A priori estimates} We consider the nonlinear elliptic equations of second order $$u_{z\bar z}-\operatorname{Re}[Qu_{zz}+A_{1}u_{z}]-\hat{A}_2u=A_3,\label{e2.1}$$ where $\hat{A}_2=A_2+|u|^\sigma$, $\sigma$ is a positive number, and assume that the above equation satisfies Condition (C). \begin{theorem} \label{thm2.1} Let \eqref{e2.1} satisfy Condition {\rm (C)}. Then any solution of Problem {\rm (P)} for \eqref{e2.1} satisfies the estimates $$\begin{gathered} \hat{C}_\beta[u,\overline{D}]=C^1_\beta[|u|^{\sigma+1},\overline{D}]\le M_1,\quad \|u\|_{W^2_{p_0,2}(D)}\le M_1, \\ \hat{C}_\beta[u,\overline{D}]\le M_2(k_1+k_2), \end{gathered} \label{e2.2}$$ in which $k=(k_0,k_1,k_2)$, $\beta$ $(0<\beta\le\alpha)$, $M_{1}=M_1(q_0,p_0, \beta,k,D)$, $M_2=M_2(q_0,p_0,\beta, k_0,p,D)$ are non-negative constants. \end{theorem} \begin{proof} Using the reduction to absurdity, we shall prove that any solution $u(z)$ of Problem (P) satisfies the estimate $$\hat{C}[u,\overline{D}]=C[|u|^{\sigma+1},\overline{D}]+C[u_z,\overline{D}]\le M_3, \label{e2.3}$$ where $M_3=M_3(q_0,p_0,\alpha,k,p,D)$ is a non-negative constant. Suppose that \eqref{e2.3} is not true, then there exist sequences of coefficients $\{A^{(m)}_j \}$ $(j=1,2,3)$, $\{Q^{(m)}\}$, $\{\lambda^{(m)}(z)\},\{c_j^{(m)}\}$ $(j=1,2)$, $\{b_j^{(m)}\}(j=0,1,\dots,N_0)$, which satisfy the same conditions of Condition (C) and \eqref{e1.6}--\eqref{e1.8}, such that $\{A^{(m)}_j\}$ $(j=1,2,3)$, $\{Q^{(m)}\}$, $\{\lambda^{(m)}(z)\}$, $\{c_j^{(m)}\}$ $(j=1,2)$ and $\{b_j^{(m)}\}$ $(j=0,1,\dots,N_0)$ in $\overline{D},\Gamma$ weakly converge or uniformly converge to $A^{(0)}_j$ $(j=1,2,3)$, $Q^{(0)}$, $\lambda^{(0)}(z)$, $c_j^{(0)}(j=1,2)$, $b_j^{(0)}$ $(j=0,1,\dots,N_0)$, and the corresponding boundary-value problem $$u_{z\bar z}-\operatorname{Re}[Q^{(m)}u_{zz}+A^{(m)}_{1}u_{z}]-\hat{A}^{(m)}_2u= A^{(m)}_3,\hat{A}_2^{(m)}=A^{(m)}_2+|u|^\sigma,\label{e2.4}$$ and $$\frac12\frac{\partial u}{\partial\nu}+a^{(m)}_1(z)u=c^{(m)}_2(z)\quad \text{on } \Gamma,\;u(a_j)=b_j,\,j=0,1,\dots,N_0, \label{e2.5}$$ have the solutions $\{u^{(m)}(z)\}$, where $\hat{C}[u^{(m)}(z),\overline D]\,(m=1,2,\dots)$ are unbounded. Hence we can choose a subsequence of $\{u^{(m)}(z)\}$ denoted by $\{u^{(m)}(z)\}$ again, such that $h_m=\hat{C}[u^{(m)}(z),\overline{G}]\to\infty$ as $m\to\infty$. We can assume $h_m\ge\max[k_1,k_2,1]$. It is obvious that $\tilde u^{(m)}(z)=u^{(m)}(z)/h_m$ $(m=1,2,\dots)$ are solutions of the boundary-value problems $$\tilde{u}_{z\bar z}-\operatorname{Re}[Q^{(m)}\tilde{u}_{zz}+A^{(m)}_{1}\tilde{u}_{z}]-\hat{A}^{(m)}_2 \tilde{u}=A^{(m)}_3/h_m,\label{e2.6}$$ and $$\frac12\frac{\partial\tilde{u}}{\partial\nu}+c^{(m)}_1(z)\tilde{u}=c^{(m)}_2(z)/h_m \quad\text{on }\Gamma,\; \tilde{u}(a_j)=b^{(m)}_j,\; j=0,1,\dots,N_0. \label{e2.7}$$ We can see that the functions in the above equation and boundary conditions satisfy condition (C), \eqref{e1.6}--\eqref{e1.8}, and $$\begin{gathered} |u|^{\sigma+1}/h_m\le1,\quad L_{p,2}[A^{(m)}_3/h_m,\overline{D}]\le 1,\\ |c_2^{(m)}/h_m|\le1,\quad |b_j^{(m)}/h_m|\le1,\quad j=0,1,\dots,N_0, \end{gathered} \label{e2.8}$$ hence from \cite[Theorem 4.10, Chapter III]{w1}, we obtain the estimate $$\hat{C}_{\beta}[\tilde{u}^{(m)}(z),\overline{D}]\le M_4,\|\tilde{u}^{(m)}(z)\|_ {W^2_{p_0,2}(D)}\le M_4,$$ in which $M_4=M_4(q_0,p_0,\beta,k,D)$ is a non-negative constant. Thus from the sequence of functions $\{\tilde{u}^{(m)}(z)\}$, we can choose the subsequence denoted by $\{\tilde{u}^{(m)}(z)\}$, which converges uniformly to $\tilde{u}^{(0)}(z)$ in $\overline{D}$, and their partial derivatives $\tilde{u}^{(m)}_x,\tilde{u}^{(m)}_y$ in $\overline{D}$ are uniformly convergent and $\tilde{u}^{(m)}_{xx},\tilde{u}^{(m)}_{yy},\tilde{u}^{(m)}_{xy}$ in $\overline{D}$ weakly convergent. This shows $\tilde{u}_0(z)$ is a solution of the boundary-value problem $$\tilde{u}_{0z\bar z}-\operatorname{Re}[Q^{(0)}\tilde{u}_{0zz}+A^{(0)}_{1}\tilde{u}_{0z}] -\hat{A}^{(0)}_2\tilde{u}_0=0, \label{e2.9}$$ and $$\frac12\frac{\partial\tilde{u}_0}{\partial\nu}+c^{(0)}_1(z)\tilde{u}_0 =0\quad\text{on }\Gamma, \; u_0(a_j)=0,\; j=0,1,\dots,N_0. \label{e2.10}$$ We see that \eqref{e2.9} possesses the condition $A^{(0)}_3=0$ and \eqref{e2.10} is the homogeneous boundary condition. On the basis of Theorem \ref{thm1.1}, the solution satisfies $\tilde{u}_0(z)=0$. However, from $\hat{C}[\tilde{u}^{(m)}(z),\overline{D}]=1$, we can derive that there exists a point $z^*\in\overline{D}$, such that $[|\tilde{u}_0(z)|^{\sigma+1}+|\tilde{u}_{0z}|]_{z=z^*}\ne0$, which is impossible. This shows the first of two estimates in \eqref{e2.2} is true. It is not difficult to verify the third estimate in \eqref{e2.2}. \end{proof} \section{Solvability} By the above estimates and the Leray-Schauder theorem, we can prove the existence of solutions of Problem (P) for equation \eqref{e1.1}. We first introduce the nonlinear elliptic equation of second order \begin{aligned} u_{z\bar z} &=f_m(z,u,u_z,u_{zz}),\,f_m(z,u,u_z,u_{zz}) \\ &=\operatorname{Re}[Q_mu_{zz}+A_{1m}u_z]+\hat{A}_{2m}u+A_3\quad\text{in }D, \end{aligned} \label{e3.1} with the coefficients \begin{gather*} Q_m=\begin{cases} Q &\text{in } D_m \\ 0 &\text{in } \mathbb{C}\setminus D_m \end{cases} \quad A_{jm}=\begin{cases} A_j & \text{in } D_m \\ 0 &\text{in } \mathbb{C}\setminus D_m \end{cases} \quad j=1,3,\\ \hat{A}_{2m}=\begin{cases}\hat{A}_2 &\text{in } D_m\\ 0 & \text{in } \mathbb{C}\setminus D_m \end{cases} \end{gather*} where $D_m=\{z\in D: \text{dist}(z,\Gamma\cup\{\infty\})\ge1/m\}$, $m$ is a positive integer. \begin{theorem} \label{thm3.1} If \eqref{e3.1} satisfies Condition {\rm (C)}, and $u(z)$ is any solution of Problem {\rm (P)} for equation \eqref{e3.1}, then $u(z)$ can be expressed in the form $$u(z)=U(z)+\tilde{v}(z)=U(z)+\hat v(z)+v(z),$$ where $\tilde{v}(z)=\hat v(z)+v(z)$ is a solution of \eqref{e3.1} with the homogeneous Dirichlet boundary condition $$\tilde{v}(z)=0\quad\text{on }\,\partial D_0=\{|z|=1\}.$$ Here $$v(z)=Hf_m=\frac2\pi\int\int_{D_0}\,\frac{f_m(1/\zeta)}{|\zeta|^4} \ln\big|\frac{1 -\zeta z}\zeta\big|d\sigma_\zeta,$$ in which $D_0$ is the image under the mapping $z=1/\zeta$, $U(z)$ is a solution of the Dirichlet boundary-value problem for $U_{z\bar z}=0$ in $D$, and $U(z)$ and $\tilde{v}(z)$ satisfy the estimates $$\hat{C}^1_{\beta}[U,\overline{D}]+\|U|_{W^2_{p_0,2}(D)}\le M_5,\quad \hat{C}^1_{\beta} [\tilde{v},\overline{D_0}]+\|\tilde{v}\|_{W^2_{p_0,2}(D_0)}\le M_6, \label{e3.2}$$ where $\beta(>0),\,M_j=M_j(q_0,p_0,\beta,k,D_m)$ $(j=5,6)$ are non-negative constants. \end{theorem} \begin{proof} It is clear that the solution $u(z)$ can be expressed as before. On the basis of Theorem \ref{thm2.1}, it is easy to see that $\tilde{v}$ satisfies the second estimate in \eqref{e3.2}, and then we know that $U(z)$ satisfies the first estimate of \eqref{e3.2}. \end{proof} \begin{theorem} \label{thm3.2} If \eqref{e1.1} satisfies Condition {\rm (C)}, then Problem {\rm (P)} for equation \eqref{e1.1} has a solution. \end{theorem} \begin{proof} To prove the existence of solutions of Problem (P) for \eqref{e3.1} by using the Leray-Schauder theorem, we introduce the equation with the parameter $t\in[0,1]$: $$V_{z\bar z}=tf_m(z,u,u_z,(U+V)_{zz})\quad\text{in }D.\label{e3.3}$$ Denote by $B_{M}$ a bounded open set in the Banach space $B=\hat W^{2}_{p_0,2}(D_0)=\hat{C}^1_{\beta}(\overline{D_0})\cap W_{p_0,2}^2(D_0)(0<\beta\le\alpha)$, the elements of which are real functions $V(z)$ satisfying the inequalities \hat{C}_\beta^{1}[V(z),\overline{D_0}]+\|V\|_{W^2_{p_0,2}(D_0)}