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E. A. Mazepa

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 4 (2014), pp. 36-46.

Résumé

In this paper we study the solvability of certain boundary and external boundary value problems for semilinear elliptic equations (5) $$ \Delta u=\phi(|u|)u, $$ where $\phi(\xi)$ — nonnegative, nondecreasing continuously differentiable function for $\xi\geq 0$ on arbitrary non-compact Riemannian manifolds. In this article we compare the behavior of nonbounded functions "at infinity". In our research we use a new approach wich is based on the consideration of equivalence classes of functions on $M$ (this approach for bounded solutions has been realized in [8]). Let $M$ be an arbitrary smooth connected noncompact Riemannian manifold without boundary and let $\{B_k\}_{k=1}^{\infty}$ be an exhaustion of $M$, i.e., a sequence of precompact open subsets of $M$ such that $\overline{B_k}\subset B_{k+1}$ and $M=\bigcup_{k=1}^{\infty}B_k$. Throughout the sequel, we assume that boundaries $\partial B_k$ are $C^1$-smooth submanifolds. Let $f_1$ and $f_2$ be arbitrary continuous functions on $M$. Say that $f_1$ and $f_2$ are equivalent on $M$ and write $f_1\sim f_2$ if for some exhaustion $\{B_k\}_{k=1}^{\infty}$ of $M$ we have $$ \lim_{k\to\infty}\sup_{M\setminus B_k}|f_1-f_2|=0. $$ It is easy to verify that the relation «$\sim$» is an equivalence which does not depend on the choice of the exhaustion of the manifold and so partitions the set of all continuous functions on $M$ into equivalence classes. Denote the equivalence class of a function $f$ by $[f]$. Let $B\subset M$ be an arbitrary connected compact subset and the boundary of $B$ is a $C^1$-smooth submanifold. Assume that the interior of $B$ is non-empty and $B\subset B_k$ for all $k$. Observe that if the manifold $M$ has compact boundary or there is a natural geometric compactification of $M$ (for example, on manifolds of negative sectional curvature or spherically symmetric manifolds) which adds the boundary at infinity, then this approach leads naturally to the classical statement of the Dirichlet problem. Denote by $v_k$ the solution of equation (5) in $B_k\setminus B$ which satisfies to conditions $$ \left.v_k\right|_{\partial{B}}=1,\qquad \left.v_k\right|_{\partial{B_k}}=0. $$ Using the maximum principle, we can easily verify that the sequence $v_k$ is uniformly bounded on $M\setminus B$ and so is compact in the class of twice continuously differentiable functions over every compact subset $G\subset M\setminus B$. Moreover, as $k\to\infty$ this sequence increases monotonically and converges on $M\setminus B$ to a solution of equation (5) $$ v=\lim_{k\to\infty}v_k,\qquad 0\leq 1,\qquad \left.v\right|_{\partial{B}}=1. $$ Also, note that the function $v$ is independent of the choice of exhaustion $\{B_k\}_{k=1}^{\infty}$. We call $v$ the $L$-potential of the compact set $B$ relative to $M$. For the Laplace-Beltrami equation, the function $v$ is nothing but the capacity potential of the compact set $B$ relative to the manifold $M$ (see [14]). Call the manifold $M$ $L$-strict if for some compact set $B\subset M$ there is an $L$-potential $v$ of $B$ such that $v\in[0]$ (see [8]). A function $f$ is called asymptotically nonnegative whenever there exists a continuous function $w\geq 0$ on $M$ with $w\sim f$. Say that a boundary value problem for (5) is solvable on $M$ with boundary conditions of class $[f]$ whenever there exists a solution $u(x)$ to (5) on $M$ with $u\in [f]$. Say that for a continuous function $\Phi(x)$ on $\partial B$ the exterior boundary value problem for (5) is solvable on $M\setminus B$ with boundary conditions of class $[f]$ whenever on $M \setminus B$ there exists a solution $u(x)$ to (5) with $u\in[f]$ and $u|_{\partial B}=\Phi|_{\partial B}$. Similarly we can state boundary value problems on arbitrary noncompact Riemannian manifolds for (1), (2), as well as a series of other second order elliptic differential equations (see [8;15]). Let the function $\phi(\xi)$ is bounded for $\xi\geq 0$, i.e. there is a constant $\lambda>0$ such that $0\leq\phi(\xi)\leq\lambda$ with $\xi\geq 0 $. We put in the equation (2) $c(x)\equiv\lambda$. Also let $f$ is an asymptotically nonnegative function on $M$. We now formulate the main result. Theorem 1. Let $M$ be an $\Delta$-strict manifold. Suppose that for every positive constant the exterior boundary value problems for the equations (1) and (2) are solvable on $M \setminus B$ with boundary conditions of class $[f]$. Then: for every continuous function $\Phi(x)\geq 0$ on $\partial B$ the exterior boundary value problem for (5) is solvable on $M \setminus B$ with boundary conditions of class $[f]$; the boundary value problem for (5) is solvable on $M$ with boundary conditions of class $[f]$.

Détail

Keywords: semilinear elliptic equation, boundary value problem, noncompact Riemannian manifolds, the Dirichlet problem.
Mots-clés : nonnegative solution

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