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 $[f]$.
