CONTINUOUS DEPENDENCE OF BOUNDARY VALUES FOR SEMIINFINITE INTERVAL ORDINARY DIFFERENTIAL EQUATIONS

Certain elliptic equations arising in catalysis theory can be transformed into ordinary differential equations on the interval (0,∞). The solutions to these problems usually depend on parameters ρ∈ℝn, say u(t,ρ). For certain types of nonlinearities, we show that the boundary value u˙(∞,ρ) is continuous on compact sets of the variable ρ. As a consequence, bifurcation results for the elliptic equation are obtained.

Some problems in catalysis theory (in two spatial dimensions) are modeled by (1.5)-(1.6)with the boundary condition 6( 0o The classic example is the case -I f(,u) u(l + u) The limiting case, f(0,u) u, gives us the Gelfand problem which can be solved explicitly in terms of elementary functions.
The methods for proving continuous dependence are also applicable to other types of nonlinearities where the bifurcation results (using f(0,u)) are much dif- ferent than in the above problem.
-E But can be chosen arbitrarily small, so lim L(g) _< L(g); that is, L(E) is upper semicontinuous at g. Since g was also arbitrary, L(E) is upper semicontinuous on the interval [O,g0].
LEMMA 5. Let C be a compact subset of D. Then there exists a number 6(C) > 0 such that L(g)m(p) __<_ 2 6 for all p C.
The last inequality implies that L(e n) and m(p n) are positive.By lemma 4, it is true that 2-6 < L(gn)m(Pn) < 2. Thus, lim L(gn)m(pn) 2. But by lemmas 2 and n n-o 3, we have that 2 li'--e(gn)m(pn which is a contradiction.Thus, there exists a 6 > 0 such that L(g)m(p) 2 6 for all p e C.
3. THE MAIN RESULT.We now show that the function m(p) is actually continuous on compact sets of the variable p.
THEOREM.Let C be a compact subset of D.  In all cases, there is a (0) such that h(t,0) < 2 1/26 on (%,m) and % is chosen as small as possible.
On the interval [0,*], by continuous dependence of u and by continuity of f where K is a uniform bound (again by continuous dependence of solutions u on compact sets in the variable (t,0)).
In the equation (2.4) we had m(p) 8-f%e-2texp[f(g,u(t,0))] dt.Since the integrand is continuous on [0,oo)xC and is uniformly bounded on the set C by the in- tegrable function K exp (-6t), m(0) is a continuous function on C.

APPLICATIONS.
Consider the Dirichlet problem where Q is the unit ball of 2 with center 0, and where A is the Laplace operator.
section of C 5. OBSERVATIONS AND CONCLUSIONS.
The condition f(g,u) u as g 0 was only needed to illustrate the example above.Similar results could be obtained if there is knowledge of a bifurcation result for other nonlinearities.For example, in Eberly [3], the nonlinearity u e -i is analyzed with similar results, although there are an infinite number of branches of solutions to the condition 6(o0) 0.

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Making the change of variables r e we have5 + e-2texp[f(e,u)] 0, 0 < r < Using a result by Gidas, Ni, and Nirenberg [i], all solutions