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unique solvabily for large {to In consequence, all branches of (5 different from Ifl are
bounded and bounded away from {O} x C(M).
Next, we study the local ~tructure of Ifl near a point ({to, wo) E Ifl\ {(O, O)}. Consider
the self-adjoint elliptic eigenvalue problem
(6)
-div(k grad 1j;)(x) + [ILOQ(X)02a(X, wo(x)) + g'(wo(x))]1j;(x) = c(x)>.1j;(x)
(x EM). We know that (6) has a principal eigenvalue, say >'o(lLo, wo), in the following
sense: >'o(lLo, wo) is a simple real eigenvalue with a positive eigenfunction and every other
eigenvalue is greater than >'o(lLo,wo). If we label the eigenvalues of (6) by magnitude
and multiplicity, we obtain a sequence (>'j(lLo, WO))jEZ+, which converges to 00.
We want to apply the implicit function theorem to the semilinear operator equation
Aw = F(IL,w) at (lLo,wo). To this end we consider A and F as mappings from R+ x
dom(A) equipped with the graph norm induced by A+Id into C(M). We can employ the
implicit function theorem straightforwardly, as long as 0 is not among the eigenvalues
of (6). Assuming that for the moment we find a neighborhood (lLo - 6, lLo + 6) x W
of (lLo, wo), W an open connected set in C(M) with Wo E W, and a CI-function III :
(lLo - 6, lLo + 6) ---* W with Ifl n (lLo - 6, lLo + 6) x W = {(IL, 1lI(1L)) : IL E (lLo - 6, {to + 6)}.
We can also deal with the case >'o(lLo,w) = 0 employing a result of [Amann (1976)],
prop. 20.7, which itself is established via the implicit function theorem utilizing the
strong order monotonicity of w 1-+ (A + Id)-I 0 (F(lLo, w) + w). In that case, we find
again a neighborhood (lLo - 6, lLo + 6) x W of (lLo, wo) and CI-functions jl : (-7],7]) ---*
(lLo - 6, lLo + 6) (7] > 0) and ~ : (-7],7]) ---* W with (jl(O), ~(O)) = (lLo, wo) and Ifl n (lLo -
6,lLo + 6) x W = {(jl(O"),~(O")) : 0" E (-7],7])}. The function ~ is strongly increasing
with respect to the pointwise ordering of C(M) and jl has a strict local extremum at O.
Now let us make the following two "structural hypotheses", which are corroborated by
numerous computer simulations:
Structural Hypotheses.
(H5) >'o(lL, w) = 0 for only finitely many (IL, w) E Ifl.
Under these assumptions one can patch together the local parameterizations of ' .l3
and obtains that ' .l3 is an oriented, I-dimensional CI-submanifold with one boundary
unique solvabily for large {to In consequence, all branches of (5 different from Ifl are
bounded and bounded away from {O} x C(M).
Next, we study the local ~tructure of Ifl near a point ({to, wo) E Ifl\ {(O, O)}. Consider
the self-adjoint elliptic eigenvalue problem
(6)
-div(k grad 1j;)(x) + [ILOQ(X)02a(X, wo(x)) + g'(wo(x))]1j;(x) = c(x)>.1j;(x)
(x EM). We know that (6) has a principal eigenvalue, say >'o(lLo, wo), in the following
sense: >'o(lLo, wo) is a simple real eigenvalue with a positive eigenfunction and every other
eigenvalue is greater than >'o(lLo,wo). If we label the eigenvalues of (6) by magnitude
and multiplicity, we obtain a sequence (>'j(lLo, WO))jEZ+, which converges to 00.
We want to apply the implicit function theorem to the semilinear operator equation
Aw = F(IL,w) at (lLo,wo). To this end we consider A and F as mappings from R+ x
dom(A) equipped with the graph norm induced by A+Id into C(M). We can employ the
implicit function theorem straightforwardly, as long as 0 is not among the eigenvalues
of (6). Assuming that for the moment we find a neighborhood (lLo - 6, lLo + 6) x W
of (lLo, wo), W an open connected set in C(M) with Wo E W, and a CI-function III :
(lLo - 6, lLo + 6) ---* W with Ifl n (lLo - 6, lLo + 6) x W = {(IL, 1lI(1L)) : IL E (lLo - 6, {to + 6)}.
We can also deal with the case >'o(lLo,w) = 0 employing a result of [Amann (1976)],
prop. 20.7, which itself is established via the implicit function theorem utilizing the
strong order monotonicity of w 1-+ (A + Id)-I 0 (F(lLo, w) + w). In that case, we find
again a neighborhood (lLo - 6, lLo + 6) x W of (lLo, wo) and CI-functions jl : (-7],7]) ---*
(lLo - 6, lLo + 6) (7] > 0) and ~ : (-7],7]) ---* W with (jl(O), ~(O)) = (lLo, wo) and Ifl n (lLo -
6,lLo + 6) x W = {(jl(O"),~(O")) : 0" E (-7],7])}. The function ~ is strongly increasing
with respect to the pointwise ordering of C(M) and jl has a strict local extremum at O.
Now let us make the following two "structural hypotheses", which are corroborated by
numerous computer simulations:
Structural Hypotheses.
(H5) >'o(lL, w) = 0 for only finitely many (IL, w) E Ifl.
Under these assumptions one can patch together the local parameterizations of ' .l3
and obtains that ' .l3 is an oriented, I-dimensional CI-submanifold with one boundary