
Multiparameter Variational Eigenvalue Problems with Indefinite Nonlinearity
We consider the multiparameter nonlinear SturmLiouville problem
$$\displaylines{
u''(x)  \sum_{k=1}^m\mu_k u(x)^{p_k} + \sum_{k=m+1}^n
\mu_ku(x)^{p_k} = \lambda u(x)^q, \quad x \in I := (1,1), \cr
u(x) > 0, \quad x \in I, \cr
u(1) = u(1) = 0,\cr}$$
where $\mu = (\mu_1, \mu_2, \ldots, \mu_m, \mu_{m+1}, \ldots \mu_n)
\in \bar{R}_+^m \times R_+^{nm} \bigl(R_+ := (0, \infty)\bigr)$
and $\lambda \in R$ are parameters. We assume that
$$1 \le q \le p_1 < p_2 < \cdots < p_n < 2q + 3.$$
We shall establish an asymptotic formula of
variational eigenvalue $\lambda = \lambda(\mu,\alpha)$ obtained
by using LjusternikSchnirelman theory on general level set
$N_{\mu,\alpha} (\alpha > 0:$ parameter of level set).
Furthermore, we shall give the optimal condition of
$\{(\mu, \alpha)\}$, under which $\mu_i (m + 1 \le i \le n:
\hbox{\rm fixed})$ dominates the asymptotic behavior of
$\lambda(\mu,\alpha)$.
Categories:34B15, 34B25 