Multiparameter Variational Eigenvalue Problems with Indefinite Nonlinearity

http://dx.doi.org/10.4153/CJM-1997-053-0
Canad. J. Math. 49(1997), 1066-1088

  Published:1997-10-01
 Printed: Oct 1997

We consider the multiparameter nonlinear Sturm-Liouville 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 $\mu = (\mu_1, \mu_2, \ldots, \mu_m, \mu_{m+1}, \ldots \mu_n) \in \bar{R}_+^m \times R_+^{n-m} \bigl(R_+ := (0, \infty)\bigr)$ 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 Ljusternik-Schnirelman 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 $\lambda(\mu,\alpha)$.