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1. CJM 2006 (vol 58 pp. 449)
Existence and Multiplicity of Positive Solutions for Singular Semipositone $p$-Laplacian Equations Positive solutions are obtained for the boundary value problem
\[\begin{cases}
-( | u'| ^{p-2}u')'
=\lambda f( t,u),\;t\in ( 0,1) ,p>1\\
u( 0) =u(1) =0.
\end{cases}
\]
Here $f(t,u) \geq -M,$ ($M$ is a positive constant)
for $(t,u) \in [0\mathinner{,}1] \times (0,\infty )$.
We will show the existence of two positive
solutions by using degree theory together with the upper-lower
solution method.
Keywords:one dimensional $p$-Laplacian, positive solution, degree theory, upper and lower solution Category:34B15 |
2. CJM 1997 (vol 49 pp. 1066)
Multiparameter Variational Eigenvalue Problems with Indefinite Nonlinearity 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
$\lambda(\mu,\alpha)$.
Categories:34B15, 34B25 |