Spectrum and global bifurcation results for nonlinear second-order problem on all of $${\mathbb {R}}$$
Tóm tắt
We investigate the following second-order problem
$$\begin{aligned} \left\{ \begin{array}{ll} -u''(t) +a(t)u(t)=\lambda b(t)f(u(t)), \ \ \ \ t\in {\mathbb {R}}, \qquad \qquad \qquad \qquad (P) \\ \underset{|t|\rightarrow \infty }{\lim }u(t)=0, \end{array} \right. \end{aligned}$$
where
$$\lambda >0$$
is a parameter,
$$a \in C({\mathbb {R}}, (0,\infty )),~b\in C({\mathbb {R}}, [0,\infty ))$$
such that
$$\underset{|t|\rightarrow \infty }{\lim }\frac{b(t)}{a(t)} = 0, ~f:{\mathbb {R}}\rightarrow {\mathbb {R}}$$
is a continuous function with
$$sf(s)> 0$$
for
$$s\ne 0$$
. For the linear case, i.e.,
$$f(u)=u$$
, we investigate the existence of principal eigenvalue of (P). For the nonlinear case, depending on the behavior of f near 0 and
$$\infty$$
, we obtain asymptotic behavior and the existence of homoclinic solutions of (P). The proof of our main results is based upon bifurcation technique.
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