Nodal solutions for an elliptic equation in an annulus without the signum condition

Bull. Korean Math. Soc. Published online August 20, 2019

Tianlan Chen, Yanqiong Lu, and Ruyun Ma
Department of Mathematics, Northwest Normal University

Abstract : This paper is concerned with the global behavior of components of
radial nodal solutions of semilinear elliptic problems
$$-\Delta v=\lambda h(x, v)\ \ \text{in}\ \Omega,\ \ \
v=0\ \ \text{on}\ \partial\Omega,
$$
where $\Omega=\{x\in \mathbb{R}^N: r_1<|x|<r_2\}$ with $0<r_1<r_2,\
N\geq2$.
The nonlinear term is continuous and satisfies $h(x, 0)=h(x,
s_1(x))=h(x, s_2(x))=0$ for suitable positive, concave function
$s_1$ and negative, convex function $s_2$, as well as $sh(x, s)>0$
for $s\in\mathbb{R}\setminus\{0, s_1(x), s_2(x)\}$. Moreover, we
give the intervals for the parameter $\lambda$ which ensure the
existence and multiplicity of radial nodal solutions for the above
problem. For this, we use global bifurcation techniques to prove our
main results.