Nodal solutions for an elliptic equation in an annulus without the signum condition
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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.
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.
참고문헌 (19)
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