Multiple positive solutions to elliptic boundary blow-up problems

Abstract

We prove the existence of multiple positive radial solutions to the sign-indefinite elliptic boundary blow-up problem \[ \left\{\begin{array}{ll} Δu + \bigl(a^+(\vert x \vert) - μa^-(\vert x \vert)\bigr) g(u) = 0, & \; \vert x \vert < 1, \\ u(x) \to \infty, & \; \vert x \vert \to 1, \end{array} \right. \] where $g$ is a function superlinear at zero and at infinity, $a^{+}$ and $a^{-}$ are the positive/negative part, respectively, of a sign-changing function $a$ and $\mu >0$ is a large parameter. In particular, we show how the number of solutions is affected by the nodal behavior of the weight function $a$. The proof is based on a careful shooting-type argument for the equivalent singular ODE problem. As a further application of this technique, the existence of multiple positive radial homoclinic solutions to $$\Delta u+(a^{+}(|x|)-\mu a^{-}(|x|))g(u)=0,x\in {\mathbb{R}}^N,$$ is also considered.

References in corpus (1)

• Multibump nodal solutions for an indefinite superlinear elliptic problem