Nonexistence of stable solutions for quasilinear Schrödinger equation

Abstract

Abstract In this paper, we study the nonexistence of stable solutions for the quasilinear Schrödinger equation 0.1 −Δu−[Δ(1+u2)1/2]u2(1+u2)1/2=h(x)|u|q−1u,x∈RN, $$ -\Delta u- \bigl[\Delta\bigl(1+u^{2}\bigr)^{1/2} \bigr]\frac{ u}{2(1+u^{2})^{1/2}}=h(x) \vert u \vert ^{q-1}u,\quad x\in R^{N}, $$ where N≥3 $N\ge3$, q≥5/2 $q\ge5/2$ and the function h(x) $h(x)$ is continuous and positive in RN $R^{N}$. Under suitable assumptions on h(x) $h(x)$ and q, we prove that Eq. (0.1) has no nonnegative and stable solutions.

Keywords

• Nonexistence of solution
• Quasilinear Schrödinger equation
• Stable solution