Positive Radial Solutions for Elliptic Equations with Nonlinear Gradient Terms on an Exterior Domain

  • Yongxiang Li
  • Yonghong Ding
  • Elyasa Ibrahim


This paper deals with the existence of positive radial solutions of the elliptic equation with nonlinear gradient term:
$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u = K(|x|)\;f(|x|,\,u,\,|\nabla u|), \quad x\in \Omega ,\\ \alpha \,u+\beta \,\frac{\partial u}{\partial n}\;\big |_{\partial \Omega }=0,\\ \lim _{|x|\rightarrow \infty }u(x)=0, \end{array}\right. \end{aligned}$$
where \(\Omega =\{x\in \mathbb {R}^N:\;|x|>r_0\}\), \(N\ge 3\), \(K: [r_0,\,\infty )\rightarrow \mathbb {R}^+\) and \(f:[r_0,\,\infty )\times \mathbb {R}^+\times \mathbb {R}^+ \rightarrow \mathbb {R}^+\) are continuous, \(\mathbb {R}^+=[0,\,\infty )\). Under the assumption that the coefficient function K(r) satisfies \(0<\int _{r_0}^{\infty }r^{N-1}K(r)\,\mathrm{{d}}r<\infty \), and the conditions that the nonlinearity \(f(r,\,u,\,\eta )\) grows sub- or super-linear in u and \(\eta \), the existence results of positive radial solutions are obtained. For the superlinear case, the growth of f on \(\eta \) is restricted by a Nagumo-type condition and the coefficient function K(r) is further assumed to have the asymptotic behaviour that \(\;K(r)=O(1/r^{2(N-1)})\). The superlinear and sublinear growth of the nonlinearity f are described by inequality conditions instead of the usual upper and lower limits conditions. Our inequality conditions are weaker than the usual lower and upper limits conditions. The discussion is based on the fixed point index theory in cones.


Elliptic equation positive radial solution exterior domain cone fixed point index 

Mathematics Subject Classification

35J25 35J60 47H11 47N20 


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Authors and Affiliations

  1. 1.Department of MathematicsNorthwest Normal UniversityLanzhouPeople’s Republic of China

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