Asymptotic Behavior of Positive Solutions of a Nonlinear Combined p-Laplacian Equation

Abstract

For p > 1, we establish existence and asymptotic behavior of a positive continuous solution to the following boundary value problem $$\left\{\begin{array}{ll}\frac{1}{A} \left( A\Phi _{p}(u^{\prime})\right) ^{\prime}+a_{1}(r)u^{\alpha _{1}}+a_{2}(r)u^{\alpha _{2}}=0, \, {\rm in}\, (0,\infty ),\\ {\rm lim}_{r\rightarrow 0} A\Phi _{p}(u^{\prime})(r)=0, {\rm lim}_{r\rightarrow \infty } u(r)=0,\end{array}\right.$$ where $${\alpha _{1}, \alpha _{2} < p -1, \Phi _{p}(t) = t|t| ^{p-2},A}$$ is a positive differentiable function and a 1, a 2 are two positive functions in $${C_{\rm loc}^{\gamma}((0, \infty )), 0 < \gamma < 1,}$$ satisfying some appropriate assumptions related to Karamata regular variation theory. Also, we obtain an uniqueness result when $${\alpha _{1}, \alpha _{2} \in (1-p,p-1)}$$ . Our arguments combine a method of sub and supersolutions with Karamata regular variation theory.