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

For p > 1, we establish existence and asymptotic behavior of a positive continuous solution to the following boundary value problem $$left{begin{array}{ll}frac{1}{A} left( APhi _{p}(u^{prime})right) ^{prime}+a_{1}(r)u^{alpha _{1}}+a_{2}(r)u^{alpha _{2}}=0, , {rm in}, (0,infty ), {rm lim}_{rrightarrow 0} APhi _{p}(u^{prime})(r)=0, {rm lim}_{rrightarrow infty } u(r)=0,end{array}right.$$ 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.