Existence of radial solution for a quasilinear equation with singular nonlinearity

Abstract

We prove that the equation \begin{eqnarray*} -Δ_p u =λ\Big( \frac{1} {u^δ} + u^q + f(u)\Big)\;\text{ in } \, B_R(0) u =0 \,\text{ on} \; \partial B_R(0), \quad u>0 \text{ in } \, B_R(0) \end{eqnarray*} admits a weak radially symmetric solution for $\lambda >0$ sufficiently small, $0<\delta <1$ and $p-1<q<p^{\ast }-1$. We achieve this by combining a blow-up argument and a Liouville type theorem to obtain a priori estimates for the regularized problem. Using a variant of a theorem due to Rabinowitz we derive the solution for the regularized problem and then pass to the limit.

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