Existence and multiplicity results on a class of quasilinear elliptic problems with cylindrical singularities involving multiple critical exponents

Abstract

This work deals with the existence of at least two positive solutions for the class of quasilinear elliptic equations with cylindrical singularities and multiple critical nonlinearities that can be written in the form \begin{align*} -\operatorname{div}\left[\frac{|\nabla u|^{p-2}}{|y|^{ap}}\nabla u\right] -\mu\,\frac{u^{p-1}}{|y|^{p(a+1)}} = h\,\frac{u^{p^*(a,b)-1}}{|y|^{bp^*(a,b)}} +\lambda g\,\frac{u^{q-1}}{|y|^{cp^*(a,c)}}, \qquad (x,y) \in \mathbb{R}^{N-k}\times\mathbb{R}^k. \end{align*} We consider $N \geqslant 3$, $\lambda >0$, $p < k \leqslant N$, $1<p<N$, $0 \leqslant \mu <\bar{\mu} \equiv \left\{[k-p(a+1)]/p\right\}^p$, $0 \leqslant a < (k-p)/p$, $a \leqslant b < a+1$, $a \leqslant c < a+1$, $1\leqslant q <p$, $p^*(a,b)=Np/[N-p(a+1-b)]$, and $p^*(a,c) \equiv Np/[N-p(a+1-c)]$; in particular, if $\mu = 0$ we can include the cases $(k-p)/p \leqslant a < k(N-p)/Np$ and $a < b< c<k(N-p(a+1))/p(N-k) < a+1$. We suppose that $g\in L_{\alpha}^{r}(\mathbb{R}^{N})$, where $r = p^*(a,c)/[p^*(a,c)-q]$ and $\alpha = c(p^*(a,c)-q)$, is positive in a ball and that it can change sign; we also suppose that $h \in L^\infty(\mathbb{R}^k)$ and that it has a finite, positive limit $h_0$ at the origin and at infinity. To prove our results we use the Nehari manifold methods and we establish sufficient conditions to overcome the lack of compactness.