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

TL;DR

This paper proves the existence of multiple positive solutions for a class of quasilinear elliptic equations with cylindrical singularities and multiple critical exponents, using variational methods and overcoming compactness issues.

Contribution

It introduces new existence results for solutions to elliptic problems with complex singularities and critical nonlinearities, expanding the understanding of such equations.

Findings

Existence of at least two positive solutions under certain conditions.

Development of Nehari manifold techniques for problems with singularities.

Conditions established to handle lack of compactness in the variational framework.

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.