arXiv subjects
Publications and source records attributed to Giuseppina Vannella.
In this paper we obtain uniformly locally $L^{\infty}$-estimate of solutions to non-autonomous quasilinear system involving operators in divergence form and a family of nonlinearities that are allowed to grow also critically.
Let us consider the quasilinear problem \[ (P_\varepsilon) \ \ \left\{ \begin{array}{ll} - \varepsilon^p Δ_{p}u + u^{p-1} = f(u) & \hbox{in} \ Ω \newline u>0 & \hbox{in} \ Ω \newline u=0 & \hbox{on} \ \partial Ω \end{array} \right. \] where $Ω$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary, $N\geq 2$, $1< p < 2$, $\varepsilon >0$ is a parameter and $f: \mathbb{R} \to \mathbb{R}$ is a continuous function with $f(0)=0$, having a subcritical growth. We prove that there exists $\varepsilon^* >0$ such that, for every $\varepsilon \in (0, \varepsilon^*)$, $(P_\varepsilon)$ has at least $2{\mathcal P}_1(Ω)-1$ solutions, possibly counted with their multiplicities, where ${\mathcal P}_t(Ω)$ is the Poincaré polynomial of $Ω$. Using Morse techniques, we furnish an interpretation of the multiplicity of a solution, in terms of positive distinct solutions of a quasilinear equation on $Ω$, approximating $(P_\varepsilon)$.
The existence of a nontrivial solution is proved for a class of quasilinear elliptic equations involving, as principal part, either the p-Laplace operator or the operator related to the p-area functional, and a nonlinearity with p-linear growth at infinity. To this aim, Morse theory techniques are combined with critical groups estimates.