We consider a compact, connected, orientable, boundaryless Riemannian manifold $(M,g)$ of class $C^\infty$ where $g$ denotes the metric tensor. Let $n= \dim M \geq 3$. Using Morse techniques, we prove the existence of $2{\mathcal P}_1(M) -1$ non-costant solutions $u\in H^{1,p}(M)$ to the quasilinear problem \[ (P_\epsilon) \left\{ \begin{array}{l} -\epsilon^p \, \Delta_{p,g} u +u^{p-1}=u^{q-1} \\ u>0 \end{array} \right. \label{eqab} \] for $\varepsilon>0$ small enough, where $2 \leq p<n$, $p < q <p^*$, $p^* = np/(n-p)$ and $\Delta_{p,g} u = \textrm{div}_g (|\nabla u|_g^{p-2}\nabla u)$ is the $p$-laplacian associated to $g$ of $u$ (note that $\Delta_{2,g} = \Delta_g$) and ${\mathcal P}_t(M)$ denotes the Poincar\'e Polynomial of $M$. We also establish results of genericity of nondegenerate solutions for the quasilinear elliptic problem $(P_\varepsilon)$.
Multiplicity and nondegeneracy of positive solutions to quasilinear equations on compact Riemannian manifolds
Cingolani Silvia;
2015-01-01
Abstract
We consider a compact, connected, orientable, boundaryless Riemannian manifold $(M,g)$ of class $C^\infty$ where $g$ denotes the metric tensor. Let $n= \dim M \geq 3$. Using Morse techniques, we prove the existence of $2{\mathcal P}_1(M) -1$ non-costant solutions $u\in H^{1,p}(M)$ to the quasilinear problem \[ (P_\epsilon) \left\{ \begin{array}{l} -\epsilon^p \, \Delta_{p,g} u +u^{p-1}=u^{q-1} \\ u>0 \end{array} \right. \label{eqab} \] for $\varepsilon>0$ small enough, where $2 \leq pI documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
