We consider the following problem $$\label{eq.abs} \left\{\begin{array}{l} \Delta_p u +\la u +f(u,r)=0 \\ u>0 \; \textrm{ in B, } \quad \textrm{ and } \quad u=0 \textrm{ on \; \partial B.} \end{array} \right.$$ where $B$ is the unitary ball in $\RR^n$. Merle and Peletier considered the classical Laplace case $p=2$, and proved the existence of a unique value $\la_0^*$ for which a radial singular positive solution exists, assuming $f(u,r)=u^{q-1}$ and $q>2^*:=\frac{2n}{n-2}$. Then Dolbeault and Flores proved that, if $q>2^*$ but $q$ is smaller than the Joseph-Lundgren exponent $\sigma^*$, then there is an unbounded sequence of radial positive classical solutions for (\ref{eq.abs}), which accumulate at $\la=\la_0^*$, again for $p=2$. \\ We extend both Merle-Peletier and Dolbeault-Flores results to the $p$-Laplace setting with the technical restriction $1<p \le 2$, and to more general non-linearities $f$, which may have more complicated dependence on $u$ and may be spatially non-homogenous. Then we reproduce the results also for similar bifurcation problems where the linear term $\la u$ is replaced by a superlinear and subcritical term of the form $\la r^{\eta} u^{Q-1}$. Our analysis relies on a generalized Fowler transformation and profits of invariant manifold theory, and it allows to discuss radial nodal solutions too.

### Phase plane analysis for radial solutions to supercritical quasilinear elliptic equations in a ball

#### Abstract

We consider the following problem $$\label{eq.abs} \left\{\begin{array}{l} \Delta_p u +\la u +f(u,r)=0 \\ u>0 \; \textrm{ in B, } \quad \textrm{ and } \quad u=0 \textrm{ on \; \partial B.} \end{array} \right.$$ where $B$ is the unitary ball in $\RR^n$. Merle and Peletier considered the classical Laplace case $p=2$, and proved the existence of a unique value $\la_0^*$ for which a radial singular positive solution exists, assuming $f(u,r)=u^{q-1}$ and $q>2^*:=\frac{2n}{n-2}$. Then Dolbeault and Flores proved that, if $q>2^*$ but $q$ is smaller than the Joseph-Lundgren exponent $\sigma^*$, then there is an unbounded sequence of radial positive classical solutions for (\ref{eq.abs}), which accumulate at $\la=\la_0^*$, again for $p=2$. \\ We extend both Merle-Peletier and Dolbeault-Flores results to the $p$-Laplace setting with the technical restriction \$1

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2015
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11566/227884
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