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We present some monotonicity and symmetry results for positive solutions of the equation $-\text{div}\left({\left|Du\right|}^{p-2}Du\right)=f\left(u\right)$ satisfying an homogeneous Dirichlet boundary condition in a bounded domain $\mathrm{\Omega }$. We assume 1 < p < 2 and $f$ locally Lipschitz continuous and we do not require any hypothesis on the critical set of the solution. In particular we get that if $\mathrm{\Omega }$ is a ball then the solutions are radially symmetric and strictly radially decreasing.

In this Note we consider the following problem $$\left\{\begin{array}{cc}-\mathrm{△}u=N\left(N-2\right)u^{p_{ϵ}}-\lambda u\hfill & \text{in }\mathrm{\Omega }\hfill \\ u>0\hfill & \text{in }\mathrm{\Omega }\hfill \\ u=0\hfill & \text{on }\partial \mathrm{\Omega }.\hfill \end{array}\right.$$ where $\mathrm{\Omega }$ is a bounded smooth starshaped domain in ${\mathbb{R}}^N$, $N\ge 3$, $p_{ϵ}=\frac{N+2}{N-2}-ϵ$, $ϵ>0$, and $\lambda \ge 0$. We prove that if $u_{ϵ}$ is a solution of Morse index $m>0$ than $u_{ϵ}$ cannot have more than $m$ maximum points in $\mathrm{\Omega }$ for $ϵ$ sufficiently small. Moreover if $\mathrm{\Omega }$ is convex we prove that any solution of index one has only one critical point and the level sets are starshaped for $ϵ$ sufficiently small.

We consider the semilinear Lane–Emden problem $$ where $p>1$ and $\Omega$ is a smooth bounded domain of ${ℝ}^2$. The aim of the paper is to analyze the asymptotic behavior of sign changing solutions of ${(}_p)$, as $p\to +\infty$. Among other results we show, under some symmetry assumptions on $\Omega$, that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as $p\to +\infty$, and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of the Liouville...

We consider the Yamabe type family of problems $\left(P_{\epsilon }\right):-\Delta u_{\epsilon }=u_{\epsilon }^{(n+2)/\left(n-2\right)}$, $u_{\epsilon }>0$ in $A_{\epsilon }$, $u_{\epsilon }=0$ on $\partial A_{\epsilon }$, where $A_{\epsilon }$ is an annulus-shaped domain of ${ℝ}^n$, $n\ge 3$, which becomes thinner as $\epsilon \to 0$. We show that for every solution $u_{\epsilon }$, the energy ${\int }_{A_{\epsilon }}|\nabla u_{|}^2$ as well as the Morse index tend to infinity as $\epsilon \to 0$. This is proved through a fine blow up analysis of appropriate scalings of solutions whose limiting profiles are regular, as well as of singular solutions of some elliptic problem on ${ℝ}^n$, a half-space or an infinite strip. Our argument also involves a Liouville type theorem...