Supercritical biharmonic equations with power-type nonlinearity

The biharmonic supercritical equation $\Delta^2u=|u|^{p-1}u$, where $n>4$ and $p>(n+4)/(n-4)$, is studied in the whole space $\mathbb{R}^n$ as well as in a modified form with $\lambda(1+u)^p$ as right-hand-side with an additional eigenvalue parameter $\lambda>0$ in the unit ball, in the latter case together with Dirichlet boundary conditions. As for entire regular radial solutions we prove oscillatory behaviour around the explicitly known radial {\it singular} solution, provided $p\in((n+4)/(n-4),p_c)$, where $p_c\in ((n+4)/(n-4),\infty]$ is a further critical exponent, which was introduced in a recent work by Gazzola and the second author. The third author proved already that these oscillations do not occur in the complementing case, where $p\ge p_c$. Concerning the Dirichlet problem we prove existence of at least one singular solution with corresponding eigenvalue parameter. Moreover, for the extremal solution in the bifurcation diagram for this nonlinear biharmonic eigenvalue problem, we prove smoothness as long as $p\in((n+4)/(n-4),p_c)$

Genomic signatures accompanying the dietary shift to phytophagy in polyphagan beetles.

The diversity and evolutionary success of beetles (Coleoptera) are proposed to be related to the diversity of plants on which they feed. Indeed, the largest beetle suborder, Polyphaga, mostly includes plant eaters among its approximately 315,000 species. In particular, plants defend themselves with a diversity of specialized toxic chemicals. These may impose selective pressures that drive genomic diversification and speciation in phytophagous beetles. However, evidence of changes in beetle gene repertoires driven by such interactions remains largely anecdotal and without explicit hypothesis testing. We explore the genomic consequences of beetle-plant trophic interactions by performing comparative gene family analyses across 18 species representative of the two most species-rich beetle suborders. We contrast the gene contents of species from the mostly plant-eating suborder Polyphaga with those of the mainly predatory Adephaga. We find gene repertoire evolution to be more dynamic, with significantly more adaptive lineage-specific expansions, in the more speciose Polyphaga. Testing the specific hypothesis of adaptation to plant feeding, we identify families of enzymes putatively involved in beetle-plant interactions that underwent adaptive expansions in Polyphaga. There is notable support for the selection hypothesis on large gene families for glutathione S-transferase and carboxylesterase detoxification enzymes. Our explicit modeling of the evolution of gene repertoires across 18 species identifies putative adaptive lineage-specific gene family expansions that accompany the dietary shift towards plants in beetles. These genomic signatures support the popular hypothesis of a key role for interactions with plant chemical defenses, and for plant feeding in general, in driving beetle diversification