Context : Giant galactic H ii regions surrounding central young clusters show compressed molecular shells , which have broken up into clumps , filaments , and elephant trunks interacting with UV light from central OB stars . Tiny , dense clumps of sub-solar mass , called globulettes , form in this environment . Aims : We observe and explore the nature and origin of the infrared emission and extinction in such cool , dusty shell features and globulettes in one such H ii region , the Rosette Nebula , and search for associated newborn stars . Methods : We imaged the northwestern quadrant of the Rosette Nebula in the near-infrared ( NIR ) through wide-band J H K s filters and narrow-band \mathrm { H _ { 2 } } 1–0 S ( 1 ) and P \beta plus continuum filters using SOFI at the New Technology Telescope ( NTT ) at ESO . NIR images were used to study the surface brightness of the globulettes and associated bright rims . NIR J H K s photometry was used to create a visual extinction map and to search for objects with NIR excess emission . In addition , archival images from Spitzer IRAC and MIPS 24 \mu { m } and Herschel PACS observations , covering several bands in the mid-infrared ( MIR ) and far-infrared ( FIR ) , were used to further analyze the stellar population , to examine the structure of the trunks and other shell structures and to study this Rosette Nebula photon-dominated region in more detail . Results : The globulettes and elephant trunks have bright rims in the K s band , which are unresolved in our images , on the sides facing the central cluster . Analysis of 21 globulettes where surface brightness in the \mathrm { H _ { 2 } } 1–0 S ( 1 ) line at 2.12 \mu { m } is detected shows that approximately a third of the surface brightness observed in the K s filter is due to this line : the observed average of the \mathrm { H _ { 2 } } / K s surface brightness is 0.26 \pm 0.02 in the globulettes cores and 0.30 \pm 0.01 in the rims . The estimated \mathrm { H _ { 2 } } 1–0 S ( 1 ) surface brightness of the rims is \sim 3–8 \times 10 ^ { -8 } Wm ^ { -2 } sr ^ { -1 } \mu { m } ^ { -1 } . The ratio of the surface brightnesses support fluorescence instead of shocks as the \mathrm { H _ { 2 } } excitation mechanism . The globulettes have number densities of n ( { \hbox { $ \mathrm { H _ { 2 } } $ } } ) \sim 10 ^ { -4 } cm ^ { -3 } or higher . Masses of individual globulettes were estimated and compared to the results from previous optical and radio molecular line surveys . We confirm that the larger globulettes contain very dense cores , that the density is high also farther out from the core , and that their mass is sub-solar . Two NIR protostellar objects were found in an elephant trunk and one in the most massive globulette in our study . Conclusions :