Asymptotic properties of solutions of the singular differential equation ${\left(p\right(t\left){u}^{\prime}\right(t\left)\right)}^{\prime}=p\left(t\right)f\left(u\right(t\left)\right)$ are described. Here, $f$ is Lipschitz continuous on $\mathbb{R}$ and has at least two zeros 0 and $L>0$. The function p is continuous on $[0,\infty )$ and has a positive continuous derivative on $(0,\infty )$ and $p\left(0\right)=0$. Further conditions for f and p under which the equation has oscillatory solutions converging to 0 are given.

## References

V. Bongiorno, L. E. Scriven, and H. T. Davis, “Molecular theory of fluid interfaces,”

*Journal of Colloid and Interface Science*, vol. 57, pp. 462–475, 1967. V. Bongiorno, L. E. Scriven, and H. T. Davis, “Molecular theory of fluid interfaces,”*Journal of Colloid and Interface Science*, vol. 57, pp. 462–475, 1967. H. Gouin and G. Rotoli, “An analytical approximation of density profile and surface tension of microscopic bubbles for Van Der Waals fluids,”

*Mechanics Research Communications*, vol. 24, no. 3, pp. 255–260, 1997. 0899.76064 H. Gouin and G. Rotoli, “An analytical approximation of density profile and surface tension of microscopic bubbles for Van Der Waals fluids,”*Mechanics Research Communications*, vol. 24, no. 3, pp. 255–260, 1997. 0899.76064 P. C. Fife,

*Mathematical Aspects of Reacting and Diffusing Systems*, vol. 28 of*Lecture Notes in Biomathematics*, Springer, Berlin, Germany, 1979. MR527914 0403.92004 P. C. Fife,*Mathematical Aspects of Reacting and Diffusing Systems*, vol. 28 of*Lecture Notes in Biomathematics*, Springer, Berlin, Germany, 1979. MR527914 0403.92004 F. F. Abraham,

*Homogeneous Nucleation Theory*, Academic Press, New York, NY, USA, 1974. 0337.90055 F. F. Abraham,*Homogeneous Nucleation Theory*, Academic Press, New York, NY, USA, 1974. 0337.90055 G. H. Derrick, “Comments on nonlinear wave equations as models for elementary particles,”

*Journal of Mathematical Physics*, vol. 5, pp. 1252–1254, 1964. MR174304 10.1063/1.1704233 G. H. Derrick, “Comments on nonlinear wave equations as models for elementary particles,”*Journal of Mathematical Physics*, vol. 5, pp. 1252–1254, 1964. MR174304 10.1063/1.1704233 F. Dell'Isola, H. Gouin, and G. Rotoli, “Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations,”

*European Journal of Mechanics, B/Fluids*, vol. 15, no. 4, pp. 545–568, 1996. 0887.76008 F. Dell'Isola, H. Gouin, and G. Rotoli, “Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations,”*European Journal of Mechanics, B/Fluids*, vol. 15, no. 4, pp. 545–568, 1996. 0887.76008 G. Kitzhofer, O. Koch, P. Lima, and E. Weinmüller, “Efficient numerical solution of the density profile equation in hydrodynamics,”

*Journal of Scientific Computing*, vol. 32, no. 3, pp. 411–424, 2007. 1179.76062 MR2335787 10.1007/s10915-007-9141-0 G. Kitzhofer, O. Koch, P. Lima, and E. Weinmüller, “Efficient numerical solution of the density profile equation in hydrodynamics,”*Journal of Scientific Computing*, vol. 32, no. 3, pp. 411–424, 2007. 1179.76062 MR2335787 10.1007/s10915-007-9141-0 P. M. Lima, N. B. Konyukhova, A. I. Sukov, and N. V. Chemetov, “Analytical-numerical investigation of bubble-type solutions of nonlinear singular problems,”

*Journal of Computational and Applied Mathematics*, vol. 189, no. 1-2, pp. 260–273, 2006. 1100.65066 MR2202978 10.1016/j.cam.2005.05.004 P. M. Lima, N. B. Konyukhova, A. I. Sukov, and N. V. Chemetov, “Analytical-numerical investigation of bubble-type solutions of nonlinear singular problems,”*Journal of Computational and Applied Mathematics*, vol. 189, no. 1-2, pp. 260–273, 2006. 1100.65066 MR2202978 10.1016/j.cam.2005.05.004 H. Berestycki, P.-L. Lions, and L. A. Peletier, “An ODE approach to the existence of positive solutions for semilinear problems in ${\mathbb{R}}^{N}$,”

*Indiana University Mathematics Journal*, vol. 30, no. 1, pp. 141–157, 1981. 0522.35036 MR600039 10.1512/iumj.1981.30.30012 H. Berestycki, P.-L. Lions, and L. A. Peletier, “An ODE approach to the existence of positive solutions for semilinear problems in ${\mathbb{R}}^{N}$,”*Indiana University Mathematics Journal*, vol. 30, no. 1, pp. 141–157, 1981. 0522.35036 MR600039 10.1512/iumj.1981.30.30012 D. Bonheure, J. M. Gomes, and L. Sanchez, “Positive solutions of a second-order singular ordinary differential equation,”

*Nonlinear Analysis*, vol. 61, no. 8, pp. 1383–1399, 2005. 1109.34310 MR2135816 D. Bonheure, J. M. Gomes, and L. Sanchez, “Positive solutions of a second-order singular ordinary differential equation,”*Nonlinear Analysis*, vol. 61, no. 8, pp. 1383–1399, 2005. 1109.34310 MR2135816 M. Conti, L. Merizzi, and S. Terracini, “Radial solutions of superlinear equations on ${\mathbb{R}}^{N}$. I. A global variational approach,”

*Archive for Rational Mechanics and Analysis*, vol. 153, no. 4, pp. 291–316, 2000. MR1773818 10.1007/s002050050015 M. Conti, L. Merizzi, and S. Terracini, “Radial solutions of superlinear equations on ${\mathbb{R}}^{N}$. I. A global variational approach,”*Archive for Rational Mechanics and Analysis*, vol. 153, no. 4, pp. 291–316, 2000. MR1773818 10.1007/s002050050015 I. Rach\accent23unková and J. Tomeček, “Bubble-type solutions of nonlinear singular problems,”

*Mathematical and Computer Modelling*, vol. 51, no. 5-6, pp. 658–669, 2010. MR2594716 1190.34029 I. Rach\accent23unková and J. Tomeček, “Bubble-type solutions of nonlinear singular problems,”*Mathematical and Computer Modelling*, vol. 51, no. 5-6, pp. 658–669, 2010. MR2594716 1190.34029 I. Rach\accent23unková and J. Tomeček, “Strictly increasing solutions of a nonlinear singular differential equation arising in hydrodynamics,”

*Nonlinear Analysis*, vol. 72, no. 3–4, pp. 2114–2118, 2010. MR2577608 I. Rach\accent23unková and J. Tomeček, “Strictly increasing solutions of a nonlinear singular differential equation arising in hydrodynamics,”*Nonlinear Analysis*, vol. 72, no. 3–4, pp. 2114–2118, 2010. MR2577608 I. Rach\accent23unková and J. Tomeček, “Homoclinic solutions of singular nonautonomous second-order differential equations,”

*Boundary Value Problems*, vol. 2009, Article ID 959636, 21 pages, 2009. MR2552066 1190.34028 I. Rach\accent23unková and J. Tomeček, “Homoclinic solutions of singular nonautonomous second-order differential equations,”*Boundary Value Problems*, vol. 2009, Article ID 959636, 21 pages, 2009. MR2552066 1190.34028 I. Rach\accent23unková, J. Tomeček, and J. Stryja, “Oscillatory solutions of singular equations arising in hydrodynamics,”

*Advances in Difference Equations*, vol. 2010, Article ID 872160, 13 pages, 2010. MR2652448 1203.34058 I. Rach\accent23unková, J. Tomeček, and J. Stryja, “Oscillatory solutions of singular equations arising in hydrodynamics,”*Advances in Difference Equations*, vol. 2010, Article ID 872160, 13 pages, 2010. MR2652448 1203.34058 J. S. W. Wong, “Second-order nonlinear oscillations: a case history,” in

*Differential & Difference Equations and Applications*, pp. 1131–1138, Hindawi Publishing Corporation, New York, NY, USA, 2006. 1147.34024 MR2309447 J. S. W. Wong, “Second-order nonlinear oscillations: a case history,” in*Differential & Difference Equations and Applications*, pp. 1131–1138, Hindawi Publishing Corporation, New York, NY, USA, 2006. 1147.34024 MR2309447 C. H. Ou and J. S. W. Wong, “On existence of oscillatory solutions of second order Emden-Fowler equations,”

*Journal of Mathematical Analysis and Applications*, vol. 277, no. 2, pp. 670–680, 2003. 1027.34039 MR1961253 10.1016/S0022-247X(02)00617-0 C. H. Ou and J. S. W. Wong, “On existence of oscillatory solutions of second order Emden-Fowler equations,”*Journal of Mathematical Analysis and Applications*, vol. 277, no. 2, pp. 670–680, 2003. 1027.34039 MR1961253 10.1016/S0022-247X(02)00617-0 P. J. Y. Wong and R. P. Agarwal, “Oscillatory behavior of solutions of certain second order nonlinear differential equations,”

*Journal of Mathematical Analysis and Applications*, vol. 198, no. 2, pp. 337–354, 1996. 0855.34039 MR1376268 10.1006/jmaa.1996.0086 P. J. Y. Wong and R. P. Agarwal, “Oscillatory behavior of solutions of certain second order nonlinear differential equations,”*Journal of Mathematical Analysis and Applications*, vol. 198, no. 2, pp. 337–354, 1996. 0855.34039 MR1376268 10.1006/jmaa.1996.0086 W.-T. Li, “Oscillation of certain second-order nonlinear differential equations,”

*Journal of Mathematical Analysis and Applications*, vol. 217, no. 1, pp. 1–14, 1998. 0893.34023 MR1492076 10.1006/jmaa.1997.5680 W.-T. Li, “Oscillation of certain second-order nonlinear differential equations,”*Journal of Mathematical Analysis and Applications*, vol. 217, no. 1, pp. 1–14, 1998. 0893.34023 MR1492076 10.1006/jmaa.1997.5680 M. R. S. Kulenović and Ć. Ljubović, “All solutions of the equilibrium capillary surface equation are oscillatory,”

*Applied Mathematics Letters*, vol. 13, no. 5, pp. 107–110, 2000. MR1760271 M. R. S. Kulenović and Ć. Ljubović, “All solutions of the equilibrium capillary surface equation are oscillatory,”*Applied Mathematics Letters*, vol. 13, no. 5, pp. 107–110, 2000. MR1760271 L. F. Ho, “Asymptotic behavior of radial oscillatory solutions of a quasilinear elliptic equation,”

*Nonlinear Analysis*, vol. 41, no. 5-6, pp. 573–589, 2000. 0962.34019 MR1780633 L. F. Ho, “Asymptotic behavior of radial oscillatory solutions of a quasilinear elliptic equation,”*Nonlinear Analysis*, vol. 41, no. 5-6, pp. 573–589, 2000. 0962.34019 MR1780633 M. Bartušek, M. Cecchi, Z. Došlá, and M. Marini, “On oscillatory solutions of quasilinear differential equations,”

*Journal of Mathematical Analysis and Applications*, vol. 320, no. 1, pp. 108–120, 2006. 1103.34016 MR2230460 10.1016/j.jmaa.2005.06.057 M. Bartušek, M. Cecchi, Z. Došlá, and M. Marini, “On oscillatory solutions of quasilinear differential equations,”*Journal of Mathematical Analysis and Applications*, vol. 320, no. 1, pp. 108–120, 2006. 1103.34016 MR2230460 10.1016/j.jmaa.2005.06.057