A problem of this type may be viewed as a prototype of pattern formation in biology and is related to the steady-state problem for a chemotactic aggregation model introduced by Keller and Segel. Such an equation also plays an important role in the study of activator–inhibitor systems modeling biological pattern formation, as proposed by Gierer and Meinhardt. Problems of this type, as well as the associated evolution equations, describe super-diffusivities phenomena. Such models have been proposed by de Gennes to describe long range van der Waals interactions in thin films spread on solid surfaces.

References:

[1] E.F. Keller, L.A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol. 26 (1970) 399–415.

[2] A. Gierer, H. Meinhardt, A theory of biological pattern formation, Kybernetik 12 (1972) 30–39.

[3] P.G. de Gennes, Wetting: statics and dynamics, Rev. Modern Phys. 57 (1985) 827–863.

[4] V. Radulescu, D. Repovs, Combined effects in nonlinear problems arising in the study of anisotropic continuous media, Nonlinear Analysis 75 (2012) 1524–1530.

This equation also describes stationary (i.e. time-independent) solutions for a field theory with the Lagrangian L=(1/2)u_t^2-(1/2)|grad u|^2-u^(p+1)/(p+1)+u^(q+1)/(q+1) (unless neither p nor q equal to -1; in this case the Lagrangian would contain logarithmic terms). For p=3 and q=1 this is an important toy Lagrangian describing the so-called spontaneous symmetry breaking, see e.g. http://en.wikipedia.org/wiki/Spontaneous_symmetry_breaking. This is closely related to the celebrated Higgs mechanism (for which this year's Nobel prize in physics was awarded) ensuring the mass of elementary particles in the standard model.

p=q is the Laplace equation which appears in electrostatics and many other problems. p=0 and q=1 is an inhomogeneous Helmholtz equation. The general solution is the sum of the solution to the inhomogeneous Helmholtz equation that meets the boundary conditions at infinity and a solution to the homogeneous Helmholtz equation \nabla^2 u + u = 0 that vanishes at infinity. The homogeneous equation arises in the solution of the wave equation.

The case p=3, q=1 gives stationary solutions to the Cahn-Allen equation. This equation arises in the theory of the kinetics of phase transitions with a non-conserved order parameter.

It might also be worth mentioning that -f''-2f^3=-f is the well-known nonlinear Schrödinger equation whose solution is f(x)=1/cosh(x).