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We propose a new mechanism for stabilization of confined modes in lasers andsemiconductor microcavities holding exciton-polariton condensates, withspatially uniform linear gain, cubic loss, and cubic self-focusing ordefocusing nonlinearity. We demonstrated that the commonly known backgroundinstability driven by the linear gain can be suppressed by a combination of aharmonic-oscillator trapping potential and effective diffusion. Systematicnumerical analysis of one- and two-dimensional (1D and 2D) versions of themodel reveals a variety of stable modes, including stationary ones, breathers,and quasi-regular patterns filling the trapping area in the 1D case. In 2D, theanalysis produces stationary modes, breathers, axisymmetric and rotatingcrescent-shaped vortices, stably rotating complexes built of up to $8$individual vortices, and, in addition, patterns featuring vortex turbulence.Existence boundaries for both 1D and 2D stationary modes are found in an exactanalytical form, and an analytical approximation is developed for the fullstationary states.

One- and two-dimensional modes in the complex Ginzburg-Landau equation with a trapping potential