Electron. J. Diff. Equ., Vol. 2016 (2016), No. 196, pp. 1-22.

Stable solutions for equations with a quadratic gradient term

Joana Terra

Abstract:
We consider positive solutions to the non-variational family of equations
$$ -\Delta u -b(x)|\nabla u|^2= \lambda g(u) \quad \text{in }\Omega, $$
where $\lambda\geq 0$, b(x) is a given function, g is an increasing nonlinearity with g(0)> 0 and $\Omega\in\mathbb{R}^n$ is a bounded smooth domain. We introduce the definition of stability for non-variational problems and establish existence and regularity results for stable solutions. These results generalize the classical results obtained when $b(x)=b$ is a constant function making the problem variational after a suitable transformation.

Submitted July 30, 2015. Published July 21, 2016.
Math Subject Classifications: 35B35, 35J61.
Key Words: Non-variational problem; stable solution.

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Joana Terra
Departamento de Matemática, FCEyN
and IMAS-CONICET, UBA (1428)
Buenos Aires, Argentina
email: joanamterra@gmail.com

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