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Bifurcation for a class of singular elliptic problems with quadratic convection term
Ghergu, Marius ; Rǎdulescu, Vicenţiu, 1958-
We study the bifurcation problem ▫$-\Delta u = g(u) + \lambda|\nabla u|^2 + \mu$▫ in ▫$\Omega$▫, ▫$u=0$▫ on ▫$\partial\Omega$▫, where ▫$\lambda, \mu \geq 0$▫ and ▫$\Omega$▫ is a smooth bounded domain ...in ▫${\Bbb R}^N$▫. The singular character of the problem is given by the nonlinearity $g$ which is assumed to be decreasing and unbounded around the origin. In this note we prove that the above problem has a positive classical solution (which is unique) if and only if ▫$\lambda (a + \mu) < \lambda_1$▫, where ▫$a = \lim_{t \to +\infty} g(t)$▫ and ▫$\lambda_1$▫ is the first eigenvalue of the Laplace operator in ▫$H^1_0(\Omega)$▫. We also describe the decay rate of this solution, as well as a blow-up result around the bifurcation parameter.
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