Abstract |
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The thesis presents the results of our research on symmetry for some semilinear elliptic problems and on existence of solution for quasilinear problems involving singularities. The text is composed of two parts, each of which begins with a specific introduction.
The first part is devoted to symmetry and symmetry-breaking results. We study a class of partial differential equations involving radial weights on balls, annuli or $R^N$ --where these weights are unbounded--. We show in particular that on unbounded domains, focusing on symmetric functions permits to recover compactness, which implies existence of solutions. Then, we stress
the fact that symmetry-breaking occurs on bounded domains, depending both on the weights and on the nonlinearity of the equation. We also show that for the considered class of problems, the multibumps-solution phenomenon appears on the annulus as well as on the ball.
The second part of the thesis is devoted to partial and ordinary differential equations with singularities. Using concentration-compactness
tools, we show that a rather large class of functionals is lower semi-continuous, leading to the existence of a ground state solution. We also focus on the unicity of solutions for such a class of problems. |