| Abstract: |
| We observe that most semi--linear problems of the form $-\Delta u(x)=F(x,u(x)),x\in\Omega$ have been studied intensively in the literature when $F$ is a given function. However, some relevant models in physics can be expressed as $-\Delta u(x)=\mu(x,u(x))$ in $\mathcal{D}^{\prime}(\Omega)$ where $\mu(x,u)$ is a measure of Radon that depends on $x$, but also on the solution $u$. One of these examples involving measures depending on the unknown of the problem corresponds to the interior (Daniel) Bernoulli problems type. This type of problems appear, for example, in the study of the dynamics of incompressible non--viscous ideal fluids and with horizontal flow or also in problems of type {\it sharp problem} that appear in nuclear fusion. The stationary case gives rise to an elliptical problem. A semi-implicit time is introduced to obtain a family of elliptical problems that converge to the associated evolution problem. Our aim is to study the existence of solutions for this type of nonlinear problems. We will apply a general principle of Mountain Passage due to Ghoussoub-Preiss to find a weak solution associated with this family of problems. Finally, going to the limit, we obtain a weak solution for the original problem. |
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