Study of Some Nonlinear Partial Differential Equations without Compactness

Résumé

The objective of this thesis is the study of some nonlinear partial differential equations on a regular bounded domain, which have a variational structure, and which present a compacitance defect. In this thesis, through the problems discussed in Chapters 2,3,4, we have shown how these difficulties can be overcome and we have also obtained new results concerning the existence and regularity of solutions. Specifically, in Chapter 2, we study a Dirichlet problem involving the operator $p-$Laplacian with weights, a critical nonlinearity in the Sobolev sense and a parameter $ \lambda. $ We discuss results of existence and non-existence of solutions which depend, among other things, on the position of $p^2$ with respect to the dimension of the space $N, $ on the behavior of the weight in the vicinity of its minima and on the parameter $ \lambda. $ In chapter 3, we study a nonlinear minimization problem with two weakly coupled unknowns which gives rise to critical Sobolev exponents Using variational methods combined with a perturbation method, we overcome the lack of regularity of the energy functional and we prove the existence of solutions. Finally, in chapter 4 of this report, we consider a problem of minimization of certain types of nonlinear polyharmonic elliptic equations of the $2r$-th order involving an operator $ (- \Delta)^r$. We establish existence results of solutions and under suitable conditions on $ \varphi, $ we obtain $ S_{\theta,r}(\varphi)< S_{0,r}(\varphi) $ and in other conditions, we obtain $ S_{\theta,r}(\varphi) = S_{0,r}(\varphi). $

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