Study of Some Elliptic and Parabolic Problems of Dirichlet or Neumann Type in Different Settings

Résumé

Over the last decade, a large literature describes various aspects of PDEs whose main part of the operator has power-type growth with the leading example of the $p$-Laplacian. There is a wide range of directions in which the polynomial growth case has been developed, including variable exponent, weighted, double phase with variable exponents and approaches involving $p(x)$-Laplacian-like operator and $p(x)$-Kirchhoff-Laplacian operator. In the following monograph we deal with the questions from existence and uniqueness theory to elliptic and parabolic problems of Dirichlet or Neumann type in different settings. The originality of this work consists of the presence of a class of studied operators allowing to look the importance of the functional framework involves weighted Lebesgue-Sobolev spaces and variable exponent Lebesgue-Sobolev spaces. This thesis covers two main parts : The first part concerns the study the existence and uniqueness of weak solution to certain Dirichlet problems governed by nonlinear degenerate elliptic equation in the setting of weighted Sobolev spaces with the right-hand side term in $L^{p'}(\Omega,\omega^{1-p'})$ or in $L^{p'}(\Omega,\omega^{1-p'})+\prod\limits_{j=1}^{n}L^{p'}(\Omega,\omega^{1-p'})$. Our main tool , in this part, is based on the Browder-Minty theorem and the theory of weighted Sobolev spaces. The second part of this thesis is devoted to study two classes of nonlinear problems, the first classe of problems that we discuss in this part are Dirichlet or Neumann boundary value problems involving the the $p(x)$-Laplacian-like operator or the $p(x)$-Kirchhoff-Laplacian operator or $(p(x),q(x))$-Laplacian operator with nonstandard growth conditions. Under suitable assumptions, we establish new several results concerning the existence and uniqueness of weak solution in the setting of variable exponent Sobolev spaces. These results are obtained by combining the theory of the variable exponent Sobolev spaces and the topological degree theory for a class of demicontinuous operator of generalized $(S_{+})$ type. The second classe of problem is a parabolic problem associated with the equation : \begin{equation*} \frac{\partial u}{\partial t}-{\rm{div}}\;\mathcal{A}(x,t,\nabla u)=\phi(x,t)+{\rm{div}}\;\mathcal{B}(x,t,u,\nabla u), \end{equation*} this problem aim to present an existence result of a weak solution in the spaces $L^p(0,T;W_{0}^{1,p}(\Omega,\omega))$ by using a topological degree theory for operators of the type $\mathcal{T}+\mathcal{S}$, where $\mathcal{S}$ is a bounded demicontinuous map of class $(S_{+})$ and $\mathcal{T}$ is a linear densely defined maximal monotone map with respect to a domain of $\mathcal{T}$.

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