Existence and regularity theory in weighted Sobolev spaces and applications

In the thesis we discuss several questions related to the study of degenerate, possibly nonlinear PDEs of elliptic type. At first we discuss the equivalent conditions between the validity of weighted Poincar\'e inequalities, structure of the functionals on weighted Sobolev spaces, isoperimetric inequalities and the existence and uniqueness of solutions to the degenerate nonlinear elliptic PDEs with nonhomogeneous boundary condition, having the form:\begin{eqnarray}\label{eqn:abs}\left\{\begin{array}{lll}{\rm div} \left( \rho (x)|\nabla u|^{p-2}\nabla u\right) =x^*,\\~~~~~~~~~~~~u-w \in W^{1,p}_{\rho,0} (\Omega),\end{array}\right.\end{eqnarray}involving any given $x^*\in (W^{1,p}_{\rho,0} (\Omega))^*$ and $w\in W^{1,p}_{\rho} (\Omega)$, where $u\in W^{1,p}_{\rho} (\Omega)$ and $W^{1,p}_{\rho} (\Omega)$ denotes certain weighted Sobolev space, $W^{1,p}_{\rho,0} (\Omega)$ is the completion of $\mathcal{C}_{0}^{\infty}(\Omega)$. As a next step, we undertake a natural question how to interpret the nonhomogenous boundary conditions in weighted Sobolev spaces, when the natural analytical tools, like trace embedding theorems, are missing. Our further goal is to contribute to solvability and uniqueness for degenerate elliptic PDEs with nonhomogenous boundary condition being the extension of~\eqref{eqn:abs}. In addition to the monotonicity method used in the first step of our discussion for the problem~\eqref{eqn:abs}, we also exploit Lax-Miligram theorem to treat the linear problem like:\begin{equation*}\begin{cases}-{\rm div} (A(x)\nabla u(x)) + B(x)\cdot\nabla u(x) + C(x)u(x) = x^{*}\ \ \text{for a.e.}\ x\in \Omega, \\~~~~~~~~~~~~~~~~~~ u(x) = g(x) \ \ \text{for a.e. }\ x\in \partial\Omega ,\end{cases}\end{equation*}as well as Ekeland's Variational Principle and Boccardo-Murat techniques to consider problem like:\begin{align*} \begin{cases} - {\rm div} \left( \rho (x)|\nabla u|^{p-2}\nabla u\right) - \lambda\, b(x)| u|^{p-2} u = x^*,\\~~~~~~~~~~~~~~~~~~~u-z \in X , \end{cases}\end{align*}where $p>1,\ \lambda>0$, and the operator $\mathcal{L}_{\lambda} u:= - {\rm div} \left( \rho (x)|\nabla u|^{p-2}\nabla u\right) - \lambda\, b(x)| u|^{p-2} u $ is non-monotone.For the study of the nonhomogeneous BVPs, we apply recent results due to Ka\l{}amajska and myself, where we constructed trace extension operator from weighted Orlicz-Slobodetskii spaces defined on the boundary of the domain to weighted Orlicz-Sobolev spaces in the domain. Information on the spectrum of the corresponding differential operator is also derived. Moreover, some nonexistence and nonuniqueness results are also analyzed.