On Degenerate Nonlocal Carrier’s Problems

The aim of this thesis is to present some original results of existence, nonexistence and multiplicity of solutions for a class of nonlocal degenerate Carrier's problems \begin{equation}\label{problemabstract} \left \{\begin{array}{cl} -a\left(\displaystyle\int_{\Omega}u^{q}dx\right)\Delta_{p} u = f(u) & \textrm{in}\ \Omega,\\ u>0 & \textrm{in} \ \Omega,\\ u=0 & \textrm{on}\ \partial\Omega, \end{array} \right. \end{equation} where $\Omega$ is a bounded domain in $\R^{N}$, $a$ and $f$ are continuous functions fulfilling some additional conditions, and $q\geq 1$. The main issue of this class of problems lies in the loss of variational structure. This feature forces us to gain our results thought new techniques. In particular, we provide some general results for existence and multiplicity of positive solutions for the nonlocal elliptic problem \eqref{problemabstract} where the degenerate weight function could be sign-changing with a known number of positive bumps. Our results will be extended to problems of type \eqref{problemabstract} driven by non-homogeneous operators as $p(x)$-Laplacian and Double Phase operator and for problems with $x$-dependence on the reaction term. A careful analysis will show that similar results can be provided also for Neumann problems. Moreover, some nonexistence results permit us to enrich the scenario. The main approach is based on the following steps: \begin{itemize} \item[•]first we freeze the nonlocal term in the positive bumps of the weight degenerate function $a$ and we obtain a variational elliptic problem; \item[•]then, we provide, through sub-super solutions methods and variational tools, the existence of at least one positive solution of the frozen problem; \item[•]next, when it is the case, we provide the uniqueness of the solution for the frozen problem or in any case, the existence of the unique smallest solution in a suitable interval; \item[•]finally, we build a one-dimensional fixed point problem that allows us to provide the existence of at least two fixed points. In particular, each fixed point of this map will be a positive solution of our original problem \eqref{problemabstract}. \end{itemize} We emphasize that, in the first part of this work, we obtain our results by imposing a monotonicity assumption involving the reaction term, i.e., we assume that the map \[ t\mapsto \frac{f(t)}{t^{p-1}} \] is strictly decreasing in a suitable interval. This hypothesis, exploiting a D\'iaz-Sa\'a-type argument, permits us to obtain the uniqueness of solution of our problem with frozen nonlocal term. In the second part, we are able to remove this monotonicity constrain. This new challenge require a deeper analysis and the introduction of a suitable set-valued map, in order to provide the existence of the unique smallest solution in an appropriate interval. In both cases a sophisticated combination of sub-super solutions methods, variational tools and fixed point theory allows us to come back to the solutions of our original problem.