We study a Dirichlet problem, investigating existence and uniqueness for semipositone and superlinear nonlinearities. We make use of Pohozaev identities, energy arguments, and bifurcation from a simple eigenvalue.

We study positive solutions to nonlinear elliptic systems of the form: \begin{eqnarray*} -\Delta u =\lambda f(v) \mbox{ in }\Omega\\-\Delta v =\lambda g(u) \mbox{ in }\Omega\\\quad~~ u=0=v \mbox{ on }\partial\Omega \end{eqnarray*} where $\Delta u$ is the Laplacian of $u$, $\lambda$ is a positive parameter and $\Omega$ is a bounded domain in $R^n$ with smooth boundary $\partial\Omega$. In particular, we will analyze the combined effects of the nonlinearities on the existence and multiplicity of positive solutions. We also study systems with multiparameters and stronger coupling. We extend our results to $p$-$q$-Laplacian systems and to $n\times n$ systems. We mainly use sub- and super-solutions to prove our results.

In this dissertation, we establish new existence, multiplicity, and uniqueness results on positive radial solutions for classes of steady state reaction diffusion equations on the exterior of a ball. In particular, for the first time in the literature, this thesis focuses on the study of solutions that satisfy a general class of nonlinear boundary conditions on the interior boundary while they approach zero at infinity (far away from the interior boundary). Such nonlinear boundary conditions occur naturally in various applications including models in the study of combustion theory. We restrict our analysis to reactions terms that grow slower than a linear function for large arguments. However, we allow all types of behavior of the reaction terms at the origin (cases when it is positive, zero, as well as negative). New results are also added to ecological systems with Dirichlet boundary conditions on the interior boundary (this is the case when the boundary is cold). We establish our existence and multiplicity results by the method of sub and super solutions and our uniqueness results via deriving a priori estimates for solutions.

Made available in DSpace on 2016-06-02T20:27:36Z (GMT). No. of bitstreams: 1 1610.pdf: 953018 bytes, checksum: 71de779ec49ee3cef03c3060c45a97f3 (MD5) Previous issue date: 2007-11-20 / Financiadora de Estudos e Projetos / In this work, we will study the existence and nonexistence of positive weak solutions for two classes of elliptic systems with weights. The first class will involve nonlinearities of the type positone and semipositone. We will prove a strong maximum principle, and we will obtain some properties of the first eigenfunction of the eigenvalue problem associated to our operator, and also we will prove the sub and supersolution method. The second class will involve a nonlinear perturbation. We will use the variational methods to study the subcritical and critical situations, and under certain hypotheses, we will show the existence of a second weak solution. / Neste trabalho, estudaremos a existência e inexistência de solução fraca positiva para duas classes de sistemas elípticos com pesos. A primeira classe envolverá não linearidades do tipo positônico e semipositônico. Provaremos um princípio de máximo forte, e obteremos algumas propriedades da primeira autofunção do problema de autovalor associado ao nosso operador, e também provaremos o método de sub e supersolução. A segunda classe que consideraremos terá uma perturbação não linear. Usaremos os métodos variacionais para estudar tanto a situação subcrítica quanto à situação crítica, e sob certas hipóteses, mostraremos a existência de uma segunda solução fraca.