34,960 research outputs found
Study on a class of Schrödinger elliptic system involving a nonlinear operator
This paper considers a class of Schrödinger elliptic system involving a nonlinear operator. Firstly, under the simple condition on and ', we prove the existence of the entire positive bounded radial solutions. Secondly, by using the iterative technique and the method of contradiction, we prove the existence and nonexistence of the entire positive blow-up radial solutions. Our results extend the previous existence and nonexistence results for both the single equation and systems. In the end, we give two examples to illustrate our results
Positive solutions of nonlinear elliptic boundary value problems
This dissertation focuses on the study of positive steady states to classes of nonlinear reaction diffusion (elliptic) systems on bounded domains as well as on exterior domains with Dirichlet boundary conditions. In particular, we study such systems in the challenging case when the reaction terms are negative at the origin, referred in the literature as semipositone problems. For the last 30 years, study of elliptic partial differential equations with semipositone structure has flourished not only for the semilinear case but also for quasilinear case. Here we establish several results that directly contribute to and enhance the literature of semipositone problems. In particular, we discuss existence, non-existence and multiplicity results for classes of superlinear as well as sublinear systems. We establish our results via the method of sub-super solutions, degree theory arguments, a priori bounds and energy analysis
Solvability of nonlinear elliptic boundary value problems
This dissertation focuses on the study of steady states of reaction diffusion problems that are motivated by applications. In particular, we focus on elliptic boundary value problems where the nonlinear reaction may appear in the interior or on the boundary of a domain in the Euclidean space. First, we study linear elliptic problems with nonlinear reaction on the boundary. In this case, we establish the existence of maximal and minimal solutions for both monotone and non monotone cases. We then extend these results to the systems case. Next, we prove the existence, nonexistence, multiplicity and global bifurcation results of positive solutions of superlinear problems. To support our analytical results we numerically approximate solutions using finite difference methods including existence and stability analysis. Second, we study problems that are nonlinear inside the domain and linear on the boundary in the context of a model arising in mathematical ecology. To begin with we perform computational simulations for the problem in the one dimensional setting. Then, motivated by the bifurcation diagrams that are obtained, we prove several analytical results such as existence, uniqueness and nonexistence
Unilateral global bifurcation for a class of quasilinear elliptic systems and applications
In this paper we establish a unilateral bifurcation result for a class of quasilinear elliptic system strongly coupled, extending the bifurcation theorem of [J. López-Gómez. Nonlinear eigenvalues and global bifurcation application to the search of positive solutions for general Lotka-Volterra reaction diffusion systems with two species. Differential Integral Equations, 7(5-6):1427–1452, 1994]. To this aim, we use several results, such that Fredholm operator of index 0 theory, eigenvalues of elliptic operators and the Krein-Rutman theorem. Lastly, we apply this result to some particular systems arising from population dynamics and determine a region of existence of coexistence states.Ministerio de EconomÃa y Competitividad (MINECO). EspañaEuropean Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER)Conselho Nacional de Desenvolvimento CientÃfico e Tecnológico (Brasil
Nonzero radial solutions for a class of elliptic systems with nonlocal BCs on annular domains
We provide new results on the existence, non-existence, localization and
multiplicity of nontrivial solutions for systems of Hammerstein integral
equations. Some of the criteria involve a comparison with the spectral radii of
some associated linear operators. We apply our results to prove the existence
of multiple nonzero radial solutions for some systems of elliptic boundary
value problems subject to nonlocal boundary conditions. Our approach is
topological and relies on the classical fixed point index. We present an
example to illustrate our theory.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1404.139
Hamiltonian elliptic systems: a guide to variational frameworks
Consider a Hamiltonian system of type where is a power-type nonlinearity, for instance , having subcritical growth, and is a bounded domain
of , . The aim of this paper is to give an overview of
the several variational frameworks that can be used to treat such a system.
Within each approach, we address existence of solutions, and in particular of
ground state solutions. Some of the available frameworks are more adequate to
derive certain qualitative properties; we illustrate this in the second half of
this survey, where we also review some of the most recent literature dealing
mainly with symmetry, concentration, and multiplicity results. This paper
contains some original results as well as new proofs and approaches to known
facts.Comment: 78 pages, 7 figures. This corresponds to the second version of this
paper. With respect to the original version, this one contains additional
references, and some misprints were correcte
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