Existence of solutions for non-linear boundary value problems

Resumo

This Thesis, compiled under the title “Existence of solutions for non-linear boundary valueproblems”, contains a detailed collection of the different results proved by the author in herpredoctoral stage.The interest of the non-linear differential equations is well-known. This is due to their ap-plications in different fields, such as physics, economy, medicine, biology or chemistry.It is very important to make a precise study of the existence of solutions for this kind ofproblems, as well as their uniqueness or multiplicity. In this work, we focus on the qualitativeanalysis of diverse boundary value problems, both linear and non-linear ones. Indeed, in mostof the cases our fundamental interest is to ensure the existence of constant sign solutions in theirdefinition interval. This interest comes from the constant sign of many of the magnitudes whichare modelled by this kind of problems.Even though in the title we only refer to non-linear problems; in many cases, studying themis not possible without doing a previous study of an associated linear problem. So, the firstchapters are devoted to linear boundary value problems. Using the previously obtained results,Chapters 6 and 7 contain the study of different non-linear boundary value problems.Despite the fact that the linear problems have been presented as a tool in the study of non-linear problems, this kind of problems have interest by themselves. Indeed, along the first fivechapters, we will see a wide number of examples where the usefulness of the results is shown.Along the seven chapters of this Thesis, we have proved a broad number or results for whichthe applicability is proved by means of the different examples.One of the most important advances is the possibility of characterising the parameter setfor constant sign solution without knowing its expression. Such an expression is usually hardto tackle, indeed, for the case of problems with non-constant coefficients we do not even ha-ve ensured being able of calculating its expression. Nevertheless, the calculus of the differenteigenvalues is relatively easy. For the constant coefficient case it reduces to solving a linearsystem of equations and, for problems with non-constant coefficients, we can solve it by usingsoftware, such asMathematica, with numerical methods. Thus, the different characterisationsobtained along the first five chapters are useful and practical.A possible problem to consider in the future is to generalise the results here proved fordifferent boundary conditions, such as the periodic ones. Even though, there are still cases tostudy, the amount of problems which can be analysed by using the results here collected isconsiderable, there are problems of all the orders with a high variety of boundary conditions,for instance in fourth order there are 40 possible boundary conditions to consider, including thewell-known simply supported beam or the clamped beam boundary conditions.In Chapter 6, using the results of previous chapters, we prove the existence of one or multipleconstant sign solutions for a wide range of non-linear boundary value problems. Indeed, if theresult from Chapters 3 and 4 were generalised for different boundary conditions, then the resultsfrom Chapter 6 would be directly generalised.In Chapter 7, we introduce a different technique to achieve existence results. This is due tothe different structure of the studied problems. Along this chapter, using the variational approachcoupled with different results which ensure the existence of critical points for suitable operators,we arrive to some existence results.A problem we will study in the future is to combine techniques of variational approach andthe Green’s function constant sign to achieve results which make sure the existence of solutionsfor different non-linear boundary value problems.