Existence and multiplicity of solutions for elliptic problems with critical growth in the gradient

Abstract : In this thesis, we provide existence, non-existence, uniqueness and multiplicity results for partial differential equations with critical growth in the gradient. The principal techniques employed in our proofs are variational techniques, lower and upper solution theory, a priori estimates and bifurcation theory. The thesis consists of six chapters. In chapter 0, we introduce the topic of the thesis and we present the main results. Chapter 1 deals with a p-Laplacian type equation with critical growth in the gradient. This equation will depend on a real parameter. Depending on the interval where this parameter lives, we obtain the existence and uniqueness of one solution or we prove the existence and multiplicity of solutions. In chapters 2 and 3, we continue our study in the case where the operator is the Laplacian. However, unlike chapter 1, we study the case where the coefficient functions may change sign. We obtain again existence and multiplicity results. In chapter 4, we study non-local problems of fractional Laplacian type with different non-local gradient terms. We prove existence and non-existence results for different equations of this type. Finally, in chapter 5, we present some open problems related to the content of the thesis and some research perspectives.
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Antonio J. Fernández Sánchez. Existence and multiplicity of solutions for elliptic problems with critical growth in the gradient. Analysis of PDEs [math.AP]. Université de Valenciennes et du Hainaut-Cambresis, 2019. English. ⟨NNT : 2019VALE0020⟩. ⟨tel-02299049⟩

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