Chapter 1. Introduction 1.1. The fractional Laplacian 1.2. The Mountain Pass Theorem 1.3. The Concentration-Compactness Principle Chapter 2. The problem studied in this monograph 2.1. Fractional critical problems 2.2. An extended problem and statement of the main results Chapter 3. Functional analytical setting 3.1. Weighted Sobolev embeddings 3.2. A Concentration-Compactness Principle Chapter 4. Existence of a minimal solution and proof of Theorem 2.2.2 4.1. Some convergence results in view of Theorem 2.2.2 4.2. Palais-Smale condition for F ε 4.3. Proof of Theorem 2.2.2 Chapter 5. Regularity and positivity of the solution 5.1. A regularity result 5.2. A strong maximum principle and positivity of the solutions Chapter 6. Existence of a second solution and proof of Theorem 2.2.4 6.1. Existence of a local minimum for I ε 6.2. Some preliminary lemmata towards the proof of Theorem 2.2.4 6.3. Some convergence results in view of Theorem 2.2.4 6.4. Palais-Smale condition for I ε 6.5. Bound on the minimax value 6.6. Proof of Theorem 2.2.4 Bibliography
Delayed neurologic deterioration from vasospasm remains the greatest cause of morbidity and mortality following subarachnoid hemorrhage. The authors assess the incidence and clinical course of symptomatic vasospasm following subarachnoid hemorrhage using a uniform management protocol over a 24-month period. One hundred eighteen consecutive patients were admitted to the neurovascular surgery service within 2 weeks of subarachnoid hemorrhage not attributed to trauma, tumor, or vascular malformation (113 patients had aneurysms). Early surgery was performed whenever possible, and hypertensive hypervolemk hemodilution therapy was instituted at the first sign of clinical vasospasm. Forty-two patients (
The goal of this paper is to study the effect of the Hardy potential on the existence and summability of solutions to a class of nonlocal elliptic problemsin Ω,Lipschitz boundary such that 0 ∈ Ω and N > 2s. We will mainly consider the solvability in two cases:(1) The linear problem, that is, f (x, t) = f (x), where according to the summability of the datum f and the parameter λ we give the summability of the solution u.(2) The problem with a nonlinear term f (x, t) = h(x) t σ for t > 0. In this case, existence and regularity will depend on the value of σ and on the summability of h. Looking for optimal results we will need a weak Harnack inequality for elliptic operators with singular coefficients that seems to be new.
Abstract:The aim of this paper is to study the solvability of the problemwhere is a bounded smooth domain of R N , N > 2s, M 2 f0; 1g, 0 < s < 1, > 0, > 0, p > 1 and f is a nonnegative function. We distinguish two cases: -For M D 0, we prove the existence of a solution for every > 0 and > 0.-For M D 1, we consider f Á 1 and we find a threshold ƒ such that there exists a solution for every 0 < < ƒ, and there does not for > ƒ. To Daniela Giachetti, in occasion of her 60th birthday, with our friendship.
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