Nonlinear equations with natural growth terms
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This thesis concerns the study of a class of second order quasilinear elliptic differential operators. For 1 < p < ∞, the model equation we consider is: (1) L(u) = -Δpu - σ∣u∣p-2u Here the potential is a function (or distribution), and the di erential operator pu is the p-Laplacian. Such operators are said to have `natural growth' terms. When p = 2, the operator reduces to the linear time independent Schrödinger operator. We will study the operator under minimal conditions on , where classical regularity theory for the operator L breaks down. Our focus will be on two heavily studied problems: 1. An existence and regularity theory for positive solutions of L(u) = 0, under the sole condition of form boundedness on the real-valued potential : (2) ∣⟨|h|p,σ|⟩ ≤ C ∫Ω |∇h|p dx, for all h ∈ C∞0 (Ω) Here is assumed to lie in the local dual Sobolev space L-41,p'loc(Ω), and the pairing in display (2) is the natural dual pairing. 2. The pointwise behavior of fundamental solutions of the operator L. Here we will be concerned with positive solutions of L(u) = 0 with a prescribed isolated singularity. The techniques developed to attack these two related problems will be quite different in nature. The first problem relies on a study of the doubling properties of nonnegative functions satisfying a weak reverse Hölder inequality, along with certain weak convergence arguments. The second problem is approached via certain nonlinear integral equations involving Wol 's potential, and makes use of tools from non-homogeneous harmonic analysis.