M.Sc Thesis
M.Sc Student Regev Netanel Criticality Theory of Half-Linear Equations with the (p,A)-Laplacian Department of Mathematics Professor Yehuda Pinchover

Abstract

Let Ω be a domain in n, n≥2 , and 1<p<∞.  We consider the functional

QA,V(φ):=ʃΩ((|φ|A)pV(x)|φ|p)dx

and its associated Euler-Lagrange equation

Q A,V(u):=1/pQ'A,V(u)=-((|u|A)p-2A(x) u) V(x)|u|p-2u=0,

where A is a symmetric, measurable, locally bounded, and locally uniformly positive definite matrix in Ω, (|u|A)2:=<A(x)u,u >, and V is locally bounded in Ω.

It is assumed that ԚA,V ≥0 on C(Ω) with compact support in Ω. We prove that either there is a nonzero, nonnegative continuous function W such that QA,V (φ)≥ʃΩW|φ|pdx for all φC(Ω) with compact support in Ω, or there is a sequence {0≤φk}C(Ω) with compact support in Ω such that QA,V (φk)→0, and φϕ locally in Lp(Ω), where ϕ is the unique positive supersolution of the equation QA,V(u) =0 in Ω (such a function is called a ground state).  We also prove that if the operator admits a ground state ϕ, then one has for QA,V  a Poincaré type inequality: there exists a positive continuous function W in Ω such that for every ψC(Ω) with compact support in Ω satisfying ʃΩψϕdx≠0, there exists a constant C>0 such that the following inequality holds: QA,V (φ)C| ʃΩψφdx |pC-1ʃΩW|φ|pdx for all φC(Ω) with compact support in Ω.

As a result, we prove further positivity properties of the functional QA,V , and we generalize the Liouville-type comparison principle. Namely, given two nonnegative functionals QA0,V0  and QA1,V1, if one of them has a ground state and the other has a subsolution, then under certain comparison conditions on both functions and their gradients, we conclude that the subsolution is actually a ground state.

Finally, we study the behavior of positive solutions of the equation QA,V(u)=0 near an isolated singularity, removable singularity results for such solutions, and the behavior of the set of all positive solutions of the above equation under perturbations.