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Existence and qualitative properties of solutions to a quasilinear elliptic equation involving the Hardy-Leray potential. (English) Zbl 1291.35082

Summary: In this work, we deal with the existence and qualitative properties of the solutions to a supercritical problem involving the \(-\Delta p(\cdot )\) operator and the Hardy-Leray potential. Assuming \(0\in\varOmega\), we study the regularizing effect due to the addition of a first order nonlinear term, which provides the existence of solutions with a breaking of resonance. Once we have proved the existence of a solution, we study the qualitative properties of the solutions such as regularity, monotonicity and symmetry.

MSC:

35J62 Quasilinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35J70 Degenerate elliptic equations
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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