Radial solutions of equations and inequalities involving the -Laplacian

Abstract

Several problems for the differential equation Lu=g(r,u) with Lpu=r-(rlu'lP-2u') are considered. For ot N-1, the operator L is the radially symmetric p-Laplacian in JR". For various uniqueness conditions the initial value problem with given data u(ro) uo u' (ro) u and counterexamples to uniqueness are given. For the case where g is increasing in u, a sharp comparison theorem is established; it leads to maximal solutions, nonuniqueness and uniqueness results, among others. Using these results, a strong comparison principle for the boundary value problem and a number of properties of blow-up solutions are proved under weak assumptions on the nonlinearity g(r, u).

Sturm–Liouville Type Problems for the p-Laplacian under Asymptotic Non-resonance Conditions

“…Thus, the cases (i) for all p, (ii)(R) for 1<p 2 and (iii)(R) for p>2 follow by the classical Lipschitz Carathe odory uniqueness. Case (ii)(S) for 1<p 2 follows from Theorem 4 case ( ;)(iii) in [13]. The homogeneous case (ii)(S) for p>2 is proved in Walter [15], and precisely the same proof holds for (ii)(R) and p>2.…”

“…Examples of uniqueness conditions for the above initial value problem are given in Theorem 4 of [13].…”

A counterexample to the Fredholm alternative for the 𝑝-Laplacian

“…II, §4]. This is a consequence of compactness from the proof of Proposition 2.1 in [13]. The continuous dependence can be stated as follows.…”

Infinitely many radial solutions to a boundary value problem in a ball

“…I. Perov-A. I. Povolotskii-P. P. Zabreiko [27,Section 15], B. L. Shekhter [34,Section 15] for a more classical approach in the ODE's case. For other results, in the PDE's setting, we also refer to V. Moroz [29], P. Omari-F. Zanolin [31], E. W. C. Van Groesen [35], M. Willem [36], mainly for the case a-1.…”