On a variant of the maximum principle involving radial $p$-Laplacian with applications to~nonlinear eigenvalue problems and nonexistence results

Abstract

We obtain the variant of maximum principle for radial solutions of p-harmonic equation −a∆p(w) = φ(w). As a consequence of this result we prove monotonicity of constant sign solutions, analyze the support of the solutions and study their oscillations. The results are applied to various type nonlinear eigenvalue problems and nonexistence theorems.

Maximum principles and nonexistence results for radial solutions to equations involvingp-Laplacian

“…If h ≡ 0 one can find some other nonexistence results in [7] derived from the radial variant of Derrick-Pokhozhaev identity.…”

On maximum principles for solutions to nonlinear elliptic ODEs and radial solutions to nonlinear elliptic PDE's via Opial‐type inequalities

“…In particular in Kalamajska and Stryjek, 3 the authors dealt with a linear variant of Equation () ( ) and investigated some special functions like Legendre, Jacobi polynomials, Laguerre polynomials, or hypergeometric functions. The two subsequent papers 1,2 focused on the application of that method to p ‐harmonic problems. The authors have shown that, under some assumptions, the local maxima of the modulus of any radial solution form monotone sequence, which is a variant of the maximum principle.…”

“…The authors have shown that, under some assumptions, the local maxima of the modulus of any radial solution form monotone sequence, which is a variant of the maximum principle. In Adamowicz and Kalamajska, 1 the authors deal with h ≡ 0, while in Adamowicz and Kalamajska, 2 in some results, it is assumed that for a.e. the function h (· , · , ·) satisfies the following pointwise estimate (see Theorem 2.1): where We contribute by proving similar type results when h (· , · , ·) satisfies different pointwise estimates: involving some nonnegative measurable function q (·)…”

A note on the dependence of solutions on functional parameters for nonlinear Sturm-Liouville problems

“…[4][5][6][7][8][9][10][11][12][13][17][18][19][20][21] and references therein) the authors investigate mainly the existence of solutions for (1.1) under a variety of boundary conditions. Moreover, in the last fifty years, we could observe increasing interest in investigating sufficient conditions for the oscillation or nonoscillation of solutions of various classes of ODEs ( [1][2][3][5][6][7][8][9], and references therein). We have to also recall results due 838 Aleksandra Orpel to G. Vidossich who considered the continuous dependence of solutions for general boundary value problems (see [20,Theorem 1]).…”