For $10 ext{ in }Omega,quadpartial_ u u=0 ext{ on }partialOmega, $$ where $Omegasubsetmathbb R^N$ is either a ball or an annulus. The nonlinearity $f$ is possibly supercritical in the sense of Sobolev embeddings; in particular our assumptions allow to include the prototype nonlinearity $f(s)=-s^{p-1}+s^{q-1}$ for every $q>p$. We use the shooting method to get existence and multiplicity of non-constant radial solutions. With the same technique, we also detect the oscillatory behavior of the solutions around the constant solution $uequiv1$. In particular, we prove a conjecture proposed in [D. Bonheure, B. Noris, T. Weth, {it Ann. Inst. H. Poincar'e Anal. Non Lin'aire} vol. 29, pp. 573-588 (2012)], that is to say, if $p=2$ and $f'(1)>lambda_{k+1}^{ extnormal{rad}}$, there exists a radial solution of the problem having exactly $k$ intersections with $uequiv1$ for a large class of nonlinearities.

Multiple positive solutions for a class of p-Laplacian Neumann problems without growth conditions

F. Colasuonno;
2018

Abstract

For $10 ext{ in }Omega,quadpartial_ u u=0 ext{ on }partialOmega, $$ where $Omegasubsetmathbb R^N$ is either a ball or an annulus. The nonlinearity $f$ is possibly supercritical in the sense of Sobolev embeddings; in particular our assumptions allow to include the prototype nonlinearity $f(s)=-s^{p-1}+s^{q-1}$ for every $q>p$. We use the shooting method to get existence and multiplicity of non-constant radial solutions. With the same technique, we also detect the oscillatory behavior of the solutions around the constant solution $uequiv1$. In particular, we prove a conjecture proposed in [D. Bonheure, B. Noris, T. Weth, {it Ann. Inst. H. Poincar'e Anal. Non Lin'aire} vol. 29, pp. 573-588 (2012)], that is to say, if $p=2$ and $f'(1)>lambda_{k+1}^{ extnormal{rad}}$, there exists a radial solution of the problem having exactly $k$ intersections with $uequiv1$ for a large class of nonlinearities.
2018
A. Boscaggin, F. Colasuonno, B. Noris
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/771349
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