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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.