For $1<p<$, we consider the following problem $$\begin{array}{c}\hfill {}_{\Delta}pu=f\left(u\right),\phantom{\rule{4pt}{0ex}}u>0\phantom{\rule{4pt}{0ex}}\mathrm{in}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\Omega ,\phantom{\rule{4pt}{0ex}}{\partial}_{\nu}u=0\phantom{\rule{4pt}{0ex}}\mathrm{on}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}\partial \phantom{\rule{0.166667em}{0ex}}\Omega \end{array}$$
where $\Omega \subset {\mathbb{R}}^{N}$ 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 $f\left(s\right)=-{s}^{p-1}+{s}^{q-1}$ 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 $u\equiv 1$. In particular, we prove a conjecture proposed in [D. Bonheure, B. Noris and T. Weth, Ann. Inst. Henri Poincaré Anal. Non Linéaire 29 (2012) 573−588], that is to say, if $p=2$ and $f\text{'}\left(1\right)>{\lambda}_{k+1}^{\mathrm{rad}}$, with ${\lambda}_{k+1}^{\mathrm{rad}}$ the $(k+1)$-th radial eigenvalue of the Neumann Laplacian, there exists a radial solution of the problem having exactly $k$ intersections with $u\equiv 1$, for a large class of nonlinearities.
Keywords: Quasilinear elliptic equations, Shooting method, Sobolev-supercritical nonlinearities, Neumann boundary, conditions
@article{COCV_2018__24_4_1625_0, author = {Boscaggin, Alberto and Colasuonno, Francesca and Noris, Benedetta}, title = {Multiple positive solutions for a class of {p-Laplacian} {Neumann} problems without growth conditions}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1625--1644}, publisher = {EDP-Sciences}, volume = {24}, number = {4}, year = {2018}, doi = {10.1051/cocv/2017074}, zbl = {1419.35072}, mrnumber = {3922442}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2017074/} }
TY - JOUR AU - Boscaggin, Alberto AU - Colasuonno, Francesca AU - Noris, Benedetta TI - Multiple positive solutions for a class of p-Laplacian Neumann problems without growth conditions JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1625 EP - 1644 VL - 24 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2017074/ DO - 10.1051/cocv/2017074 LA - en ID - COCV_2018__24_4_1625_0 ER -
%0 Journal Article %A Boscaggin, Alberto %A Colasuonno, Francesca %A Noris, Benedetta %T Multiple positive solutions for a class of p-Laplacian Neumann problems without growth conditions %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1625-1644 %V 24 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2017074/ %R 10.1051/cocv/2017074 %G en %F COCV_2018__24_4_1625_0
Boscaggin, Alberto; Colasuonno, Francesca; Noris, Benedetta. Multiple positive solutions for a class of p-Laplacian Neumann problems without growth conditions. ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 4, pp. 1625-1644. doi : 10.1051/cocv/2017074. http://archive.numdam.org/articles/10.1051/cocv/2017074/
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