A priori bounds and multiplicity of positive solutions for pp-Laplacian Neumann problems with sub-critical growth

Let $1<p<+\infty$ and let $\Omega\subset\mathbb R^N$ be either a ball or an annulus. We continue the analysis started in [Boscaggin, Colasuonno, Noris, ESAIM Control Optim. Calc. Var. (2017)], concerning quasilinear Neumann problems of the type -\Delta_p u = f(u), \quad u>0 \mbox{ in } \Omega, \quad \partial_\nu u = 0 \mbox{ on } \partial\Omega. We suppose that $f(0)=f(1)=0$ and that $f$ is negative between the two zeros and positive after. In case $\Omega$ is a ball, we also require that $f$ grows less than the Sobolev-critical power at infinity. We prove a priori bounds of radial solutions, focusing in particular on solutions which start above 1. As an application, we use the shooting technique to get existence, multiplicity and oscillatory behavior (around 1) of non-constant radial solutions.Comment: 26 pages, 3 figure

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

For $1<p<\infty$, we consider the following problem $$-\Delta_p u=f(u),\quad u>0\text{ in }\Omega,\quad\partial_\nu u=0\text{ on }\partial\Omega,$$ 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 $f(s)=-s^{p-1}+s^{q-1}$ 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 $u\equiv1$. 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 $f'(1)>\lambda_{k+1}^{rad}$, there exists a radial solution of the problem having exactly $k$ intersections with $u\equiv1$ for a large class of nonlinearities.Comment: 22 pages, 4 figure

Existence of radial solution for a quasilinear equation with singular nonlinearity

We prove that the equation \begin{eqnarray*} -\Delta_p u =\lambda\Big( \frac{1} {u^\delta} + u^q + f(u)\Big)\;\text{ in } \, B_R(0) u =0 \,\text{ on} \; \partial B_R(0), \quad u>0 \text{ in } \, B_R(0) \end{eqnarray*} admits a weak radially symmetric solution for $\lambda>0$ sufficiently small, $0<\delta<1$ and $p-1<q<p^{*}-1$. We achieve this by combining a blow-up argument and a Liouville type theorem to obtain a priori estimates for the regularized problem. Using a variant of a theorem due to Rabinowitz we derive the solution for the regularized problem and then pass to the limit.Comment: 16 page