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

Examples of sharp asymptotic profiles of singular solutions to an elliptic equation with a sign-changing non-linearity

The first two authors [Proc. Lond. Math. Soc. (3) {\bf 114}(1):1--34, 2017] classified the behaviour near zero for all positive solutions of the perturbed elliptic equation with a critical Hardy--Sobolev growth $$-\Delta u=|x|^{-s} u^{2^\star(s)-1} -\mu u^q \hbox{ in }B\setminus\{0\},$$ where $B$ denotes the open unit ball centred at $0$ in $\mathbb{R}^n$ for $n\geq 3$, $s\in (0,2)$, $2^\star(s):=2(n-s)/(n-2)$, $\mu>0$ and $q>1$. For $q\in (1,2^\star-1)$ with $2^\star=2n/(n-2)$, it was shown in the op. cit. that the positive solutions with a non-removable singularity at $0$ could exhibit up to three different singular profiles, although their existence was left open. In the present paper, we settle this question for all three singular profiles in the maximal possible range. As an important novelty for $\mu>0$, we prove that for every $q\in (2^\star(s) -1,2^\star-1)$ there exist infinitely many positive solutions satisfying $|x|^{s/(q-2^\star(s)+1)}u(x)\to \mu^{-1/(q-2^\star(s)+1)}$ as $|x|\to 0$, using a dynamical system approach. Moreover, we show that there exists a positive singular solution with $\liminf_{|x|\to 0} |x|^{(n-2)/2} u(x)=0$ and $\limsup_{|x|\to 0} |x|^{(n-2)/2} u(x)\in (0,\infty)$ if (and only if) $q\in (2^\star-2,2^\star-1)$.Comment: Mathematische Annalen, to appea