This paper deals with the existence and multiplicity of positive solutions to the following quasilinear problem subject to the Neumann boundary condition: \[ \begin{cases} -\Delta_p u +u^{p-1}= g(u) &\text{in } B_R,\\ u>0 &\text{in }B_R,\\ \partial_\nu u =0 &\text{on }\partial B_R, \end{cases} \tag{1} \] where \(B_R\subset\mathbb{R}^N\) (\(N\geq 1\)) is the ball of radius \(R\) centered at the origin, \(1<p<\infty\), \(\Delta_p u:=\operatorname{div}(|\nabla u|^{p-2}\nabla u)\) denotes the \(p\)-Laplacian, \(\nu\) is the outer unit normal to \(\partial B_R\), and \(g\in C([0,\infty))\cap C^1((0,\infty))\) is allowed to have supercritical growth in the sense of Sobolev embeddings, particularly the prototype nonlinearity \(g(u)=u^{q-1}\) for every \(q>p\).
In this paper, the authors summarize and give an outline of proofs of recent existence results for problem (1) which does not have to assume any subcritical condition on \(g\). By employing the shooting method for ODE’s, the authors obtain existence and multiplicity of non-constant radial solutions to problem (1). This result is contained in [A. Boscaggin et al., ESAIM: Control Optim. Calc. Var. 24, No. 4, 1625–1644 (2018; Zbl 1419.35072)]. For the case \(p\geq2\), the authors also show the existence of a non-constant, radial, radially nondecreasing solution of problem (1). In order to do this, by working in the cone of nonnegative, nondecreasing radial functions of \(W^{1,p}(B_R)\) and using a truncation of \(g\), the authors convert problem (1) into the subcritical problem and apply a variational argument. This result is contained in [D. Bonheure et al., Ann. Inst. Henri Poincaré Anal. Non Linéaire 29, No. 4, 573–588 (2012; Zbl 1248.35079)] for the case \(p=2\) and in [the authors, Discrete Contin. Dyn. Syst. 37, No. 6, 3025–3057 (2017; Zbl 1360.35077)] for the case \(p>2\).
Some numerical simulations are also performed to get insights into the features of the solutions of (1) when \(p\geq 2\).
For the entire collection see [Zbl 1411.35007].