We construct countably infinitely many non-radial singular solutions of the problem

of the form

where v(σ) depends only on σ ∈𝕊N−1. To this end we construct countably infinitely many solutions of

using ordinary differential equation techniques.

We study global solution curves and prove the existence of infinitely many positive solutions for three classes of self-similar equations with p-Laplace operator. In the p = 2 case these are well-known problems involving the Gelfand equation, the equation modelling electrostatic micro-electromechanical systems (MEMS), and a polynomial nonlinearity. We extend the classical results of Joseph and Lundgren to the case in which p ≠ 2, and we generalize the main result of Guo and Wei on the equation modelling MEMS.

We prove the existence of infinitely many solutions $u\in W_{0}^{1,2}(\unicode[STIX]{x1D6FA})$ for the Kirchhoff equation

$$\begin{eqnarray}\displaystyle -\biggl(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}\int _{\unicode[STIX]{x1D6FA}}|\unicode[STIX]{x1D6FB}u|^{2}\,dx\biggr)\unicode[STIX]{x1D6E5}u=a(x)|u|^{q-1}u+\unicode[STIX]{x1D707}f(x,u)\quad \text{in }\unicode[STIX]{x1D6FA}, & & \displaystyle \nonumber\end{eqnarray}$$

$\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{N}$

$a(x)$

$0<q<1$

$\unicode[STIX]{x1D6FC}>0$

$\unicode[STIX]{x1D6FD}\geq 0$

$\unicode[STIX]{x1D707}>0$

$f$

In this paper we study a quasi-linear elliptic problem coupled with Dirichlet boundary conditions. We propose a new set of assumptions ensuring the existence of infinitely many solutions.

We discuss the multiplicity of nonnegative solutions of a parametric one-dimensional mean curvature problem. Our main effort here is to describe the configuration of the limits of a certain function, depending on the potential at zero, that yield, for certain values of the parameter, the existence of infinitely many weak nonnegative and nontrivial solutions. Moreover, thanks to a classical regularity result due to Lieberman, this sequence of solutions strongly converges to zero in $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}C^1([0,1])$. Our approach is based on recent variational methods.

We consider the prescribed boundary mean curvature problem in ${{\mathbb{B}}^{N}}$ with the Euclidean metric

$$\{_{\frac{\partial u}{\partial v}+\frac{N-2}{2}u=\frac{N-2}{2}\tilde{K}\left( x \right){{u}^{{{2}^{\#-1}}}}\,\,\,\,\,\,\text{on}{{\mathbb{S}}^{N-1}},}^{-\Delta u=0,\,\,\,\,\,\,u>0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{in}{{\mathbb{B}}^{N}},}$$

where $\tilde{K}\left( x \right)$ is positive and rotationally symmetric on ${{\mathbb{S}}^{N-1}},{{2}^{\#}}=\frac{2\left( N-1 \right)}{N-2}$. We show that if $\tilde{K}\left( x \right)$ has a local maximum point, then this problem has infinitely many positive solutions that are not rotationally symmetric on ${{\mathbb{S}}^{N-1}}$.