De Pablo et al. [Proc. Roy. Soc. Edinburgh Sect. A 138 (2008), 513-530] considered a nonlinear boundary value problem for a porous medium equation with a convection term, and they classified exponents of nonlinearities which lead either to the global-in-time existence of solutions or to a blow-up of solutions. In their analysis they left open the case of a certain critical range of exponents. The purpose of this note is to fill this gap.

Let $\Omega \subset {ℝ}^n$ be a bounded starshaped domain and consider the $(p,q)$-Laplacian problem $$\begin{array}{c}\hfill -{\Delta }_{p}u-{\Delta }_{q}u=\lambda \left(𝐱\right){\left|u\right|}^{p^{☆}-2}u+\mu {\left|u\right|}^{r-2}u\end{array}$$ where $\mu$ is a positive parameter, $1<q\le p<n$, $r\ge p^{☆}$ and $p^{☆}:=\frac{np}{n-p}$ is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the $(p,q)$-Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.

We discuss the existence of solutions for a system of elliptic equations involving a coupling nonlinearity containing a critical and subcritical Sobolev exponent. We establish the existence of ground state solutions. The concentration of solutions is also established as a parameter λ becomes large.

We investigate the solvability of a singular equation of Caffarelli-Kohn-Nirenberg type having a critical-like nonlinearity with a sign-changing weight function. We shall examine how the properties of the Nehari manifold and the fibering maps affect the question of existence of positive solutions.

In this paper, we are interested in multiple positive solutions for the Kirchhoff type problem ⎧$-(a+b{\int }_{\Omega }\left|\nabla u\right|²dx)\Delta u=u⁵+\lambda u^{q-1}/{\left|x\right|}^{\beta }$ in Ω ⎨ ⎩ u = 0 on ∂Ω, where Ω ⊂ ℝ³ is a smooth bounded domain, 0∈Ω, 1 < q < 2, λ is a positive parameter and β satisfies some inequalities. We obtain the existence of a positive ground state solution and multiple positive solutions via the Nehari manifold method.

We consider the Neumann problem involving the critical Sobolev exponent and a nonhomogeneous boundary condition. We establish the existence of two solutions. We use the method of sub- and supersolutions, a local minimization and the mountain-pass principle.

Let Ω be a smooth bounded domain in ${𝐑}^n$, n > 1, let a and f be continuous functions on $\overline{\Omega }$, $1^{☆}=\frac{n}{n-1}$. We are concerned here with the existence of solution in $BV\left(\Omega \right)$, positive or not, to the problem:
$$\left\{\begin{array}{cccc}\hfill -\mathrm{div}\sigma +a\left(x\right)signu& amp;={f\left|u\right|}^{1^{☆}-2}u\sigma .\nabla u\hfill & amp;=\left|\nabla u\right|\mathrm{in}\Omega u\mathrm{is}\mathrm{not}\mathrm{identically}\mathrm{zero},& amp;-\sigma .n\left(u\right)=\left|u\right|\mathrm{on}\partial \Omega .\end{array}\right.$$ This problem is closely related to the extremal functions for the problem of the best constant of $W^{1,1}\left(\Omega \right)$ into $L^{\frac{N}{N-1}}\left(\Omega \right)$.