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 $L^{\frac{N}{N-1}}\left(\Omega \right)$.

We investigate the solvability of the Neumann problem (1.1) involving a critical Sobolev exponent and lower order perturbations in bounded domains. Solutions are obtained by min max methods based on a topological linking. A nonlinear perturbation of a lower order is allowed to interfere with the spectrum of the operator -Δ with the Neumann boundary conditions.

This paper deals with the global well-posedness of the $3$D axisymmetric Euler equations for initial data lying in critical Besov spaces $B_{p,1}^{1+\frac{3}{p}}$. In this case the BKM criterion is not known to be valid and to circumvent this difficulty we use a new decomposition of the vorticity .

We consider the Neumann problem for an elliptic system of two equations involving the critical Sobolev nonlinearity. Our main objective is to study the effect of the coefficient of the critical Sobolev nonlinearity on the existence and nonexistence of least energy solutions. As a by-product we obtain a new weighted Sobolev inequality.

We establish the existence of multiple solutions of an asymptotically linear Neumann problem. These solutions are obtained via the mountain-pass principle and a local minimization.

We consider the Neumann problem for the equation $-\Delta u-\lambda u={Q\left(x\right)\left|u\right|}^{2*-2}u$, u ∈ H¹(Ω), where Q is a positive and continuous coefficient on Ω̅ and λ is a parameter between two consecutive eigenvalues ${\lambda }_{k-1}$ and ${\lambda }_k$. Applying a min-max principle based on topological linking we prove the existence of a solution.

In this paper we investigate the solvability of some Neumann problems involving the critical Sobolev and Hardy exponents.

This review is dedicated to recent results on the 2d parabolic-elliptic Patlak-Keller-Segel model, and on its variant in higher dimensions where the diffusion is of critical porous medium type. Both of these models have a critical mass $M_c$ such that the solutions exist globally in time if the mass is less than $M_c$ and above which there are solutions which blowup in finite time. The main tools, in particular the free energy, and the idea of the methods are set out. A number of open questions are also...

We study the existence of solutions for a p-biharmonic problem with a critical Sobolev exponent and Navier boundary conditions, using variational arguments. We establish the existence of a precise interval of parameters for which our problem admits a nontrivial solution.