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This article is devoted to the study of the combined effects of logarithmic and critical nonlinearities for the Kirchhoff-Poisson system −M∫Ω∣∇Hu∣2dξΔHu+μϕu=λ∣u∣q−2uln∣u∣2+∣u∣2uinΩ,−ΔHϕ=u2inΩ,u=ϕ=0on∂Ω,\left\{\begin{array}{ll}-M\left(\mathop{\displaystyle \int }\limits_{\Omega }| {\nabla }_{H}u{| }^{2}{\rm{d}}\xi \right){\Delta }_{H}u+\mu \phi u=\lambda | u{| }^{q-2}u\mathrm{ln}| u{| }^{2}+| u{| }^{2}u& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ -{\Delta }_{H}\phi ={u}^{2}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=\phi =0& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where ΔH{\Delta }_{H} is the Kohn-Laplacian operator in the first Heisenberg group H1{{\mathbb{H}}}^{1}, Ω\Omega is a smooth bounded domain of H1{{\mathbb{H}}}^{1}, q∈(2θ,4)q\in \left(2\theta ,4), μ∈R\mu \in {\mathbb{R}}, and λ>0\lambda \gt 0 are some real parameters. Under suitable assumptions on the Kirchhoff function MM, which cover the degenerate case, we prove the existence of nontrivial solutions for the above problem when λ>0\lambda \gt 0 is sufficiently large. Moreover, our results are new even in the Euclidean case.

The paper deals with the existence of solutions for (p,Q)(p,Q) coupled elliptic systems in the Heisenberg group, with critical exponential growth at infinity and singular behavior at the origin. We derive existence of nonnegative solutions with both components nontrivial and different, that is solving an actual system, which does not reduce into an equation. The main features and novelties of the paper are the presence of a general coupled critical exponential term of the Trudinger-Moser type and the fact that the system is set in ℍn{{\mathbb{H}}}^{n}.

In this paper, we study the existence of ground state solutions for the nonlinear Schrödinger–Bopp–Podolsky system with critical Sobolev exponent

We consider the elliptic equation -Δ⁢u=uq⁢|∇⁡u|p{-\Delta u=u^{q}|\nabla u|^{p}} in ℝn{\mathbb{R}^{n}} for any p>2{p>2} and q>0{q>0}. We prove a Liouville-type theorem, which asserts that any positive bounded solution is constant. The proof technique is based on monotonicity properties for the spherical averages of sub- and super-harmonic functions, combined with a gradient bound obtained by a local Bernstein argument. This solves, in the case of bounded solutions, a problem left open in [2], where the case 0

This paper deals with the existence and the asymptotic behavior of nontrivial solutions for some classes of stationary Kirchhoff problems driven by a fractional integro-differential operator and involving a Hardy potential and different critical nonlinearities. In particular, we cover the delicate degenerate case, that is, when the Kirchhoff function M is zero at zero. To overcome the difficulties due to the lack of compactness as well as the degeneracy of the models, we have to make use of different approaches.

In this paper we establish some variants of the celebrated concentration-compactness principle of Lions - CC principle briefly - in the classical and fractional Folland-Stein spaces. In the first part of the paper, following the main ideas of the pioneering papers of Lions, we prove the CC principle and its variant, that is the CC principle at infinity of Chabrowski, in the classical Folland- Stein space, involving the Hardy-Sobolev embedding in the Heisenberg setting. In the second part, we extend the method to the fractional Folland-Stein space. The results proved here will be exploited in a forthcoming paper to obtain existence of solutions for local and nonlocal subelliptic equations in the Heisenberg group, involving critical nonlinearities and Hardy terms. Indeed, in this type of problems a triple loss of compactness occurs and the issue of finding solutions is deeply connected to the concentration phenomena taking place when considering sequences of approximated solutions. [ABSTRACT FROM AUTHOR]

Abstract: By the Mountain Pass Theorem and the constrained minimization method existence of positive or compactly supported radial ground states for quasilinear singular elliptic equations with weights are established. The paper also includes the discussion of regularity and the validity of useful qualitative properties of the solutions, which seems of independent interest. Finally, a Pohozaev type identity is produced to deduce some non-existence results. [Copyright &y& Elsevier]

We consider divergence structure quasilinear singular elliptic partial differential equations on domains of ȑn and show that there exist solutions with dead cores and, furthermore, solutions which involve both a dead core and bursts within the core. The results are obtained under appropriate monotonicity conditions on both the nonlinearity and the elliptic operator. Important special cases treated here are the p-Laplace and the mean curvature operators. We also study related problems for p-Laplace equations with weights, which include the Matukuma equation as a prototype. While it is usually thought that dead cores arise due to loss of smoothness of the underlying equation, we show by examples that they can occur equally for analytic p-Laplace equations. [ABSTRACT FROM AUTHOR]