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This article is devoted to the study of a critical Choquard-Kirchhoff pp-sub-Laplacian equation on the entire Heisenberg group Hn{{\mathbb{H}}}^{n}, where the Kirchhoff function KK can be zero at zero, i.e., the equation can be degenerate, and involving a nonlinearity, which is critical in the sense of the Hardy-Littlewood-Sobolev inequality. We first establish the concentration-compactness principle for the pp-sub-Laplacian Choquard equation on the Heisenberg group, and we then prove existence results.

The article deals with the existence of a pair of nontrivial nonnegative and nonpositive solutions for a nonlinear weighted quasilinear equation in RN{{\mathbb{R}}}^{N}, which involves a double-phase general variable exponent elliptic operator A{\bf{A}}. More precisely, A{\bf{A}} has behaviors like ∣ξ∣q(x)−2ξ{| \xi | }^{q\left(x)-2}\xi if ∣ξ∣| \xi | is small and like ∣ξ∣p(x)−2ξ{| \xi | }^{p\left(x)-2}\xi if ∣ξ∣| \xi | is large. Existence is proved by the Cerami condition instead of the classical Palais-Smale condition, so that the nonlinear term f(x,u)f\left(x,u) does not necessarily have to satisfy the Ambrosetti-Rabinowitz condition.

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.

This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity:

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 article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents,

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 of entire nontrivial solutions for critical quasilinear systems (𝓢) in the Heisenberg group ℍn, driven by general (p, q) elliptic operators of Marcellini types. The study of (𝓢) requires relevant topics of nonlinear functional analysis because of the lack of compactness. The key step in the existence proof is the concentration–compactness principle of Lions, here proved for the first time in the vectorial Heisenberg context. Finally, the constructed solution has both components nontrivial and the results extend to the Heisenberg group previous theorems on quasilinear (p, q) systems.

This paper deals with the existence of nontrivial solutions for critical Hardy–Schrödinger–Kirchhoff systems driven by the fractional p-Laplacian operator. Existence is derived as an application of the mountain pass theorem and the Ekeland variational principle. The main features and novelty of the paper are the presence of the Hardy terms as well as critical nonlinearities.

In this paper we derive and solve nonlocal elasticity a model describing the elastic behavior of composite materials, involving the fractional Laplacian operator. In dimension one we consider in (($\mathcal{D}$)) the case of a nonlocal elastic rod restrained at the ends, and we completely solve the problem showing the existence of a unique weak solution and providing natural sufficient conditions under which this solution is actually a classical solution of the problem. For the model (($\mathcal{D}$)) we also perform numerical simulations and a parametric analysis, in order to highlight the response of the rod, in terms of displacements and strains, according to different values of the mechanical characteristics of the material. The main novelty of this approach is the extension of the central difference method by the numerical estimate of the fractional Laplacian operator through a finite-difference quadrature technique. For higher dimensions N≥2{N\geq 2} we study more general problems for which the existence of weak solutions is proved via variational methods. The obtained results provide an original contribute in the knowledge of composite materials with properties of nonlocal elasticity.

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.

The purpose of this paper is mainly to investigate the existence of entire solutions of the stationary Kirchhoff type equations driven by the fractional p-Laplacian operator in ℝN. By using variational methods and topological degree theory, we prove multiplicity results depending on a real parameter λ and under suitable general integrability properties of the ratio between some powers of the weights. Finally, existence of infinitely many pair of entire solutions is obtained by genus theory. Last but not least, the paper covers a main feature of Kirchhoff problems which is the fact that the Kirchhoff function M can be zero at zero. The results of this paper are new even for the standard stationary Kirchhoff equation involving the Laplace operator.

In this paper, we prove the existence of a nontrivial (weak) solution of the Brézis–Nirenberg equation for the $ (m, p) $ Laplacian in the whole space $ \mathbb R^N $. Critical problems have been intensively studied in the last decades, starting with the pioneering paper by Brézis and Nirenberg for the Dirichlet Laplacian problems in bounded domains of $ \mathbb R^N $. Even if we obtain existence of solutions for the $ (m, p) $ Laplacian Brézis–Nirenberg equation in the whole space via standard variational techniques, the last step of the proof is pretty involved and requires new ideas and a delicate use of the Talenti functions. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in a class of critical nonlocal problems with variable exponents of the form: { M (T p (⋅ , ⋅) (u)) [ (− Δ) p (x , y) s u + | u | p ~ (x) − 2 u ] = λ f (x , u) + | u | q (x) − 2 u , in R N , u ∈ W s , p (⋅ , ⋅) (R N) , p ~ (x) = p (x , x) , where T p (⋅ , ⋅) (u) := ∬ R 2 N | u (x) − u (y) | p (x , y) p (x , y) | x − y | N + s p (x , y) d x d y + ∫ R N 1 p ~ (x) | u | p ~ (x) d x , M is the Kirchhoff function, λ is a real parameter and f is a continuous function. We also assume that { x ∈ R N : q (x) = p s ∗ (x) } ≠ ∅ , where p s ∗ (x) = N p ~ (x) / (N − s p ~ (x)) is the critical Sobolev exponent for variable exponents. The strategy of the proof for these results is to approach the problem variationally by using the mountain pass theorem and the concentration-compactness principles for fractional Sobolev spaces with variable exponents. In addition, we obtain the existence and multiplicity of nontrivial solutions for the above problem in non-degenerate and degenerate cases. [ABSTRACT FROM AUTHOR]

In the present paper we propose a model describing the nonlocal behavior of an elastic body using a peridynamical approach. Indeed, peridynamics is a suitable framework for problems where discontinuities appear naturally, such as fractures, dislocations, or, in general, multiscale materials. In particular, the regional fractional Laplacian is used as the nonlocal operator. Moreover, a combination of the fractional and classical Laplacian operators is used to obtain a better description of the phenomenological response in elasticity. We consider models with linear and nonlinear perturbations. In the linear case, we prove the existence and uniqueness of the solution, while in the nonlinear case the existence of at least two nontrivial solutions of opposite sign is proved. The linear and nonlinear problems are also solved by a numerical approach which estimates the regional fractional Laplacian by means of its singular integral representation. In both cases, a numerical estimation of the solutions is obtained, using in the nonlinear case an approach involving a random variation of an initial guess of the solution. Moreover, in the linear case a parametric analysis is made in order to study the effects of the parameters involved in the model, such as the order of the fractional Laplacian and the mixture law between local and nonlocal behavior. [ABSTRACT FROM AUTHOR]

This paper is devoted to a subject that Professor Csaba Varga suggested during his frequent visits to the University of Perugia and in my regular stays at the "Babeș-Bolyai" University. More specifically, continuing the work started in jointly with Professor Varga, here we establish the existence of two nontrivial (weak) solutions of some one parameter eigenvalue (p,q)-Laplacian problems under homogeneous Dirichlet boundary conditions in bounded domains of RN. [ABSTRACT FROM AUTHOR]