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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 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}.

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.

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 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]

The paper deals with the existence of nontrivial solutions for (p , Q) equations in the Heisenberg group H n with critical exponential growth at infinity and a singular behavior at the origin. The main features and novelty of the paper are the above generality on the right-hand side of the equation, the (p , Q) growth of the elliptic operator and the fact that the equation is studied in the entire Heisenberg group. [ABSTRACT FROM AUTHOR]

In this paper we establish the existence of at least two (weak) solutions for the following fractional Kirchhoff problem involving singular and exponential nonlinearities where is a smooth bounded domain of , , , is a real parameter, and.The paper covers the so called degenerate Kirchhoff case andthe existence proofs rely on the Nehari manifold techniques. [ABSTRACT FROM AUTHOR]

The paper is concerned with existence of nonnegative solutions of a Schrödinger–Choquard–Kirchhoff-type fractional p-equation. As a consequence, the results can be applied to the special case (a + b ⁢ ∥ u ∥ s p ⁢ (θ - 1) ) ⁢ [ (- Δ) p s ⁢ u + V ⁢ (x) ⁢ | u | p - 2 ⁢ u ] = λ ⁢ f ⁢ (x , u) + (∫ ℝ N | u | p μ , s * | x - y | μ ⁢ 𝑑 y ) ⁢ | u | p μ , s * - 2 ⁢ u in ⁢ ℝ N , (a+b\|u\|_{s}^{p(\theta-1)})[(-\Delta)^{s}_{p}u+V(x)|u|^{p-2}u]=\lambda f(x,u)% +\Bigg{(}\int_{\mathbb{R}^{N}}\frac{|u|^{p_{\mu,s}^{*}}}{|x-y|^{\mu}}\,dy% \Biggr{)}|u|^{p_{\mu,s}^{*}-2}u\quad\text{in }\mathbb{R}^{N}, where ∥ u ∥ s = ( ∬ ℝ 2 ⁢ N | u ⁢ (x) - u ⁢ (y) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y + ∫ ℝ N V ⁢ (x) ⁢ | u | p ⁢ 𝑑 x ) 1 p , \|u\|_{s}=\Bigg{(}\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}% \,dx\,dy+\int_{\mathbb{R}^{N}}V(x)|u|^{p}\,dx\Biggr{)}^{\frac{1}{p}}, a , b ∈ ℝ 0 + {a,b\in\mathbb{R}^{+}_{0}} , with a + b > 0 {a+b>0} , λ > 0 {\lambda>0} is a parameter, s ∈ (0 , 1) {s\in(0,1)} , N > p ⁢ s {N>ps} , θ ∈ [ 1 , N / (N - p ⁢ s)) {\theta\in[1,N/(N-ps))} , (- Δ) p s {(-\Delta)^{s}_{p}} is the fractional p-Laplacian, V : ℝ N → ℝ + {V:\mathbb{R}^{N}\rightarrow\mathbb{R}^{+}} is a potential function, 0 < μ < N {0<\mu

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. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with the existence of nontrivial solutions for critical Hardy quasilinear systems (S) driven by general (p , q) elliptic operators of Marcellini types. Existence is derived as an application of the concentration-compactness principle of Lions via the mountain pass geometry. The constructed solution has both components nontrivial, that is it solves the actual system, which does not reduce into an equation. We present also a simplified version (S ′) of the main system (S), which is anyway interesting. We exhibit a new proof for the existence of nontrivial solutions of (S), which is more direct and elegant. However, the assumptions for both systems (S) and (S ′) are milder and in any case much different from the usual requests granted in related problems. Finally, the results improve or complement previous theorems for the quasilinear (p , q) scalar as well as vectorial problems. [ABSTRACT FROM AUTHOR]

The purpose of this paper is mainly to investigate the existence of weak solutions of fractional Laplacian equations with sublinear growth and oscillatory behavior. By using variational methods and abstract critical point theory, we want to establish the existence of multiplicity positive, multiplicity negative, and in particular, of multiplicity sign changing solutions. The multiplicity results depend on the real parameter. The key point is the choice of the framework to study the existence of weak solutions. In the suitable framework, by verifying that the conditions in abstract critical point theory are satisfied, we are able to fulfill our strategy. [ABSTRACT FROM AUTHOR]

In this paper we establish existence and multiplicity of nontrivial non-negative entire (weak) solutions of a stationary Kirchhoff eigenvalue problem, involving a general nonlocal integro-differential operator. The model under consideration depends on a real parameter λ and involves two superlinear nonlinearities, one of which could be critical or even supercritical. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of two nontrivial weak solutions of possibly degenerate nonlinear eigenvalue problems involving the p-polyharmonic Kirchhoff operator in bounded domains. The p-polyharmonic operators were recently introduced in [F. Colasuonno and P. Pucci, Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal. 74 (2011) 5962-5974] for all orders L and independently, in the same volume of the journal, in [V. F. Lubyshev, Multiple solutions of an even-order nonlinear problem with convex-concave nonlinearity, Nonlinear Anal. 74 (2011) 1345-1354] only for L even. In Sec. 3, the results are then extended to non-degeneratep(x)-polyharmonic Kirchhoff operators. The main tool of the paper is a three critical points theorem given in [F. Colasuonno, P. Pucci and Cs. Varga, Multiple solutions for an eigenvalue problem involving p-Laplacian type operators, Nonlinear Anal. 75 (2012) 4496-4512]. Several useful properties of the underlying functional solution space , endowed with the natural norm arising from the variational structure of the problem, are also proved both in the homogeneous case p ≡ and in the non-homogeneous case p = p(x). In the latter some sufficient conditions on the variable exponent p are given to prove the positivity of the infimum λ1 of the Rayleigh quotient for the p(x)-polyharmonic operator . [ABSTRACT FROM AUTHOR]

Abstract: We study the existence and multiplicity of solutions for elliptic equations in , driven by a non-local integro-differential operator, which main prototype is the fractional Laplacian. The model under consideration, denoted by , depends on a real parameter λ and involves two superlinear nonlinearities, one of which could be critical or even supercritical. The main theorem of the paper establishes the existence of three critical values of λ which divide the real line in different intervals, where admits no solutions, at least one nontrivial non-negative entire solution and two nontrivial non-negative entire solutions. [Copyright &y& Elsevier]

Abstract: In this paper we are concerned with a new class of anisotropic quasilinear elliptic equations with a power-like variable reaction term. One of the main features of our work is that the differential operator involves partial derivatives with different variable exponents, so that the functional-analytic framework relies upon anisotropic Sobolev and Lebesgue spaces. Existence and nonexistence results are deeply influenced by the competition between the growth rates of the anisotropic coefficients. Our main results point out some striking phenomena related to the existence of a continuous spectrum in several distinct situations. [Copyright &y& Elsevier]

The paper deals with existence and multiplicity of nonnegative weak solutions for quasilinear Schrödinger equations involving nonlinearities with possibly supercritical growth at infinity and indefinite sign. An appropriate change of variables reduces the quasilinear problem into a semilinear one. Variational and sub- super-methods are applied in order to obtain the main results of the paper. [ABSTRACT FROM AUTHOR]

This paper deals with the existence and the asymptotic behavior of non-negative solutions for a class of stationary Kirchhoff problems driven by a fractional integro-differential operator L K and involving a critical nonlinearity. In particular, we consider the problem − M ( | | u | | 2 ) L K u = λ f ( x , u ) + | u | 2 s ∗ − 2 u in Ω , u = 0 in R n ∖ Ω , where Ω ⊂ R n is a bounded domain, 2 s ∗ is the critical exponent of the fractional Sobolev space H s ( R n ) , the function f is a subcritical term and λ is a positive parameter. The main feature, as well as the main difficulty, of the analysis is the fact that the Kirchhoff function M could be zero at zero, that is the problem is degenerate . The adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature. [ABSTRACT FROM AUTHOR]

The paper deals with the existence of nonnegative solutions for (p , N) systems in R N involving critical exponential growth nonlinearities. The constructed solution has both components nontrivial and different, that is it solves the actual system, which does not reduce into an equation. The main feature and novelty of the paper is the presence of a general coupled critical exponential term of the Trudinger–Moser type, set in R N. [ABSTRACT FROM AUTHOR]