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This article deals with existence and multiplicity of normalized solutions for critical Schrödinger-Poisson system involving the p -subLaplacian in the Heisenberg group. Under appropriate assumptions, together with the truncation technique, the concentration-compactness principle, the genus theory, we prove that the existence and multiplicity of the normalized solutions in the L p -subcritical case. As far as we know, this study seems to be the first contribution regarding existence of normalized solutions for the critical p -subLaplacian Schrödinger-Poisson system in the Heisenberg group. Moreover, the results of the paper are completely new even in the Euclidean case. [ABSTRACT FROM AUTHOR]

In this paper we study the blow-up phenomenon of the initial boundary value problem for the periodic rotation-two-component Camassa–Holm (R2CH) system. The results include a precise blow-up criterion, when σ > 0 , and two new blow-up statements and the exact blow-up rate for strong solutions of the system. [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]

This paper deals with the existence of nontrivial nonnegative solutions of Schrödinger–Hardy systems driven by two possibly different fractional ℘ -Laplacian operators, via various variational methods. The main features of the paper are the presence of the Hardy terms and the fact that the nonlinearities do not necessarily satisfy the Ambrosetti–Rabinowitz condition. Moreover, we consider systems including critical nonlinear terms, as treated very recently in literature, and present radial versions of the main theorems. Finally, we briefly show how to extend the previous results when the fractional Laplacian operators are replaced by more general elliptic nonlocal integro–differential operators. [ABSTRACT FROM AUTHOR]

The paper deals with Kirchhoff type equations on the whole space R N , driven by the p -fractional Laplace operator, involving critical Hardy–Sobolev nonlinearities and nonnegative potentials. We present different variational approaches to overcome the lack of compactness at critical levels, due to the presence of critical terms as well as the possibly degenerate nature of the Kirchhoff problem. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we establish the existence of two nontrivial weak solutions of some eigenvalue problems, involving a general elliptic operator in divergence form. First we study a one parameter problem under homogeneous Dirichlet boundary conditions in bounded domains, and then a two parameters problem under inhomogeneous nonlinear Robin boundary conditions in regular bounded domains. These results complete in several directions some recent papers, where a Ricceri type critical point theorem was used. The main argument of this paper is based on two recent theorems due to Arcoya and Carmona (2007) in which we use in a slightly modified version, see Section . [Copyright &y& Elsevier]

Abstract: In this paper we establish the existence of an unbounded sequence of solutions for a class of quasilinear elliptic -polyharmonic Kirchhoff equations, including the new delicate degenerate case, not yet covered in the literature. The main tool is the symmetric mountain pass theorem of Ambrosetti and Rabinowitz. [Copyright &y& Elsevier]

Abstract: We are interested in the study of the global non-existence of solutions of hyperbolic nonlinear problems, governed by the -Kirchhoff operator, under dynamic boundary conditions, when with . The systems involve nonlinear external forces and may be affected by a perturbation of the type . Several models already treated in the literature are covered in special subcases, and concrete examples are provided for the source term and the external nonlinear boundary damping . [Copyright &y& Elsevier]

In this paper, we consider a class of the critical Kirchhoff–Poisson systems in the Heisenberg group. Under suitable assumptions on the Kirchhoff function and on the nonlinear terms, the existence of multiple solutions is obtained by using the symmetric mountain pass theorem. Moreover, our results are new even in the Euclidean case. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we give sufficient conditions for the nonexistence of nonnegative nontrivial entire weak solutions of class of -Laplacian elliptic inequalities with possibly singular weights. In order to get the results a new Omori–Yau type principle is used. We complement our nonexistence results by establishing existence of infinitely many positive radial solutions each of which blows up at some finite . Finally, a criterium for the existence of positive entire large radial solutions of class is also established. [Copyright &y& Elsevier]

Abstract: We study the asymptotic stability for solutions of the nonlinear damped Kirchhoff system, with homogeneous Dirichlet boundary conditions, under fairly natural assumptions on the external force and the distributed damping . Then the results are extended to a more delicate problem involving also an internal dissipation of higher order, the so called strongly damped Kirchhoff system. Finally, the study is further extended to strongly damped Kirchhoff–polyharmonic systems, which model several interesting problems of the Woinowsky–Krieger type. [Copyright &y& Elsevier]

Abstract: Using the Mountain-Pass Theorem of Ambrosetti and Rabinowitz we prove that admits a positive weak solution in of class , whenever , and . The technique is based on the existence of extremals of some Hardy–Sobolev type embeddings of independent interest. We also show that if is a weak solution in of , then when either , or and u is also of class . [Copyright &y& Elsevier]

In this paper we first introduce an innovative equivalent norm in the Musielak-Orlicz Sobolev spaces in a very general setting and we then present a new result on the boundedness of the solutions of a wide class of nonlinear Neumann problems, both of independent interest. Moreover, we study a variable exponent double phase problem with a nonlinear boundary condition and prove the existence of multiple solutions under very general assumptions on the nonlinearities. To be more precise, we get constant sign solutions (nonpositive and nonnegative) via a mountain-pass approach and a sign-changing solution by using an appropriate subset of the corresponding Nehari manifold along with the Brouwer degree and the Quantitative Deformation Lemma. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of entire nontrivial solutions for fractional (p , q) systems with critical Hardy terms in R N. Existence of solutions for system (S) is derived via the mountain pass lemma, whose application is pretty involved and requires some variants of the concentration–compactness principle of Lions and of the concentration–compactness principle at infinity of Chabrowski for vectorial fractional spaces, here proved. Indeed, the "triple loss of compactness" in (S), caused by the simultaneous presence of Hardy and critical terms in the whole R N , forces to study the exact behaviour of the P S sequences, in the spirit of Lions. We also present a simplified version (S ˜) of the main system (S), which is anyway interesting, since the existence proof for (S ˜) is simpler and more direct. Finally, both existence theorems improve or complement previous results for fractional (p , q) scalar and vectorial problems. [ABSTRACT FROM AUTHOR]

This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity: (E) (− Δ) s u = a (x) | u | 2 s ∗ − 2 u + f (x) in R N , u ∈ H ̇ s (R N) , where s ∈ (0 , 1) , N > 2 s , 2 s ∗ ≔ 2 N N − 2 s , 0 < a ∈ L ∞ (R N) and f is a nonnegative nontrivial functional in the dual space of H ̇ s (R N) i.e., ( H ̇ s) ′ 〈 f , u 〉 H ̇ s ≥ 0 , whenever u is a nonnegative function in H ̇ s (R N). We prove existence of a positive solution whose energy is negative. Further, under the additional assumption that a is a continuous function, a (x) ≥ 1 in R N , a (x) → 1 as | x | → ∞ and ‖ f ‖ H ̇ s (R N) ′ is small enough (but f ⁄ ≡ 0), we establish existence of at least two positive solutions to (E). [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]

In this paper we give sufficient conditions both for existence and for nonexistence of nontrivial nonnegative entire solutions of nonlinear elliptic inequalities with gradient terms on the Heisenberg group. The picture is completed with the presentation of a uniqueness result which is, as far as we know, the first attempt for general equations with gradient terms on the Heisenberg group. [ABSTRACT FROM AUTHOR]

Abstract: We establish some maximum and comparison principles for weak distributional solutions of anisotropic elliptic inequalities in divergence form, both in the homogeneous and non-homogeneous cases. The main prototypes we have in mind are inequalities involving the -Laplace operator and the generalized mean curvature operator. [Copyright &y& Elsevier]

Abstract: We prove maximum and comparison principles for weak distributional solutions of quasilinear, possibly singular or degenerate, elliptic differential inequalities in divergence form on complete Riemannian manifolds. A new definition of ellipticity for nonlinear operators on Riemannian manifolds is introduced, covering the standard important examples. As an application, uniqueness results for some related boundary value problems are presented. [Copyright &y& Elsevier]

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]