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This paper deals with global compactness and the multiplicity of positive solutions to problems of the type −Δ픹Nu − λu = a(x)|u|2∗−2u + f(x)in 픹N, u ∈ H1(픹N), where 픹N denotes the ball model of the hyperbolic space of dimension N ≥ 4, 2∗ = 2N N−2, N(N−2) 4 < λ < (N−1)2 4 and f ∈ H−1(픹N) (f≢0) is a nonnegative functional in the dual space of H1(픹N). The potential a ∈ L∞(픹N) is assumed to be strictly positive, such that limd(x,0)→∞a(x) = 1, where d(x, 0) denotes the geodesic distance. We establish profile decomposition of the functional associated with the above equation. We show that concentration occurs along two different profiles, namely, hyperbolic bubbles and localized Aubin–Talenti bubbles. For f = 0 and a ≡ 1, profile decomposition was studied in Bhakta and Sandeep [Poincaré–Sobolev equations in the hyperbolic space, Calc. Var. Partial Differential Equations 44 (2012) 247–269]. However, due to the presence of the potential a(.), an extension of profile decomposition to the present set-up is highly nontrivial and requires several delicate estimates and geometric arguments concerning the isometry group (Möbius group) of the hyperbolic space. Further, using the decomposition result, we derive various energy estimates involving the interacting hyperbolic bubbles and hyperbolic bubbles with localized Aubin–Talenti bubbles. Finally, combining these estimates with topological and variational arguments, we establish a multiplicity of positive solutions in the cases: a ≥ 1 and a < 1 separately. The equation studied in this paper can be thought of as a variant of a scalar-field equation with a critical exponent in the hyperbolic space, although such a critical exponent problem in the Euclidean space ℝN has only a trivial solution when f ≡ 0, a(x) ≡ 1 and λ < 0. [ABSTRACT FROM AUTHOR]

We study the existence and non-existence of positive solutions for the following class of nonlinear elliptic problems in the hyperbolic space − Δ B N u − λ u = a (x) u p − 1 + ε u 2 ∗ − 1 in B N , u ∈ H 1 (B N) , where B N denotes the hyperbolic space, 2 < p < 2 ∗ : = 2 N N − 2 , if N ⩾ 3 ; 2 < p < + ∞ , if N = 2 , λ < (N − 1) 2 4 , and 0 < a ∈ L ∞ (B N). We first prove the existence of a positive radially symmetric ground-state solution for a (x) ≡ 1. Next, we prove that for a (x) ⩾ 1 , there exists a ground-state solution for ε small. For proof, we employ "conformal change of metric" which allows us to transform the original equation into a singular equation in a ball in R N . Then by carefully analysing the energy level using blow-up arguments, we prove the existence of a ground-state solution. Finally, the case a (x) ⩽ 1 is considered where we first show that there is no ground-state solution, and prove the existence of a bound-state solution (high energy solution) for ε small. We employ variational arguments in the spirit of Bahri–Li to prove the existence of high energy-bound-state solutions in the hyperbolic space. [ABSTRACT FROM AUTHOR]

We prove an improved version of Poincaré-Hardy inequality in suitable subspaces of the Sobolev space on the hyperbolic space via Bessel pairs. As a consequence, we obtain a new Hardy type inequality with an improved constant (than the usual Hardy constant). Furthermore, we derive a new kind of improved Caffarelli-Kohn-Nirenberg inequality on the hyperbolic space. [ABSTRACT FROM AUTHOR]

We study Hardy-type inequalities associated to the quadratic form of the shifted Laplacian − Δ H N − ( N − 1 ) 2 / 4 on the hyperbolic space H N , ( N − 1 ) 2 / 4 being, as it is well-known, the bottom of the L 2 -spectrum of − Δ H N . We find the optimal constant in a resulting Poincaré–Hardy inequality, which includes a further remainder term which makes it sharp also locally: the resulting operator is in fact critical in the sense of [17] . A related improved Hardy inequality on more general manifolds, under suitable curvature assumption and allowing for the curvature to be possibly unbounded below, is also shown. It involves an explicit, curvature dependent and typically unbounded potential, and is again optimal in a suitable sense. Furthermore, with a different approach, we prove Rellich-type inequalities associated with the shifted Laplacian, which are again sharp in suitable senses. [ABSTRACT FROM AUTHOR]

We study existence and multiplicity of positive solutions of the following class of nonlocal scalar field equations: P${\mathcal {P}}$(−Δ)su+u=a(x)|u|p−1u+f(x)inRNu∈Hs(RN),$$\begin{equation} \hspace*{5pc} {\left\lbrace \begin{aligned} (-\Delta)^s u + u &= a(x) |u|^{p-1}u+f(x)\;\;\text{in}\;\mathbb {R}^{N}\\ u &\in H^{s}{\big(\mathbb {R}^{N}\big)}, \end{aligned} \right.} \end{equation}$$where s∈(0,1)$s \in (0,1)$, N>2s$N>2s$, 1

In this article, we will study the nondegeneracy properties of positive finite energy solutions of the equation -Δ픹u - λu = |u|p-1u in the hyperbolic space. We will show that the degeneracy occurs only in an N-dimensional subspace. We will prove that the positive solutions are nondegenerate in the case of geodesic balls. [ABSTRACT FROM AUTHOR]

We provide a complete classification with respect to asymptotic behaviour, stability and intersections properties of radial smooth solutions to the equation − Δ g u = e u on Riemannian model manifolds (M , g) in dimension N ≥ 2. Our assumptions include Riemannian manifolds with sectional curvatures bounded or unbounded from below. Intersection and stability properties of radial solutions are influenced by the dimension N in the sense that two different kinds of behaviour occur when 2 ≤ N ≤ 9 or N ≥ 10 , respectively. The crucial role of these dimensions in classifying solutions is well-known in Euclidean space; here the analysis highlights new properties of solutions that cannot be observed in the flat case. [ABSTRACT FROM AUTHOR]

In this article we will study the existence, multiplicity and Morse index of sign changing solutions for the Hardy-Sobolev-Maz'ya (HSM) equation in bounded domain and involving critical growth. We obtain infinitely many sign changing solutions for HSM equation. We also establish an estimate on the Morse index for the sign changing solutions. [ABSTRACT FROM AUTHOR]

We prove second and fourth order improved Poincaré type inequalities on the hyperbolic space involving Hardy-type remainder terms. Since theirs l.h.s. only involve the radial part of the gradient or of the laplacian, they can be seen as stronger versions of the classical Poincaré inequality. We show that such inequalities hold true on model manifolds as well, under suitable curvature assumptions and sharpness of some constants is also discussed. [ABSTRACT FROM AUTHOR]

Our main aim is to study Green function for the fractional Hardy operator P : = (− Δ) s − θ | x | 2 s in R N , where 0 < θ < Λ N , s and Λ N , s is the best constant in the fractional Hardy inequality. Using Green function, we also show that the integral representation of the weak solution holds. [ABSTRACT FROM AUTHOR]

We present certain Liouville properties of eigenfunctions of secondorder elliptic operators with real coefficients, via an approach that is based on stochastic representations of positive solutions, and criticality theory of secondorder elliptic operators. These extend results of Y. Pinchover to the case of nonsymmetric operators of Schrödinger type. In particular, we provide an answer to an open problem posed by Pinchover in [Comm. Math. Phys. 272 (2007), pp. 75-84, Problem 5]. In addition, we prove a lower bound on the decay of positive supersolutions of general second-order elliptic operators in any dimension, and discuss its implications to the Landis conjecture. [ABSTRACT FROM AUTHOR]

We investigate the possibility of improving the p -Poincaré inequality ‖ ∇ H N u ‖ p p ≥ Λ p ‖ u ‖ p p on the hyperbolic space, where p > 1 and Λ p : = [ ( N − 1 ) / p ] p is the best constant for which such inequality holds. We prove several different, and independent, improved inequalities, one of which is a Poincaré–Hardy inequality, namely an improvement of the best p -Poincaré inequality in terms of the Hardy weight r − p , r being geodesic distance from a given pole. Certain Hardy–Maz’ya-type inequalities in the Euclidean half-space are also obtained. [ABSTRACT FROM AUTHOR]

We consider a partially hinged rectangular plate and its normal modes. The dynamical properties of the plate are influenced by the spectrum of the associated eigenvalue problem. In order to improve the stability of the plate, we place a certain amount of denser material in appropriate regions. If we look at the partial differential equation appearing in the model, this corresponds to insert a suitable weight coefficient inside the equation. A possible way to locate such regions is to study the eigenvalue problem associated to the aforementioned weighted equation. In this paper, we focus our attention essentially on the first eigenvalue and on its minimization in terms of the weight. We prove the existence of minimizing weights inside special classes and we try to describe them together with the corresponding eigenfunctions. [ABSTRACT FROM AUTHOR]