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This paper is concerned with the bifurcation properties of the standing wave solutions for the Schrödinger–Poisson system with doubly critical case The study of system (P ) is motivated by its important applications in many physical models, such as the quantum mechanical systems under external influences. Here, 3 ≤ N ≤ 6 , 0 < α < N , λ ∈ R , g is a nonnegative weight function, and 2 α ♯ and 2 α ∗ are the lower and upper Hardy–Littlewood–Sobolev critical exponents, respectively. Moreover, when N = 6 and 0 < α < 2 existence of the (weak) solutions of the system under consideration is also proved via the global bifurcation theorem due to Rabinowitz. [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]

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]

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]

We complete the study started in the paper [P. Pucci, L. Temperini, On the concentration-compactness principle for Folland-Stein spaces and for fractional horizontal Sobolev spaces, Math. Eng. 5 (2023), Paper no. 007], giving some applications of its abstract results to get existence of solutions of certain critical equations in the entire Heinseberg group. In particular, different conditions for existence are given for critical horizontal p-Laplacian equations. [ABSTRACT FROM AUTHOR]

In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents, − (a + b ∫RN |∇u|2dx) Δu = αk(x)|u|q−2u + β (∫RN |u(y)|2*μ/|x−y|μ dy) |u|2*μ−2u, x ∈ RN, where a > 0, b ≥ 0, 0 < μ < N, N ≥ 3, α and β are positive real parameters, 2*μ = (2 N − μ) / (N − 2) is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, k ∈ Lr(ℝN), with r = 2∗/(2∗ − q) if 1 < q < 2* and r = ∞ if q ≥ 2∗. According to the different range of q, we discuss the multiplicity of solutions to the above equation, using variational methods under suitable conditions. In order to overcome the lack of compactness, we appeal to the concentration compactness principle in the Choquard-type setting. [ABSTRACT FROM AUTHOR]

It is investigated the role of the state–dependent source–term for the localization by means of the kinetic energy of radially symmetric states for the stationary p–Laplace diffusion. It is shown that the oscillatory behavior of the source–term, with respect to the state amplitude, yields multiple possible states, located in disjoint energy bands. The mathematical analysis makes use of critical point theory in conical shells and of a version of Pucci–Serrin three–critical point theorem for the intersection of a cone with a ball. A key ingredient is a Harnack type inequality in terms of the energetic norm. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of nontrivial solutions for critical possibly degenerate Kirchhoff fractional (p, q) systems. For clarity, the results are first presented in the scalar case, and then extended into the vectorial framework. The main features and novelty of the paper are the (p, q) growth of the fractional operator, the double lack of compactness as well as the fact that the systems can be degenerate. As far as we know the results are new even in the scalar case and when the Kirchhoff model considered is non–degenerate. [ABSTRACT FROM AUTHOR]

We consider the elliptic and parabolic superquadratic diffusive Hamilton–Jacobi equations: Δ u + | ∇ u | p = 0 and u t = Δ u + | ∇ u | p , with p > 2 and homogeneous Dirichlet conditions. For the elliptic problem in a half-space, we prove a Liouville-type classification, or symmetry result, which asserts that any solution has to be one-dimensional. This turns out to be an efficient tool to study the behavior of boundary gradient blow-up (GBU) solutions of the parabolic problem in general bounded domains of R n with smooth boundaries. Namely, we show that in a neighborhood of the boundary, at leading order, solutions display a global ODE type behavior of the form u ν ν ∼ − u ν p , with domination of the normal derivatives upon the tangential derivatives. This leads to the existence of a universal, sharp blow-up profile in the normal direction at any GBU point, and moreover implies that the behavior in the tangential direction is more singular. A description of the space-time profile is also obtained. The ODE type behavior and its connection with the Liouville-type theorem can be considered as an analog of the well-known results of Merle and Zaag (1998) for the subcritical semilinear heat equation, with the significant difference that for the latter, the ODE behavior is in the time direction (instead of the normal spatial direction). On the other hand, it is known that any GBU solution admits a weak continuation, under the form of a global viscosity solution. As another consequence, we show that these viscosity solutions generically lose boundary conditions after GBU. Namely, solutions without loss of boundary conditions after GBU are exceptional and can be characterized as thresholds between global classical solutions and GBU solutions which lose boundary conditions. This result, as well as the above GBU profile, was up to now essentially known only in one space-dimension. Finally, in the case of elliptic Dirichlet problems, we deduce from our Liouville theorem an optimal Bernstein-type estimate, which gives a partial improvement of a local estimate of P.-L. Lions (1985). [ABSTRACT FROM AUTHOR]

In this paper we complete the study started in [P. Pucci, L. Temperini, Existence for (p,q) critical systems in the Heisenberg group, Adv. Nonlinear Anal. 9 (2020), 895-922] on some variants of the concentration-compactness principle in bounded PS domains of the Heisenberg group Hn. The concentration-compactness principle is a basic tool for treating nonlinear problems with lack of compactness. The results proved here can be exploited when dealing with some kind of elliptic systems involving critical nonlinearities and Hardy terms. [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]

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

In this paper we complete the study started in [Existence of entire solutions for quasilinear equations in the Heisenberg group, Minimax Theory Appl. 4 (2019)] on entire solutions for a quasilinear equation (ελ) in ℍn, depending on a real parameter λ, which involves a general elliptic operator Α in divergence form and two main nonlinearities. Here, in the so called sublinear case, we prove existence for all λ > 0 and, for special elliptic operators Α, existence of infinitely many solutions (uk)k. [ABSTRACT FROM AUTHOR]

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]

We establish a maximum principle for the weighted (p, q)-Laplacian, which extends the general Pucci-Serrin strong maximum principle to this quasilinear abstract setting. The feature of our main result is that it does not require any monotonicity assumption on the nonlinearity. The proof combines a local analysis with techniques on nonlinear differential equations. [ABSTRACT FROM AUTHOR]

In this paper we investigate boundary blow-up solutions of the problem { − Δ p ( x ) u + f ( x , u ) = ± K ( x ) | ∇ u | m ( x ) in Ω , u ( x ) → + ∞ as d ( x , ∂ Ω ) → 0 , where Δ p ( x ) u = div ( | ∇ u | p ( x ) − 2 ∇ u ) is called the p ( x ) -Laplacian. Our results extend the previous work [25] of Y. Liang, Q.H. Zhang and C.S. Zhao from the radial case to the non-radial setting, and [46] due to Q.H. Zhang and D. Motreanu from the assumption that K ( x ) | ∇ u ( x ) | m ( x ) is a small perturbation, to the case in which ± K ( x ) | ∇ u | m ( x ) is a large perturbation. We provide an exact estimate of the pointwise different behavior of the solutions near the boundary in terms of d ( x , ∂ Ω ) and in terms of the growth of the exponents. Furthermore, the comparison principle is no longer applicable in our context, since f ( x , ⋅ ) is not assumed to be monotone in this paper. [ABSTRACT FROM AUTHOR]

This paper is devoted to the question of global and local asymptotic stability for nonlinear damped Kirchhoff systems, with homogeneous Dirichlet boundary conditions, under fairly natural assumptions on the external force f = f ( t , x , u ) , the distributed damping Q = Q ( t , x , u , u t ) , the perturbation term μ | u | p − 2 u and the dissipative term ϱ ( t ) M ( [ u ] s p ) | u t | p − 2 u t , with ϱ ≥ 0 and in L loc 1 ( R 0 + ) , when the initial data are in a special region. Here u = ( u 1 , … , u N ) = u ( t , x ) represents the vectorial displacement, with N ≥ 1 . Particular attention is devoted to the asymptotic behavior of the solutions in the linear case specified in Section 5 . Finally, the results are extended to problems where the fractional p -Laplacian is replaced by a more general elliptic nonlocal integro-differential operator. The paper extends in several directions recent theorems and covers also the so-called degenerate case, that is the case in which M is zero at zero. [ABSTRACT FROM AUTHOR]

The paper focuses on the existence of nontrivial solutions of a nonlinear eigenvalue perturbed problem depending on a real parameter under homogeneous boundary conditions in bounded domains with Lipschitz boundary. The problem involves a weighted fractional -Laplacian operator. Denoting by a sequence of eigenvalues obtained via mini-max methods and linking structures we prove the existence of (weak) solutions both when there exists such that and when . The paper is divided into two parts: in the first part existence results are determined when the perturbation is the derivative of a globally positive function whereas, in the second part, the case when the perturbation is the derivative of a function that could be either locally positive or locally negative at is taken into account. In the latter case, it is necessary to extend the main results reported in [A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional -Laplacian problems via Morse theory, Adv. Calc. Var. 9(2) (2016) 101-125]. In both cases, the existence of solutions is achieved via linking methods. [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 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]

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]

In this paper, we study the initial boundary value problem of space-fractional diffusion equations. First, we introduce a family of potential wells. Then we show the existence of global weak solutions, provided that the initial energy J(u0) is positive and less than the potential well depth d. Finally, we establish the vacuum isolating and blow up of strong solutions. [ABSTRACT FROM AUTHOR]

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 ℝ N. 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. [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 investigate the existence of multiple solutions for the nonhomogeneous fractional p-Laplacian equations of Schrödinger-Kirchhoff type in $${\mathbb {R}}^N$$ , where $$(-\varDelta )^s_p$$ is the fractional p-Laplacian operator, with $$0

Abstract: The paper deals with the existence of entire solutions for a quasilinear equation in , depending on a real parameter λ, which involves a general variable exponent elliptic operator A in divergence form and two main nonlinearities. The competing nonlinear terms combine each other. Under some conditions, we prove the existence of a critical value with the property that admits nontrivial nonnegative entire solutions if and only if . Furthermore, under the further assumption that the potential of A is uniform convex, we give the existence of a second independent nontrivial nonnegative entire solution of , when . Our results extend the previous work of Autuori and Pucci (2013) [6] from the case of constant exponents p, q and r to the case of variable exponents. More interesting, we weaken the condition to the simple request that . Furthermore, we extend the previous work of Alama and Tarantello (1996) [2] from Dirichlet Laplacian problems in bounded domains of to the case of a general variable exponent differential equation in the entire , and also remove the assumption . Hence the results of this paper are new even in the canonical case . [Copyright &y& Elsevier]

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 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]

We establish the existence of an unbounded sequence of solutions for a class of quasilinear elliptic equations involving the anisotropic [image omitted]-Laplace operator, on a bounded domain with smooth boundary. We work on the anisotropic variable exponent Sobolev spaces and our main tool is the symmetric mountain-pass theorem of Ambrosetti and Rabinowitz. [ABSTRACT FROM AUTHOR]

Abstract: By applying a main comparison theorem of Pucci and Serrin (2007) we cover, for general equations of p-Laplace type, the open cases of Theorems B, D, E of Farina and Serrin (submitted for publication) as described in Problems 2 and 3 of Section 12 of Farina and Serrin (submitted for publication) . Moreover, we provide significant improvements of Theorem C and Theorem 5 of Farina and Serrin (submitted for publication) , the latter in the context of mean curvature type operators, see Theorem 1.3 and Theorems 5.2–5.4 below. Finally, Theorem 1.1 provides a new Liouville theorem outside the context of work in Farina and Serrin (submitted for publication) . [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 give sufficient conditions for the existence and nonexistence of nonnegative nontrivial entire weak solutions of p-Laplacian elliptic inequalities, with possibly singular weights and gradient terms, of the form div{g(|x|)|Du|p-2Du} ≥ h(|x|)f(u)ℓ(|Du|). We achieve our conclusions by using a generalized version of the well-known Keller–Ossermann condition, first introduced in [2] for the generalized mean curvature case, and in [11, Sec. 4] for the nonweighted p-Laplacian equation. Several existence results are also proved in Secs. 2 and 3, from which we deduce simple criteria of independent interest stated in the Introduction. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of non-continuation of solutions of dissipative nonlinear Kirchhoff systems, involving the p( x)-Laplacian operator and governed by nonlinear driving forces f = f ( t, x, u), as well as nonlinear external damping terms Q = Q( t, x, u, u t), both of which could significantly dependent on the time t. The theorems are obtained through the study of the natural energy Eu associated to the solutions u of the systems. Thanks to a new approach of the classical potential well and concavity methods, we show the nonexistence of global solutions, when the initial energy is controlled above by a critical value; that is, when the initial data belong to a specific region in the phase plane. Several consequences, interesting in applications, are given in particular subcases. The results are original also for the scalar standard wave equation when p ≡ 2 and even for problems linearly damped. [ABSTRACT FROM AUTHOR]

Abstract: This Note deals with a nonlinear eigenvalue problem involving the polyharmonic operator on a ball in . The main result of this Note establishes the existence of a continuous spectrum of eigenvalues such that the least eigenvalue is isolated. [Copyright &y& Elsevier]

Abstract: In this paper we give sufficient conditions for the nonexistence of nonnegative nontrivial entire weak solutions of p-Laplacian elliptic inequalities, with possibly singular weights and gradient terms, of the form , under the main request that h and are continuous on . We achieve our conclusions introducing a generalized version of the well-known Keller–Osserman condition. [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: We study the question of asymptotic stability, as time tends to infinity, of solutions of dissipative anisotropic Kirchhoff systems, involving the -Laplacian operator, governed by time-dependent nonlinear damping forces and strongly nonlinear power-like variable potential energies. This problem had been considered earlier for potential energies which arise from restoring forces, whereas here we allow also the effect of amplifying forces. Global asymptotic stability can then no longer be expected, and should be replaced by local stability. The results are further extended to the more delicate problem involving higher order damping terms. [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]

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]

Non-existence results for non-negative distribution entire solutions of singular quasilinear elliptic differential inequalities with weights are established. Such inequalities include the capillarity equation with varying gravitational field h, as well as the general p-Poisson equation of radiative cooling with varying heat conduction coefficient g and varying radiation coefficient h. Since we deal with inequalities and positive weights, it is not restrictive to assume h radially symmetric. Theorem 1 extends in several directions previous results and says that solely entire large solutions can exist, while Theorem 2 shows that in the p-Laplacian case positive entire solutions cannot exist. The results are based on some qualitative properties of independent interest. [ABSTRACT FROM AUTHOR]

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]

Existence and non-existence results are established for quasilinear elliptic problems with nonlinear boundary conditions and lack of compactness. The proofs combine variational methods with the geometrical feature, due to the competition between the different growths of the nonlinearities. [ABSTRACT FROM AUTHOR]

Abstract: We study the nonlinear boundary value problem in Ω, on ∂Ω, where is a bounded domain with smooth boundary, λ is a positive real number, and the continuous functions and q satisfy and for any and any . By analyzing the growth of the functions and q we prove in this Note several existence results in Sobolev spaces with variable exponents. To cite this article: M. Mihăilescu et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007). [Copyright &y& Elsevier]

Abstract: We establish several qualitative properties for solutions of singular quasilinear elliptic differential inequalities on complete Riemannian manifolds, such as the validity of the compact support principle, of the strong maximum principle, existence of solutions to exterior Dirichlet problems, existence of dead core solutions. [Copyright &y& Elsevier]