Showing papers in "Advanced Nonlinear Studies in 2008"
TL;DR: In this paper, the existence of semiclassical states for a nonlinear Schrödinger-Poisson system that concentrate near critical points of the external potential and of the density charge function was studied.
Abstract: Abstract We study the existence of semiclassical states for a nonlinear Schrödinger-Poisson system that concentrate near critical points of the external potential and of the density charge function. We use a perturbation scheme in a variational setting, extending the results in [1]. We also discuss necessary conditions for concentration.
129 citations
TL;DR: In this paper, the authors give an alternative, shorter proof of the classical result of Berestycki and Cazenave on the instability by blow-up for the standing waves of some nonlinear Schr¨ odinger equations.
Abstract: In this note we give an alternative, shorter proof of the classical result of Berestycki and Cazenave on the instability by blow-up for the standing waves of some nonlinear Schr¨ odinger equations.
82 citations
TL;DR: In this article, a positive ground state solution and also a nodal solution were obtained by combining variational techniques and the concentration-compactness principle, and the proofs rely on localizing the infimum of associated functional constrained to Nehari type sets.
Abstract: We deal with the nonlinear Schrodinger equation −∆u + V (x)u = f(u) in R , where V is a (possible) sign changing potential satisfying mild assumptions and the nonlinearity f ∈ C(R,R) is a subcritical and superlinear function. By combining variational techniques and the concentration-compactness principle we obtain a positive ground state solution and also a nodal solution. The proofs rely in localizing the infimum of the associated functional constrained to Nehari type sets. 1991 Mathematics Subject Classification. 35J20, 35J60, 35B38.
67 citations
TL;DR: In this article, the authors consider nonlinear Neumann problems driven by p-Laplacian type operators which are not homogeneous in general and prove an existence and a multiplicity result for such problems.
Abstract: Abstract We consider nonlinear Neumann problems driven by p-Laplacian-type operators which are not homogeneous in general. We prove an existence and a multiplicity result for such problems. In the existence theorem, we assume that the right hand side nonlinearity is p-superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. In the multiplicity result, when specialized to the case of the p-Laplacian, we allow strong resonance at infinity and resonance at 0.
59 citations
TL;DR: In this article, the existence of solitary waves for Abelian gauge theories was analyzed and it was shown that the lower order term W is positive and that the coupling between matter and electromagnetic field is small.
Abstract: Abelian gauge theories consist of a class of field equations which provide a model for the interaction between matter and electromagnetic fields. In this paper we analyze the existence of solitary waves for these theories. We assume that the lower order term W is positive and we prove the existence of solitary waves if the coupling between matter and electromagnetic field is small. We point out that the positiveness assumption on W implies that the energy is positive: this fact makes these theories more suitable to model physical phenomena.
50 citations
TL;DR: In this article, the existence and multiplicity of solutions depending on the parameters p and λ were investigated in a smooth bounded domain with p = 0, ϕ ∈ H1(Ω) and Ω ⊂ ℝ3.
Abstract: Abstract In this note we consider the problem where u, ϕ ∈ H01(Ω) and Ω ⊂ ℝ3 is a smooth bounded domain. We are interested in existence and multiplicity of solutions depending on the parameters p and λ. Our results extend previous work made in ℝ3.
48 citations
TL;DR: In this paper, the authors give two short and elementary proofs of a characterization of constants function by Brezis, which are based either on convolutions or on a sort of nonsmooth Mean Value Theorem which is new to our knowledge.
Abstract: Abstract We give two short and elementary proofs of a characterization of constants function by Brezis. Whereas the original proof involves some refined arguments on Sobolev spaces and BV functions, ours are based either on convolutions or on a sort of nonsmooth Mean Value Theorem which is new to our knowledge.
17 citations
TL;DR: In this article, the authors studied relations between functional-analytic properties of the functional Q and properties of positive solutions of the equation Q'(u) = 0 on C-0(infinity) (Omega).
Abstract: Let Omega be a domain in R-d, d >= 2, and 1 = 0 on C-0(infinity)(Omega). In a previous paper [221 we discussed relations between the absence of weak coercivity of the functional Q on C-0(infinity) (Omega) and the existence of a generalized ground state. In the present paper we study further relationships between functional-analytic properties of the functional Q and properties of positive solutions of the equation Q'(u) = 0.
16 citations
TL;DR: In this paper, it was shown that there exists a residual subset R (with respect to the C0 topology) of d-dimensional linear differential systems based in a μ-invariant flow and with transition matrix evolving in GL(d,ℝ) such that if A ∈ R, then, for μ-a, the Oseledets splitting along the orbit is dominated (uniform projective hyperbolicity) or else the Lyapunov spectrum is trivial.
Abstract: Abstract We prove that there exists a residual subset R (with respect to the C0 topology) of d-dimensional linear differential systems based in a μ-invariant flow and with transition matrix evolving in GL(d,ℝ) such that if A ∈ R, then, for μ-a.e. point, the Oseledets splitting along the orbit is dominated (uniform projective hyperbolicity) or else the Lyapunov spectrum is trivial. Moreover, in the conservative setting, we obtain the dichotomy: dominated splitting versus zero Lyapunov exponents.
14 citations
TL;DR: In this paper, the existence of infinitely many radial positive solutions to the Hopf-Cole type change problem has been shown, and the main idea is to get a relation between radial positive solution and a suitable quasilinear family of problems with measures data.
Abstract: Abstract The main result of this work is to get the existence of infinitely many radial positive solutions to the problem -Δpu = |▽u|q + λf(x) in Ω, u|aΩ = 0, where Ω = B1(0) and f is a radial positive function. Since, in general when q ≠ p, a Hopf-Cole type change can not be used, we will consider just the existence and multiplicity of positive radial solutions. The main idea is to get a relation between radial positive solutions of the above equation and a suitable quasilinear family of problems with measures data that we will make precise later.
13 citations
TL;DR: In this paper, the L p and L q norms of coe−cients and some known Sobolev constants were used to define ellipticity criteria for planar linear Hamiltonian systems with periodic coe‐cients.
Abstract: In this paper we will establish two difierent classes of ellipticity criteria, called the L p criteria and the L p -L q criteria respectively, for planar linear Hamiltonian systems with periodic coe‐cients. The criteria are explicitly expressed using the L p and L q norms of coe‐cients and some known Sobolev constants. These results can be considered as the extensions of the famous Lyapunov stability criterion for Hill’s equations. 2000 Mathematics Subject Classiflcation. Primary: 34L40; Secondary: 34L15, 34D20, 93D05.
TL;DR: In this article, the existence of rotating solitary waves (vortices) with enough largely prescribed L2 norm for the nonlinear Klein-Gordon equation with nonnegative potential was proved, which makes the equation suitable for physical models and guarantees the wellposedness of the corresponding Cauchy problem.
Abstract: Abstract We prove the existence of rotating solitary waves (vortices) with enough largely prescribed L2 norm for the nonlinear Klein-Gordon equation with nonnegative potential, which makes the equation suitable for physical models and guarantees the well-posedness of the corresponding Cauchy problem. This is done by finding nonnegative cylindrical solutions to the standing equation where x = (y, z) ∈ ℝk × RN–k, N > k ≥ 0 and μ > 0 is the fundamental eigenvalue, namely, the Lagrange multiplier of the minimization problem (with infimum over a suitable subspace of H1(ℝN)).
TL;DR: In this article, the instability of the ground state solitary wave solution to the Klein-Gordon-Zakharov system in three dimensions with c ≥ 1 was studied, which is a model to describe the Langmuir turbulence in plasma.
Abstract: Abstract This paper deals with the instability of the ground state solitary wave solution to the Klein-Gordon-Zakharov system in three space dimensions with c ≥ 1, which is a model to describe the Langmuir turbulence in plasma. First we construct a suitable constrained variational problem and obtain the existence of the standing waves with ground state by using variational calculus and scaling argument. Then by defining invariant sets and applying some priori estimates, we prove the orbital instability of the ground state in the following sense: in each neighborhood of it, there exists a solution whose energy diverges in finite or infinite time.
TL;DR: In this article, the mountain pass theorem combined with the principle of symmetric criticality was used to establish multiplicity of solutions for the class of quasilinear elliptic equations.
Abstract: Abstract We use the mountain-pass theorem combined with the principle of symmetric criticality to establish multiplicity of solutions for the class of quasilinear elliptic equations -Δu + V(z)u - Δ(u2)u = h(u) in ℝN where N ≥ 4, the potential V : ℝN → ℝ is positive and bounded away from zero and satisfies appropriate periodic and symmetric conditions. The nonlinear term h(u) has subcritical growth and satisfies a condition of the Ambrosetti-Rabinowitz type. Schrödinger equations of this type have been studied as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics.
TL;DR: In this article, the definition of homology for contact forms defined in [1] was extended to the case when the "dual" form β is not a contact form anymore, which is the case in this paper.
Abstract: Abstract We extend in this paper the definition of the homology for contact forms defined in [1] to the case when the ”dual” form β is not a contact form anymore.
TL;DR: In this paper, the authors derived the exact asymptotic behavior of the unique classical solution near the boundary to a singular Dirichlet problem, where the weight b is positive in Ω and singular on the boundary.
Abstract: Abstract By Karamata regular variation theory, we first derived the exact asymptotic behavior of the local solution to the problem -φʹʹ(s) = g(φ(s)), φ(s) > 0, s ∈ (0, a) and φ(0) = 0. Then, by a perturbation method and constructing comparison functions, we derived the exact asymptotic behavior of the unique classical solution near the boundary to a singular Dirichlet problem -Δu = b(x)g(u) + λ|▽u|q, u > 0, x ∈ Ω, u|aΩ = 0, where Ω is a bounded domain with smooth boundary, λ ∈ ℝ, q ∈ [0, 2]; g ∈ C1((0, ∞), (0, ∞)), is decreasing in (0, ∞) with lims→0 + g(s) = +∞; the weight b is positive in Ω and singular on the boundary.
TL;DR: In this paper, the authors consider nonlinear Dirichlet problems driven by the p-Laplacian differential operator and prove the existence of at least three nontrivial smooth solutions.
Abstract: Abstract In this paper, we consider nonlinear Dirichlet problem driven by the p-Laplacian differential operator. Using variational methods based on the critical point theory and truncation techniques, we prove the existence of at least three nontrivial smooth solutions. The hypotheses on the nonlinearity incorporate in our framework of analysis both coercive and noncoercive problems. For the semilinear problem (p = 2), using Morse theory, we show the existence of four nontrivial smooth solutions.
TL;DR: In this article, the authors studied the dynamics of the periodic asymmetric oscillator and proved that the exponential growth rate χ(x) := lim t→+∞ 1 t log √ (x(t))2 + (x′(t)), for each non-zero solution x(t) of the oscillator.
Abstract: In this paper we will study the dynamics of the periodic asymmetric oscillator x′′ + q(t)x+ + q−(t)x− = 0, where q, q− ∈ L(R/2πZ) and x+ = max(x, 0), x− = min(x, 0) for x ∈ R. It will be proved that the exponential growth rate χ(x) := lim t→+∞ 1 t log √ (x(t))2 + (x′(t))2 does exist for each non-zero solution x(t) of the oscillator. The properties of these rates, or the Lyapunov exponents, will be given using the induced circle diffeomorphism of the oscillator. The proof is extensively based on the Denjoy theorem in topological dynamics and the unique ergodicity theorem in ergodic theory. 2000 Mathematics Subject Classification. Primary: 34D08; Secondary: 37E10, 37A25.
TL;DR: In this paper, the existence of solutions for the semilinear elliptic equation where 1 < p <, Q(x) is a sign-changing function such that both the supports of Q+ and Q- may have infinite measure was studied.
Abstract: Abstract We study the existence of solutions for the semilinear elliptic equation where 1 < p < , Q(x) is a sign-changing function such that both the supports of Q+ and Q- may have infinite measure. We show that the problem has at least one solution.
TL;DR: In this paper, the authors give necessary and sufficient conditions for bifurcation of T periodic solutions for the equation ẋ = Ax + f(t, x) + eg(t, x, e) from a k-parameterized family of T -periodic solutions of the unperturbed equation corresponding to e = 0.
Abstract: Let A : D(A) → E, D(A) ⊂ E, be an infinitesimal generator either of an analytic compact semigroup or of a contractive C0-semigroup of linear operators acting in a Banach space E. In this paper we give both necessary and sufficient conditions for bifurcation of T periodic solutions for the equation ẋ = Ax + f(t, x) + eg(t, x, e) from a k-parameterized family of T -periodic solutions of the unperturbed equation corresponding to e = 0. We show that by means of a suitable modification of the classical Mel’nikov approach we can construct a bifurcation function and to formulate the conditions for the existence of bifurcation in terms of the topological index of the bifurcation function. To do this, since the perturbation term g is only Lipschitzian we need to extend the classical LyapunovSchmidt reduction to the present nonsmooth case. 2000 Mathematics Subject Classification. 34G05, 37G15, 47D05.
TL;DR: It is proved that in order to define the homology of [3], the hypothesis that there exists a vector field in the kernel of the contact form which defines a dual form with the same orientation is not essential.
Abstract: Abstract In this paper we prove that in order to define the homology of [3], the hypothesis that there exists a vector field in the kernel of the contact form which defines a dual form with the same orientation is not essential. The technique is quantitative: as we introduce a large amount of rotation near the zeroes of the vector field in the kernel, we track down the modification of the variational problem and provide bounds on a key quantity (denoted by τ).
TL;DR: In this paper, a system of equations arising from angiogenesis which contains a nonregular term that vanishes below a certain threshold was studied and the existence, uniqueness and permanence of a positive solution for the time-dependent problem was established.
Abstract: We study a system of equations arising from angiogenesis which contains a nonregular term that vanishes below a certain threshold. This loss of regularity forces one to modify the usual methods of bifurcation theory. Nevertheless, we obtain results on the existence, uniqueness and permanence of a positive solution for the time-dependent problem; and the existence and uniqueness of a positive solution for the stationary one.
TL;DR: In this article, it was shown that the linear part at a singular point p0 of a C4 differential system Y0 in ℝ4 has eigenvalues ±αi and ±βi such that β/α = 1/3.
Abstract: Abstract Assume that the linear part at a singular point p0 of a C4 differential system Y0 in ℝ4 has eigenvalues ±αi and ±βi such that β/α = 1/3. In the main result of the paper we exhibit a one-parameter family of systems Yε for ε ∈ (-δ0;+δ0) where is shown that the original vector field around p0 can bifurcate in 0, 1, 2, 3 or 4 one-parameter families of periodic orbits. The tool for proving such a result is the averaging theory for non-C1 differentiable system. Moreover, assuming now that Yε is a one-parameter family of ℤ2- reversible polynomial vector fields of degree 5, we show that it can bifurcate in 0 or 2 one-parameter families of periodic orbits.
TL;DR: In this paper, a doubly degenerate parabolic equation with absorption (un)t = (|ux|m-1ux)x - ηup in (0, 1) ×(0, T) subject to the boundary source qc is determined under different dominations to identify global and non-global solutions.
Abstract: Abstract This paper deals with a doubly degenerate parabolic equation with absorption (un)t = (|ux|m-1ux)x - ηup in (0, 1) × (0, T) subject to the boundary source (|ux|m-1ux)(1, t) = uq(1, t). The critical boundary source exponent qc is determined under different dominations to identify global and non-global solutions. A complete classification for all of the four nonlinear exponents m, n, p, q with the coefficient η draws a very clear picture of interactions among the multi-nonlinearities. It is shown that qc depends on the absorption exponent p if and only if p > n. It is found as well that qc is related to small m only: qc relies on m if and only if m < p when p > n, also qc relies on m if and only if m < n when p ≤ n. The behavior of solutions in the critical case of q = qc is interesting. The absorption coefficient η affects the critical property of solutions with q = qc if and only if p > n: the case q = qc belongs to the global existence situation (regardless of the absorption coefficient η) if p ≤ n; while for q = qc with p > n, the solutions may be both global and non-global, depending on the size of the absorption coefficient.
TL;DR: In this paper, the semilinear heat equation with absorption and a power-type potential was considered in the subcritical range, where the problem admits a finite set of self-similar very singular solutions (VSSs).
Abstract: Abstract The semilinear heat equation with absorption and a power-type potential, ut = Δu - |x|β|u|p-1u in ℝN × ℝ+, with p > 1, β > -2, is considered. In the subcritical range , this problem is shown to admit a finite set of self-similar very singular solutions (VSSs), where f(y) has exponential decay at infinity and solves the semilinear elliptic equation . Some local bifurcation results are extended to the fourth-order equation ut = -Δ2u - |x|β|u|p-1u, with p > 1, β > -4.
TL;DR: The well-known Belousov-Zhabotinskii system can be characterized in this article, where the authors characterize its global analytic first integrals in terms of the first order integrals.
Abstract: Abstract The well-known Belousov-Zhabotinskii system can be written as ẋ = s(x + y - qx2 - xy), ẏ = s-1(-y + fz - xy), ż = w(x - z) with f, q, s, w ∈ ℝ and s ≠ 0. In this paper we characterize its global analytic first integrals.
TL;DR: In this article, the authors proved the nondegeneracy of least energy solutions to the above problem as Ɛ → 0 under certain assumptions, which is an extension of the recent work of Grossi [9].
Abstract: Abstract Consider the problem -∆u = N(N - 2)up + εk(x)u in Ω, u > 0 in Ω, u|∂Ω = 0 where Ω is a smooth bounded domain in ℝN (N ≥ 6), p = (N +2)/(N - 2), ε > 0 and k ∈ C2(Ω̅). Under certain assumptions, we prove the nondegeneracy of least energy solutions to the above problem as Ɛ →0. This is an extension of the recent work of Grossi [9].
TL;DR: In this article, a functional at infinity associated to a contact form on a three-dimensional manifold was studied and it was shown that the Morse index can be decomposed into two parts along the characteristic manifold and along the normal directions.
Abstract: Abstract In this paper we study a functional at infinity associated to a contact form on a three dimensional manifold. The Morse index of this functional at infinity can be decomposed into two parts, one along the characteristic manifold and the other along the normal directions. We prove that we can redistribute the negative directions between the two subspaces through a local deformation of the contact form near a critical point at infinity.
TL;DR: In this paper, the Navier boundary conditions on ∆ ∆ are applied to the biharmonic equation, where ∆ is an open bounded domain of ℝn, with n ≥ 5, having a smooth boundary, and ∆ ⊂ Ω is either a set of finite number of points of Ω or a smooth closed closed submanifold of ∆ without boundary.
Abstract: Abstract In this paper, we are interested in the following biharmonic equation: with Navier boundary conditions on ∂Ω; where Ω is an open bounded domain of ℝn, with n ≥ 5, having a smooth boundary, and ∑ ⊂ Ω is either a set of finite number of points of Ω or a smooth closed submanifold of Ω without boundary. We construct positive solutions to this problem which are singular at ∑.
TL;DR: In this paper, the Morse indices of iterates of the two closed geodesics on the 2-sphere with Katok's metric were computed and used to give a direct proof for the Poincaré series of the free loop space on S2.
Abstract: Abstract In this paper, we compute precisely Morse indices of iterates of the two closed geodesics on the 2-sphere with Katok’s metric. This information is used to give a direct proof for the Poincaré series of the free loop space on S2.