Multiple positive solutions of nonhomogeneous elliptic systems with strongly indefinite structure and critical Sobolev exponents

In this paper, we study the existence of multiple positive solutions to some Hamiltonian elliptic systems −Δv=λu+u^p+εf(x), −Δu=μv+v^q+δg(x) in Ω;u,v>0 in Ω; u=v=0 on ∂Ω, where Ω is a bounded domain in R^N (N?3); 0?f, g∈L∞(Ω); 1/(p+1)+1/(q+1)=(N−2)/N, p,q>1; λ,μ>0. Using sub- and supersolution method and based on an adaptation of the dual variational approach, we prove the existence of at least two nontrivial positive solutions for all λ,μ∈(0,λ₁) and ε,δ∈(0,δ₀), where λ₁ is the first eigenvalue of the Laplace operator −Δ with zero Dirichlet boundary conditions and δ₀ is a positive number.     

Boundary behaviour of the unique solution to a singular Dirichlet problem with a convection term

By Karamata regular variation theory and constructing comparison functions, we derive that the boundary behaviour of the unique solution to a singular Dirichlet problem −Δu=b(x)g(u)+λq|∇u|, u>0, x∈Ω, u_|∂Ω=0, which is independent of λq|∇uλ|, where Ω is a bounded domain with smooth boundary in RN, λ∈R, q∈(0,2], lims→⁰⁺g(s)=+∞, and b is non-negative on Ω, which may be vanishing on the boundary.     

Asymptotical Formulas For Solutions Of Linear Differential Systems Of The First Order

In this paper a system of differential equations y′ ? A(·,λ)y = 0 is considered on the finite interval [a,b] where λ ∈ C, A(·, λ):= λ A₁+ A ₀ +λ _?1A_?1(·,λ) and A ₁,A ₀, A _{? 1} are n × n matrix-functions. The main assumptions: A ₁ is absolutely continuous on the interval [a, b], A ₀ and A _{- 1}(·,λ) are summable on the same interval when ¦λ¦ is sufficiently large; the roots φ₁(x),…,φ_n (x) of the characteristic equation det (φ E — A ₁) = 0 are different for all x ∈ [a,b] and do not vanish; there exists some unlimited set Ω ? C on which the inequalities Re(λφ₁(x)) ≤ … ≤ Re (λφ_n(x)) are fulfilled for all x ∈ [a,b] and for some numeration of the functions φ_j(x). The asymptotic formula of the exponential type for a fundamental matrix of solutions of the system is obtained for sufficiently large ¦λ¦. The remainder term of this formula has a new type dependence on properties of the coefficients A ₁ (x), A _o (x) and A _{- 1} (x).     

Reduced Weyl asymptotics for pseudodifferential operators on bounded domains II. The compact group case

Let G⊂O(n) be a compact group of isometries acting on n-dimensional Euclidean space Rn, and X a bounded domain in Rn which is transformed into itself under the action of G. Consider a symmetric, classical pseudodifferential operator A₀ in L²(Rn) that commutes with the regular representation of G, and assume that it is elliptic on X. We show that the spectrum of the Friedrichs extension A of the operator is discrete, and using the method of the stationary phase, we derive asymptotics for the number Nχ(λ) of eigenvalues of A equal or less than λ and with eigenfunctions in the χ-isotypic component of L²(X) as λ→∞, giving also an estimate for the remainder term for singular group actions. Since the considered critical set is a singular variety, we recur to partial desingularization in order to apply the stationary phase theorem.     

Isolated singularities of solutions to quasi-linear elliptic equations with absorption

We study the problem of removability of isolated singularities for a general second-order quasi-linear equation in divergence form −divA(x,u,∇u)+a₀(x,u)+g(x,u)=0 in a punctured domain Ω?{0}, where Ω is a domain in Rn, n?3. The model example is the equation −Δpu+gu|u|^p−2+u|u|^q−1=0, q>p−1>0, p<n. Assuming that the lower-order terms satisfy certain non-linear Kato-type conditions, we prove that for all point singularities of the above equation are removable, thus extending the seminal result of Brezis and Véron.     

Reduced Weyl asymptotics for pseudodifferential operators on bounded domains I. The finite group case

Let G⊂O(n) be a compact group of isometries acting on n-dimensional Euclidean space Rn, and X a bounded domain in Rn which is transformed into itself under the action of G. Consider a symmetric, classical pseudodifferential operator A₀ in L²(Rn) with G-invariant Weyl symbol, and assume that it is semi-bounded from below. We show that the spectrum of the Friedrichs extension A of the operator is discrete, and derive asymptotics for the number Nχ(λ) of eigenvalues of A less or equal λ and with eigenfunctions in the χ-isotypic component of L²(X) as λ→∞, giving also an estimate for the remainder term in case that G is a finite group. In particular, we show that the multiplicity of each unitary irreducible representation in L²(X) is asymptotically proportional to its dimension.     

The Rotation Number Approach to the Periodic Fuc caron ik Spectrum

In this paper, we study the Fu?ik spectrum of the problem: (^*) ?+(λ₊+q₊(t))x₊+(λ₋+q₋(t))x₋=0 with the 2π-periodic boundary condition, where q_±(t) are 2π-periodic. After introducing a rotation number function ρ(λ₊, λ₋) for (^*), we prove using the Hamiltonian structure and the positive homogeneity of (^*) that for any positive integer n, the two boundary curves of the domain ρ⁻¹(n/2) in the (λ₊, λ₋)-plane are Fu?ik curves of (^*). The result obtained in this paper shows that such a spectrum problem is much like that of the higher dimensional Fu?ik spectrum with the Dirichlet condition. In particular, it remains open if the Fu?ik spectrum of (^*) is composed of only these curves.     

Complete quenching for singular parabolic problems

We prove finite time extinction of the solution of the equation ut−Δu+χ{u>0}(u−β−λf(u))=0 in Ω×(0,∞) with boundary data u(x,t)=0 on ∂Ω×(0,∞) and initial condition u(x,0)=u₀(x) in Ω, where Ω⊂RN is a bounded smooth domain, 0<β<1 and λ>0 is a parameter. For every small enough λ>0 there exists a time t₀>0 such that the solution is identically equal to zero.     

An elliptic system with bifurcation parameters on the boundary conditions

In this paper we consider the elliptic system Δu=a(x)upvq, Δv=b(x)urvs in Ω, a smooth bounded domain, with boundary conditions , on ∂Ω. Here λ and μ are regarded as parameters and p,s>1, q,r>0 verify (p−1)(s−1)>qr. We consider the case where a(x)?0 in Ω and a(x) is allowed to vanish in an interior subdomain Ω₀, while b(x)>0 in . Our main results include existence of nonnegative nontrivial solutions in the range 0<λ<λ₁?∞, μ>0, where λ₁ is characterized by means of an eigenvalue problem, and the uniqueness of such solutions. We also study their asymptotic behavior in all possible cases: as both λ,μ→0, as λ→λ₁<∞ for fixed μ (respectively μ→∞ for fixed λ) and when both λ,μ→∞ in case λ₁=∞.     

Explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions

We consider transfer operators acting on spaces of holomorphic functions, and provide explicit bounds for their eigenvalues. More precisely, if Ω is any open set in Cd, and L is a suitable transfer operator acting on Bergman space A²(Ω), its eigenvalue sequence {λn(L)} is bounded by |λn(L)|?Aexp(−an1/d), where a,A>0 are explicitly given.     

Iterative solution of nonlinear equations of accretive and pseudocontractive types

Let E be a real uniformly smooth Banach space. Let A:D(A)=E→2E be an accretive operator that satisfies the range condition and A⁻¹(0)≠∅. Let {λ_n} and {θ_n} be two real sequences satisfying appropriate conditions, and for z∈E arbitrary, let the sequence {x_n} be generated from arbitrary x₀∈E by x_n+1=x_n−λ_n(u_n+θ_n(x_n−z)), u_n∈Ax_n, n?0. Assume that {u_n} is bounded. It is proved that {x_n} converges strongly to some x∗∈A−1(0). Furthermore, if K is a nonempty closed convex subset of E and T:K→K is a bounded continuous pseudocontractive map with F(T):={Tx=x}≠∅, it is proved that for arbitrary z∈K, the sequence {x_n} generated from x₀∈K by x_n+1=x_n−λ_n((I−T)x_n+θ_n(x_n−z)), n?0, where {λ_n} and {θ_n} are real sequences satisfying appropriate conditions, converges strongly to a fixed point of T.     

On global attractor for m-Laplacian parabolic equation with local and nonlocal nonlinearity

In this paper, we study the long-time behavior of solutions for m-Laplacian parabolic equation in Ω×(0,∞) with the initial data u(x,0)=u₀(x)∈Lq, q?1, and zero boundary condition in ∂Ω. Two cases for a(x)?a₀>0 and a(x)?0 are considered. We obtain the existence and Lp estimate of global attractor A in Lp, for any p?max{1,q}. The attractor A is in fact a bounded set in if a(x)?a₀>0 in Ω, and A is bounded in if a(x)?0 in Ω.     

Friedrichs extensions of Schrödinger operators with singular potentials

The Friedrichs extension for the generalized spiked harmonic oscillator given by the singular differential operator −d²/dx²+Bx²+Ax⁻²+λx^−α (B>0, A?0) in L₂(0,∞) is studied. We look at two different domains of definition for each of these differential operators in L₂(0,∞), namely C₀^∞(0,∞) and D(T_2,F)∩D(M_λ,α), where the latter is a subspace of the Sobolev space W_2,2(0,∞). Adjoints of these differential operators on C₀^∞(0,∞) exist as result of the null-space properties of functionals. For the other domain, convolutions and Jensen and Minkowski integral inequalities, density of C₀^∞(0,∞) in D(T_2,F)∩D(M_λ,α) in L₂(0,∞) lead to the other adjoints. Further density properties C₀^∞(0,∞) in D(T_2,F)∩D(M_λ,α) yield the Friedrichs extension of these differential operators with domains of definition D(T_2,F)∩D(M_λ,α).     

On linear systems with quasiperiodic coefficients and bounded solutions

For a discrete dynamical system ω _n =ω₀+αn, where a is a constant vector with rationally independent coordinates, on thes-dimensional torus Ω we consider the setL of its linear unitary extensionsx _n+1=A(ω₀+αn)x _n , whereA (Ω) is a continuous function on the torus Ω with values in the space ofm-dimensional unitary matrices. It is proved that linear extensions whose solutions are not almost periodic form a set of the second category inL (representable as an intersection of countably many everywhere dense open subsets). A similar assertion is true for systems of linear differential equations with quasiperiodic skew-symmetric matrices.     

Large time behavior of solutions to degenerate parabolic equations with weights

We study the degenerate parabolic equation t_∂u=a(δ(x))upΔu−g(u) in Ω×(0,∞), where Ω⊂RN (N?1) is a smooth bounded domain, p?1, δ(x)=dist(x,∂Ω) and a is a continuous nondecreasing function such that a(0)=0. Under some suitable assumptions on a and g we prove the existence and the uniqueness of a classical solution and we study its asymptotic behavior as t→∞.     

Faber-Krahn Inequality for Anisotropic Eigenvalue Problems with Robin Boundary Conditions

In this paper we study the main properties of the first eigenvalue λ ₁(Ω) and its eigenfunctions of a class of highly nonlinear elliptic operators in a bounded Lipschitz domain Ω ? ? _n , assuming a Robin boundary condition. Moreover, we prove a Faber-Krahn inequality for λ ₁(Ω).     

On an eigenvalue problem involving variable exponent growth conditions

We study the problem in Ω, u=0 on ∂Ω, where Ω is a bounded domain in R^N, is a continuous function and λ and ε are two positive constants. We prove that for any ε>0 each λ∈(0,λ₁) is an eigenvalue of the above problem, where λ₁ is the principal eigenvalue of the Laplace operator on Ω. Moreover, for each eigenvalue λ∈(0,λ₁) it corresponds a unique eigenfunction. The proofs will be based on the Banach fixed point theorem combined with adequate variational techniques.     

Elliptic equation with a singular potential in a domain with a conic point

This paper deals with the behavior of the nonnegative solutions of the problem $$- \Delta u = V(x)u, \left. u \right|\partial \Omega = \varphi (x)$$ in a conical domain Ω ? ? ⁿ , n ≥ 3, where 0 ≤ V (x) ∈ L₁(Ω), 0 ≤ ?(x) ∈ L₁(?Ω) and ?(x) is continuous on the boundary ?Ω. It is proved that there exists a constant C _*(n) = (n ? 2)²/4 such that if V ₀(x) = (c + λ ₁)|x|^?2, then, for 0 ≤ c ≤ C _*(n) and V(x) ≤ V ₀(x) in the domain Ω, this problem has a nonnegative solution for any nonnegative boundary function ?(x) ∈ L ₁(?Ω); for c > C _*(n) and V(x) ≥ V ₀(x) in Ω, this problem has no nonnegative solutions if ?(x) > 0.     

Critical potentials of the eigenvalues and eigenvalue gaps of Schrödinger operators

Let M be a compact Riemannian manifold with or without boundary, and let −Δ be its Laplace-Beltrami operator. For any bounded scalar potential q, we denote by λi(q) the ith eigenvalue of the Schrödinger type operator −Δ+q acting on functions with Dirichlet or Neumann boundary conditions in case ∂M≠∅. We investigate critical potentials of the eigenvalues λi and the eigenvalue gaps Gij=λj−λi considered as functionals on the set of bounded potentials having a given mean value on M. We give necessary and sufficient conditions for a potential q to be critical or to be a local minimizer or a local maximizer of these functionals. For instance, we prove that a potential q∈L^∞(M) is critical for the functional λ₂ if and only if q is smooth, λ₂(q)=λ₃(q) and there exist second eigenfunctions f₁,…,fk of −Δ+q such that . In particular, λ₂ (as well as any λi) admits no critical potentials under Dirichlet boundary conditions. Moreover, the functional λ₂ never admits locally minimizing potentials.     

A remark on uniqueness of large solutions for elliptic systems of competitive type

We prove that the semilinear system Δu=a(x)upvq, Δv=b(x)urvs in a smooth bounded domain Ω⊂RN has a unique positive solution with the boundary condition u=v=+∞ on ∂Ω, provided that p,s>1, q,r>0 and (p−1)(s−1)−qr>0. The main novelty is imposing a growth on the possibly singular weights a(x), b(x) near ∂Ω, rather than requiring them to have a precise asymptotic behavior.