Infinitely many solutions to elliptic systems with critical exponents and Hardy potentials

In this paper, we consider the following elliptic systems with critical Sobolev growth and Hardy potentials: where N ≥ 3, η > 0, λ₁,λ₂ ∈ [0,Λ_N), and is the best Hardy constant. is the critical Sobolev exponent. a₁, a₂, b₁, and b₂ are positive parameters, and α,β > 1 satisfy 2 < α + β < 2*. h(x) ? 0, h(x) ≥ 0, , , and with . By means of the concentration–compactness principle and R. Kajikiya's new version of symmetric mountain pass lemma, we obtain infinitely many solutions that tend to zero. Copyright © 2013 John Wiley & Sons, Ltd.     

Existence of positive solutions for nonlinear Kirchhoff type problems in with critical Sobolev exponent

In this paper, we study the following nonlinear problem of Kirchhoff type with critical Sobolev exponent: where a, b > 0 are constants. Under certain assumptions on the sign‐changing function f(x,u), we prove the existence of positive solutions by variational methods. Our main results can be viewed as a partial extension of a recent result of He and Zou in [Journal of Differential Equations, 2012] concerning the existence of positive solutions to the nonlinear Kirchhoff problem where ϵ > 0 is a parameter, V (x) is a positive continuous potential, and with 4 < p < 6 and satisfies the Ambrosetti–Rabinowitz type condition. Copyright © 2013 John Wiley & Sons, Ltd.     

Solutions for perturbed biharmonic equations with critical nonlinearity

In this paper, we study the perturbed biharmonic equations where Δ² is the biharmonic operator, is the Sobolev critical exponent, p ∈ (2,2 ^{* *} ), P(x), and Q(x) are bounded positive functions. Under some given conditions on V, we prove that the problem has at least one nontrivial solution provided that and that for any , it has at least n ^* pairs solutions if , where and are sufficiently small positive numbers. Moreover, these solutions u_ε → 0 in as ε → 0. Copyright © 2013 The authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons, Ltd.     

Positive solutions for asymptotically linear Schrödinger–Kirchhoff‐type equations

In this paper, we focus on the Schrödinger–Kirchhoff‐type equation (SK) where a,b > 0 are constants, may not be radially symmetric, and f(x,u) is asymptotically linear with respect to u at infinity. Under some technical assumptions on V and f, we prove that the problem (SK) has a positive solution. Copyright © 2013 John Wiley & Sons, Ltd.     

The existence of a nontrivial solution to p‐Laplacian equations in with supercritical growth

In this paper, we consider the p‐Laplacian equations in with supercritical growth where △ _pu = div( | ? u | ^{p ? 2} ? u),1 < p < N is the p‐Laplacian operator. Under certain assumptions on V (x) and f(u) that will be given in Section 1, we prove that the problem has at least a nontrivial solution by using variational methods combined with perturbation arguments. The solutions to subcritical p‐Laplacian equations are estimated applying the L ^∞ norm. Copyright © 2012 John Wiley & Sons, Ltd.     

Multiple solutions for a semilinear elliptic system in

In this paper, we study the following semilinear elliptic system where N > 2, f(x,t) and g(x,t) are continuous functions and satisfy additional conditions. By using critical point theory of strongly indefinite functionals, we obtain a positive ground state solution and infinitely many geometrically distinct solutions when f(x,t) and g(x,t) are periodic in X and odd in t. Copyright © 2013 John Wiley & Sons, Ltd.     

Global existence and blow‐up solution for doubly degenerate parabolic system with nonlocal sources and inner absorptions

This paper deals with the following doubly degenerate parabolic system with null Dirichlet boundary conditions in a smooth bounded domain Ω ? R^N, where m, n ≥ 1, p, q ≥ 2, r₁, r₂, s₁, s₂ ≥ 1, α, β < 0. Under appropriate hypotheses, we prove that the solution either exists globally or blows up in finite time. Copyright © 2013 John Wiley & Sons, Ltd.     

Extending LSQR methods to solve the generalized Sylvester‐transpose and periodic Sylvester matrix equations

It is well known that the least‐squares QR‐factorization (LSQR) algorithm is a powerful method for solving linear systems Ax = b and unconstrained least‐squares problem min_x | | Ax ? b | | . In the paper, the LSQR approach is developed to obtain iterative algorithms for solving the generalized Sylvester‐transpose matrix equation the minimum Frobenius norm residual problem and the periodic Sylvester matrix equation Numerical results are given to illustrate the effect of the proposed algorithms. Copyright © 2013 John Wiley & Sons, Ltd.     

Existence and multiplicity results for singular quasilinear elliptic equation on unbounded domain

In this paper, we are interested in the existence and multiplicity results of solutions for the singular quasilinear elliptic problem with concave–convex nonlinearities (0.1) where is an unbounded exterior domain with smooth boundary ?Ω, 1 < p < N,0 ≤ a < (N ? p) ∕ p,λ > 0,1 < s < p < r < q = pN ∕ (N ? pd),d = a + 1 ? b,a ≤ b < a + 1. By the variational methods, we prove that problem 0.1 admits a sequence of solutions u_k under the appropriate assumptions on the weight functions H(x) and H(x). For the critical case, s = q,h(x) = | x | ^{? bq}, we obtain that problem 0.1 has at least a nonnegative solution with p < r < q and a sequence of solutions u_k with 1 < r < p < q and J(u_k) → 0 as k → ∞ , where J(u) is the energy functional associated to problem 0.1 . Copyright © 2013 John Wiley & Sons, Ltd.     

Homoclinic solutions for second‐order discrete Hamiltonian systems with asymptotically quadratic potentials

In this paper, we study the existence of infinitely many homoclinic solutions for the second‐order self‐adjoint discrete Hamiltonian system , where , and are unnecessarily positive definites for all . By using the variant fountain theorem, we obtain an existence criterion to guarantee that the aforementioned system has infinitely many homoclinic solutions under the assumption that W(n,x) is asymptotically quadratic as | x | → + ∞ . Copyright © 2013 John Wiley & Sons, Ltd.     

Positive solutions for a fourth order discrete p‐Laplacian boundary value problem

In this paper, we study the existence and multiplicity of positive solutions for the following fourth order nonlinear discrete p‐Laplacian boundary value problem where φ_p(s) = | s | ^{p ? 2}s, p > 1, is continuous, T is an integer with T ≥ 5 and . By virtue of Jensen's discrete inequalities, we use fixed point index theory to establish our main results. Copyright © 2013 John Wiley & Sons, Ltd.     

Existence of nontrivial solutions for fractional Schrödinger‐Poisson system with sign‐changing potentials

In this paper, we consider the following fractional Schrödinger‐Poisson system where 0 < t ≤ α < 1, , and 4α+2t ≥3 and the functions V(x), K(x) and f(x) have finite limits as |x|→∞. By imposing some suitable conditions on the decay rate of the functions, we prove that the above system has two nontrivial solutions. One of them is positive and the other one is sign‐changing. Recent results from the literature are generally improved and extended.     

Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source

We study a quasilinear parabolic–elliptic Keller–Segel system involving a source term of logistic type u_t = ? ? (?(u) ? u) ? χ ? ? (u ? v) + g(u), ? Δv = ? v + u in Ω × (0,T), subject to nonnegative initial data and the homogeneous Neumann boundary condition in a bounded domain with smooth boundary, n ≥ 1, χ > 0, ? ≥ c₁s^p for s ≥ s₀ > 1, and g(s) ≤ as ? μs² for s > 0 with a,g(0) ≥ 0, μ > 0. There are three nonlinear mechanisms included in the chemotaxis model: the nonlinear diffusion, aggregation and logistic absorption. The interaction among the triple nonlinearities shows that together with the nonlinear diffusion, the logistic absorption will dominate the aggregation such that the unique classical solution of the system has to be global in time and bounded, regardless of the initial data, whenever , or, equivalently, , which enlarge the parameter range , or , required by globally bounded solutions of the quasilinear K‐S system without the logistic source. Copyright © 2013 John Wiley & Sons, Ltd.     

Self‐organized criticality of cellular automata model; absorbtion in finite‐time of supercritical region into the critical one

In this work, it is studied the evolution and time behavior of solutions to nonlinear diffusion equation in where , d ≥ 1, and H is the Heaviside function. For d = 1,2,3, this equation describes the dynamics of self‐organizing sandpile process with critical state ρ_c. The main conclusion is that the supercritical region is absorbed in a finite‐time in the critical region . Copyright © 2013 John Wiley & Sons, Ltd.     

Multiple solutions of a quasilinear elliptic problem involving nonlinear boundary condition on exterior domain

In this paper, we study the multiplicity of non‐negative solutions for the quasilinear p‐Laplacian equation with the nonlinear boundary condition (1) where Δ_p denotes the p‐Laplacian operator, defined by △ _pu = div( | ? u | ^{p ? 2} ? u),1 < p < N, Ω is a smooth exterior domain in . is the outward normal derivative, . The parameters p,q,r are either or . The weight functions a(x),h(x),g(x) satisfy some suitable conditions. Using the decomposition of the Nehari manifold and the variational methods, we prove that problem (1) has at least two positive solutions provided 0 < | λ | < λ₁ for some λ₁. Copyright © 2012 John Wiley & Sons, Ltd.     

Infinitely many peak solutions for a biharmonic equation involving critical exponent

In this paper, we study the following biharmonic equation where , K(1) > 0,K′(1) > 0, B₁(0) is the unit ball in (N≥6). We show that the aforementioned problem has infinitely many peak solutions, whose energy can be made arbitrarily large. Copyright © 2016 John Wiley & Sons, Ltd.     

Extinction and decay estimates of solutions for a polytropic filtration equation with the nonlocal source and interior absorption

This paper deals with the extinction properties of solutions for the homogeneous Dirichlet boundary value problem with the nonlocal source and interior absorption where m,λ,k,q > 0, 0 < m(p ? 1) < 1, r ≤ 1, and . By using L^p‐integral norm estimate method, we obtain the sufficient conditions of extinction solutions. Moreover, we also give the precise decay estimates of the extinction solutions. Copyright © 2012 John Wiley & Sons, Ltd.     

Small data global well‐posedness for the nonlinear wave equation with nonlocal nonlinearity

In this paper, we investigate the Cauchy problem of the nonlinear wave equation , where V(u) = μ|·|^?γ ? |u|², , 0 < γ < min(4, n) and n ≥ 3. We prove small data global well‐posedness for the radial data and for the general data with angular regularity. We also give an improved result of the Hartree equation with negative critical regularity. Copyright © 2012 John Wiley & Sons, Ltd.     

Solutions of nonperiodic super quadratic Hamiltonian systems

This paper concerns solutions for the Hamiltonian system: , where H(t,u) = 1 ∕ 2Lu ? u + W(t,u), L is a 2N × 2N symmetric matrix, and . We consider the case that 0 ? σ_c( ? (Jd ∕ dt + L)) and W satisfies some new generalized super quadratic condition different from the type of Ambrosetti–Rabinowitz. The method is variational: By virtue of some auxiliary system related to the ‘limit equation’ of the Hamiltonian system, we first establish that (C)_c‐condition holds true for all c less than the least energy of the limit equation. Then, using some weak linking theorem recently developed, we obtain one least energy solution of the Hamiltonian system. Copyright © 2013 John Wiley & Sons, Ltd.     

Variational approach to solutions for a class of fractional Hamiltonian systems

In this paper, we investigate the existence of infinitely many solutions for the following fractional Hamiltonian systems: ( FHS ) where α ∈ (1 ∕ 2,1), , , and are symmetric and positive definite matrices for all , , and ? W is the gradient of W at u. The novelty of this paper is that, assuming L is coercive at infinity, and W is of subquadratic growth as | u | → + ∞ , we show that (FHS) possesses infinitely many solutions via the genus properties in the critical theory. Recent results in the literature are generalized and significantly improved. Copyright © 2013 John Wiley & Sons, Ltd.