Showing papers in "Nonlinear Analysis-theory Methods & Applications in 2015"
TL;DR: In this paper, a Harnack inequality for minimisers of a class of non-autonomous functionals with non-standard growth conditions is proved for the case where the energy density switches between two types of different degenerate phases.
Abstract: We prove a Harnack inequality for minimisers of a class of non-autonomous functionals with non-standard growth conditions. They are characterised by the fact that their energy density switches between two types of different degenerate phases.
283 citations
TL;DR: In this paper, the qualitative analysis of solutions to nonlinear elliptic problems of the type { − div A (x, ∇ u ) = λ | u | q (x ) − 2 u in Ω u = 0 on ∂ Ω, where Ω is a bounded or an exterior domain of R N and q is a continuous positive function.
Abstract: In this survey paper, by using variational methods, we are concerned with the qualitative analysis of solutions to nonlinear elliptic problems of the type { − div A ( x , ∇ u ) = λ | u | q ( x ) − 2 u in Ω u = 0 on ∂ Ω , where Ω is a bounded or an exterior domain of R N and q is a continuous positive function. The results presented in this paper extend several contributions concerning the Lane–Emden equation and we focus on new phenomena which are due to the presence of variable exponents.
218 citations
TL;DR: In this article, the existence and the asymptotic behavior of non-negative solutions for a class of stationary Kirchhoff problems driven by a fractional integro-differential operator LK and involving a critical nonlinearity were analyzed.
Abstract: This paper deals with the existence and the asymptotic behavior of non-negative solutions for a class of stationary Kirchhoff problems driven by a fractional integro-differential operator LK and involving a critical nonlinearity. In particular, we consider the problem
−M(||u||2)LKu=λf(x,u)+|u|2s∗−2uin Ω,u=0in Rn∖Ω,
where Ω⊂Rn is a bounded domain, 2s∗ is the critical exponent of the fractional Sobolev space Hs(Rn), the function f is a subcritical term and λ is a positive parameter. The main feature, as well as the main difficulty, of the analysis is the fact that the Kirchhoff function M could be zero at zero, that is the problem is degenerate. The adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature.
166 citations
TL;DR: In this article, it has been shown that the FitzHugh-Nagumo model admits a stable traveling pulse solution for sufficiently small ϵ > 0.1, both for circular axons and axons of infinite length.
Abstract: The FitzHugh–Nagumo model is a reaction–diffusion equation describing the propagation of electrical signals in nerve axons and other biological tissues. One of the model parameters is the ratio ϵ of two time scales, which takes values between 0.001 and 0.1 in typical simulations of nerve axons. Based on the existence of a (singular) limit at ϵ = 0 , it has been shown that the FitzHugh–Nagumo equation admits a stable traveling pulse solution for sufficiently small ϵ > 0 . Here we prove the existence of such a solution for ϵ = 0.01 , both for circular axons and axons of infinite length. This is in many ways a completely different mathematical problem. In particular, it is non-perturbative and requires new types of estimates. Some of these estimates are verified with the aid of a computer. The methods developed in this paper should apply to many other problems involving homoclinic orbits, including the FitzHugh–Nagumo equation for a wide range of other parameter values.
78 citations
TL;DR: In this paper, a general Adams-Moser-Trudinger type inequality for the embedding of Besselpotential spaces H n p, p ( Ω ) into Orlicz spaces for an arbitrary domain Ω with finite measure was proved.
Abstract: Extending several works, we prove a general Adams–Moser–Trudinger type inequality for the embedding of Bessel-potential spaces H n p , p ( Ω ) into Orlicz spaces for an arbitrary domain Ω with finite measure. In particular we prove sup u ∈ H n p , p ( Ω ) , ‖ ( − Δ ) n 2 p u ‖ L p ( Ω ) ≤ 1 ∫ Ω e α n , p | u | p p − 1 d x ≤ c n , p | Ω | , for a positive constant α n , p whose sharpness we also prove. We further extend this result to the case of Lorentz-spaces (i.e. ( − Δ ) n 2 p u ∈ L ( p , q ) ). The proofs are simple, as they use Green functions for fractional Laplace operators and suitable cut-off procedures to reduce the fractional results to the sharp estimate on the Riesz potential proven by Adams and its generalization proven by Xiao and Zhai. We also discuss an application to the problem of prescribing the Q -curvature and some open problems.
75 citations
TL;DR: In this article, the existence of infinitely many solutions for Kirchhoff type equations involving nonlocal integro-differential operators with homogeneous Dirichlet boundary conditions was studied.
Abstract: In this paper, we use the Fountain Theorem and the Dual Fountain Theorem to study the existence of infinitely many solutions for Kirchhoff type equations involving nonlocal integro-differential operators with homogeneous Dirichlet boundary conditions. A model for these operators is given by the fractional Laplacian of Kirchhoff type: { M ( ∬ R 2 N | u ( x ) − u ( y ) | 2 | x − y | N + 2 s d x d y ) ( − Δ ) s u ( x ) − λ u = f ( x , u ) in Ω u = 0 in R N ∖ Ω , where Ω is a smooth bounded domain of R N , ( − Δ ) s is the fractional Laplacian operator with 0 s 1 and 2 s N , λ is a real parameter, M is a continuous and positive function and f is a Caratheodory function satisfying the Ambrosetti–Rabinowitz type condition.
65 citations
TL;DR: In this paper, the authors considered the non-steady Navier-Stokes equations with Dirichlet boundary conditions in thin tube structures and derived asymptotic partial domain decomposition.
Abstract: The non-steady Navier–Stokes equations with Dirichlet boundary conditions are considered in thin tube structures. These domains are connected finite unions of thin finite cylinders (in the 2 D case respectively thin rectangles). The complete asymptotic expansion of the solution is constructed in the case without boundary-layer-in-time. The estimates for the difference of the exact solution and its J -th asymptotic approximation is proved. The method of asymptotic partial domain decomposition is formulated and justified for the non-steady Navier–Stokes equations in a tube structure. It gives the asymptotically exact interface conditions of coupling of the 1D and 3D models of the flow. Note that the obtained results hold true for the important in applications case of time periodic flows.
63 citations
TL;DR: In this article, the existence of a weak solution and its continuous dependence on the data are proved using a suitable setting for the conservation of a total mass in the bulk plus the boundary.
Abstract: The well-posedness of a system of partial differential equations and dynamic boundary conditions, both of Cahn–Hilliard type, is discussed. The existence of a weak solution and its continuous dependence on the data are proved using a suitable setting for the conservation of a total mass in the bulk plus the boundary. A very general class of double-well like potentials is allowed. Moreover, some further regularity is obtained to guarantee the strong solution.
62 citations
TL;DR: For the subcritical case in high dimensions, it has been believed that the basic tools of its analysis are Kato's lemma on ordinary differential inequalities and the rescaling argument in the functional method as discussed by the authors.
Abstract: We are interested in the upper bound of the lifespan of solutions of semilinear wave equations. For the sub-critical case in high dimensions, it has been believed that the basic tools of its analysis are Kato’s lemma on ordinary differential inequalities and the rescaling argument in the functional method. But there is a small lack of delicate analysis and no published paper about this. Here we give a simple alternative proof by means of improved Kato’s lemma without any rescaling argument.
62 citations
TL;DR: In this paper, the authors extend the global compactness result of Gerard (1998) to any fractional Sobolev spaces H s ( Ω ), for 0 s N / 2 and Ω ⊂ R N a bounded domain with smooth boundary.
Abstract: We extend the global compactness result by Struwe (1984) to any fractional Sobolev spaces H s ( Ω ) , for 0 s N / 2 and Ω ⊂ R N a bounded domain with smooth boundary. The proof is a simple direct consequence of the so-called profile decomposition of Gerard (1998).
50 citations
TL;DR: In this paper, the authors studied the existence of positive ground state solutions for the nonlinear Kirchhoff type problem and showed that the problem has at least a positive ground-state solution.
Abstract: In this paper, we study the existence of positive ground state solutions for the nonlinear Kirchhoff type problem { − ( a + b ∫ R 3 | ∇ u | 2 ) △ u + V ( x ) u = f ( u ) in R 3 , u ∈ H 1 ( R 3 ) , u > 0 in R 3 , where a , b > 0 are constants, f ∈ C ( R , R ) is subcritical near infinity and superlinear near zero and satisfies the Berestycki–Lions condition. By using an abstract critical point theorem established by Jeanjean and a new global compactness lemma, we show that the above problem has at least a positive ground state solution. Our result generalizes the results of Li and Ye (2014) concerning the nonlinearity f ( u ) = | u | p − 1 u with p ∈ ( 2 , 5 ) .
TL;DR: In this article, the non-steady Navier-Stokes equations with Dirichlet boundary conditions are considered in thin tube structures, and the complete asymptotic expansion of the solution is constructed.
Abstract: The non-steady Navier–Stokes equations with Dirichlet boundary conditions are considered in thin tube structures. These domains are connected finite unions of thin finite cylinders (in the 2 D case respectively thin rectangles). The complete asymptotic expansion of the solution is constructed. It contains a regular part and three types of the boundary layer correctors: “in-space”, “in-time” and “in-space-and-in-time”. The estimates for the difference of the exact solution and its J th asymptotic approximation are proved.
TL;DR: In this article, the authors study the higher regularity of the free boundary for the elliptic Signorini problem with respect to the Legendre transform and show that it is real analytic.
Abstract: In this paper we study the higher regularity of the free boundary for the elliptic Signorini problem. By using a partial hodograph–Legendre transformation we show that the regular part of the free boundary is real analytic. The first complication in the study is the invertibility of the hodograph transform (which is only C 0 , 1 / 2 ) which can be overcome by studying the precise asymptotic behavior of the solutions near regular free boundary points. The second and main complication in the study is that the equation satisfied by the Legendre transform is degenerate. However, the equation has a subelliptic structure and can be viewed as a perturbation of the Baouendi–Grushin operator. By using the L p theory available for that operator, we can bootstrap the regularity of the Legendre transform up to real analyticity, which implies the real analyticity of the free boundary.
TL;DR: In this article, an impulsive neutral stochastic fractional integro-differential equation with infinite delays in an arbitrary separable Hilbert space is considered and the existence of mild solution is obtained by using resolvent operator and fixed point theorems.
Abstract: In this work, we consider an impulsive neutral stochastic fractional integro-differential equation with infinite delays in an arbitrary separable Hilbert space. The existence of mild solution is obtained by using resolvent operator and fixed point theorems. An example is considered to illustrate the theory and conclusion is also provided at the end of the manuscript.
TL;DR: In this article, the Liouville theorem for α -harmonic functions under a weaker condition was proved using potential theory and Fourier analysis, which is a corollary of the fact that affine functions are affine.
Abstract: In this paper, we prove the following result. Let α be any real number between 0 and 2 . Assume that u is a solution of { ( − △ ) α / 2 u ( x ) = 0 , x ∈ R n , lim ¯ ∣ x ∣ → ∞ u ( x ) ∣ x ∣ γ ≥ 0 , for some 0 ≤ γ ≤ 1 and γ α . Then u must be constant throughout R n . This is a Liouville Theorem for α -harmonic functions under a much weaker condition. For this theorem we have two different proofs by using two different methods: One is a direct approach using potential theory. The other is by Fourier analysis as a corollary of the fact that the only α -harmonic functions are affine.
TL;DR: In this paper, it was shown that internal resonances, which depend on the bridge structure only, are the origin of torsional instability in suspension bridges and obtained both theoretical and numerical estimates of the thresholds of instability.
Abstract: We consider a mathematical model for the study of the dynamical behavior of suspension bridges. We show that internal resonances, which depend on the bridge structure only, are the origin of torsional instability. We obtain both theoretical and numerical estimates of the thresholds of instability. Our method is based on a finite dimensional projection of the phase space which reduces the stability analysis of the model to the stability of suitable Hill equations. This gives an answer to a long-standing question about the origin of torsional instability in suspension bridges.
TL;DR: In this article, it was shown that α ∗ is achieved by a positive, radially symmetric and strictly decreasing function provided 0 s N 2, 0 α 2 s.
Abstract: In this paper, we show that the minimizing problem (1) Λ s , α = inf u ∈ H s ( R N ) , u ≢ 0 ∫ R N | ( − Δ ) s 2 u ( x ) | 2 d x ( ∫ R N | u ( x ) | 2 s , α ∗ | x | α d x ) 2 2 s , α ∗ is achieved by a positive, radially symmetric and strictly decreasing function provided 0 s N 2 , 0 α 2 s .
TL;DR: In this article, the authors considered the barotropic compressible quantum Navier-Stokes equations with a linear density dependent viscosity and its limit when the scaled Planck constant vanishes.
Abstract: In this paper we consider the barotropic compressible quantum Navier–Stokes equations with a linear density dependent viscosity and its limit when the scaled Planck constant vanishes. Following recent works on degenerate compressible Navier–Stokes equations, we prove the global existence of weak solutions by the use of a singular pressure close to vacuum. With such singular pressure, we can use the standard definition of global weak solutions which also allows to justify the limit when the scaled Planck constant denoted by e tends to 0 .
TL;DR: In this article, the authors present the proof of existence of renormalized solutions to a nonlinear parabolic problem ∂ t u − div a ( ⋅, D u ) = f with right-hand side f and initial data u 0 in L 1, where growth and coercivity conditions on the monotone vector field a are prescribed by a generalized N -function M which is anisotropic and inhomogeneous with respect to the space variable.
Abstract: We will present the proof of existence of renormalized solutions to a nonlinear parabolic problem ∂ t u − div a ( ⋅ , D u ) = f with right-hand side f and initial data u 0 in L 1 . The growth and coercivity conditions on the monotone vector field a are prescribed by a generalized N -function M which is anisotropic and inhomogeneous with respect to the space variable. In particular, M does not have to satisfy an upper growth bound described by a Δ 2 -condition. Therefore we work with generalized Musielak–Orlicz spaces which are not necessarily reflexive. Moreover we provide a weak sequential stability result for a more general problem: ∂ t β ( ⋅ , u ) − div ( a ( ⋅ , D u ) + F ( u ) ) = f , where β is a monotone function with respect to the second variable and F is locally Lipschitz continuous. Within the proof we use truncation methods, Young measure techniques, the integration by parts formula and monotonicity arguments which have been adapted to nonreflexive Musielak–Orlicz spaces.
TL;DR: In this paper, the authors considered wave propagation in the Friedmann-Lemaitre-Robertson-Walker spacetimes and showed the global in time existence in the energy class of solutions of the Cauchy problem.
Abstract: We consider waves, which obey the semilinear Klein–Gordon equation, propagating in the Friedmann–Lemaitre–Robertson–Walker spacetimes. The equations in the de Sitter and Einstein–de Sitter spacetimes are the important particular cases. We show the global in time existence in the energy class of solutions of the Cauchy problem.
TL;DR: In this article, the existence and the regularity of non-trivial T -periodic solutions to the Schrodinger equation (0.1) with m = 0 were studied.
Abstract: We study the existence and the regularity of non trivial T -periodic solutions to the following nonlinear pseudo-relativistic Schrodinger equation (0.1) ( − Δ x + m 2 − m ) u ( x ) = f ( x , u ( x ) ) in ( 0 , T ) N where T > 0 , m is a non negative real number, f is a regular function satisfying the Ambrosetti–Rabinowitz condition and a polynomial growth at rate p for some 1 p 2 ♯ − 1 . We investigate such problem using critical point theory after transforming it to an elliptic equation in the infinite half-cylinder ( 0 , T ) N × ( 0 , ∞ ) with a nonlinear Neumann boundary condition. By passing to the limit as m → 0 in (0.1) we also prove the existence of a non trivial T -periodic weak solution to (0.1) with m = 0 .
TL;DR: In this paper, a reaction fractional diffusion equation with additive noise on the entire space R n is studied and the existence of a random attractor in L 2 ( R n ) is established.
Abstract: We study a reaction–fractional diffusion equation with additive noise on the entire space R n with particular interest in the asymptotic behavior of solutions. We first transform the equation into a random equation whose solutions generate a random dynamical system. A priori estimates for solutions are derived when the nonlinearity satisfies certain growth conditions. Using estimates for far-field values of solutions and a cut-off technique, asymptotic compactness is proved. Thus, the existence of a random attractor in L 2 ( R n ) is established.
TL;DR: In this paper, Cheng et al. derived Cheng-Yau, Li and Hamilton estimates for Riemannian manifolds with Bakry-Emery-Ricci curvatures bounded from below, and also global and local upper bounds for the Hessian of positive and bounded solutions of the weighted heat equation.
Abstract: In this paper we derive Cheng–Yau, Li–Yau, Hamilton estimates for Riemannian manifolds with Bakry–Emery–Ricci curvature bounded from below, and also global and local upper bounds, in terms of Bakry–Emery–Ricci curvature, for the Hessian of positive and bounded solutions of the weighted heat equation on a closed Riemannian manifold.
TL;DR: In this paper, a multiparameter global bifurcation theorem for differential inclusions with the periodic condition is proved by using the method of integral guiding functions, which can be applied to the study of the two-parameter global partitioning of periodic solutions for a class of differential variational inequalities.
Abstract: In this paper, by using the method of integral guiding functions a multiparameter global bifurcation theorem for differential inclusions with the periodic condition is proved. It is shown how the abstract result can be applied to the study of the two-parameter global bifurcation of periodic solutions for a class of differential variational inequalities.
TL;DR: In this article, the dispersion relation of water waves and a nonlinearity of the shallow water equations was modified to permit the effects of surface tension and constant vorticity.
Abstract: We study modulational stability and instability in the Whitham equation, combining the dispersion relation of water waves and a nonlinearity of the shallow water equations, and modified to permit the effects of surface tension and constant vorticity. When the surface tension coefficient is large, we show that a periodic traveling wave of sufficiently small amplitude is unstable to long wavelength perturbations if the wave number is greater than a critical value, and stable otherwise, similarly to the Benjamin–Feir instability of gravity waves. In the case of weak surface tension, we find intervals of stable and unstable wave numbers, whose boundaries are associated with the extremum of the group velocity, the resonance between the first and second harmonics, the resonance between long and short waves, and a resonance between dispersion and the nonlinearity. For each constant vorticity, we show that a periodic traveling wave of sufficiently small amplitude is unstable if the wave number is greater than a critical value, and stable otherwise. Moreover it can be made stable for a sufficiently large vorticity. The results agree with those based upon numerical computations or formal multiple-scale expansions to the physical problem.
TL;DR: In this paper, the existence of at least one nonnegative nonsmooth weak solution in D 1, p (R N ) ∩ D 1, q ( R N ) for the equation − Δ p u − Δ q u + a (x ) | u | p − 2 u + b (x)| u | q − 2u = f ( x, u ), x ∈ R N, where 1 q p q ⋆ : = N q N − q, p N, Δ m u : = div ( | ∇ u | m
Abstract: In this paper we prove the existence of at least one nonnegative nontrivial weak solution in D 1 , p ( R N ) ∩ D 1 , q ( R N ) for the equation − Δ p u − Δ q u + a ( x ) | u | p − 2 u + b ( x ) | u | q − 2 u = f ( x , u ) , x ∈ R N , where 1 q p q ⋆ : = N q N − q , p N , Δ m u : = div ( | ∇ u | m − 2 ∇ u ) is the m -Laplacian operator, the coefficients a and b are continuous, coercive and positive functions, and the nonlinearity f is a Caratheodory function satisfying some hypotheses which do not include the Ambrosetti–Rabinowitz condition.
TL;DR: In this paper, the Dirichlet BVP for second-order Hamiltonian systems was studied and the existence of homoclinic solutions for BVP on the finite interval T of R was proved.
Abstract: In this paper, we present some new results of homoclinic solutions for second-order Hamiltonian systems u − λ L ( t ) u + W u ( t , u ) = 0 ; here λ > 0 is a parameter, L ∈ C ( R , R N × N ) and W ∈ C 1 ( R × R N , R ) . Unlike most other papers on this problem, we require that L ( t ) is a positive semi-definite symmetric matrix for all t ∈ R , that is, L ( t ) ≡ 0 is allowed to occur in some finite interval T of R . Under some suitable assumptions on W , we prove the existence of two different homoclinic solutions u λ ( 1 ) , u λ ( 2 ) , which both vanish on R ∖ T as λ → ∞ , and converge to u 0 ( 1 ) , u 0 ( 2 ) in H 1 ( R ) , respectively; here u 0 ( 1 ) ≠ u 0 ( 2 ) ∈ H 0 1 ( T ) are two nontrivial solutions of the Dirichlet BVP for Hamiltonian systems on the finite interval T .
TL;DR: In this paper, the existence of weak weak solutions for the nonlocal fractional Laplace operator problem is proved under two different types of conditions on the functions a and f, by using a famous critical point theorem in the presence of splitting established by Brezis and Nirenberg.
Abstract: In this paper we consider problems modeled by the following nonlocal fractional equation
{(−Δ)su+a(x)u=μf(u)in Ωu=0in Rn∖Ω,
where s∈(0,1) is fixed, Ω is an open bounded subset of Rn, n>2s, with Lipschitz boundary, (−Δ)s is the fractional Laplace operator and μ is a real parameter.
Under two different types of conditions on the functions a and f, by using a famous critical point theorem in the presence of splitting established by Brezis and Nirenberg, we obtain the existence of at least two nontrivial weak solutions for our problem.
TL;DR: In this article, the authors study the asymptotic behavior of a time-dependent parabolic equation with nonlocal diffusion and nonlinear terms with sublinear growth, obtaining existence, uniqueness, and continuity results, analyzing the stationary problem and decay of the solutions of the evolutionary problem.
Abstract: This paper is devoted to study the asymptotic behavior of a time-dependent parabolic equation with nonlocal diffusion and nonlinear terms with sublinear growth. Namely, we extend some previous results from the literature, obtaining existence, uniqueness, and continuity results, analyzing the stationary problem and decay of the solutions of the evolutionary problem, and finally, under more general assumptions, ensuring the existence of pullback attractors for the associated dynamical system in both L 2 and H 1 norms. Relationships among these objects are established using regularizing properties of the equation.
TL;DR: In this article, a suitably integrated maximal-dissipation principle is devised to select force-driven local solutions and eliminate solutions with "too-early jumps" as it may occur in energy-driven ones.
Abstract: The system of two inclusions ∂ u E ( t , u ( t ) , z ( t ) ) ∋ 0 and ∂ R ( z ) + ∂ z E ( t , u ( t ) , z ( t ) ) ∋ 0 with the dissipation potential R degree-1 homogeneous and with the stored energy E ( t , ⋅ , ⋅ ) separately convex is considered. The relation between conventional weak solutions and local solutions is shown, and a suitably integrated maximal-dissipation principle is devised to select force-driven local solutions and eliminate solutions with “too-early jumps” as it may occur in energy-driven ones. This is illustrated on scalar examples. An approximation by a simple and efficient semi-implicit time discretization of the fractional-step type is shown to converge to local solutions. On the scalar examples, the approximate solutions are shown to satisfy the integrated maximal-dissipation principle asymptotically, while in general it is devised only to serve as an a-posteriori tool to justify (or possibly adaptively adjust) thus obtained approximate solutions as, in fact, force driven. Applications of such solutions are illustrated on specific examples from continuum mechanics at small strains involving inelastic processes in a bulk or on a surface, namely damage and delamination.