Showing papers in "Advanced Nonlinear Studies in 2014"
TL;DR: In this paper, the concentration and multiplicity of weak solutions to the Kirch-hoff type equation with critical Sobolev growth was studied, and the authors showed that weak solutions can be found in a large number of locations.
Abstract: We study the concentration and multiplicity of weak solutions to the Kirch- hoff type equation with critical Sobolev growth
154 citations
TL;DR: In this paper, a new super-quadratic condition was derived for the case where both V and f are periodic in x and 0 belongs to a spectral gap of −△ + V.
Abstract: Consider the semilinear Schrodinger equation { −△u+ V (x)u = f(x, u), x ∈ R , u ∈ H(R ), where f is a superlinear, subcritical nonlinearity. We mainly study the case where both V and f are periodic in x and 0 belongs to a spectral gap of −△ + V . Based on work of Szulkin and Weth (2009) [17], we develop a new technique to show the boundedness of Cerami sequences and derive a new super-quadratic condition that there exists a θ0 ∈ (0, 1) such that 1− θ 2 tf(x, t) ≥ ∫ t θt f(x, s)ds, ∀ θ ∈ [0, θ0] for the existence a “ground state solution” which minimizes the corresponding energy among all nontrivial solutions. Our result unifies and improves some known ones and the recent ones of Szulkin and Weth (J Funct Anal 257: 3802-3822, 2009) and Liu (Calc. Var. 45: 1-9, 2012).
89 citations
TL;DR: In this article, the precompactness of minimizing sequences under multiconstraint conditions is studied in a coupled nonlinear Schrödinger system and the orbital stability of the set of all minimizers is obtained.
Abstract: Abstract In this paper, the precompactness of minimizing sequences under multiconstraint conditions are discussed. This minimizing problem is related to a coupled nonlinear Schrödinger system which appears in the field of nonlinear optics. As a consequence of the compactness of each minimizing sequence, the orbital stability of the set of all minimizers is obtained.
72 citations
TL;DR: In this article, the existence and multiplicity results for a class of second-order difference equations are established, and the existence of at least one positive solution without requiring any asymptotic condition at infinity on the nonlinear term is presented.
Abstract: Abstract In this paper, existence and multiplicity results for a class of second-order difference equations are established. In particular, the existence of at least one positive solution without requiring any asymptotic condition at infinity on the nonlinear term is presented and the existence of two positive solutions under a superlinear growth at infinity of the nonlinear term is pointed out. The approach is based on variational methods and, in particular, on a local minimum theorem and its variants. It is worth noticing that, in this paper, some classical results of variational methods are opportunely rewritten by exploiting fully the finite dimensional framework in order to obtain novel results for discrete problems.
54 citations
TL;DR: In this paper, a family of integral operators in the Kantorovich sense acting on functions defined on locally compact topological groups is introduced and studied, and convergence results for the above operators with respect to the pointwise and uniform convergence and in the setting of Orlicz spaces are obtained.
Abstract: We introduce and study a family of integral operators in the Kantorovich sense acting onfunctions defined on locally compact topological groups. We obtain convergence resultsfor the above operators with respect to the pointwise and uniform convergence and in thesetting of Orlicz spaces with respect to the modular convergence. Moreover, we showhow our theory applies to several classes of integral and discrete operators, as sampling,convolution and Mellin type operators in the Kantorovich sense, thus obtaining a simulta-neous approach for discrete and integral operators. Further, we obtain general convergenceresults in particular cases of Orlicz spaces, as L p −spaces, interpolation spaces and expo-nential spaces. Finally we construct some concrete examples of our operators and we showsome graphical representations. 2000 Mathematics Subject Classification . 41A35, 46E30, 47A58, 47B38, 94A12, 94A20 . Keywords . Orlicz spaces, modular convergence, Kantorovich sampling type operators, Kantorovich convolution type operators, KantorovichMellin type operators, estimates, pointwise convergence, uniform convergence.
51 citations
TL;DR: In this article, a class of nonlocal fractional Laplacian problems with three real parameters is studied and the existence of weak solutions for fractional equations via a recent abstract critical point result for differentiable and parametric functionals recently proved by Ricceri.
Abstract: Abstract This article concerns a class of nonlocal fractional Laplacian problems depending of three real parameters. More precisely, by using an appropriate analytical context on fractional Sobolev spaces due to Servadei and Valdinoci (in order to correctly encode the Dirichlet boundary datum in the variational formulation of our problem) we establish the existence of three weak solutions for fractional equations via a recent abstract critical point result for differentiable and parametric functionals recently proved by Ricceri.
43 citations
TL;DR: In this paper, an improvement of Moser-Trudinger inequality involving L p norm for a bounded domain in n dimensions was proposed, where ¯ ( ) = inf w2H 1;n 0 ( );w.0 krwk n kwkn
Abstract: The paper is concerned about an improvement of Moser-Trudinger inequality involving L p norm for a bounded domain in n dimensions. Let ¯ ( ) = inf w2H 1;n 0 ( );w.0 krwk n kwkn
41 citations
TL;DR: In this paper, it was shown that u is a solution of the Dirichet problem iff it is a critical point of the corresponding non-smooth action functional, and applications concerning nontrivial solutions for a class of such Dirichhet problems depending on a parameter are provided.
Abstract: is a smooth open bounded set and f : Ω × ℝ → ℝ is a Caratheodory function. We show that u is a solution of the above problem iff it is a critical point of the corresponding non-smooth action functional. Applications concerning nontrivial solutions for a class of such Dirichet problems depending on a parameter are provided.
35 citations
TL;DR: In this paper, the authors considered a general boundary value problem involving operators of the form div(a(·, ∇u(·)) in which a is a Carathéodory function satisfying a p(·)-growth condition.
Abstract: Abstract We consider a general boundary value problem involving operators of the form div(a(·, ∇u(·)) in which a is a Carathéodory function satisfying a p(·)-growth condition. We are interested on the weak solvability of the problem and, to this end, we start by introducing the Lebesgue and Sobolev spaces with variable exponent, together with their main properties. Then, we state and prove our main existence and uniqueness result, Theorem 3.1. The proof is based on a Weierstrass-type argument. We also consider two antiplane contact problems for nonhomogenous elastic materials of Hencky-type. The contact is frictional and it is modelled with a regularized version of Tresca’s friction law and a power-law friction, respectively. We prove that the problems cast in the abstract setting, then we use Theorem 3.1 to deduce their unique weak solvability.
24 citations
TL;DR: In this article, the authors studied and carried out the theory of symbolic calculus for multi-parameter and multi-linear pseudo-differential operators, including the symbol estimates of the adjoints, asymptotic behavior, kernel estimates and boundedness properties.
Abstract: Abstract Since the work of Hörmander on linear pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, harmonic analysis, theory of several complex variables and other branches of modern analysis (e.g., they are used to construct parametrices and establish the regularity of solutions to PDEs such as the ∂̅ problem, etc.). The work of Coifman and Meyer on multi-linear Fourier multipliers and pseudo-differential operators has stimulated further such applications. In [2], the authors developed a fairly satisfactory theory of symbolic calculus for multi-linear pseudo-differential operators. Motivated by this work [2] and Lp estimates of [34, 35] on multi-parameter and multi-linear Fourier multipliers and of [12] on multi-parameter and multi-linear pseudo-differential operators, we study and carry out the theory of symbolic calculus for multi-parameter and multi-linear pseudo-differential operators. Our results include the symbol estimates of the adjoints, asymptotic behavior, kernel estimates and boundedness properties and extend those in [2] to the multi-parameter and multi-linear setting. The estimates of the distributional kernel of associated multi-parameter and multilinear pseudo-differential operators can be found useful in establishing the boundedness of such multi-parameter and multi-linear pseudo-differential operators.
24 citations
TL;DR: In this paper, the notion of an exponential dichotomy with respect to a sequence of norms is introduced and characterized in terms of the admissibility in pairs of spaces (lp, lq ), with p and q not necessarily equal.
Abstract: For a nonautonomous dynamics defined by a sequence of linear operators, we consider the notion of an exponential dichotomy with respect to a sequence of norms and we characterize it completely in terms of the admissibility in pairs of spaces (lp , lq ), with p and q not necessarily equal. This includes the notion of a nonuniform exponential dichotomy as a very special case. Moreover, we consider a general non- invertible dynamics and a strong nonuniform exponential behavior. The latter is the typical situation from the point of view of smooth ergodic theory.
TL;DR: In this article, a nonlinear parametric elliptic equation (nonlinear eigenvalue problem) driven by a nonhomogeneous differential operator was studied and it was shown that for λ > 0 (the parameter) sufficiently large the problem has at least three nontrivial smooth solutions whereby one is positive, one is negative and the last one has changing sign (nodal).
Abstract: Abstract We study a nonlinear parametric elliptic equation (nonlinear eigenvalue problem) driven by a nonhomogeneous differential operator. Our setting incorporates equations driven by the p-Laplacian, the (p, q)-Laplacian, and the generalized p-mean curvature differential operator. Applying variational methods we show that for λ > 0 (the parameter) sufficiently large the problem has at least three nontrivial smooth solutions whereby one is positive, one is negative and the last one has changing sign (nodal). In the particular case of (p, 2)-equations, using Morse theory, we produce another nodal solution for a total of four nontrivial smooth solutions.
TL;DR: In this paper, the existence and nonexistence of strictly positive solutions for nonlinear elliptic problems of the form $-\Delta u=m\left(x\right) f\leftleft(u\right), $u=0$ on ∆-partial-Omega was studied.
Abstract: Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$ and let $m$ be a possibly discontinuous and unbounded function that changes sign in $\Omega$. Let $f:\left[ 0,\infty\right) \rightarrow\left[ 0,\infty\right) $ be a continuous function such that $k_{1}\xi^{p}\leq f\left(\xi\right) \leq k_{2}\xi^{p}$ for all $\xi\geq0$ and some $k_{1},k_{2}>0$ and $p\in\left(0,1\right) $. We study existence and nonexistence of strictly positive solutions for nonlinear elliptic problems of the form $-\Delta u=m\left(x\right) f\left(u\right) $ in $\Omega$, $u=0$ on $\partial\Omega$.
TL;DR: In this paper, an existence result for a least energy nodal (or sign-changing) solution for the class of p&q problems given by where Ω is a smooth bounded domain in ℝN, N ≥ 3 and 2 ≤ p < N.
Abstract: Abstract In this paper we prove an existence result for a least energy nodal (or sign-changing) solution for the class of p&q problems given by where Ω is a smooth bounded domain in ℝN, N ≥ 3 and 2 ≤ p < N. The function a : ℝ+ → ℝ+ grows like as t → +∞ for some p ≤ q < N, the case q = p meaning that a is bounded away from zero and infinity. The nonlinearity f : ℝ → ℝ grows like |t|m-1 at infinity with Moreover, we find that u has exactly two nodal domains or changes sign exactly once in Ω. The functions f and a satisfy suitable additional growth and monotonicity conditions which allow this result to extend previous ones to a larger class of p&q type problems. The proof is based on a minimization argument and a variant of quantitative deformation lemma.
TL;DR: In this paper, the Hardy-Sobolev exponent p2(α) := N + 2+ 2α N − 2, where N ≥ 3, is shown to be an optimal result.
Abstract: and in the case of N = 2 there is no solution for any p > 1. We further remark that this is an optimal result. We now give a brief background on the case where α is nonzero, for more details see [11]. We define the Hardy-Sobolev exponent p2(α) := N + 2+ 2α N − 2 , where N ≥ 3. It is known, see [7], that if α −2. The case of radial solutions is completely understood, see [7] and [1], where they show there exists a positive classical radial solution of (3) if and only if p ≥ p2(α). This result suggests the following: Conjecture 1. Suppose that α > −2. If 1 < p < p2(α) then (3) has no classical bounded solution. Note that p2(α) ≤ N+2 N−2 exactly when α ≤ 0. Also note that the term |x|α changes monotonicity when α changes sign. For these reasons the methods available to prove this conjecture greatly depend on the sign of α. Until recently the best known results concerning (3), apart from the radial case, were
TL;DR: This work considers the Fučik problem for a family of weights depending on ε with Dirichlet and Neumann boundary conditions and obtains the rate of convergence of the first non-trivial curve of the spectrum.
Abstract: Fil: Salort, Ariel Martin. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matematica; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas; Argentina
TL;DR: In this article, a jumping problem for singular semilinear elliptic equations is considered and the existence of two distinct solutions is proved via a truncation argument and exploiting minimax methods to prove the existence.
Abstract: Abstract We consider a jumping problem for singular semilinear elliptic equations. Existence of two distinct solutions is proved via a truncation argument and exploiting minimax methods to prove the existence of solutions to the truncated problem. In the final step, to pass to the limit, a refined analysis about the behavior of the solutions is required, because of the singular nature of the problem.
TL;DR: In this article, a thorough description of blow-up and global in time oscillatory solutions in the subcritical range in (0.1) is given, based on various analytic methods including nonlinear capacity, variational, category, fibering, and invariant manifold techniques.
Abstract: It is well known from the seminal paper by Fujita [22] for 1 p0 there exists a class of sufficiently "small" global in time solutions. This fundamental result from the 1960-70s (see also [39] for related contributions), was a cornerstone of further active blow-up research. Nowadays, similar Fujita-type critical exponents p0, as important characteristics of stability, unstability, and blow-up of solutions, have been calculated for various nonlinear PDEs. The above blow-up conclusion does not include solutions of changing sign, so some of them may remain global even for p ≤ p0. Our goal is a thorough description of blow-up and global in time oscillatory solutions in the subcritical range in (0.1) on the basis of various analytic methods including nonlinear capacity, variational, category, fibering, and invariant manifold techniques. Two countable sets of global solutions of changing sign are shown to exist. Most of them are not radially symmetric in any dimension N ≥ 2 (previously, only radial such solutions in ℝN or in the unit ball B1 ⊆ ℝN were mostly studied). A countable sequence of critical exponents, at which the whole set of global solutions changes its structure, is detected: pl = 1 + N 2 +l, 1 = 0, 1, 2, See [47, 48] for earlier interesting contributions on sign changing solutions.
TL;DR: In this paper, the asymptotic behavior of positive first eigenfunctions for a class of inhomogeneous eigenvalue problems is studied in the setting of Orlicz-Sobolev spaces.
Abstract: Abstract The asymptotic behavior of the sequence {un} of positive first eigenfunctions for a class of inhomogeneous eigenvalue problems is studied in the setting of Orlicz-Sobolev spaces. After possibly extracting a subsequence, we prove that un → u∞ uniformly in Ω as n→∞, where u∞ is a nontrivial viscosity solution of a nonlinear PDE involving the ∞-Laplacian.
TL;DR: In this paper, the existence of nonzero solution for the quasilinear Schrödinger equation −Δu + V(x)u − Δ(u2)u = g(x, u), x ∈ ℝN, u ∈ H1(ℝ N), where V is a positive potential and the nonlinearity g(s, s) behaves like K0(x), s at the origin and like K∞(x)|s|p, 1 ≤ p ≤ 3, at infinity.
Abstract: Abstract We deal with the existence of nonzero solution for the quasilinear Schrödinger equation −Δu + V(x)u − Δ(u2)u = g(x, u), x ∈ ℝN, u ∈ H1(ℝN), where V is a positive potential and the nonlinearity g(x, s) behaves like K0(x)s at the origin and like K∞(x)|s|p, 1 ≤ p ≤ 3, at infinity. In the proofs we apply minimization methods.
TL;DR: In this paper, an existence result for a fourth order elliptic equation where the associated functional does not satisfy the Palais-Smale condition is proved. But the existence result is not applicable to the case where the functional is a truncation.
Abstract: Abstract We prove an existence result for a fourth order elliptic equation where the associated functional does not satisfy the Palais-Smale condition. We use some truncations argument and L∞-norm estimates.
TL;DR: In this article, it was shown that any positive solution of (E) ∂tu − Δu + uq = 0 in ℝN × (0, ∞) admits an initial trace which is a nonnegative Borel measure, outer regular with respect to the fine topology associated to the Bessel capacity, and absolutely continuous.
Abstract: Abstract Let . We prove that any positive solution of (E) ∂tu − Δu + uq = 0 in ℝN × (0, ∞) admits an initial trace which is a nonnegative Borel measure, outer regular with respect to the fine topology associated to the Bessel capacity in ℝN (qʹ = q/(q − 1)) and absolutely continuous with respect to this capacity. If ν is a nonnegative Borel measure in ℝN with the above properties we construct a positive solution u of (E) with initial trace ν and we prove that this solution is the unique σ-moderate solution of (E) with such an initial trace. Finally we prove that every positive solution of (E) is σ-moderate.
TL;DR: In this article, a subspace of the space of Legendrian loops is studied and it is shown that the injection of this subspace into the full loop space is an S-equivariant homotopy equivalence.
Abstract: In this paper we study a subspace of the space of Legendrian loops and we show that the injection of this space into the full loop space is an S-equivariant homotopy equivalence. This space can be also seen as the space of zero Maslov index Legendrian loops and it shows up as a suitable space of variations in contact form geometry.
TL;DR: In this paper, it was shown that the validity of the higher-order chain rule for superposition operators in the supercritical case can be guaranteed in the case of arbitrary Sobolev spaces.
Abstract: Abstract This paper is a continuation of the work begun in [6] on superposition operators, (Ngu) (x) = g(u(x)), between two arbitrary Sobolev spaces. Sufficient conditions which ensure the well-definedness, the continuity and the validity of the higher-order chain rule for such operators are given in the supercritical case (see Remark 1.1). As a consequence of these properties, it is proved that Ng (Wm,p (Ω) ∩ W0k,p (Ω)) ⊂ W0l,q (Ω).
TL;DR: In this article, the authors considered the multi-valued parabolic variational inequalities involving quasilinear parabolic operators, and multivalued integral terms over the underlying parabolic cylinder as well as over parts of the lateral parabolic boundary.
Abstract: Abstract In this paper we study multi-valued parabolic variational inequalities involving quasilinear parabolic operators, and multi-valued integral terms over the underlying parabolic cylinder as well as over parts of the lateral parabolic boundary, where the multi-valued functions involved are assumed to be upper semicontinuous only. Note, since lower semicontinuous multi-valued functions allow always for a Carathéodory selection, this case can be considered as the trivial case, and therefore will be omitted. Our main goal is threefold: First, we provide an analytical frame work and an existence theory for the problems under consideration. Unlike in recent publications on multi-valued parabolic variational inequalities, the closed convex set K representing the constraints is not required to possess a nonempty interior. Second, we prove enclosure and comparison results based on a recently developed sub-supersolution method due to the authors. Third, we consider classes of relevant generalized parabolic variational-hemivariational inequalities that will be shown to be special cases of the multi-valued parabolic variational inequalities under consideration.
TL;DR: In this paper, a nonlinear Schrödinger equation with magnetic fields involving subcritical growth was studied and it was shown that the equation has multi-peak solutions under some suitable conditions.
Abstract: Abstract In this paper, we study a nonlinear Schrödinger equation with magnetic fields involving subcritical growth. Applying the finite reduction method, we prove that the equation has multi-peak solutions under some suitable conditions which are given in section 1.
TL;DR: In this article, the existence of positive bound states for the nonlinear Schrödinger equation −ε2Δu + V(x)u = up, u > 0, u ∈ H1(ℝN), where N ≥ 3, 1 < p < (N + 2)/(N − 2), and V is a nonnegative potential with compact support.
Abstract: Abstract We consider the existence of positive bound states for the nonlinear Schrödinger equation −ε2Δu + V(x)u = up, u > 0, u ∈ H1(ℝN), where N ≥ 3, 1 < p < (N + 2)/(N − 2), and V is a nonnegative potential with compact support. For arbitrary positive integer k ∈ Z+, we construct higher energy solutions with exactly k peaks which interact with each other and cluster around a local maximum point of V when ε is sufficiently small. The main part of the solutions decays exponentially but the perturbation part decays algebraically at infinity.
TL;DR: In this article, it was shown that if Σ ⊂ R 2n is a symmetric compact convex hypersurface, then there are infinitely many closed characteristics on it and at least 2[n/2] nonhyperbolic closed characteristics.
Abstract: Abstract In this paper, let Σ ⊂ R2n be a symmetric compact convex hypersurface. We prove that either there are infinitely many closed characteristics or there exist at least 2[n/2] nonhyperbolic closed characteristics on Σ. Due to the example of weakly non-resonant ellipsoids, this estimate is sharp when n is even. Moreover, we prove that if Σ carries exactly n closed characteristics, then at least n − 2 of them possess irrational mean indices.
TL;DR: In this article, the authors considered the semilinear elliptic problem with Dirichlet boundary condition and gave upper and lower bounds for the extremal parameter λ∗ and extremal solution u∗.
Abstract: Abstract We consider the semilinear elliptic problem −Δu = f (x, u), posed in a smooth bounded domain Ω of ℝN with Dirichiel data u|∂Ω = 0, where f : Ω × [0, αf ) → ℝ+ (0 < αf ≤ +∞) is a function of appropriate regularity which blows up at αf. We give pointwise lower bounds for the supersolutions under some appropriate conditions on f , and apply them to eigenvalue problem −Δu = λ f (x, u), by giving upper and lower bounds for the extremal parameter λ∗ and the extremal solution u∗. To demonstrate the sharpness of our results, we consider the eigenvalue problem −Δu = λ f (up) (p ≥ 1) with Dirichlet boundary condition, and show that for every increasing, convex and superlinear C2 function f: ℝ+→ℝ+ with , where ψΩ is the maximum of the torsion function of Ω. Also, we consider the eigenvalue problem −Δu = λρ(x) f (u), where f is either a regular singularity such as f (u) = eu, or a singular one such as and give explicit estimates on λ∗ and u∗, that improve and extend several results in the literature, by Payne[17], Sperb [21], Brezis-Vasquez [3], Guo-Pan-Ward [11], Ghoussoub-Guo [10], Cowan-Ghoussoub [6], and others.
TL;DR: In this article, the authors consider a class of nonlinear stationary Schrödinger-type equations and obtain conditions for the existence of infinitely many solutions with high (resp., small) energies.
Abstract: Abstract We consider a class of nonlinear stationary Schrödinger-type equations and we are concerned with sufficient properties that guarantee the existence of multiple solutions in a suitable Sobolev space with variable exponents. We first establish that in the case of small perturbations, the problem admits at least two weak solutions. Next, in the case of convexconcave nonlinearities, we obtain conditions for the existence of infinitely many solutions with high (resp., small) energies. The arguments combine variational techniques with a careful analysis of the energy levels.