Monotonicity formula and \varepsilon -regularity of stable solutions to supercritical problems and applications to finite Morse index solutions

We prove a monotonicity formula and an \(\varepsilon \) -regularity theorem for stable solutions to a class of weighted supercritical semilinear elliptic equations. We then use them to study the behavior of finite Morse index solutions and obtain some sharp results, which improve those of Dancer et al. (J Differ Equ 250:3281–3310, 2011) and Du and Guo (Adv Differ Eqns 2013) and completely answer the questions left open there.     

Rigidity of manifolds with Bakry–émery Ricci curvature bounded below

Let M be a complete Riemannian manifold with Riemannian volume vol _g and f be a smooth function on M. A sharp upper bound estimate on the first eigenvalue of symmetric diffusion operator ${\Delta_f = \Delta- \nabla f \cdot \nabla}$ was given by Wu (J Math Anal Appl 361:10?C18, 2010) and Wang (Ann Glob Anal Geom 37:393?C402, 2010) under a condition that finite dimensional Bakry?Cémery Ricci curvature is bounded below, independently. They propounded an open problem is whether there is some rigidity on the estimate. In this note, we will solve this problem to obtain a splitting type theorem, which generalizes Li?CWang??s result in Wang (J Differ Geom 58:501?C534, 2001, J Differ Geom 62:143?C162, 2002). For the case that infinite dimensional Bakry?CEmery Ricci curvature of M is bounded below, we do not expect any upper bound estimate on the first eigenvalue of ?? _f without any additional assumption (see the example in Sect. 2). In this case, we will give a sharp upper bound estimate on the first eigenvalue of ?? _f under the additional assuption that ${|\nabla f|}$ is bounded. We also obtain the rigidity result on this estimate, as another Li?CWang type splitting theorem.     

Elliptic Equations and Systems with Subcritical and Critical Exponential Growth Without the Ambrosetti–Rabinowitz Condition

In this paper, we prove the existence of nontrivial nonnegative solutions to a class of elliptic equations and systems which do not satisfy the Ambrosetti–Rabinowitz (AR) condition where the nonlinear terms are superlinear at 0 and of subcritical or critical exponential growth at ∞. The known results without the AR condition in the literature only involve nonlinear terms of polynomial growth. We will use suitable versions of the Mountain Pass Theorem and Linking Theorem introduced by Cerami (Istit. Lombardo Accad. Sci. Lett. Rend. A, 112(2):332–336, 1978 Ann. Mat. Pura Appl., 124:161–179, 1980). The Moser–Trudinger inequality plays an important role in establishing our results. Our theorems extend the results of de Figueiredo, Miyagaki, and Ruf (Calc. Var. Partial Differ. Equ., 3(2):139–153, 1995) and of de Figueiredo, do Ó, and Ruf (Indiana Univ. Math. J., 53(4):1037–1054, 2004) to the case where the nonlinear term does not satisfy the AR condition. Examples of such nonlinear terms are given in Appendix A. Thus, we have established the existence of nontrivial nonnegative solutions for a wider class of nonlinear terms.     

Changing-sign bubble solutions for an anisotropic sinh-Poisson equation

We consider the following anisotropic sinh-Poisson equation $${\rm div} (a(x) \nabla u)+ 2\varepsilon^2 a(x) {\rm sinh}\,u=0\ \ {\rm in}\ \Omega, \quad u=0 \ \ {\rm on}\ \partial \Omega,$$ where ${\Omega \subset \mathbb{R}^2}$ is a bounded smooth domain and a(x) is a positive smooth function. We investigate the effect of anisotropic coefficient ${a(x)}$ on the existence of bubbling solutions. We show that there exists a family of solutions u _?? concentrating positively and negatively at ${\bar{x}}$ , a given local critical point of a(x), for ?? sufficiently small, for which with the property $$2\varepsilon^2a(x){\rm sinh} u_\varepsilon \rightharpoonup 8\pi\sum\limits_{j=1}^{m}b_j\delta_{\bar{x}},$$ where ${b_j=\pm 1}$ . This result shows a striking difference with the isotropic case (a(x) ?? Constant) in Bartolucci and Pistoia (IMA J Appl Math 72(6):706?C729, 2007), Jost et?al. (Calc Var Partial Differ Equ 31:263?C276, 2008) and Esposito and Wei (Calc Var Partial Differ Equ 34:341?C375, 2009).     

Global dynamics above the ground state energy for the cubic NLS equation in 3D

We extend the result in Nakanishi and Schlag (J Differ Equ 250:2299–2333, 2011) on the nonlinear Klein–Gordon equation to the nonlinear Schr?dinger equation with the focusing cubic nonlinearity in three dimensions, for radial data of energy at most slightly above that of the ground state. We prove that the initial data set splits into nine nonempty, pairwise disjoint regions which are characterized by the distinct behaviors of the solution for large time: blow-up, scattering to 0, or scattering to the family of ground states generated by the phase and scaling freedom. Solutions of this latter type form a smooth center-stable manifold, which contains the ground states and separates the phase space locally into two connected regions exhibiting blow-up and scattering to 0, respectively. The special solutions found by Duyckaerts and Roudenko (Rev Mater Iberoam 26(1):1–56, 2010), following the seminal work on threshold solutions by Duyckaerts and Merle (Funct Anal 18(6):1787–1840, 2009), appear here as the unique one-dimensional unstable/stable manifolds emanating from the ground states. In analogy with Nakanishi and Schlag (J Differ Equ 250:2299–2333, 2011), the proof combines the hyperbolic dynamics near the ground states with the variational structure away from them. The main technical ingredient in the proof is a “one-pass” theorem which precludes “almost homoclinic orbits”, i.e., those solutions starting in, then moving away from, and finally returning to, a small neighborhood of the ground states. The main new difficulty compared with the Klein–Gordon case is the lack of finite propagation speed. We need the radial Sobolev inequality for the error estimate in the virial argument. Another major difference between Nakanishi and Schlag (J Differ Equ 250:2299–2333, 2011) and this paper is the need to control two modulation parameters.     

Uniqueness for the coagulation-fragmentation equation with strong fragmentation

The existence of at least one mass-conserving solution for continuous coagulation-fragmentation equation has been established by Escobedo et?al. (J Differ Equ 195:143?C174, 2003) for a large class of coagulation kernels under strong binary fragmentation. In this work, uniqueness of mass-conserving solutions is demonstrated with some additional restrictions on the fragmentation kernels.     

Decay Rates to Equilibrium for Nonlinear Plate Equations with Degenerate, Geometrically-Constrained Damping

We analyze the convergence to equilibrium of solutions to the nonlinear Berger plate evolution equation in the presence of localized interior damping (also referred to as geometrically constrained damping). Utilizing the results in (Geredeli et al. in J. Differ. Equ. 254:1193–1229, 2013), we have that any trajectory converges to the set of stationary points $\mathcal{N}$ . Employing standard assumptions from the theory of nonlinear unstable dynamics on the set $\mathcal{N}$ , we obtain the rate of convergence to an equilibrium. The critical issue in the proof of convergence to equilibria is a unique continuation property (which we prove for the Berger evolution) that provides a gradient structure for the dynamics. We also consider the more involved von Karman evolution, and show that the same results hold assuming a unique continuation property for solutions, which is presently a challenging open problem.     

Termination time of characteristics of Hamilton-Jacobi equations

This paper concerns with Hamilton-Jacobi equations of n space variables, where the Hamiltonians are convex and the initial data are admitted to be unbounded. First, we study the characteristics for the general case of initial data being Lipschitz by using the Hopf formula. Sufficient and necessary conditions are established for guaranteeing a characteristic to start from y ₀ at t = 0 with direction DH(P ₀) and for a characteristic never terminating on a singularity of the solution. Next, in the case of initial data being C ², we prove that the set of singularities consists of at most countable path-connected components, which is an extension of (Zhao et al. in J Hyperbolic Differ Equ 5(3):663–680, 2008) and (Li in Sci Sinica 22(9):979–990, 1979).     

Non-Decaying Solutions to the Navier Stokes Equations in Exterior Domains

We establish a new theorem of existence (and uniqueness) of solutions to the Navier-Stokes initial boundary value problem in exterior domains. No requirement is made on the convergence at infinity of the kinetic field and of the pressure field. These solutions are called non-decaying solutions. The first results on this topic dates back about 40 years ago see the references (Galdi and Rionero in Ann. Mat. Pures Appl. 108:361–366, 1976, Arch. Ration. Mech. Anal. 62:295–301, 1976, Arch. Ration. Mech. Anal. 69:37–52, 1979, Pac. J. Math. 104:77–83, 1980; Knightly in SIAM J. Math. Anal. 3:506–511, 1972). In the articles Galdi and Rionero (Ann. Mat. Pures Appl. 108:361–366, 1976, Arch. Ration. Mech. Anal. 62:295–301, 1976, Arch. Ration. Mech. Anal. 69:37–52, 1979, Pac. J. Math. 104:77–83, 1980) it was introduced the so called weight function method to study the uniqueness of solutions. More recently, the problem has been considered again by several authors (see Galdi et al. in J. Math. Fluid Mech. 14:633–652, 2012, Quad. Mat. 4:27–68, 1999, Nonlinear Anal. 47:4151–4156, 2001; Kato in Arch. Ration. Mech. Anal. 169:159–175, 2003; Kukavica and Vicol in J. Dyn. Differ. Equ. 20:719–732, 2008; Maremonti in Mat. Ves. 61:81–91, 2009, Appl. Anal. 90:125–139, 2011).     

Error estimates for series solutions to a class of nonlinear integral equations of mixed type

In this paper, we prove that the accelerated Adomian polynomials formula suggested by Adomian (Nonlinear Stochastic Systems: Theory and Applications to Physics, Kluwer, Dordrecht, 1989) and the accelerated formula suggested by El-Kalla (Int. J. Differ. Equs. Appl. 10(2):225?C234, 2005; Appl. Math. E-Notes 7:214?C221, 2007) are identically the same. The Kalla-iterates exhibit the same faster convergence exhibited by Adomian??s accelerated iterates with the additional advantage of absence of any derivative terms in the recursion, thereby allowing for ease of computation. Moreover, the formula of El-Kalla is used directly to prove the convergence of the series solution to a class of nonlinear two dimensional integral equations. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of the Adomian series solution.     

Ground states for quasilinear Schrödinger equations with critical growth

For a class of quasilinear Schrödinger equations with critical exponent we establish the existence of both one-sign and nodal ground states of soliton type solutions by the Nehari method. The method is to analyze the behavior of solutions for subcritical problems from our earlier work (Liu et al. Commun Partial Differ Equ 29:879–901, 2004) and to pass limit as the exponent approaches to the critical exponent.     

Entropy formulation of degenerate parabolic equation with zero-flux boundary condition

We consider the general degenerate hyperbolic-parabolic equation: $$u_t + {\rm div} f(u) - \Delta \phi(u) = 0\; {\rm in} Q = (0, T) \times \Omega, \quad T > 0, \quad \Omega \subset \mathbb{R}^N;$$ with initial condition and the zero flux boundary condition. Here ${\phi}$ is a continuous non-decreasing function. Following Bürger et al. (J Math Anal Appl 326:108–120, 2007), we assume that f is compactly supported (this is the case in several applications), and we define an appropriate notion of entropy solution. Using vanishing viscosity approximation, we prove existence of entropy solution for any space dimension N ≥ 1 under a partial genuine nonlinearity assumption on f. Uniqueness is shown for the case N = 1, using the idea of Andreianov and Bouhsiss (J Evol Equ 4:273–295, 2004), nonlinear semigroup theory and a specific regularity result for one dimension.     

Continuity of the solution mapping to parametric generalized vector equilibrium problems

In this paper, two kinds of parametric generalized vector equilibrium problems in normed spaces are studied. The sufficient conditions for the continuity of the solution mappings to the two kinds of parametric generalized vector equilibrium problems are established under suitable conditions. The results presented in this paper extend and improve some main results in Chen and Gong (Pac J Optim 3:511–520, 2010), Chen and Li (Pac J Optim 6:141–152, 2010), Chen et al. (J Glob Optim 45:309–318, 2009), Cheng and Zhu (J Glob Optim 32:543–550, 2005), Gong (J Optim Theory Appl 139:35–46, 2008), Li and Fang (J Optim Theory Appl 147:507–515, 2010), Li et al. (Bull Aust Math Soc 81:85–95, 2010) and Peng et al. (J Optim Theory Appl 152(1):256–264, 2011).     

Geometric rigidity for analytic estimates of Müller–?verák

In the paper Müller–?verák (J Differ Geom 42(2):229–258, 1995) conformally immersed surfaces with finite total curvature were studied. In particular it was shown that surfaces with total curvature ${\int_{\Sigma} |A|^2 < 8 \pi}$ in dimension three were embedded and conformal to the plane with one end. Here, using techniques from Kuwert–Li (W ^2,2-conformal immersions of a closed Riemann surface into R ⁿ . arXiv:1007.3967v2 [math.DG], 2010), we will show that if the total curvature ${ \int_{\Sigma}|A|^2\leq8\pi}$ , then we are either embedded and conformal to the plane, isometric to a catenoid or isometric to Enneper’s minimal surface. In fact the technique of our proof shows that if we are conformal to the plane, then if n?≥ 3 and ${ \int_{\Sigma} | A|^{2}\leq 16 \pi }$ then Σ is embedded or Σ is the image of a generalized catenoid inverted at a point on the catenoid. In order to prove these theorems, we prove a Gauss–Bonnet theorem for surfaces with complete ends and isolated finite area singularities which extends a theorem of Jorge-Meeks (Topology 22(2):203–221, 1983). Using this theorem, we then prove an inversion formula for the Willmore energy.     

Generators for cubic surfaces with two skew lines over finite fields

Let S be a smooth cubic surface defined over a field K. As observed by Segre [5] and Manin [3, 4], there is a secant and tangent process on S that generates new K-rational points from old ones. It is natural to ask for the size of a minimal generating set for S(K). In a recent paper, for fields K with at least 13 elements, Siksek [7] showed that if S contains a skew pair of K-lines, then S(K) can be generated from one point. In this paper we prove the corresponding version of this result for fields K having at least 4 elements, and slightly milder results for # K = 2 or 3.     

An existence result for a class of quasilinear elliptic eigenvalue problems in unbounded domains

We consider a nonlinear eigenvalue problem under Robin boundary conditions in a domain with (possibly noncompact) smooth boundary. The problem involves a weighted p–Laplacian operator and subcritical nonlinearities satisfying Ambrosetti–Rabinowitz type conditions. Using Morse theory and a cohomological local splitting as in Degiovanni et al. (Commun Contemp Math 12:475–486, 2010), we prove the existence of a nontrivial weak solution for all (real) values of the eigenvalue parameter. Our result is new even in the semilinear case p = 2 and complements some recent results obtained in Autuori et al. (Adv Anal Equ 18:1–48, 2013).     

Directional Lipschitzness of Minimal Time Functions in Hausdorff Topological Vector Spaces

In a general Hausdorff topological vector space E, we associate to a given nonempty closed set S???E and a bounded closed set Ω???E, the minimal time function T _S,Ω defined by $T_{S,\Omega}(x):= \inf \{ t> 0: S\cap (x+t\Omega)\not = \emptyset\}$ . The study of this function has been the subject of various recent works (see Bounkhel (2012, submitted, 2013, accepted); Colombo and Wolenski (J Global Optim 28:269–282, 2004, J Convex Anal 11:335–361, 2004); He and Ng (J Math Anal Appl 321:896–910, 2006); Jiang and He (J Math Anal Appl 358:410–418, 2009); Mordukhovich and Nam (J Global Optim 46(4):615–633, 2010) and the references therein). The main objective of this work is in this vein. We characterize, for a given Ω, the class of all closed sets S in E for which T _S,Ω is directionally Lipschitz in the sense of Rockafellar (Proc Lond Math Soc 39:331–355, 1979). Those sets S are called Ω-epi-Lipschitz. This class of sets covers three important classes of sets: epi-Lipschitz sets introduced in Rockafellar (Proc Lond Math Soc 39:331–355, 1979), compactly epi-Lipschitz sets introduced in Borwein and Strojwas (Part I: Theory, Canad J Math No. 2:431–452, 1986), and K-directional Lipschitz sets introduced recently in Correa et al. (SIAM J Optim 20(4):1766–1785, 2010). Various characterizations of this class have been established. In particular, we characterize the Ω-epi-Lipschitz sets by the nonemptiness of a new tangent cone, called Ω-hypertangent cone. As for epi-Lipschitz sets in Rockafellar (Canad J Math 39:257–280, 1980) we characterize the new class of Ω-epi-Lipschitz sets with the help of other cones. The spacial case of closed convex sets is also studied. Our main results extend various existing results proved in Borwein et al. (J Convex Anal 7:375–393, 2000), Correa et al. (SIAM J Optim 20(4):1766–1785, 2010) from Banach spaces and normed spaces to Hausdorff topological vector spaces.     

Exponential or polynomial decay of solutions to a viscoelastic equation with nonlinear localized damping

We consider the nonlinear viscoelastic equation $$u_{tt}-\Delta u+\int_{0}^{t}g(t-\tau)\Delta u(\tau)\,d\tau +a(x)|u_{t}|^{m}u_{t}+b|u|^{\gamma }u=0$$ in a bounded domain and establish exponential or polynomial decay result which depend on the rate of the decay of the relaxation function g. This result improves an earlier one given by Berrimi and Messaoudi (Electron. J. Differ. Equ. (88):1–10, 2004).