nodal type bound states of schrodinger equations via invariant set and minimax methods for nonlinear schrodinger equations in the entire space we present new results on invariant sets of the gradient flows of the corresponding variational functionals the structure of the invariant sets will be built into minimax procedures to construct nodal type bound state solutions of nonlinear schrodinger type equations 2004 elsevier inc all rights reserved

elliptic problems with nonlinearities indefinite in sign abstract let r n n 3 be a bounded open set with smooth boundary and r we study the dirichlet problem formula with 1 q p and h k l nonnegative functions we prove existence nonexistence and multiplicity results depending on and according to the integrability properties of the ratio k p 1 h q 1

existence and multiplicity of solutions for a nonlocal problem with critical sobolev exponent abstract in this work we are interested in considering the following nonlocal problem a b u 2 d x u u 2 2 u u q 2 u x u 0 x where r n n 4 is a smooth bounded domain a 0 b 0 1 q 2 0 and 2 2 n n 2 is the critical sobolev exponent by using the variational method and the critical point theorem some existence and multiplicity results are obtained

nontrivial solutions of kirchhoff type problems via the yang index abstract we obtain nontrivial solutions of a class of nonlocal quasilinear elliptic boundary value problems using the yang index and critical groups

existence of a positive solution to kirchhoff type problems without compactness conditions abstract the existence of a positive solution to a kirchhoff type problem on r n is proved by using variational methods and the new result does not require usual compactness conditions a cut off functional is utilized to obtain the bounded palais smale sequences

computer aided experiments on the hopf bifurcation of the fitzhugh nagumo nerve model abstract a space clamped fitzhugh nagumo fhn nerve model subjected to a stimulating electrical current i is investigated by a combination of perturbation and numerical methods our goal is to trace out the path of periodic solutions initiated by a hopf bifurcation especially when the fhn model presents a slow recovery mechanism denoted here by the small control parameter it is shown in the computed period diagram that in addition to the two hopf bifurcation points i and i there are another two critical points i m and i n satisfying i i m i n i and forming the points of maximum period for fhn models with single steady state while satisfying i m i i i n and forming the turning points for models with multiple steady states if is sufficiently small the results are accompanied with cusp formation at i m and i n this fact indicates a discontinuous transition between oscillations of different characters further evidences are given by other bifurcation diagrams for fhn models with multiple steady states a similar hysteresis phenomenon is also observed for periodic solutions

existence and bifurcation behavior of positive solutions for a class of kirchhoff type problems abstract we study the existence of positive solutions for the nonlinear kirchhoff type equation a b u 2 d x u u u 3 with dirichlet boundary condition where r are two real parameters we provide a description of a two dimensional set in the plane which correspond to the existence of positive solutions for the above kirchhoff type equation

multiple positive solutions to a class of kirchhoff equation on r3 with indefinite nonlinearity abstract we study the existence and multiplicity of positive solutions to a class of kirchhoff equation 1 b r 3 u x 2 d x u x u x v x u x p 2 u x k x u x u x u h 1 r 3 when v x is sign changing in r 3 we prove that there is 0 such that the kirchhoff equation possesses at least one positive solution for 0 1 and at least two positive solutions for 1 1 here 1 is the first eigenvalue of u u k x u u h 1 r 3 a positive eigenfunction corresponding to 1 is denoted by e 1 an interesting phenomena is that the term b r 3 u 2 d x u can be used to compete with the indefinite nonlinearity and we do not assume the condition r 3 v x e 1 p d x 0 which has been shown to be a necessary condition to the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearity alama and tarantello 1993

optimizing the encounter rate in biological interactions ballistic versus levy versus brownian strategies bartumeus et al phys rev lett 88 097901 2002 compared the efficiency of a levy random walk predator strategy with a brownian random walk strategy in a periodic one dimensional domain with nonrevisitable moving targets their findings from numerical simulations conclude that a levy search strategy is the best option in some but not all cases for a random search process using the same methodology we show that the simplest random search strategy of all ballistic motion in a random direction outperforms a levy strategy in almost every case we further show that in the small set of cases where the ballistic strategy is not optimal the periodic model does not capture the more realistic nonperiodic case in the nonperiodic case the ballistic strategy again outperforms the levy strategy

invariant sets of descending flow in critical point theory with applications to nonlinear differential equations abstract properties of invariant sets of descending flow defined by a pseudogradient vector field of a functional in a banach space are studied in this way several critical points can be found by constructing different invariant sets on which the functional is bounded below under suitable conditions the existence of at least four critical points of a functional is proved each critical point being in a certain invariant set the theoretical results are applied to nonlinear elliptic boundary value problems and nonlinear systems of ordinary differential equations in variant cases at least four solutions are obtained for these equations

existence and asymptotic behavior of positive ground state solutions for coupled nonlinear fractional kirchhoff type systems abstract this paper is concerned with the following linearly coupled fractional kirchhoff type system a b r 3 2 u 2 d x u u f u v in r 3 c d r 3 2 v 2 d x v v g v u in r 3 u v h r 3 where a c 0 b d 0 are constants 3 4 1 and 0 is a coupling parameter under the general berestycki lions conditions on the nonlinear terms f and g we prove the existence of positive vector ground state solutions of poho aev type for the above system via variational methods moreover the asymptotic behavior of these solutions as 0 is explored as well recent results from the literature are generally improved and extended

on ground states for the kirchhoff type problem with a general critical nonlinearity abstract in this paper we consider the following kirchhoff type problem a b r 3 u 2 u f u in r 3 u h 1 r 3 u 0 in r 3 where a b 0 are constants and f has a critical growth the aim of this paper is to study the existence of ground state solutions for kirchhoff type equations with a general nonlinearity in the critical growth without the assumption of the monotonicity of the function t f t t 3 moreover we will show that the mountain pass value gives the least energy level and also obtain a mountain pass solution

existence of solutions for a higher order kirchhoff type problem with exponential critical growth abstract a higher order kirchhoff type equation r 2 m m u 2 0 m 1 a x u 2 d x m u 0 m 1 1 a x u f x u x h x in r 2 m is considered in this paper we assume that the nonlinearity of the equation has exponential critical growth and prove that for a positive which is small enough there are two distinct nontrivial solutions to the equation when 0 we also prove that the equation has a nontrivial mountain pass type solution

infinitely many solutions for a fractional kirchhoff type problem via fountain theorem 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 s u x u f x u in u 0 in r n where is a smooth bounded domain of r n s is the fractional laplacian operator with 0 s 1 and 2 s n is a real parameter m is a continuous and positive function and f is a caratheodory function satisfying the ambrosetti rabinowitz type condition

robustness of optimal random searches in fragmented environments the random search problem is a challenging and interdisciplinary topic of research in statistical physics realistic searches usually take place in nonuniform heterogeneous distributions of targets e g patchy environments and fragmented habitats in ecological systems here we present a comprehensive numerical study of search efficiency in arbitrarily fragmented landscapes with unlimited visits to targets that can only be found within patches we assume a random walker selecting uniformly distributed turning angles and step lengths from an inverse power law tailed distribution with exponent our main finding is that for a large class of fragmented environments the optimal strategy corresponds approximately to the same value opt 2 moreover this exponent is indistinguishable from the well known exact optimal value opt 2 for the low density limit of homogeneously distributed revisitable targets surprisingly the best search strategies do not depend or depend only weakly on the specific details of the fragmentation finally we discuss the mechanisms behind this observed robustness and comment on the relevance of our results to both the random search theory in general as well as specifically to the foraging problem in the biological context

adaptive levy walks can outperform composite brownian walks in non destructive random searching scenarios abstract recently it has been found that composite brownian walk searches are more efficient than any levy walk when searching is non destructive and when the levy walks are not responsive to conditions found in the search here a new class of adaptive levy walk searches is presented that encompasses composite brownian walks as a special case in these new models bouts of levy walk searching alternate with bouts of more intensive brownian walk searching switching from extensive to intensive searching is prompted by the detection of a target and here switching back to extensive searching arises if a target is not located after travelling a distance equal to the giving up distance it is found that adaptive levy walks outperform composite brownian walks when searching for sparsely distributed resources consequently there is stronger selection pressures for levy processes when resources are sparsely distributed within unpredictable environments the findings reconcile levy walk search theory with the ubiquity of two modes of searching by predators and with their switching search mode immediately after finding a prey

properties of levy flights on an interval with absorbing boundaries we consider a l evy yer of orderthat starts from a point x0 on an interval 0 l with absorbing boundaries we 7nd a closed form expression for an arbitrary average quantity char acterizing the trajectory of the yer such as mean 7rst passage time average total path length probability to be absorbed by one of the boundaries using fractional di erential equations with a riesz kernel we 7nd exact analytical expressions for these quantities in the continuous limit we 7nd numerically the eigenfunctions and the eigenvalues of these equations we study how the results of monte carlo simulations of the l evy ights with di erent ight length distribu tions converge to the continuous approximations we show that if x0 is placed in the vicinity of absorbing boundaries the average total path length has a minimum near 1 corresponding to the cauchy distribution we discuss the relevance of these results to the problem of biological foraging and transmission of light through cloudy atmosphere c 2001 elsevier science b v all rights reserved

positive and negative solutions of a class of nonlocal problems abstract the existence of nontrivial solutions of kirchhoff type equations is an important nonlocal quasilinear problem in this paper still by using the invariant sets of descent flow we obtain positive and negative solutions of a class of nonlocal quasilinear elliptic boundary value problems as follows a b u 2 u f x u in u 0 on

positive solutions for a quasilinear elliptic equation of kirchhoff type this paper is concerned with the existence of positive solutions to the class of nonlocal boundary value problems of the type m w u 2dx du f x u email protected u 0 email protected email protected where w is a smooth bounded domain of n m is a positive function and f has subcritical growth

semiclassical states for nonlinear schrodinger equations with sign changing potentials we establish the existence and multiplicity of semiclassical bound states of the following nonlinear schrodinger equation e2 u v x u g x u for x rn u x 0as x where v changes sign and g is superlinear with critical or supercritical growth as u

infinitely many radial solutions for kirchhoff type problems in rn abstract in this paper we concern with a class of kirchhoff type problems in r n by using the fountain theorem we obtain three existence results of infinitely many radial solutions for the problem

multiple positive solutions for kirchhoff type of problems with singularity and critical exponents abstract in this paper we study multiplicity of positive solutions for a class of kirchhoff type of equations with the nonlinearity containing both singularity and critical exponents we obtain two positive solutions via the variational and perturbation methods

existence and concentrating behavior of solutions for kirchhoff type equations with steep potential well abstract in this paper we consider the following kirchhoff type equations 0 1 a b r 3 u 2 u v x u q x f u in r 3 u h 1 r 3 where a b 0 v c r 3 r is a potential well q x is a positive bounded function f s is either asymptotically linear or asymptotically 3 linear in s at infinity under some other suitable conditions on v q and f the existence nonexistence and concentrating behavior of solutions to problem 0 1 are obtained by using variational methods we mainly extend the results in j sun and t wu 2014 26 which dealt with kirchhoff type equations with positive potential well to kirchhoff type equations with sign changing potential well

hopf bifurcation in a reaction diffusion system governed by the fitz hugh nagumo equations abstract a combination of analytical and numerical methods is proposed to deal with the hopf bifurcation in a reaction diffusion system modeled by the fitz hugh nagumo fhn equations finite differencing is applied to the diffusion term to transform the fhn equations into an autonomous ode system perturbation techniques based on poincare normal form and center manifold theory are used to trace a bifurcate periodic solution branch from the onset up to a reasonable extent fourier spectral method and continuation techniques are used to trace the subsequent development of this branch the capability of this approach is demonstrated by examples in which the pursued periodic solution becomes highly unstable and with period tending towards infinity

infinitely many positive solutions for kirchhoff type problems abstract this paper is concerned with the existence of infinitely many positive solutions to a class of kirchhoff type problem a b u 2 d x u f x u in and u 0 on where is a smooth bounded domain of r n a b 0 0 and f r r is a caratheodory function satisfying some further conditions we obtain a sequence of a e positive weak solutions to the above problem tending to zero in l with f being more general than that of k perera z zhang nontrivial solutions of kirchhoff type problems via the yang index j differential equations 221 2006 246 255 z zhang k perera sign changing solutions of kirchhoff type problems via invariant sets of descent flow j math anal appl 317 2006 456 463

bound state solutions of kirchhoff type problems with critical exponent abstract in the present paper we consider the following kirchhoff type problems a b r 3 u 2 d x u v x u u 5 in r 3 u d 1 2 r 3 where a 0 b 0 and v l 3 2 r 3 is a given nonnegative function if v 3 2 is suitable small we prove that this problem has at least one bound state solution

ground state sign changing solutions for kirchhoff type problems in bounded domains abstract in the present paper we consider the existence of ground state sign changing solutions for a class of kirchhoff type problems 0 1 a b u 2 d x u f u x u 0 x where r n is a bounded domain with a smooth boundary n 1 2 3 a 0 b 0 and f c r r under some weak assumptions on f with the aid of some new analytical skills and non nehari manifold method we prove that 0 1 possesses one ground state sign changing solution u b and its energy is strictly larger than twice that of the ground state solutions of nehari type furthermore we establish the convergence property of u b as the parameter b 0 our results improve and generalize some results obtained by w shuai 2015 34

optimal search behavior and classic foraging theory random walk methods and diffusion theory pervaded ecological sciences as methods to analyze and describe animal movement consequently statistical physics was mostly seen as a toolbox rather than as a conceptual framework that could contribute to theory on evolutionary biology and ecology however the existence of mechanistic relationships and feedbacks between behavioral processes and statistical patterns of movement suggests that beyond movement quantification statistical physics may prove to be an adequate framework to understand animal behavior across scales from an ecological and evolutionary perspective recently developed random search theory has served to critically re evaluate classic ecological questions on animal foraging for instance during the last few years there has been a growing debate on whether search behavior can include traits that improve success by optimizing random stochastic searches here we stress the need to bring together the general encounter problem within foraging theory as a mean for making progress in the biological understanding of random searching by sketching the assumptions of optimal foraging theory oft and by summarizing recent results on random search strategies we pinpoint ways to extend classic oft and integrate the study of search strategies and its main results into the more general theory of optimal foraging

multiplicity of positive solutions for some quasilinear elliptic equation in rnwith critical sobolev exponent consider the equation pu g x u p 2u f x u p 2u 1 in rn where 1 0 be the principal eigenvalue of pu g x u p 2uinrn rng x u p 0 2 withu 1 0 the associated eigenfunction we prove i equation 1 has at least one positive solution if 0 1 ii suppose rn f x u 1 p 1such that 1 has at least one positive solution for 1 0 moreover ifp 2 there exists 0 1such that 1 has at least two positive solutions for 1 0

existence and concentration of positive solutions for kirchhoff type problems with a steep well potential abstract the existence and concentration of positive solutions for a class of kirchhoff type problems with a nonnegative steep well potential in r 3 are obtained via variational methods

remarks on multiple nontrivial solutions for quasi linear resonant problems in this paper morse theory and local linking are used to study the existence of multiple nontrivial solutions for a class of dirichlet boundary value problems with double resonance at infinity and at 0

existence of three solutions for some equation of kirchhoff type involving variable exponents abstract in this paper we are concerned with some nonlocal equation involving variable exponents the equation studied could be regarded as an eigenvalue problem a result of existence of three nonnegative solutions is established using variational technique

multiple solutions for a kirchhoff type problem involving the p x laplacian operator abstract the aim of this paper is to obtain weak solutions for a kirchhoff type problem involving the p x laplacian operator the main tool used for obtaining our result is a recent three critical points theorem established by ricceri

exotic dynamic behavior of the forced fitzhugh nagumo equations abstract space clamped fitzhugh nagumo nerve model subjected to a stimulating electrical current of form i o i cos t is investigated via poincare map and numerical continuation if i 0 it is known that hopf bifurcation occurs when i o is neither too small nor too large given such an i o if is chosen close to the natural frequency of the hopf bifurcated oscillation a series of exotic phenomena varying with i are observed numerically let 2 denote the generic period we watched then the scenario consists of two categories of period adding bifurcation the first category consists of a sequence of hysteretic 2 period adding starting with 1 at i 0 and ending at some finite i say i as the second category contains multiple levels of period adding bifurcation the top level consists of a sequence of 1 period adding starting with 2 at i i from this sequence a hierarchy of m m n n period adding in between are derived such a regular pattern is sometimes interrupted by a series of chaos this category of bifurcation also terminates at some finite i harmonic resonance sets in afterwards lyapunov exponents power spectra and fractal dimensions are used to assist these observations

existence and multiplicity of non trivial solutions for schrodinger kirchhoff type equations with radial potential abstract in the present paper the following schrodinger kirchhoff type problem p a b r n u 2 d x u v x u q x f u in r n u x 0 as x is studied four existence theorems of nontrivial solutions and a sequence of high energy solutions for problem p are obtained by the mountain pass theorem and symmetric mountain pass theorem

multiplicity result for some nonlocal anisotropic equation via nonsmooth critical point theory approach abstract in this paper we study quasilinear problems involving variable exponent growth conditions and nonlocal terms on the whole space r n a multiplicity result is established all the coefficients involved in the terms of the equation depend both on the variable x and the unknown function u our main argument is nonsmooth critical point theory

the influence of the environment on levy random search efficiency fractality and memory effects an open problem in the field of random searches relates to optimizing the search efficiency in fractal environments here we address this issue through a systematic study of levy searches in landscapes encompassing several degrees of target aggregation and fractality for scarce resources non destructive searches with unrestricted revisits to targets are shown to present universal optimal behavior irrespective of the general scaling properties of the spatial distribution of targets in contrast no such universal behavior occurs in the destructive case with forbidden revisits in which the optimal strategy strongly depends on the degree of target aggregation we also investigate how the presence of memory and learning skills of the searcher affect the search efficiency by considering a limiting model in which the searcher learns through recent experience to recognize food rich areas we find that a statistical memory of previous encounters does not necessarily increase the rate of target findings in random searches instead there is an optimal extent of memory dependent on specific details of the search space and stochastic dynamics which maximizes the search efficiency this finding suggests a more general result namely that in some instances there are actual advantages to ignoring certain pieces of partial information while searching for objects

on obtaining global nonlinear system characteristics through interpolated cell mapping abstract period doubling bifurcations and fractal basin boundaries are investigated by means of interpolated cell mapping icm icm is a new method to determine the continuous time trajectory of a nonlinear system by spatially discretizing the system and employing an interpolation procedure to estimate the system s response at any arbitrary point in phase space the parameter values at which period doubling occurs for a duffing s oscillator is found by both conventional time integration and two mapping techniques and the results compared the icm method is shown to very accurately determine the points of bifurcation the dimension of a fractal basin boundary is determined by icm and compared to an exact determination the icm procedure produces the same fractal dimension determination as the exact analysis but only requires one thousandth of the computation time

multiple positive solutions for a class of kirchhoff type problems involving critical sobolev exponents abstract in this paper we study the multiplicity results of positive solutions for a class of kirchhoff type problems involving critical growth terms with the help of nehari manifold and ljusternik schnirelmann category theory we investigate how the coefficient g x of the critical nonlinearity affects the number of positive solutions of that problem furthermore we obtain a relationship between the number of positive solutions and the topology of the global maximum set of g

berestycki lions conditions on ground state solutions for kirchhoff type problems with variable potentials based on the strategy introduced by chen and tang adv nonlinear anal 9 496 515 2020 and some new tricks we prove that the nonlinear problem of kirchhoff type a b r3 u 2dx u v x u f u x r3 in h1 r3 admits two classes of ground state solutions under the general berestycki lions assumptions on the nonlinearity f which are almost necessary conditions as well as some weak assumptions on the potential v moreover we also give a simple minimax characterization of the ground state energy our results improve and extend recent theorems in several directions

steep potential well may help kirchhoff type equations to generate multiple solutions abstract in this paper we consider a nonlinear kirchhoff type problem with steep potential well a r 3 u 2 d x b u v x u f x u p 2 u in r 3 u h 1 r 3 where a b 0 2 p 4 v c r 3 r and f l r 3 r such problem cannot be studied by applying variational methods in a standard way even by restricting its corresponding energy functional on the nehari manifold because palais smale sequences may not be bounded in this paper we introduce a novel constraint method to prove the existence of one and two positive solutions under the different assumptions on v respectively we conclude that steep potential well v may help kirchhoff type equations to generate multiple solutions which has never been involved before

on fractional regularity methods for a class of nonlocal problems abstract in the past years the phenomenon of fractional regularity has been addressed for a large class of linear and or quasilinear differential operators mostly in terms of certain besov spaces as it turned out for equations governed by the p laplacian in general the regularity of solutions appears in terms of functional spaces with nonlinear order of smoothness moreover despite its own interest fractional regularity methods may be used as a tool for the investigation of some partial differential equations which are not usually addressed in this manner thus the purpose of the present paper is to exploit such methods in order to provide some results regarding existence and regularity of solutions to a class nonlocal elliptic equations which are linked to the p laplacian this is done by means of explicit a priori estimates regarding lebesgue and nikolskii spaces which are part of the present contribution as a consequence this approach allows a relaxation on some of the standard conditions employed in this class of problems

on the solvability of an indefinite nonlinear kirchhoff equation via associated eigenvalue problems abstract we study the non existence existence and multiplicity of positive solutions to the following nonlinear kirchhoff equation m r n u 2 d x u v x u q x u p 2 u f x u in r n u h 1 r n where n 3 2 p 2 2 n n 2 m t a t b a b 0 the potential v is a nonnegative function in r n and the weight function q l r n with changes sign in v 0 we mainly prove the existence of at least two positive solutions in the cases that i 2 p min 4 2 and 0 1 2 4 p 4 2 p 1 f i i p 4 1 f and near 1 f for 0 sufficiently large where 1 f is the first eigenvalue of in h 0 1 with weight function f f whose corresponding positive principal eigenfunction is denoted by 1 furthermore we also investigated the non existence and existence of positive solutions if a belongs to different intervals

singular perturbed kirchhoff type problem with critical exponent abstract in the present paper we consider the following kirchhoff type problem with critical exponent e 2 a e b r 3 u 2 d x u v x u f u in r 3 u h 1 r 3 by variational methods we show the existence of the positive solutions concentrating around global minima of the potential v x as e 0 we do not need the monotonicity of the function f 3

sign changing solutions of a class of nonlocal quasilinear elliptic boundary value problems abstract this paper is concerned with nonlocal kirchhoff s equation a b u 2 u f x u in u 0 on via variational methods and invariant sets of descent flow we obtain existence of signed and sign changing solutions with asymptotically 3 linear bounded nonlinearity

levy flights and random searches in this work we discuss some recent contributions to the random search problem our analysis includes superdiffusive levy processes and correlated random walks in several regimes of target site density mobility and revisitability we present results in the context of mean field like and closed form average calculations as well as numerical simulations we then consider random searches performed in regular lattices and lattices with defects and we discuss a necessary criterion for distinguishing true superdiffusion from correlated random walk processes we invoke energy considerations in relation to critical survival states on the edge of extinction and we analyze the emergence of levy behavior in deterministic search walks finally we comment on the random search problem in the context of biological foraging

sign changing solutions of kirchhoff type problems via invariant sets of descent flow we obtain sign changing solutions of a class of nonlocal quasilinear elliptic boundary value problems using variational methods and invariant sets of descent flow

sign changing and multiple solutions of kirchhoff type problems without the p s condition abstract the existence of nontrivial solutions of kirchhoff type equations is an important nonlocal quasilinear problem in this paper we use minimax methods and invariant sets of descent flow to prove two interesting existence theorems for the following 4 superlinear kirchhoff type problems without the p s condition one concerning the existence of a nontrivial solution and the other one concerning the existence of sign changing solutions and multiple solutions a b u 2 u f x u in u 0 on

origin of power law distributions in deterministic walks the influence of landscape geometry we investigate the properties of a deterministic walk whose locomotion rule is always to travel to the nearest site initially the sites are randomly distributed in a closed rectangular a l ifmmode times else texttimes fi l landscape and once reached they become unavailable for future visits as expected the walker step lengths present characteristic scales in one l ensuremath rightarrow 0 and two a l ensuremath sim l dimensions however we find scale invariance for an intermediate geometry when the landscape is a thin striplike region this result is induced geometrically by a dynamical trapping mechanism leading to a power law distribution for the step lengths the relevance of our findings in broader contexts of both deterministic and random walks is also briefly discussed

improvements in the statistical approach to random levy flight searches recently it has been shown that the most e0cient strategy for searching randomly located objects when the sites are randomly distributed and can be revisited any number of times leads to a power law distribution p of the ights with 2 we show analytically that the incorporation of energy considerations limits the possible range for the l evy exponent however 2 still emerges as the optimal foraging condition furthermore we show that the probability distribution of ight lengths for the short and intermediate ight length regimes depends on the details of the system c 2001 elsevier science b v all rights reserved

two positive solutions for the kirchhoff type elliptic problem with critical nonlinearity in high dimension abstract we investigate the kirchhoff type elliptic problem with critical nonlinearity 1 u 2 d x u u q u 2 1 u 0 in u 0 on where n 5 r n is a bounded domain with smooth boundary 0 r 2 2 n n 2 and q 1 2 1 we prove the existence of two solutions of it via the variational method since n 5 and 0 the uniqueness assertion for the associated limiting problem may fail this causes serious difficulties in controlling concentrating palais smale sequences we overcome these by introducing new techniques for a mountain pass type solution we utilize the limit function of the fibering maps of the concentrating palais smale sequence this tool is based on our careful setting of nehari type sets on the other hand a suitable modification to a concentrating minimizing sequence enables us to obtain a global minimum solution this is the first work which proves the multiplicity of positive solutions of the kirchhoff type critical problem in high dimension

average time spent by levy flights and walks on an interval with absorbing boundaries we consider a levy flyer of order alpha that starts from a point x 0 on an interval o l with absorbing boundaries we find a closed form expression for the average number of flights the flyer takes and the total length of the flights it travels before it is absorbed these two quantities are equivalent to the mean first passage times for levy flights and levy walks respectively using fractional differential equations with a riesz kernel we find exact analytical expressions for both quantities in the continuous limit we show that numerical solutions for the discrete levy processes converge to the continuous approximations in all cases except the case of alpha 2 and the cases of x 0 0 and x 0 l for alpha 2 when the second moment of the flight length distribution exists our result is replaced by known results of classical diffusion we show that if x 0 is placed in the vicinity of absorbing boundaries the average total length has a minimum at alpha 1 corresponding to the cauchy distribution we discuss the relevance of this result to the problem of foraging which has received recent attention in the statistical physics literature

new existence and multiplicity of nontrivial solutions for nonlocal elliptic kirchhoff type problems abstract in the present paper we study the existence and multiplicity of nontrivial solutions for nonlocal elliptic kirchhoff type problems a b u 2 d x u f x u in u 0 on where a b 0 are real numbers some new existence and multiplicity results of nontrivial solutions are obtained by means of the mountain pass theorem and a cut off technique one known result is improved

the emergence of scaling laws search dynamics in a particle swarm optimization abstract this paper investigates the search dynamics of a fundamental particle swarm optimization pso algorithm via gathering and analyzing the data of the search area during the optimization process the pso algorithm exhibits a distinct performance when optimizing different functions which induces the emergence of different search dynamics during the optimization process the simulation results show that the performance is tightly related to the search dynamics which results from the interaction between the pso algorithm and the landscape of the solved problems the levy type scaling laws search dynamics emerges from the process in which the pso algorithm shows good performance while the brownian dynamics appears after the algorithm has stagnated due to the premature convergence the levy dynamics characterized by a large number of intensive local searches punctuated by long range transfers is an indicator of good performance which allows the algorithm to achieve an efficient balance between exploration and exploitation so as to improve the search efficiency

multiple solutions for n kirchhoff type problems with critical exponential growth in rn abstract in this paper we consider a class of nonlocal n kirchhoff type problems involving a nonlinearity term having critical exponential growth by applying variational methods together with trudinger moser inequality in whole r n we obtain the existence of at least two positive solutions

existence and concentration of sign changing solutions to kirchhoff type system with hartree type nonlinearity abstract in this paper we discuss the existence and the concentration of sign changing solutions to a class of kirchhoff type systems with hartree type nonlinearity in r 3 by the minimization argument on the sign changing nehari manifold and a quantitative deformation lemma we prove that the system has a sign changing solution moreover concentration behaviors of sign changing solutions are obtained when the coefficient of the potential function tends to infinity specially our results cover general schrodinger equations kirchhoff equations and schrodinger poisson systems

a note on kirchhoff type equations with hartree type nonlinearities abstract we consider the following kirchhoff type equation in r 3 a b r 3 u 2 d x u 1 g x u 1 x u p u p 2 u where a 0 b 0 are constants 0 3 p 2 6 0 is a parameter and g x is a nonnegative continuous potential satisfying some conditions by using the nehari manifold and the concentration compactness principle we establish the existence of ground state solutions for the equation if the parameter is large enough moreover some concentration behaviors of these solutions as are discussed

and did he search for you and could not find you we discuss stochastic search mechanisms for sparse targets these mechanisms may be normal diffusion in different dimensionalities as well as anomalous diffusion namely levy walks and subdiffusion we argue that under different boundary conditions either mechanism may be beneficial for the search as a concrete example we investigate facilitated diffusion of regulating proteins in gene regulation moreover we discuss potential pitfalls in acquiring diffusion behaviour from time series analysis of single particle trajectories