existence of infinitely many solutions to a class of kirchhoff schrodinger poisson system in this paper we consider the existence of infinitely many solutions to following nonlinear kirchhoff schrodinger poisson system a b r 3 u 2 v x u 2 u v x u l x u f x u x r 3 l x u 2 x r 3 where constants a 0 b 0 and 0 when f has sublinear growth in u we obtain infinitely many solutions under certain assumption that v do not have a positive lower bound the technique we use in this paper is the symmetric mountain pass theorem established by kajikiya 2005

embedding theorems and existence results for nonlinear schrodinger poisson systems with unbounded and vanishing potentials abstract motivated by existence results for positive solutions of non autonomous nonlinear schrodinger poisson systems with potentials possibly unbounded or vanishing at infinity we prove embedding theorems for weighted sobolev spaces we both consider a general framework and spaces of radially symmetric functions when assuming radial symmetry of the potentials

continuum percolation in two dimensions monte carlo tests of scaling and universality for non interacting discs detailed results are reported for the connectivity properties of a system of discs of unit radius free to be situated anywhere within a square of area 2l2 ordinary lattke percolation would correspond to the discs being situated on the vertices of a j2l x j2l lattice computer simulations are carried out for a sequence of increasing system sizes ranging from l 20 to l 1000 for each value of l a large number of realisations are generated for 25 values of the disc concentration x we calculate a variety of estimates for the threshold parameter x as well as the critical exponents p y t and v our exponent estimates are in close agreement with accepted values for ordinary lattice percolation therefore this continuum system appears to be in the same universality class as lattice percolation

solitons and the electromagnetic field in a recent paper 4 it has been introduced a lorentz invariant equation in three space dimensions having soliton like solutions we recall that roughly speaking a soliton is a solution whose energy travels as a localized packet and which preserves this form of localization under small perturbations see 6 15 13 10 the equation introduced in 4 is the euler lagrange equation of an action functional

continuum percolation with discs having a distribution of radii it is shown that for continuum percolation with overlapping discs having a distribution of radii the net areal density of discs at percolation threshold depends non trivially on the distribution and is not bounded by any finite constant results of a monte carlo simulation supporting the argument are presented

a class of p q laplacian type equation with concave convex nonlinearities in bounded domain abstract in this paper our main purpose is to establish the existence of multiple solutions of a class of p q laplacian equation involving concave convex nonlinearities p u q u v x u r 2 u u p 2 u f x u x u 0 x where is a bounded domain in r n 0 1 r q p n and p n p n p is the critical sobolev exponent s u div u s 2 u is the s laplacian of u we prove that for any 0 0 is a constant there is a 0 such that for every 0 the above problem possesses infinitely many weak solutions we also obtain some results for the case 1 q p r p the existence results of solutions are obtained by variational methods

existence and uniqueness of multi bump bound states of nonlinear schrodinger equations with electromagnetic fields abstract in this paper we are concerned with the existence and uniqueness of multi bump bound states of the nonlinear schrodinger equations with electromagnetic potential i t i a x 2 v x p 2 x r n for sufficiently small 0 where i is the imaginary unit 2 p 2 n n 2 for n 3 and 2 p for n 1 2 v x is a bounded real function on r n and a x a 1 x a 2 x a n x is such that a j x is a bounded real function on r n for j 1 2 n for any finite collection of a 1 a 2 a k of non degenerate critical points of v x we show that there exists a solution x t exp iet u x which is a small perturbation of a sum of one bump solutions concentrated at a 1 a 2 a k and u x is unique up to rotations for small 0

from lattice to continuum percolation via clustering considers the growth of eden clusters on a square lattice about randomly distributed seeds or through continuous nucleation the author analyses the dependence of the percolation threshold pc on the seed density or ratio of nucleation to growth rates i e on parameters controlling the average size of individual clusters or islands before coalescence it is found that pc approaches a large island size continuum limit from below at a rate depending on the active zone width scaling exponent this result applies to a variety of island forming models

an optimal one way multigrid algorithm for discrete time stochastic control the numerical solution of discrete time stationary infinite horizon discounted stochastic control problems is considered for the case where the state space is continuous and the problem is to be solved approximately within a desired accuracy after a discussion of problem discretization the authors introduce a multigrid version of the successive approximation algorithm that proceeds one way from coarse to fine grids and analyze its computational requirements as a function of the desired accuracy and of the discount factor they also study the effects of a certain mixing ergodicity condition on the algorithm s performance it is shown that the one way multigrid algorithm improves upon the complexity of its single grid variant and is in a certain sense optimal

the existence of a nontrivial solution to the p q laplacian problem with nonlinearity asymptotic to up 1 at infinity in rn abstract we consider the following elliptic problem p u m u p 2 u q u n u q 2 u f x u x r n u w 1 p r n w 1 q r n where m n 0 n 3 and 1 q p n f x u u p 1 tends to a positive constant l as u we prove in this paper that the problem possesses a nontrivial solution even if the nonlinearity f x t does not satisfy the ambrosetti rabinowitz condition 0 f x u 0 u f x s d s 1 p f x u u for x u r n r

positive solutions for some non autonomous schrodinger poisson systems abstract in this paper we study the schrodinger poisson system sp u u k x x u a x u p 1 u x r 3 k x u 2 x r 3 with p 3 5 assuming that a r 3 r and k r 3 r are nonnegative functions such that lim x a x a 0 lim x k x 0 and satisfying suitable assumptions but not requiring any symmetry property on them we prove the existence of positive solutions

regularity properties of deformations with finite energy the purpose of this paper is to point out some regularity properties of a class of functions which play an important role in nonlinear elasticity

infinitely many homoclinic orbits for hamiltonian systems with indefinite sign subquadratic potentials abstract in this paper we deal with the existence and multiplicity of homoclinic solutions of the second order hamiltonian system u t l t u t w t u t 0 where l t and w t x are neither autonomous nor periodic in t under the assumption that w t x is indefinite sign and subquadratic as x and l t is a n n real symmetric positive definite matrices for all t r we establish some existence criteria to guarantee that the above system has at least one or infinitely many homoclinic solutions by using the genus properties in critical theory

positive solutions for schrodinger poisson equations with a critical exponent abstract in this paper we establish the existence of a positive solution of the schrodinger poisson equations with a critical sobolev exponent the methods used here are based on the concentration compactness principle of p l lions and methods of brezis and nirenberg

monte carlo renormalisation group approach to percolation on a continuum test of universality it is shown that a monte carlo renormalisation group technique can be employed for direct renormalisation of a system on a continuum i e without restriction to a periodic lattice for the problem of overlapping discs the authors find the critical area fraction sc 0 688 or 0 005 and the correlation length critical exponent nu 1 33 or 0 07 the latter result indicates the regular irregular lattice universality

low energy solutions for singularly perturbed coupled nonlinear systems on a riemannian manifold with boundary abstract let m g be a smooth compact riemannian manifold with smooth boundary with n dim m 2 3 we suppose the boundary m to be a smooth submanifold of m with dimension n 1 we consider a singularly perturbed nonlinear system namely klein gordon maxwell proca system or klein gordon maxwell system of schroedinger maxwell system on m we prove that the number of low energy solutions when the perturbation parameter is small depends on the topological properties of the boundary m by means of the lusternik schnirelmann category also these solutions have a unique maximum point that lies on the boundary

uncertain convex programs randomized solutions and confidence levels many engineering problems can be cast as optimization problems subject to convex constraints that are parameterized by an uncertainty or instance parameter two main approaches are generally available to tackle constrained optimization problems in presence of uncertainty robust optimization and chance constrained optimization robust optimization is a deterministic paradigm where one seeks a solution which simultaneously satisfies all possible constraint instances in chance constrained optimization a probability distribution is instead assumed on the uncertain parameters and the constraints are enforced up to a pre specified level of probability unfortunately however both approaches lead to computationally intractable problem formulations

existence of multiple positive solutions for a p q laplacian system with critical nonlinearities abstract in this paper our main purpose is to establish the existence of multiple positive solutions of a p q laplacian system with convex and sobolev critical nonlinearities by some standard variational methods whose key is to construct homotopies between and levels of the functional j and some analytical techniques

existence of infinitely many large solutions for the nonlinear schrodinger maxwell equations abstract in this paper we study the nonlinear stationary schrodinger maxwell equations u v x u u f x u in r 3 u 2 in r 3 using the variant fountain theorem introduced by zou w zou variant fountain theorems and their applications manuscripta math 104 2001 343 358 under certain assumptions on v and f we get infinitely many large solutions for

on constraint sampling in the linear programming approach to approximate dynamic programming in the linear programming approach to approximate dynamic programming one tries to solve a certain linear program the alp that has a relatively small numberk of variables but an intractable numberm of constraints in this paper we study a scheme that samples and imposes a subset ofm

on the bolza problem abstract we consider the following boundary value problem x v x 0 x 0 x 0 x t x 1 where t 0 x 0 x 1 r n are given we prove that this problem has infinitely many solutions provided v is even and superquadratic

infinitely many solutions for a class of sublinear schrodinger maxwell equations abstract in this paper we study the existence of infinitely many solutions for a class of sublinear schrodinger maxwell equations the proof is based on the variant fountain theorem established by zou recent results from the literature are extended

a multigrid approach to sdp relaxations of sparse polynomial optimization problems we propose a multigrid approach for the global optimization of polynomial optimization problems with sparse support the problems we consider arise from the discretization of infinite dimensional optimization problems such as pde optimization problems boundary value problems and some global optimization applications in many of these applications the level of discretization can be used to obtain a hierarchy of optimization models that capture the underlying infinite dimensional problem at different degrees of fidelity this approach inspired by multigrid methods has been successfully used for decades to solve large systems of linear equations however multigrid methods are difficult to apply to semidefinite programming sdp relaxations of polynomial optimization problems the main difficulty is that the information between grids is lost when the original problem is approximated via an sdp relaxation despite the loss of information we develop a multigrid approach and propose prolongation operator

multiple solutions for a nonlinear schrodinger poisson system with sign changing potential this paper deals with the existence and multiplicity of nontrivial solutions for a nonlinear schrodinger poisson system under some suitable conditions some criteria on the existence of nontrivial solutions including a ground state solution two solutions and a sequence of nontrivial solutions u k with u k 0 as k is established by applying the morse theory and variational methods some known results in the literatures are extended and improved

nonuniversality and continuity of the critical covered volume fraction in continuum percolation we establish using mathematically rigorous methods that the critical covered volume fraction cvf for a continuum percolation model with overlapping balls of random sizes is not a universal constant independent of the distribution of the size of the balls in addition we show that the critical cvf is a continuous function of the distribution of the radius random variable in the sense that if a sequence of random variables converges weakly to some random variable then the critical cvf based on these random variables converges to the critical cvf of the limiting random variable

high energy solutions for the superlinear schrodinger maxwell equations abstract in this paper we study the existence of infinitely many large energy solutions for the superlinear schrodinger maxwell equations u v x u u f x u in r 3 u 2 in r 3 via the fountain theorem in critical point theory

positive solutions of asymptotically linear schrodinger poisson systems with a radial potential vanishing at infinity in this paper we study the schrodinger poisson system sp u v x u x u k x f u inr3 u2 u 0 inr3 and prove the existence of positive solutions for system sp when the nonlinearity f has growth at most linear for small allowing the potential v x to vanish at infinity in addition also we obtain the nonexistence of a nontrivial positive solution for 14

singularly perturbed markov decision processes a multiresolution algorithm singular perturbation techniques allow the derivation of an aggregate model whose solution is asymptotically optimal for markov decision processes with strong and weak interactions we develop an algorithm that takes advantage of the asymptotic optimality of the aggregate model in order to compute the solution of the original model we derive conditions for which the proposed algorithm has better worst case complexity than conventional contraction algorithms based on our complexity analysis we show that the major benefit of aggregation is that the reduced order model is no longer ill conditioned the reduction in the number of states due to aggregation is a secondary benefit this is a surprising result since intuition would suggest that the reduced order model can be solved more efficiently because it has fewer states however we show that this is not necessarily the case our theoretical analysis and numerical experiments show that the proposed algorithm can compute the optimal solution with a redu

multiple solutions for the nonhomogeneous klein gordon equation coupled with born infeld theory on r3 abstract in this paper we prove the existence of multiple solutions for the following nonhomogeneous klein gordon equation coupled with born infeld theory u m 2 2 u u q 2 u h x in r 3 4 4 u 2 in r 3 when 2 q 6

a stochastic multiscale model for electricity generation capacity expansion abstract long term planning for electric power systems or capacity expansion has traditionally been modeled using simplified models or heuristics to approximate the short term dynamics however current trends such as increasing penetration of intermittent renewable generation and increased demand response requires a coupling of both the long and short term dynamics we present an efficient method for coupling multiple temporal scales using the framework of singular perturbation theory for the control of markov processes in continuous time we show that the uncertainties that exist in many energy planning problems in particular load demand uncertainty and uncertainties in generation availability can be captured with a multiscale model we then use a dimensionality reduction technique which is valid if the scale separation present in the model is large enough to derive a computationally tractable model we show that both wind data and electricity demand data do exhibit sufficient scale separation a numerical example using real data and a finite difference approximation of the hamilton jacobi bellman equation is used to illustrate the proposed method we compare the results of our approximate model with those of the exact model we also show that the proposed approximation outperforms a commonly used heuristic used in capacity expansion models

number and profile of low energy solutions for singularly perturbed klein gordon maxwell systems on a riemannian manifold abstract given a 3 dimensional riemannian manifold m g we investigate the existence of positive solutions of the klein gordon maxwell system e 2 g u a u u p 1 2 q v 1 2 u in m g v 1 q 2 u 2 v q u 2 in m and schrodinger maxwell system e 2 g u u u v u p 1 in m g v v q u 2 in m when p 4 6 we prove that the number of one peak solutions depends on the topological properties of the manifold m by means of the lusternik schnirelmann category

coupled klein gordon and born infeld type equations looking for solitary waves the existence of infinitely many non trivial radially symmetric solitary waves for the nonlinear klein gordon equation coupled with a born infeld type equation is established under general assumptions

existence of solitary wave solutions for the nonlinear klein gordon equation coupled with born infeld theory with critical sobolev exponent abstract in this paper we prove the existence of solutions for the nonlinear klein gordon equation coupled with born infeld theory under the electrostatic solitary wave ansatz by using variational methods

universality and cluster structures in continuum models of percolation with two different radius distributions the percolation thresholds and the fractal cluster structures for continuum models of percolation with uniform cm1 and variable radius cm2 distributions of discs and spheres are investigated and compared with the results of ordinary lattice percolation configurations of up to 250000 discs 2 dimensions and 100000 spheres 3 dimensions are numerically simulated in two dimensions the authors find distinctly different percolation concentrationss for models cm1 and cm2 in the three dimensional systems the percolation concentration for both models cannot be distinguished within their limits of accuracy the fractal dimensions of the cluster hull surface and volume are the same as in the corresponding lattice models the harris criterion for the continuum percolation problem is confirmed by their simulation

global lorentz estimates for nonuniformly nonlinear elliptic equations via fractional maximal operators abstract this article is a contribution to the study of regularity theory for nonlinear elliptic equations the aim of this article is to establish some global estimates for nonuniformly elliptic equations in divergence form as div u p 2 u a x u q 2 u div f p 2 f a x f q 2 f which arises from double phase functional problems in particular the main results provide the regularity estimates for the distributional solutions in terms of maximal and fractional maximal operators this work extends that of colombo and mingione 2016 23 and byun and oh 2017 9 by dealing with the global estimates in lorentz spaces it also extends our recent result tran and nguyen 2020 66 concerning new estimates of divergence elliptic equations using cutoff fractional maximal operators for future research the approach developed in this article will allow global estimates of distributional solutions to nonuniformly nonlinear elliptic equations to be obtained in the framework of other spaces

multiplicity of small negative energy solutions for a class of nonlinear schrodinger poisson systems this paper deals with the following nonlinear schrodinger poisson email protected v x u k x f x u h x f x u inr 3 email protected f k x u 2 inr 3 where v x k x and h x are nonnegative continuous functions under appropriate assumptions on v x k x h x and f x u we prove the existence of infinitely many small negative energy solutions by using the variant fountain theorem established by zou recent results from the literature are extended

sobolev maps with integer degree and applications to skyrme s problem let note see the image of page 197 for this formatted text phi r 3 rightarrow s 3 subset r 4 quad a phi 2 sum alpha beta 1 3 left frac partial phi partial x alpha wedge frac partial phi partial x beta right 2 and let k in z skyrme s problem consists in minimizing the energy note see the image of page 197 for this formatted text scr e phi int r 3 nabla phi 2 a phi 2 dx among maps with degree note see the image of page 197 for this formatted text k d phi frac 1 2 pi 2 int r 3 det phi nabla phi dx we show that for all phi with finite energy d phi is an integer and then obtain existence of a minimizer of scr e in the natural class of maps with finite energy

on the distributional jacobian of maps from sn into sn in fractional sobolev and holder spaces h brezis and l nirenberg proved that if gk c 0 s n s n and g2 c 0 s n s n n 1 are such that gk g in bmo s n then deggk degg on the other hand if g 2 c 1 s n s n then kronecker s formula asserts that degg 1 jsnj r sn det rg d consequently r sn det rgk d converges to r sn det rg d provided gk g in bmo s n in the same spirit we consider the quantity j g r sn det rg d for all 2 c 1 s n r and study the convergence of j gk in particular we prove that j gk converges to j g for any 2 c 1 s n r if gk converges to g in c 0 s n for some n 1 n surprisingly this result is optimal when n 1 in the case n 1 we prove that if gk g almost everywhere and lim supk 1jgk gjbmo is suciently small then j gk j g for any 2 c 1 s 1 r we also establish bounds for j g which are motivated by the works of j bourgain h brezis and h m nguyen and h m nguyen we pay special attention to the case n 1

infinitely many solutions for a nonlinear klein gordon maxwell system abstract in this paper a nonlinear klein gordon maxwell system with solitary wave solution is considered using critical point theory we establish sufficient conditions for the existence of infinitely many solitary wave solutions results obtained complement and improve the existing ones

existence of nonnegative solutions for quasilinear elliptic equations with indefinite critical nonlinearities abstract we study the quasilinear elliptic equation div u u a x f u b x g u in with dirichlet boundary condition u 0 on where is a bounded domain in r n a x b x are sign changing continuous functions and g u has critical growth at infinity with respect to the principal part a nonnegative nontrivial solution is given under appropriate growth conditions on f u g u at 0 and

existence criteria for baby skyrmions for a wide range of potentials abstract topologically stable field configurations are an important subject of study in various areas such as magnetism condensed matter physics nuclear physics liquid crystals in the baby skyrme models the two dimensional versions of the skyrme model of nuclear physics such structures are called baby skyrmions and they have been widely studied by means of numerical methods the aim of this paper is to give a simple criterion that guarantees the existence of skyrmions with unit topological charge for most baby skyrme models considered in the literature the proof of this criterion is purely analytic and it is based on the concentration of the energy of minimizing sequences

multiplicity of positive solutions for a p q laplacian system with concave and critical nonlinearities abstract in this paper we consider a p q laplacian system with concave and sobolev critical nonlinearities by using the n ehari manifold fibering maps and the lusternik schnirelman category we prove that the problem has at least cat 1 distinct positive solutions where cat denotes the lusternik schnirelman category of in itself

optimal control of molecular dynamics using markov state models a numerical scheme for solving high dimensional stochastic control problems on an infinite time horizon that appear relevant in the context of molecular dynamics is outlined the scheme rests on the interpretation of the corresponding hamilton jacobi bellman equation as a nonlinear eigenvalue problem that using a logarithmic transformation can be recast as a linear eigenvalue problem for which the principal eigenvalue and its eigenfunction are sought the latter can be computed efficiently by approximating the underlying stochastic process with a coarse grained markov state model for the dominant metastable sets we illustrate our method with two numerical examples one of which involves the task of maximizing the population of helices in an ensemble of small biomolecules alanine dipeptide and discuss the relation to the large deviation principle of donsker and varadhan

asymmetry in the percolation thresholds of fully penetrable disks with two different radii we perform simulations of continuum gradient percolation to measure the percolation threshold of systems of fully penetrable disks of two different radii as a function of volumetric proportion v the percolation threshold phic v lambda is approximately symmetric about v 0 5 for a fixed ratio lambda of the radii however the difference from symmetry is strongly statistically significant we also improve by an order of magnitude the measurement of the percolation threshold for disks of equal radius phic 0 676 347 5 6

holonomy and skyrme s model in this paper we consider two generalizations of the skyrme model one is a variational problem for maps from a compact 3 manifold to a compact lie group the other is a variational problem for flat connections we describe the path components of the configuration spaces of smooth fields for each of the variational problems we prove that the invariants separating the path components are well defined for not necessarily smooth fields with finite skyrme energy we prove that for every possible value of these invariants there exists a minimizer of the skyrme functional throughout the paper we emphasize the importance of holonomy in the skyrme model some of the results may be useful in other contexts in particular we define the holonomy of a distributionally flat l 2 loc connection the local developing maps for such connections need not be continuous

a fixed point theorem for a class of sobolev maps we prove an analog of the brouwer fixed point theorem for a map whose differential and adjoint are integrable with exponents n 1 and n n 1 respectively here is a convex bounded open subset of r n

random convex programs random convex programs rcps are convex optimization problems subject to a finite number n of random constraints the optimal objective value j of an rcp is thus a random variable we study the probability with which j is no longer optimal if a further random constraint is added to the problem violation probability v it turns out that this probability rapidly concentrates near zero as n increases we first develop a theory for rcps leading to explicit bounds on the upper tail probability of v then we extend the setup to the case of rcps with r a posteriori violated constraints rcpvs a paradigm that permits us to improve the optimal objective value while maintaining the violation probability under control explicit and nonasymptotic bounds are derived also in this case the upper tail probability of v is upper bounded by a multiple of a beta distribution irrespective of the distribution on the random constraints all results are derived under no feasibility assumptions on the problem further the relation between rcpvs and chance constrained problems ccp is explored showing that the optimal objective j of an rcpv with the generic constraint removal rule provides with arbitrarily high probability an upper bound on the optimal objective of a corresponding ccp moreover whenever an optimal constraint removal rule is used in the rcpvs then appropriate choices of n and r exist such that j approximates arbitrarily well the objective of the ccp

infinitely many solutions for schrodinger maxwell equations with indefinite sign subquadratic potentials in this paper we deal with multiplicity of solutions of a class of sublinear schrodinger maxwell equations du v x u fu f x u inr 3 d f u 2 lim x f x 0 inr 3 under more relaxed assumptions on v and f we establish some existence criteria to guarantee that the above problem has at least one or infinitely many nontrivial solutions by using the genus properties in critical theory

some properties of the degree for a class of sobolev maps we prove that a map u w loc1 1 m n whose differential and adjoint are integrable with exponents n 1 p n and n n 1 q respectively possesses an integer valued degree with the usual properties here m c 0 1 and n c 1 1 are n dimensional oriented riemannian manifolds with m possibly non empty

one dimensional variational problems whose minimizers do not satisfy the euler lagrange equation in this paper we consider the problem of minimizing i u text int a b text f x u x u x dx 1 1 in the set a of absolutely continuous function u a b satisfying the end conditions u left a right text alpha text quad u left b right text beta text 1 2 where and are given constant in 1 1 a b is a finite interval denotes frac d dx and the integrand f f x u p is assumed to be smooth nonnegative and to satisfy the regularity condition f pp text 0 text 1 3

multiplicity of positive solutions to a p q laplacian equation involving critical nonlinearity abstract in this paper our main purpose is to establish the existence of multiple solutions of a p q laplacian equation with convex and sobolev critical nonlinearity by some standard variational methods whose key is to construct homotopies between and levels of the functional i and some analytical techniques

existence of energy minimizers as stable knotted solitons in the faddeev model in this paper we study the existence of knot like solitons realized as the energy minimizing configurations in the faddeev quantum field theory model topologically these solitons are characterized by an hopf invariant q which is an integral class in the homotopy group 3 s 2 open image in new window we prove in the full space situation that there exists an infinite subset open image in new window of open image in new window such that for any m open image in new window the faddeev energy e has a minimizer among the topological class q m besides we show that there always exists a least positive energy faddeev soliton of non zero hopf invariant in the bounded domain situation we show that the existence of an energy minimizer holds for open image in new window open image in new window as a by product we obtain an important technical result which says that e and q satisfy the sublinear inequality e c q 3 4 where c 0 is a universal constant such a fact explains why knotted clustered soliton configurations are preferred over widely separated unknotted multisoliton configurations when q is sufficiently large

the gap phenomenon for variational integrals in sobolev spaces we show that a gap phenomenon occurs for general variational integrals for mappings from a domain r n into a riemannian manifold if has a non trivial topology

on the electrostatic born infeld equation with extended charges in this paper we deal with the electrostatic born infeld equation left begin array ll operatorname div left displaystyle frac nabla phi sqrt 1 nabla phi 2 right rho quad in mathbb r n displaystyle lim x to infty phi x 0 end array right quad quad quad quad mathcal bi where rho is an assigned extended charge density we are interested in the existence and uniqueness of the potential phi and finiteness of the energy of the electrostatic field nabla phi we first relax the problem and treat it with the direct method of the calculus of variations for a broad class of charge densities assuming rho is radially distributed we recover the weak formulation of mathcal bi and the regularity of the solution of the poisson equation under the same smoothness assumptions in the case of a locally bounded charge we also recover the weak formulation without assuming any symmetry the solution is even classical if rho is smooth then we analyze the case where the density rho is a superposition of point charges and discuss the results in kiessling commun math phys 314 509 523 2012 other models are discussed as for instance a system arising from the coupling of the nonlinear klein gordon equation with the born infeld theory

nontrivial solutions for kirchhoff type problems with a parameter abstract the paper deals with the following kirchhoff type problems a b r n u 2 d x u v x u f x u in r n u h 1 r n where constants a b 0 1 is a parameter and f x u is either sublinear or superlinear at infinity under relaxed assumptions on v x we establish the existence and multiplicity results without the compactness of embedding of the working space

semiclassical solutions for the nonlinear schrodinger maxwell equations abstract in this paper by using variational methods and critical point theory we study the existence of semiclassical solutions for the following nonlinear schrodinger maxwell equations e 2 u v x u u f x u in r 3 4 u 2 in r 3 where e 0 and v x 0 for all x r 3 under some weaker assumptions on v and f we prove that the system has at least one nontrivial solution for sufficient small e 0

infinitely many solutions for fourth order elliptic equations with general potentials abstract this paper is concerned with the following fourth order elliptic equations 2 u u v x u f x u in r n u h 2 r n where u r n r under general assumption on potential v x and f x u is indefinite sign and sublinear as u we establish the existence of infinitely many solutions by using the genus properties in critical point theory recent results from the literature are extended

solitons in several space dimensions derrick s problem and infinitely many solutions in this paper we study a class of lorentz invariant nonlinear field equations in several space dimensions the main purpose is to obtain soliton like solutions these equations were essentially proposed by c h derrick in a celebrated paper in 1964 as a model for elementary particles however an existence theory was not developed