existence and multiplicity of solutions for elliptic systems with nonstandard growth condition in rn abstract in this paper we study the elliptic system of gradient type with nonstandard growth conditions in r n existence and multiplicity results under two growth conditions on the reaction terms are established

qualitative properties and support compactness of solutions for quasilinear schrodinger equation with sign changing potentials abstract in this paper we deal with the following general elliptic equation involving the weigh p laplace operator p u v x u a x u q 1 u x r n where n 3 p 2 0 q and a x v x change sign in r n the main result of this work establishes some qualitative properties of the solutions of the above equation more precisely in a first part we study the compactness of support of classical solutions in the second part we study the behavior and the qualitative properties of the radial classical solutions and we give a classification of these solutions

uniqueness theorems for positive radial solutions of quasilinear elliptic equations in a ball we establish a new pohozaev type identity and use it to prove a theorem on the uniqueness of positive radial solutions to the quasilinear elliptic problem div u m 2emsp14 u f u 0 inb andu 0 on b wherebis a finite ball in rn n 3 and 1 0 and 1 p q n 2 n 2 has a unique positive solution whenn 6 this gives a complete answer to an open problem raised by brezis and nirenberg in 1983 in the casen 6 we shall also derive some partial results to the open problem in the casesn 3 4 5

entire solutions of quasilinear elliptic equations abstract we consider the existence of entire solutions of a quasilinear elliptic equation div d u p 2 d u k x f u x r n where p 1 n n conditions of the existence of entire solutions have been obtained by different authors we prove an optimality of some of these results and give new sufficient conditions for the nonexistence of entire solutions

quasilinear elliptic problems in r n involving oscillatory nonlinearities in this paper we study an elliptic problem in r n which involves the p laplacian p n 2 and the nonlinear term has an oscillatory behavior by means of a direct variational approach we establish the existence of infinitely many homoclinic solutions whose w 1 p r n norms tend to zero to infinity respectively whenever the nonlinearity oscillates at zero at infinity respectively the solutions have invariance properties with respect to certain subgroups of the orthogonal group o n

on positive radial entire solutions of second order quasilinear elliptic systems abstract we study positive radial entire solutions of second order quasilinear elliptic systems of the form p u f x u v q v g x u v x r n sufficient and or necessary conditions of f and g are obtained for to have positive radial entire solutions with prescribed asymptotic behavior at infinity we first study the upper and lower estimates for the nonlinear integral operator which is inverse of one dimensional polar form of the n dimensional m laplace operator

uniqueness theorems for negative radial solutions of k hessian equations in a ball abstract in this paper we apply a pohozaev type identity and the monotone separation techniques to prove the uniqueness of negative radial solutions of k hessian equations in a finite ball

morse index and uniqueness for positive solutions of radial p laplace equations we study the positive radial solutions of the dirichlet problem p u f u 0 in b u 0 in b u 0 on b where p u div u p 2 u p 1 is the p laplace operator b is the unit ball in r n centered at the origin and f is a c 1 function we are able to get results on the spectrum of the linearized operator in a suitable weighted space of radial functions and derive from this information on the morse index in particular we show that positive radial solutions of mountain pass type have morse index 1 in the subspace of radial functions of w 1 p 0 b we use this to prove uniqueness and nondegeneracy of positive radial solutions when f is of the type u s u q and p 2

classical and non classical solutions of a prescribed curvature equation we discuss existence and multiplicity of positive solutions of the one dimensional prescribed curvature problem u root 1 u 2 lambda f t u u 0 0 u 1 0 depending on the behaviour at the origin and at infinity of the potential integral u 0 f t s ds besides solutions in w 2 w 1 0 1 we also consider solutions in w loc 2 1 0 1 which are possibly discontinuous at the endpoints of 0 1 our approach is essentially variational and is based on a regularization of the action functional associated with the curvature problem c 2007 elsevier inc all rights reserved

exact number of solutions of a one dimensional prescribed mean curvature equation with concave convex nonlinearities abstract in this paper an exact number of solutions is obtained for the one dimensional prescribed mean curvature equation with concave convex nonlinearities in the form of u 1 u 2 u p u q u x 0 0 x 1 u 0 u 1 0 where 0 is a parameter and p q satisfy 0 p 1 q the arguments are based upon a time map method

positive versus compact support solutions to a singular elliptic problem abstract the semilinear elliptic problem u k x u q u p u 0 in u 0 on is considered in this paper where is a bounded domain of r n n 2 with c 2 boundary 0 q p 1 and k x as x we mainly study the effect of the blow up rate of k x near to the existence of positive and compact support solutions furthermore an optimal compact support principle is given for a class of elliptic differential inequalities

exact multiplicity for some quasilinear elliptic dirichlet problems where the non linearity changes sign abstract the exact multiplicity of positive solutions is obtained for a class of quasilinear elliptic dirichlet problems where the non linearity changes sign meanwhile the uniqueness of positive ground state on r n is also obtained

a lower bound for the wehrl entropy of quantum spin with sharp high spin asymptotics a lower bound for the wehrl entropy of a single quantum spin is derived the high spin asymptotics of this bound coincides with lieb s conjecture up to but not including terms of first and higher order in the inverse spin quantum number the result presented here may be seen as complementary to the verification of the conjecture in cases of lowest spin by schupp commun math phys 207 481 1999 the present result for the wehrl entropy is obtained from interpolating a sharp norm bound that also implies a sharp lower bound for the so called renyi wehrl entropy with certain indices that are evenly spaced by half of the inverse spin quantum number

indefinite weight nonlinear problems with neumann boundary conditions abstract we present a multiplicity result of positive solutions for the neumann problem associated with a second order nonlinear differential equation of the following form u a t g u 0 where the weight function a t has indefinite sign the only assumption we make for the nonlinear term g u is that its primitive g u presents some oscillations at infinity expressed by the condition involving lim g u u 2 0 lim g u u 2 as an application we obtain multiple radially symmetric solutions for neumann problems associated with u a x g u 0

on the topology of the lu attractor and related systems we use well established methods of knot theory to study the topological structure of the set of periodic orbits of the l u attractor we show that for a specific set of parameters the l u attractor is topologically different from the classical lorenz attractor whose dynamics is formed by a double cover of the simple horseshoe this argues against the similarity between the l and lorenz attractors claimed for these parameter values by some authors on the basis of non topological observations however we show that the l system belongs to the lorenz like family since by changing the values of the parameters the behaviour of the system follows the behaviour of all members

a quantitative comparison theorem for nonlinear equations abstract in the present paper we establish a quantitative comparison theorem for positive solutions of the following initial value problems formula and formula withr 0 andm 1 and show some applications of the theorem to the nonexistence problem of positive solutions of quasilinear equations basic models leading to such nonlinear equations are the degenerate laplace equation and the steady state porous medium equation

uniqueness of positive radial solutions for quasilinear elliptic equations in an annulus abstract extending a previous result of tang 1 we prove the uniqueness of positive radial solutions of p u f u 0 subject to dirichlet boundary conditions on an annulus in r n with 2 p n under suitable hypotheses on the nonlinearity f this argument also provides an alternative proof for the uniqueness of positive solutions of the same problem in a finite ball see 9 in the complement of a ball or in the whole space r n see 10 3 11

a compact support principle for singular elliptic differential inequalities with a gradient term abstract in this paper a compact support principle is established for the elliptic differential inequality u u p k x f u u 0 for x large where p 1 k x 0 and f satisfies conditions f1 f3 below the main feature of this note is the presence of the gradient term u p and the singular coefficient function k x the result is optimal in some sense related to the power p and the decaying rate of k x at and the proof is based on finding appropriate comparison functions we also give a similar result in a bounded domain of r n n 2

structure of boundary blow up solutions for quasi linear elliptic problems ii small and intermediate solutions the structure of positive boundary blow up solutions to quasi linear elliptic problems of the form pu f u u on 1 0 in z1 z2 large small and intermediate solutions are obtained for sufficiently large it is known from part i see structure of boundary blow up solutions for quasilinear elliptic problems part i large and small solutions preprint that the large solution is the unique large solution to the problem we will see that the small solution is also the unique small solution to the problem while there are infinitely many intermediate solutions our results are new even for the case p 2

positive solutions of nonlinear p laplacian equations in rn abstract this paper studies the existence of positive solutions of the nonlinear p laplacian equation p u f x u in r n existence of infinitely many positive solutions is established via super sub solution methods decaying conditions on f x u at x are imposed necessity of the conditions and nonexistence of positive solutions are illustrated through decay growth estimates of solutions at infinity special situation and examples are checked in detail

on sufficient conditions for the existence of radially symmetric solutions of the p laplace equation abstract we consider the dirichlet problem for the p laplace equation with a nonlinear source new a priori estimates for solutions are established in particular we obtain an explicit formula which shows the dependence of the existence of a weak bounded radially symmetric solution on the dimension of the problem

remarks on time map for quasilinear equations abstract we present two different generalizations of r schaaf s 1990 18 time map formula to quasilinear equations including the case of p laplacian we give conditions for monotonicity and for convexity of the time map which imply uniqueness or multiplicity results for the corresponding dirichlet boundary value problem our time map formulas can be also used for effective computations of the global solution curves

nonexistence of positive radial solutions of a quasilinear neumann problem with a critical sobolev exponent note that problem 1 1 always admits the constant solution u0 k ny2 4 in adimurthi yadava 3 and budd knaap peletier 6 it has been proved that there exists a constant k0 0 such that for 0 k k0 1 1 admits a nonconstant solution for n 4 5 6 and does not for n 3 in this paper we study problem 1 1 when n 3 7 in this case the method used in 3 and 6 breaks down due to the fact that n 2 ny 2 2 which leads to a drastic change in the behaviour of the associated pohozhaev s functionals in contrast to the case n 4 5 6 we have

uniqueness of positive solutions of δu f u 0 in ℝn n 3 we study the uniqueness of radial ground states for the semilinear elliptic partial differential equation delta u f u 0 eqno in n we assume that the function f has two zeros the origin and u 0 0 above u 0 the function f is positive is locally lipschitz continuous and satisfies convexity and growth conditions of a superlinear nature below u 0 f is assumed to be non positive non identically zero and merely continuous our results are obtained through a careful analysis of the solutions of an associated initial value problem and the use of a monotone separation theorem it is known that for a large class of functions f the ground states of are radially symmetric in these cases our result implies that possesses at most one ground state

numerical solution of a class of singular free boundary problems involving the m laplace operator for a class of singular free boundary problems with applications in electromagnetism and plasma physics an analytical numerical approach is proposed based on the asymptotic expansion of the solution in the neighborhood of the singular points this approach was already used to approximate the solution of certain classes of singular boundary value problems on bounded lima and morgado 2009 14 and unbounded domains konyukhova et al 2008 12 here one parameter families of solutions of suitable singular cauchy problems describing the behavior of the solution at the singularities are derived and based on these families numerical methods for the approximation of the solution of the free boundary problems are constructed

monotonicity up to radially symmetric cores of positive solutions to nonlinear elliptic equations local moving planes and unique continuation in a non lipschitz case we prove local monotonicity and symmetry properties for nonnegative solutions of scalar field equations with nonlinearities which are not lipschitz our main tools are a local moving plane method and a unique continuation argument

multiplicity results for sign changing bound state solutions of a semilinear equation abstract in this paper we give conditions on f so that the problem u f u 0 x r n n 2 has at least two radial bound state solutions with any prescribed number of zeros and such that u 0 belongs to a specific subinterval of 0 this property will allow us to give conditions on f so that this problem has at least any given number of radial solutions having a prescribed number of zeros

ground states and free boundary value problems for the n laplacian in n dimensional space using a new gradient estimate we prove several theorems on the existence of radial ground states for the n laplace equation div u n 2 u f u 0 in rn n 1 and the existence of positive radial solutions for the associated dirichlet neumann free boundary value problem in a ball we treat exponentially subcritical critical and supercritical nonlinearities f u which also are allowed to have singularities at zero moreover we show that the local behavior of f at zero affects the existence in a crucial way this allows us to prove the existence of ground states for a large class of functions f u without imposing any restriction on their growth for large u

on radially symmetric minima of nonconvex functionals abstract we study a minimization problem in the space w1 10 br where br is the ball of radius r with center at the origin the functional considered is not necessarily convex under suitable assumptions we prove the existence of a radially symmetric decreasing solution by strengthening the assumptions we obtain uniqueness results finally we study under which assumptions and in which sense the solutions found solve the corresponding euler equation the proofs are very direct and simple they only make use of the functions t n introduced by the author arch rational mech anal 1999

on the existence of sign changing bound state solutions of a quasilinear equation abstract in this paper we establish the existence of bound state solutions having a prescribed number of sign changes for p m u f u 0 x r n n m 1 where m u u m 2 u our result is new even for the case of the laplacian m 2

the strong maximum principle revisited in this paper we rst present the classical maximum principle due to e hopf together with an extended commentary and discussion of hopf s paper we emphasize the comparison technique invented by hopf to prove this principle which has since become a main mathematical tool for the study of second order elliptic partial dierential equations and has generated an enormous number of important applications while hopf s principle is generally understood to apply to linear equations it is in fact also crucial in nonlinear theories such as those under consideration here in particular we shall treat and discuss recent generalizations of the strong maximum principle and also the compact support principle for the case of singular quasilinear ellip tic dierential inequalities under generally weak assumptions on the quasilinear operators and the nonlinearities involved our principal interest is in necessary and sucient con ditions for the validity of both principles in exposing and simplifying earlier proofs of corresponding results and in extending the conclusions to wider classes of singular oper ators than previously considered the results have unexpected rami cations for other problems as will develop from the exposition e g i two point boundary value problems for singular quasilinear ordinary dierential equations sections 3 4 ii the exterior dirichlet boundary value problem section 5 iii the existence of dead cores and compact support solutions i e dead cores at in nity section 7 iv euler lagrange inequalities on a riemannian manifold section 9 v comparison and uniqueness theorems for solutions of singular quasilinear dierential inequalities section 10 the case of p regular elliptic inequalities is briesy considered in section 11

glueing a peak to a non zero limiting profile for a critical moser trudinger equation abstract druet 6 proved that if f is a sequence of moser trudinger type nonlinearities with critical growth and if u solves 1 and converges weakly in h 0 1 to some u then the dirichlet energy is quantified namely there exists an integer n 0 such that the energy of u converges to 4 n plus the dirichlet energy of u as a crucial step to get the general existence results of 7 it was more recently proved in 8 that for a specific class of nonlinearities see 2 the loss of compactness i e n 0 implies that u 0 in contrast we prove here that there exist sequences f of moser trudinger type nonlinearities which admit a noncompact sequence u of solutions of 1 having a nontrivial weak limit

nodal solutions for a sublinear elliptic equation we consider radial solutions of u u u 2 u 0 in rd with d 1 0 and prove by a shooting method the existence of compactly supported solutions with any given number of nodes

the generalized cusp in ads 4 x cp3 and more one loop results from semiclassical strings we evaluate the exact one loop partition function for fundamental strings whose world surface ends on a cusp at the boundary of ads 4 and has a jump in cp3 this allows us to extract the stringy prediction for the abjm generalized cusp anomalous dimension gamma abjm cusp phi theta up to nlo in sigma model perturbation theory with a similar analysis we present the exact partition functions for folded closed string solutions moving in the ads 3 parts of ads 4 x cp3 and ads 3 x s 3 x s 3 x s 1 backgrounds results are obtained applying to the string solutions relevant for the ads 4 cft3 and ads 3 cft2 correspondence the tools previously developed for their ads 5 x s 5 counterparts

asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters part iii abstract we study the asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters thanks to an additional fixed parameter we show that two different critical exponents play a crucial role in the asymptotic analysis giving an explanation of the phenomena discovered in gazzola et al asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters ann inst h poincare anal non lineaire to appear and gazzola and serrin ann inst h poincare anal non lineaire 19 2002 477