existence of sign changing solutions for a nonlinear elliptic problem on rn with a hardy term abstract let n 3 1 p n 2 n 2 and 0 n 2 2 4 we study the existence of multiple sign changing solutions of u h 1 r n u u u x 2 u p 1 u in r n which are not radially symmetric

multiple positive solutions for a quasilinear elliptic equation involving singular potential and critical sobolev exponent in this paper we are concerned with a class of quasilinear elliptic equation in which the potential is singular at the origin and the nonlinearity is p sublinear near the origin by the ekeland variational principle and mountain pass theorem we prove that there exist at least two positive solutions under suitable assumptions on the potential and the nonlinearity

some further results for a class of weighted nonlinear elliptic equations following our previous work m lin multiple positive solutions for a class of weighted nonlinear elliptic equations preprint 2006 we give an exact growth order near zero for positive solutions of a class of weighted elliptic equations and use it to give the existence of multiple positive solutions and sign changing solutions of the considered problem

comparison results between linear and quasilinear elliptic boundary value problems on considere des resultats de comparaison du type suivant on suppose que u c 2 c satisfait un probleme aux valeurs limites elliptique quasi lineaire div a x u u u b x u u dans u sur et on suppose que v est la solution classique du probleme lineaire v const dans v sur on etudie l existence de situations ou u x v x x

oscillation criteria for second order nonlinear delay dynamic equations abstract in this paper we consider the second order nonlinear delay dynamic equation r t x t p t f x t 0 on a time scale t by employing the generalized riccati technique we will establish some new sufficient conditions which ensure that every solution oscillates or converges to zero the obtained results improve the well known oscillation results for dynamic equations and include as special cases the oscillation results for differential equations some applications and examples are considered to illustrate the main results

the brezis nirenberg problem for the laplacian with a singular drift in rn and sn abstract we consider the brezis nirenberg problem for the laplacian with a singular drift for a geodesic ball in both r n and s n 3 n 5 the singular drift we consider derives from a potential which is symmetric around the center of the geodesic ball here the potential is given by a parameter say times the logarithm of the distance to the center of the ball in both cases we determine the exact region in the parameter space for which positive smooth solutions of this problem exist and the exact region for which there are no solutions the parameter space is characterized by the geodesic radius of the ball and the coupling constant of the linear term of the brezis nirenberg problem

the eigenvalue problem and infinitely many sign changing solutions for an elliptic equation with critical hardy constant abstract by li yau s idea and a more precise hardy s inequality we obtain a new lower bound of eigenvalues for an elliptic equation with critical hardy constant then using the lower bounds we prove that there are infinitely many sign changing solutions for a class of elliptic equation with critical hardy constant which extends the results in schechter and zou 2005 9 to the critical hardy constant case

on the quasilinear elliptic problems with critical sobolev hardy exponents and hardy terms abstract let r n be a smooth bounded domain such that 0 n 3 in this paper we study the critical quasilinear elliptic problems p u u p 2 u x p u p t 2 x t u u q 2 x s u u w 0 1 p with dirichlet boundary condition where p u div u p 2 u 1 p n 0 n p p p 0 0 s t p p q p s p n s n p p t p n t n p p s and p t are the critical sobolev hardy exponents via variational methods we deal with the conditions that ensure the existence of positive solutions for the equation the results depend crucially on the parameters p q s and

oscillation for nonlinear second order dynamic equations on a time scale abstract we obtain some oscillation criteria for solutions to the nonlinear dynamic equation x q t x p t f x 0 on time scales in particular no explicit sign assumptions are made with respect to the coefficients p t q t we illustrate the results by several examples including a superlinear emden fowler dynamic equation

solutions of the quasilinear elliptic problem with a critical sobolev hardy exponent and a hardy type term abstract in the present paper the quasilinear elliptic problem with a critical sobolev hardy exponent and a hardy type term is considered by means of a variational method the existence of nontrivial solutions to the problem is obtained

solutions of a quasilinear elliptic problem involving a critical sobolev exponent and multiple hardy type terms in the present paper a quasilinear elliptic problem with a critical sobolev exponent and multiple hardy type terms is considered by means of a variational method the existence of positive solutions of the problem is obtained

oscillation of certain second order nonlinear difference equations abstract in this paper we consider the second order nonlinear difference equation formula where y n y n 1 y n is the forward difference operator f c r r f is nondecreasing and uf u 0 as u 0 q n is a real sequence some new sufficient conditions for the oscillation of all solutions of 1 are obtained

multiple positive solutions for a semilinear equation with critical exponent and prescribed singularity abstract this paper is devoted to the study of a semilinear equation of the form u v x u u 2 2 u h x u h 0 1 where r n n 3 is an open bounded domain with smooth boundary 0 n 2 2 2 2 2 n n 2 h x 0 and v x has prescribed finitely many singular points using variational methods we establish some existence and multiplicity of positive solutions for the problem

oscillation of second order damped dynamic equations on time scales abstract the study of dynamic equations on time scales has been created in order to unify the study of differential and difference equations the general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so called time scale which may be an arbitrary closed subset of the reals this way results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained in this paper by employing the riccati transformation technique we will establish some oscillation criteria for second order linear and nonlinear dynamic equations with damping terms on a time scale t our results in the special case when t r and t n extend and improve some well known oscillation results for second order linear and nonlinear differential and difference equations and are essentially new on the time scales t h n h 0 t q n for q 1 t n 2 etc some examples are considered to illustrate our main results

nodal and positive solutions for singular semilinear elliptic equations with critical exponents in symmetric domains abstract in this paper we consider the semilinear elliptic problem u u x 2 u k x u 2 2 u in u 0 on where is a smooth bounded domain in r n n 4 0 n 2 2 4 2 2 n n 2 is the critical sobolev exponent k x is a continuous function when and k x are invariant under a group of orthogonal transformations we prove the existence of nodal and positive solutions for 0 1 where 1 is the first dirichlet eigenvalue of x 2 on

oscillation criteria for second order nonlinear neutral delay dynamic equations in this paper we will establish some oscillation criteria for the second order nonlinear neutral delay dynamic equation r t y t p t y t f t y t 0 on a time scale t here 0 is a quotient of odd positive integers with r t and p t real valued positive functions defined on t to the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales so this paper initiates the study

multiple positive solutions for semilinear elliptic equations involving multi singular inverse square potentials and concave convex nonlinearities abstract in this paper we deal with the existence and multiplicity of positive solutions for the multi singular semilinear elliptic equations u i 1 k i x a i 2 u u 2 2 u u q 2 u x where r n n 3 is a smooth bounded domain such that the points a i i 1 2 k k 2 are different 0 i n 2 2 2 0 1 q 2 and 2 2 n n 2 the results depend crucially on the parameters q and i for i 1 2 k

solutions to critical elliptic equations with multi singular inverse square potentials abstract let be an open bounded domain in r n n 3 with smooth boundary we are concerned with the multi singular critical elliptic problem u i 1 k i u x a i 2 q x u 2 2 u u h 0 1 where i r 2 2 n n 2 a i 1 i k and q x is a positive bounded function on using moser iteration we prove the asymptotic behavior of solutions for at points a i 1 i k by exploiting the effect of the coefficient of the critical nonlinearity we by means of a variational method establish the existence of positive and sign changing solutions for problem

on the existence and multiplicity of solutions for a class of singular elliptic problems abstract in this paper we prove the existence and multiplicity of solutions for a class of elliptic problems with singular coefficients we also prove the existence of a sign changing solution for a brezis nirenberg type problem with hardy potential and singular coefficients our proofs are based on the krasnoselskii genus the concentration compactness principle of lions and the linking theorem

asymptotic behavior of solutions of a third order nonlinear dynamic equation on time scales in this paper we will establish some sufficient conditions which guarantee that every solution of the third order nonlinear dynamic equation c t a t x t q t f x t 0 t t0 oscillates or converges to zero

some further results on a semilinear equation with concave convex nonlinearity abstract following our previous work j math anal appl 295 2004 341 we give an exact growth order near zero for positive solutions of a class of elliptic equation and use it to give the existence of multiple solutions with negative energy multiple positive solutions and sign changing solutions of the considered problem even in the particular case our results extend a previous work by ambrosetti garcia peral j funct anal 137 1996 219

existence of solutions for a nonlinear pde with an inverse square potential abstract via linking theorem and delicate energy estimates the existence of nontrivial solutions for a nonlinear pde with an inverse square potential and critical sobolev exponent is proved this result gives a partial positive answer to an open problem proposed in ferrero and gazzola j differential equations 177 2001 494

oscillation criteria for second order matrix dynamic equations on a time scale we obtain oscillation criteria for a second order self adjoint matrix differential equation on a measure chain in terms of the eigenvalues of the coefficient matrices and the graininess function we illustrate our results with some nontrivial examples

on the elliptic problems with critical weighted sobolev hardy exponents abstract let r n be a smooth bounded domain such that 0 n 3 in this paper we deal with the conditions that ensure the existence of nontrivial solutions for the elliptic equation div x 2 a u u x 2 1 a u p 2 x b p u u with dirichlet boundary condition which involving the caffarelli kohn nirenberg inequalities the results depend crucially on the parameters a b and

oscillatory behavior of second order half linear damped dynamic equations by refining the generalized riccati transformation technique we obtain new oscillation criteria for a class of second order half linear delay dynamic equations with damping on a time scale assumptions in our theorems are less restrictive these results complement and improve related contributions to the subject in particular the results obtained amend related results reported in the literature

properties of some balayage operators with applications to quadrature domains and moving boundary problems in this paper we define and derive some properties of a fairly wide class of balayage operators we also apply our results to obtain geometric information about quadrature domains and solutions of certain moving boundary problems e g for hele shaw flows actually it was our interest in these applications that was the starting point for the present investigations and the balayage operators were developed rather as a tool our balayage operators are denoted f fp fp r and they depend on an open set r c i and a measure p in r actually we always assume that p has a density also denoted p in l r what f does is replace a given measure p by the nearest one in the energy norm v f p which satisfies v 5 p in r this presupposes that fi has finite energy if this is not the case there is another definition which is the one we will actually use among all distributions u in irn satisfying u i 17 the newtonian potential of p in irn and au i p in r there is a largest one u vp then f p ai by definition the operator of interest in the above mentioned applications is f with r irn p 1 the definition of f y when u has finite energy can be formulated as an elliptic variational inequality or linear complementarity problem and this is one of the standard tools when dealing with free boundary problems of the above kind see l 4 however we think that the balayage point of view is quite natural in our context the reason for considering the balayage operators for more general r and p than r rn p 1 is firstly that we need them in the proof of one of our theorems theorem 4 1 secondly and perhaps more important is that we think that the more general operators have quite a lot of intrinsic interest in particular as they turn out to contain classical balayage of positive measures as a special case namely with r bounded and p 0 recently we also learnt that discrete or numerical versions of these operators have been developed and applied in geophysics during the last three decades by dimiter zidarov who calls the balayage process partial gravi equivalent mass scattering see 5 and also for example 6 the paper is organized as follows in section 2 we define and establish the basic properties of our operators at the end we also explain their relation to quadrature domains and certain moving boundary problems section 3 is devoted to one single theorem theorem 3 1 and its corollaries these results concern the f with r lrn p 1 and give rather good geometric

oscillation theorems for general quasilinear second order difference equations the authors consider difference equations of the form where an 0 qn 0 f and are continuous real valued functions and uf u 0 for u 0 they give oscillation results for equation e examples are included to illustrate the results

elliptic problems with critical exponents and hardy potentials this paper is devoted to the existence of positive solutions of a dirichlet problem with critical exponent and a singular potential under various assumption on the domain omega which include some kinds of unbounded domains we prove the existence of ground states and of symmetric solutions c 2002 elsevier science usa all rights reserved

on the singular critical quasilinear problems in rn abstract in this paper we study the following quasilinear problem p u u p 2 u u p 2 u x p u p 2 u f u u w r 1 p r n where n 3 2 p n p n p n p and f u satisfies some kinds of assumptions by employing the variational methods and analytic techniques we obtain the existence of a nontrivial solution to the problem under some assumptions for the parameters p n and at last we manage to prove that the problem has at least a pair of nodal solutions u k for each k n and both u k possess exactly k nodes

applications of variational inequalities to the existence theorem on quadrature domains in this paper we shall study quadrature domains for the class of subharmonic functions by using the theory of variational inequalities we shall give a new proof of the existence and uniqueness theorem as an application we deal with hele shaw flows with a free boundary and show that their two weak solutions one of which was defined by the author using quadrature domains and the other was defined by gustafsson 3 using variational inequalities are identical with each other introduction in a previous paper 7 the author has defined the quadrature domains of positive measures for the class of subharmonic functions and studied their applications to complex function theory let v be a finite positive measure on the two dimensional euclidean space r2 let sl1 q be the class of subharmonic functions in an open set q which are integrable with respect to the two dimensional lebesgue measure m a nonempty open set q is called a quadrature domain of v for class sl1 if qi v is concentrated in q namely v qc 0 where qc denotes the complement of q qii jus s dv 1 a e in a connected open set w with finite area f 0 a e in wc and jf dm m w then q fm sl1 0 and there exists a minimum domain w in q fm sl1 namely q e q jm sl1 if and only if w c q and m q w o received by the editors october 5 1981 and in revised form february 23 1982 1980 mathematics subject classification primary 3 1aos 3 1bos secondary 35a15

weak eigenfunctions for the linearization of extremal elliptic problems abstract we consider the semilinear elliptic problem formula where is a nonnegative parameter and g is a positive nondecreasing convex nonlinearity there exists a value of the parameter which is extremal in terms of existence of solution we study the linearization of the semilinear problem at the extremal weak solution corresponding to the parameter in some cases this linearized problem has discrete and positive h 1 0 spectrum however we prove that there always exists a positive weak eigenfunction in l 1 with eigenvalue zero for this linearized problem the zero l 1 eigenvalue is coherent with the nonexistence of solutions of the semilinear problem for finally we find all weak eigenfunctions and eigenvalues for the linearization of the extremal problem when is the unit ball and g u e u or g u 1 u p

multiple positive solutions for a quasilinear elliptic problem involving critical sobolev hardy exponents and concave convex nonlinearities abstract let r n n 3 be a bounded smooth domain containing the origin in this paper by using variational methods the multiplicity of positive solutions is obtained for a quasilinear elliptic problem p u u p 2 u x p u p t 2 x t u u q 2 x s u u w 0 1 p with dirichlet boundary condition where p u div u p 2 u 1 p n 0 n p p p 0 0 s t p 1 q p and p t p n t n p is the critical sobolev hardy exponent

oscillation criteria of second order half linear dynamic equations on time scales in this paper by using the riccati transformation technique chain rule and inequalitya l lab l 1 l 1 b l 0 l 1 where a and b are positive constants we will establish some oscillation criteria for the second order half linear dynamic equation p t x d t c d q t x c t 0 t a b on time scales where c 1 is an odd positive integer our results not only unify the oscillation of half linear differential and half linear difference equations but can be applied on different types of time scales and improve some well known results in the difference equation case some examples are considered here to illustrate our main results

positive solutions for neumann elliptic problems involving critical hardy sobolev exponent with boundary singularities abstract in this paper we consider the existence of positive solution for some semilinear elliptic equations with neumann boundary condition involving a critical hardy sobolev exponent and hardy terms with boundary singularities using mountain pass lemma without ps condition and the strong maximum principle we get the existence of a positive solution

combined effects of concave and convex nonlinearities in some elliptic problems abstract this paper deals with a class of semilinear elliptic dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results

hardy sobolev critical singular elliptic equations with mixed dirichlet neumann boundary conditions abstract three important inequalities the poincare hardy and generalized poincare inequalities on the mixed boundary conditions are firstly established by some analytical techniques then the existence and multiplicity of positive solutions are studied for a class of semilinear elliptic equations with mixed dirichlet neumann boundary conditions involving hardy terms and hardy sobolev critical exponents by using the variational methods

a note on the sign changing solutions to elliptic problems with critical sobolev and hardy terms let rn be a bounded domain such that 0 n 7 2 2nn 2 we obtain existence of sign changing solutions for the dirichlet problem u u x 2 u 2 2u u on u 0 on for suitable positive numbers and

existence of positive solutions for a class of degenerate or singular equations abstract in this paper we consider the class of equations div x 2 a u a x 2 a u u x b p x r n 0 where a b r n 2 a r 0 and 0 p 2 n n 2 1 a b in various range of the parameters involved we obtain both results of existence and nonexistence of positive solutions by combining the mountain pass lemma and the pohozaev type identity

solutions for a singular critical growth problem with a weight in this paper we consider some semilinear elliptic equations with hardy potential by using linking theorem in p rabinowitz minimax methods in critical points theory with applications to differential equations cbms reg conf ser math vol 65 amer math soc providence ri 1986 and analyzing the effect of nonlinearities we establish the existence of nontrivial solutions

oscillatory and nonoscillatory behavior of second order functional difference equations abstract the asymptotic and oscillatory behavior of solutions of second order difference equations with deviating argument of the form a n y n p n y n q n f y n 1 0 n 0 1 is studied examples are inserted to illustrate the results

existence of multiple positive solutions to singular elliptic systems involving critical exponents abstract this paper is devoted to investigate multiple positive solutions to a singular elliptic system where the nonlinearity involves a combination of concave and convex terms by exploiting the effect of the coefficient of the critical nonlinearity and a variational method we establish the main result which is based on the argument of the compactness

on the quasilinear elliptic problem with a critical hardy sobolev exponent and a hardy term abstract in the present paper a quasilinear elliptic problem with a critical sobolev exponent and a hardy type term is considered by means of a variational method the existence of nontrivial solutions for the problem is obtained the result depends crucially on the parameters p t s and

on the elliptic problems involving multi singular inverse square potentials and critical sobolev exponent abstract assume that r n n 3 is a smooth bounded domain a i are different points in i 1 2 k k 2 0 2 q 2 n 2 2 2 2 2 n n 2 is the critical sobolev exponent we study the existence and asymptotic behavior of solutions to the multi singular critical elliptic problem u u i 1 k x a i 2 u 2 2 u u q 2 u with dirichlet boundary condition the results depend crucially on the parameters and q

multiple positive solutions for robin problem involving critical weighted hardy sobolev exponents with boundary singularities abstract this paper deals with the existence of positive solutions for robin elliptic problems involving critical weighted hardy sobolev exponents with boundary singularities using the caffarelli kohn nirenberg inequalities and variational methods we prove the existence and multiplicity of positive solutions

oscillation of nonlinear dynamic equations on time scales by means of riccati transformation techniques we establish some new oscillation criteria for the second order nonlinear dynamic equation p t x d t d q t f x s t 0 email protected a b on time scales in terms of the coefficients and the graininess function we illustrate our results with a nontrivial example

existence and multiplicity of positive solutions for elliptic equation with critical weighted hardy sobolev exponents and boundary singularities abstract we study a semilinear elliptic equation involving critical weighted hardy sobolev exponents with boundary singularities the existence and multiplicity of positive solutions are established our method relies upon ekeland s variational principle and mountain pass lemma

hardy sobolev maz ya type equations in bounded domains abstract we study the regularity palais smale characterization and existence nonexistence of solutions of the hardy sobolev maz ya equation u u y 2 u p t 1 u y t in a bounded domain in r n where x r n is denoted as x y z r k r n k and p t n 2 2 t n 2 we show different behaviors of ps sequences depending on t 0 or t 0

solutions for semilinear elliptic equations with critical exponents and hardy potential abstract in this paper we answer affirmatively an open problem cf theorem 4 in ferrero and gazzola j differential equations 177 2001 494 let 0 be an open bounded domain r n n 5 and assume that 0 n 2 2 2 n 2 n 2 then for all 0 there exists a nontrivial solution with critical level in the range 0 1 n s n 2 for the problem u u x 2 u u 2 2 u in u 0 on

dynamic equations on time scales a survey the study of dynamic equations on time scales which goes back to its founder stefan hilger 1988 is an area of mathematics that has recently received a lot of attention it has been created in order to unify the study of differential and difference equations in this paper we give an introduction to the time scales calculus we also present various properties of the exponential function on an arbitrary time scale and use it to solve linear dynamic equations of first order several examples and applications among them an insect population model are considered we then use the exponential function to define hyperbolic and trigonometric functions and use those to solve linear dynamic equations of second order with constant coefficients finally we consider self adjoint equations and more generally so called symplectic systems and present several results on the positivity of quadratic functionals

a necessary and sufficient condition for oscillation of the sturm liouville dynamic equation on time scales in this paper we investigate oscillatory properties of the second order sturm liouville dynamic equation on a time scale r t x c t x 0 using the so called trigonometric transformation we establish a necessary and sufficient condition for oscillation of this condition is then used to derive an explicit oscillation criterion for this equation

application of variational inequalities to the moving boundary problem in a fluid model for biofilm growth vi betraktar ett problem med rorlig rand motsvarande ett gransfall i en fluid modell for biofilmstillvaxt som foreslagits av j dockery och i klapper siam j appl math 62 3 2001

existence and multiplicity of positive solutions for some hardy sobolev critical elliptic equation with boundary singularities abstract in this paper by ekeland s variational principle and strong maximum principle we consider the existence and multiplicity of positive solutions for some semilinear elliptic equation involving critical hardy sobolev exponents and hardy terms with boundary singularities

on a disconjugacy criterion for second order dynamic equations on time scales in this paper a sufficient condition for disconjugacy of second order differential equations on time scales is given for this purpose a mean value result on time scales is employed

the effect of domain topology on the number of positive solutions for singular elliptic problems involving the caffarelli kohn nirenberg inequalities abstract in this paper we consider the critical singular equation involving the caffarelli kohn nirenberg inequalities of the type p 1 div x 2 a u u x 2 1 a u q 2 u x d d u p 2 u x b p x u 0 x u 0 x here is a bounded domain with smooth boundary in r n and contains 0 in its interior 0 a 2 n 2 2 2 n 3 a b a 1 a d a 1 p p a b 2 n n 2 1 a b d d a d 2 n n 2 1 a d is a positive parameter and 2 q d by lusternik schnirelmann category theory we prove that the problem p 1 has at least cat positive solutions

the existence of solutions for quasilinear elliptic problems with combined critical sobolev hardy terms abstract in this paper we investigate the quasilinear elliptic equations with dirichlet boundary conditions and combined critical sobolev hardy terms on bounded smooth domains r n n 3 and prove the existence and multiplicity of solutions by employing ekeland s variational principle

a variational inequality approach to hele shaw flow with a moving boundary this paper is concerned with the study of a mathematical model of the injection of fluid into a finite hele shaw cell the mathematical problem is one of solving laplace s equation in an unknown region whose boundary changes with time by a transformation of the dependent variable an elliptic variational inequality formulation of the moving boundary problem is obtained the variational inequality is shown to have a unique solution up to the time at which the cell is filled regularity results for the solution of the inequality are obtained by studying a penalty approximation of the inequality

multiple positive solutions for semilinear elliptic equations with critical weighted hardy sobolev exponents abstract the existence and multiplicity of positive solutions are obtained for a class of semilinear elliptic equations with critical weighted hardy sobolev exponents and the concave convex nonlinearity by variational methods and some analysis techniques

existence and multiplicity of positive solutions for a perturbed semilinear elliptic equation with two hardy sobolev critical exponents abstract in this paper by applying ekeland s variational principle and strong maximum principle we investigate the existence and multiplicity of positive solutions for a perturbed semilinear elliptic equation with two critical hardy sobolev exponents and boundary singularities

positive solutions to the weighted critical quasilinear problems in a bounded domain with smooth boundary we consider a kind of weighted quasilinear elliptic problem which satisfies dirichlet boundary condition and involves the hardy sobolev inequality by the analytic techniques we first get the properties of the extremal functions by which the best hardy sobolev constant is achieved then by the variational methods the existence of positive solutions to the problem is verified by careful estimates and computations

an application of discrete inequality to second order nonlinear oscillation abstract by using some simple discrete inequalities oscillation criteria are provided for the second order difference equations 2 y n a n 1 y n 1 0 n n where the operator is defined by yn yn 1 yn an is a real sequence the function is such that u u 0 for u 0 and u v g u v u v for u v 0 for some nonnegative function g

existence and non existence results for quasilinear kirchhoff problems with the hardy sobolev exponent abstract in this paper we are concerned with the following quasilinear elliptic problems of kirchhoff type a b r n u p u p x p d x 1 p u u p 2 u x p u p 2 u x f x u q 2 u x where a 0 b 0 1 0 n p p p 0 p and q 1 p are constants and p u div u p 2 u with 1 p n p p n n p is the hardy sobolev exponent we establish the non existence and the existence of infinitely many nontrivial solutions when 0 for the above problem also for suitable weight function f x we prove the existence of two positive solutions by using nehari manifold and fibering map when 0 is sufficiently small

existence and multiplicity of positive solutions for a class of semilinear elliptic equations involving hardy term and hardy sobolev critical exponents abstract existence and multiplicity of positive solutions are studied for a class of semilinear elliptic equations with hardy term hardy sobolev critical exponents and sublinear nonlinearity by variational methods and some analysis techniques

existence and multiplicity of solutions to a singular elliptic system with critical sobolev hardy exponents and concave convex nonlinearities abstract this paper is concerned with a singular elliptic system which involves the caffarelli kohn nirenberg inequality and critical sobolev hardy exponents the existence and multiplicity results of positive solutions are obtained by variational methods

existence and multiplicity of solutions for semilinear elliptic equations with critical weighted hardy sobolev exponents abstract some existence results are obtained for solutions of semilinear elliptic equations with critical weighted hardy sobolev exponents and superlinear nonlinearity by the variational methods and some analysis techniques