conformal deformation of a riemannian metric to a scalar flat metric with constant mean curvature on the boundary one of the most celebrated theorems in mathematics is the riemann mapping theorem it says that an open simply connected proper subset of the plane is conformally diffeomorphic to the disk in higher dimensions very few regions are conformally diffeomorphic to the ball however we can still ask whether a domain is conformally diffeomorphic to a manifold that resembles the ball in two ways namely it has zero scalar curvature and its boundary has constant mean curvature in this paper we generalize the riemann mapping theorem to higher dimensions in that sense

on a liouville type equation with sign changing weight in this paper we study the existence nonexistence and multiplicity of non negative solutions for the family of problems u a x e f x u u h 0 where is a bounded domain in r2 and 0 is a parameter the coefficient a x is allowed to change sign the techniques used in the proofs are a combination of upper and lower solutions the trudingermoser inequality and variational methods note that when f x u 0 the equation is of liouville type

multiple positive solutions for semilinear elliptic systems with nonlinear boundary condition abstract in this paper we study the nonlinear boundary value problem u u f u u v in v v f v u v in u n g u u v v n g v u v on where is a bounded domain in r n with smooth boundary and the pairs of f u f v and g u g v are subcritical nonlinearities with the help of the nehari manifold we show existence of at least two positive solutions when one of the pairs of f u f v and g u g v is sublinear and the other one superlinear

infinitely many nodal solutions for nonlinear elliptic equations with combined nonlinearities and lack of symmetry abstract we present a new approach to studying a parametric nonlinear dirichlet problem driven by a nonhomogeneous differential operator and with a reaction which is concave i e p 1 sublinear near zero and convex i e p 1 superlinear near the reaction term is not assumed to be odd we show that for all small values of the parameter 0 the problem has infinitely many nodal solutions

convexity properties of harmonic measures it is shown that any convex combination of harmonic measures where u1 uk are relatively compact open neighborhoods of a given point d2 can be approximated by a sequence of harmonic measures such that each wn is an open neighborhood of x in u1uk this answers a question raised in connection with jensen measures moreover it implies that for every green domain x containing x the extremal representing measures for x with respect to the convex cone of potentials on x these measures are obtained by balayage of the dirac measure at x on borel subsets of x are dense in the compact convex set of all representing measures this is achieved approximating balayage on open sets by balayage on unions of balls which are pairwise disjoint and very small with respect to their mutual distances and then reducing the size of these balls in a suitable manner these results which are presented simultaneously for the classical potential theory and for the theory of riesz potentials can be sharpened if the complements or the boundaries of the open sets have a capacity doubling property the methods developed for this purpose continuous balayage on increasing families of compact sets approximation using scattered sets with small capacity finally lead to answers even in a very general potential theoretic setting covering a wide class of second order partial differential operators uniformly elliptic or in divergence form or sums of squares of vector fields satisfying hormander s condition for example sub laplacians on stratified lie algebras

local superlinearity and sublinearity for the p laplacian we study the existence nonexistence and multiplicity of positive solutions for a family of problems updelta p u f lambda x u u in w 0 1 p upomega where is a bounded domain in mathbb r n n p and 0 is a parameter the family we consider includes the well known nonlinearities of ambrosetti brezis cerami type in a more general form namely lambda a x u q b x u r where 0 leqslant q p 1 r leqslant p 1 here the coefficient a x is assumed to be nonnegative but b x is allowed to change sign even in the critical case preliminary results of independent interest include the extension to the p laplacian context of the brezis nirenberg result on local minimization in w 0 1 p text and c 0 1 a c 1 alpha estimate for equations of the form updelta p u h x u with h of critical growth a strong comparison result for the p laplacian and a variational approach to the method of upper lower solutions for the p laplacian

limits as p of p laplacian concave convex problems abstract we study the behavior as p of the sequel of positive weak solutions of the concave convex problem p div u p 2 u u q p u r p in u 0 in u 0 on where r n is a bounded domain 0 and the exponents q r satisfy lim p q p p 1 q lim p r p p 1 r with 0 q 1 r we characterize any positive uniform limit of a sequence of weak solutions of p as a viscosity solution of min u max u q u r u 0 in notice that the limit process decouples the nonlinearity we obtain existence non existence and global multiplicity of positive viscosity solutions of the limit problem in terms of the parameter

multiplicity of solutions to a nonlinear boundary value problem of concave convex type abstract problem p p u u p 2 u u r 1 u x u p 2 u u s 1 u x where r n is a bounded smooth domain is the unit outward normal at p is the p laplacian operator and 0 is a parameter was studied in sabina de lis 2011 and sabina de lis and segura de leon in press among other features it was shown there that when exponents lie in the regime 1 s p r a minimal positive solution exists if 0 for a certain finite while no positive solutions exist in the complementary range furthermore in the radially symmetric case a second positive solution exists for varying in the same full range 0 provided r p our main achievement in this work just asserts that such global multiplicity feature holds true when is a general domain to show such result the well known brezis nirenberg variational result in brezis and nirenberg 1993 must be extended to the framework of p this is the second main contribution in the present work

on semilinear elliptic equations involving concave convex nonlinearities and sign changing weight function abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of positive solutions for semilinear elliptic equations with a sign changing weight function with the help of the nehari manifold we prove that there are at least two positive solutions for eq e f in bounded domains

combined effects of concave and convex nonlinearities in elliptic equation on rn abstract in this paper we consider the existence of two positive solutions of the elliptic equation on r n with concave and convex nonlinearities

multiple positive solutions of a nonlinear boundary value problem involving a sign changing weight abstract in this paper we study the multiplicity of positive solutions for the following elliptic equation u u 0 in r n u a x u q 2 u b x u p 2 u on r n where 1 q 2 p 2 2 2 n 1 n 2 if n 3 2 if n 2 r n x x n r n 1 r x n 0 is an upper half space in r n 0 and the functions a and b are satisfying some suitable conditions using the decomposition of the nehari manifold we prove that the equation has at least two positive solutions provided is sufficiently small

nonnegative solutions of elliptic problems with sublinear indefinite nonlinearity abstract we are concerned with existence nonexistence and multiplicity of nonnegative solutions for the elliptic problem u a x u q b x u p in u 0 on where is a bounded domain in r n r 0 q 1 p 2 1 and a b are bounded functions with b x 0 and a x changes its sign

positive solutions of superlinear elliptic equations abstract in this paper we study the existence of two positive solutions of superlinear elliptic equations without assuming the conditions which have been used in the literature to deduce either the p s condition or a priori bounds of positive solutions the first solution is proved as the minimal positive solution while the second one is obtained as the limit of a gradient flow whose starting point is properly chosen the dependence of the minimal solution upon a parameter is also considered

exact multiplicity and bifurcation diagrams of positive solutions of a one dimensional multiparameter prescribed mean curvature problem abstract we study the exact multiplicity and bifurcation diagrams of positive solutions u c 2 l l c l l of the one dimensional multiparameter prescribed mean curvature problem u x 1 u x 2 u p u q l x l u l u l 0 where 0 is a bifurcation parameter l 0 the radius of the one dimensional ball l l is an evolution parameter and 0 p q are two constants we prove that the problem has at most two positive solutions for any 0 p q and l 0 in addition if 0 p q q p p 1 2 p 1 we give a classification of totally three qualitatively different bifurcation diagrams on the u plane for any l 0 for any fixed p 0 and q q p p 2 2 2 p 3 we prove that there exist positive l l such that the bifurcation diagrams on the u plane are individually qualitatively different for the cases i 0 l l or l l ii l l iii l l l

uniqueness and non uniqueness for a system of heat equations with nonlinear coupling at the boundary we study the uniqueness problem for non negative solutions of a system of two heat equations ut u vt v in a bounded smooth domain with nonlinear boundary conditions u v v u we prove that for identically zero initial data u x 0 v x 0 0 0 the zero solution is unique if and only if if pq 1 moreover in the case of non negative non trivial initial data the solution is always unique

density of extremal measures in parabolic potential theory it is shown that for the heat equation on mathbb r d times mathbb r d 1 any convex combination of harmonic caloric measures mu x u 1 dots mu x u k where u 1 u k are relatively compact open neighborhoods of a given point x can be approximated by a sequence mu x w n n in mathbb n of harmonic measures such that each w n is an open neighborhood of x in u 1 cup dots cup u k moreover it is proven that for every open set u in mathbb r d 1 containing x the extremal representing measures for x with respect to the convex cone of potentials on u these measures are obtained by balayage with respect to u of the dirac measure at x on borel subsets of u are dense in the compact convex set of all representing measures since essential ingredients for a proof of corresponding results in the classical case or more general elliptic situations see hansen and netuka in adv math 218 4 1181 1223 2008 are not available for the heat equation an approach heavily relying on the transit character of the hyperplanes mathbb r d times c c in mathbb r is developed in fact the new method is suitable to obtain convexity results for limits of harmonic measures and the density of extremal representing measures on x x times mathbb r for practically every space time structure which is given by a sub markov semigroup p t t 0 on a space x such that there are strictly positive continuous densities t x y mapsto p t x y with respect to a non atomic measure on x in particular this includes many diffusions and corresponding symmetric processes given by heat kernels on manifolds and fractals moreover the results may be applied to restrictions of the space time structure on arbitrary open subsets

a convex concave problem with a nonlinear boundary condition in this paper we study the existence of nontrivial solutions of the problem u u u p 2uin u u q 2uon with 1 q 2 n 1 n 2 and 1 p 2n n 2 in the concave convex case i e 1 q 2 p if is small there exist two positive solutions while for large there is no positive solution when p is critical and q subcritical we obtain existence results using the concentration compactness method finally we apply the implicit function theorem to obtain solutions for small near u0 1

elliptic problems with lack of compactness via a new fixed point theorem abstract this paper provides a new fixed point theorem for increasing self mappings g b b of a closed ball b x where x is a banach semilattice which is reflexive or has a weakly fully regular order cone x by means of this fixed point theorem we are able to establish existence results of elliptic problems with lack of compactness

lyapunov inequality for an elliptic problem with the robin boundary condition in this paper we study the lq lyapunov type inequalities for quasi linear elliptic problems with the robin boundary conditions in which the principal part of the equation is p laplacian operator similar problems have been considered by several authors for linear elliptic problems with the dirichlet or neumann boundary conditions we show that the critical value of the problem is n 1 p 1 where n is the dimension of the domain we reveal the relation between this critical value and the critical exponent p of the trace sobolev embedding w1 p lp

existence and multiplicity of positive solutions for neumann problems involving singularity and critical growth abstract in this paper the existence and multiplicity of positive solutions is established for a class of numann problems of the form u u 2 1 u in u 0 on here is a smooth bounded domain in r n n 3 2 2 n n 2 0 1 and 0 is a real parameter is the outer normal derivative the main technical approach is based on the variational and perturbation methods

multiple solutions for a semilinear problem with combined terms and nonlinear boundary conditions abstract we consider the problem u u f x u in u h x u q 2 u on where r n is a smooth bounded domain n 3 1 q 2 and h belongs to an appropriated lebesgue space in our main results we suppose that f is an asymptotically linear function and we obtain multiplicity of solutions when the norm of h is small we also present a multiplicity result in the case that f is nonquadratic at infinity

combined effects in nonlinear problems arising in the study ofanisotropic continuous media the paper deals with the study of a lane emden fowler equation with dirichlet boundary condition and variable potential functions the analysis developed in this paper combines monotonicity methods with variational arguments

on a dirichlet problem with p q laplacian and parametric concave convex nonlinearity abstract a homogeneous dirichlet problem with p q laplace differential operator and reaction given by a parametric p convex term plus a q concave one is investigated a bifurcation type result describing changes in the set of positive solutions as the parameter 0 varies is proven since for every admissible the problem has a smallest positive solution u both monotonicity and continuity of the map u are studied

a semilinear elliptic system involving nonlinear boundary condition and sign changing weight function abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of solutions for a semilinear elliptic system e involving nonlinear boundary condition and sign changing weight function with the help of the nehari manifold we prove that the system has at least two nontrivial nonnegative solutions when the pair of the parameters belongs to a certain subset of r 2

existence of solutions for perturbed fractional p laplacian equations abstract the purpose of this paper is to investigate the existence of weak solutions for a perturbed nonlinear elliptic equation driven by the fractional p laplacian operator as follows p s u v x u p 2 u a x u r 2 u b x u q 2 u in r n where is a real parameter p s is the fractional p laplacian operator with 0 s 1 p p r min q p s and v a b r n 0 are three positive weights using variational methods we obtain nonexistence and multiplicity results for the above mentioned equations depending on and according to the integrability properties of the ratio a q p b r p our results extend the previous work of autuori and pucci 2013 5 to the fractional p laplacian setting furthermore we weaken one of the conditions used in their paper hence the results of this paper are new even in the fractional laplacian case

critical point theory on partially ordered hilbert spaces we develop some abstract critical point theory in order to prove that boundary value problems like the model problem formula on a bounded domain rn 2 p 2n n 2 have infinitely many sign changing solutions uk k n which are not comparable that is uk ul and uk ul change sign for k l we also show that there are no subsolutions u such that u uk for some k and u is positive somewhere a corresponding nonexistence result applies to supersolutions related results on the existence of sign changing solutions hold for other classes of nonlinearities

interaction between the geometry of the boundary and positive solutions of a semilinear neumann problem with critical nonlinearity abstract we consider the problem u u un 2 n 2 u 0 in u v 0 on where is a bounded smooth domain in r n n 3 we show that for large least energy solutions of the above problem have a unique maximum point p on and the limit points of p as are contained in the set of the points of maximum mean curvature we also prove that if has k peaks then the equation has at least k solutions for large

existence of fixed points for some convex operators and applications to multi point boundary value problems existence results of fixed points for some convex operators are given by means of fixed point theorem of cone expansion and compression then they are applied to nonlinear multi point boundary value problems

the nehari manifold for a semilinear elliptic system involving sign changing weight functions abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of solutions for a semilinear elliptic system e with sign changing weight function with the help of the nehari manifold we prove that the system has at least two nontrivial nonnegative solutions when the pair of the parameters belongs to a certain subset of r 2

harmonic measures for a point may form a square abstract let x be a green domain in r d d 2 x x and let m x p x denote the compact convex set of all representing measures for x recently it has been proven that the set of harmonic measures x u u open in x x u which is contained in the set of extreme points of m x p x is dense in m x p x in this paper it is shown that m x p x is not a simplex and hence not a poulsen simplex this is achieved by constructing open neighborhoods u 0 u 1 u 2 u 3 of x such that the harmonic measures x u 0 x u 3 are pairwise different and x u 0 x u 2 x u 1 x u 3 in fact these measures form a square with respect to a natural l 2 structure since the construction is mainly based on having certain symmetries it can be carried out just as well for riesz potentials the heisenberg group or any stratified lie algebra and the heat equation or more general parabolic situations

existence of multiple solutions for singular elliptic problems with nonlinear boundary conditions in this paper the existence and multiplicity results of solutions for the following quasilinear elliptic problem 0 1 div x ap u p 2 u h x u p 2u g x u r 2u x x ap u p 2 u f x u q 2u x are established where is an exterior domain in rn with the compact and smooth boundary by the variational methods we prove that the problem 0 1 has at least two positive solutions at the last part of the paper we also consider the critical case and prove the existence of solution by concentration compactness principle

multiple solutions to p laplacian problems with concave nonlinearities in this paper we study the p laplacian type elliptic problems with concave nonlinearities using some asymptotic behavior of f at zero and infinity three nontrivial solutions are established

compact sobolev embeddings and positive solutions to a quasilinear equation with mixed nonlinearities abstract in this paper we study the existence and multiplicity results of nontrivial positive solutions to the following quasilinear elliptic equation on r n when n 2 p u k x u r 1 v x u q 1 here k x v x 0 are suitable potentials 1 p r q and 0 is a parameter to study this problem some compact embedding results regarding m v q p r n l k r r n are proved that unify and extend some recent results of the author 12 13 14

nonlinear elliptic equations with an asymptotically linear reaction term abstract we examine nonlinear elliptic problems driven by the p laplacian and with a reaction term which asymptotically at exhibits a p 1 linear growth 1 p using tools from critical point theory and from morse theory we prove existence theorems and we provide information about the sign of the solutions

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

exact multiplicity and numerical computation of solutions for two classes of non autonomous problems with concave convex nonlinearities abstract we establish the exact multiplicity of positive solutions and the global solution structure for two classes of problems on circular domains the first class involves non autonomous concave convex problems on a ball in r n and the second one deals with concave convex problems on a thin annulus in r n we illustrate our results by numerical computations using a novel algorithm which involves continuation in a global parameter

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

fixed points of generalized e concave generalized e convex operators and their applications in this paper we present the definitions of generalized e concave operators and generalized e convex operators which are the generalizations of e concave operators and e convex operators respectively without compactness or continuity assumption of generalized e concave operators and generalized e convex operators we have proved the existence uniqueness and monotone iterative techniques of their fixed points our results are even new to e concave operators and e convex operators finally we apply the results to the singular boundary value problems for second order differential equations

addendum conformal deformation of a riemannian metric to a scalar flat metric with constant mean curvature is finite on the other hand the argument at the beginning of proposition 2 1 in e shows that if al b is finite then q m om is finite jin zhiren pointed out to me that the sobolev quotient q m o9m can be oo this is the case if we delete a small geodesic ball on a compact manifold without boundary with negative scalar curvature more generally the sobolev quotient is oo if the first eigenvalue for the conformal laplacian with respect to dirichlet boundary condition is negative in order to see that let p1 be the first eigenfunction for the problem

the best sobolev trace constant in a domain with oscillating boundary abstract in this paper we study homogenization problems for the best constant for the sobolev trace embedding w 1 p l q in a bounded smooth domain when the boundary is perturbed by adding an oscillation we find that there exists a critical size of the amplitude of the oscillations for which the limit problem has a weight on the boundary for sizes larger than critical the best trace constant goes to zero and for sizes smaller than critical it converges to the best constant in the domain without perturbations

multiplicity of positive solutions for a semilinear p laplacian system with sobolev critical exponent abstract in this paper we study the semilinear p laplacian system with critical growth terms in bounded domains by using the nehari manifold and variational methods we prove that the system has at least two positive solutions when the pair of the parameters belongs to a certain subset of r 2

normalized ground states for the nls equation with combined nonlinearities abstract we study existence and properties of ground states for the nonlinear schrodinger equation with combined power nonlinearities u u u q 2 u u p 2 u in r n n 1 having prescribed mass r n u 2 a 2 under different assumptions on q p a 0 and r we prove several existence and stability instability results in particular we consider cases when 2 q 2 4 n p 2 q p i e the two nonlinearities have different character with respect to the l 2 critical exponent these cases present substantial differences with respect to purely subcritical or supercritical situations which were already studied in the literature we also give new criteria for global existence and finite time blow up in the associated dispersive equation

multiplicity of positive solutions for p laplace equation with superlinear type nonlinearity abstract in this work we consider the problem p r p u u p u q u 0 in b r 0 u 0 on b r 0 where p 2 p np n p and b r 0 is the open ball of radius r centered about the origin in r n n 3 under the condition that p 1 q p 2 n p 1 we show that there exists r 0 such that p r admits atleast two solutions for all r r and no solution if r this extends the results of nonlinear anal tma 10 8 1986 755 to the case of p laplace equation

on elliptic problems involving critical hardy sobolev exponents and sign changing function abstract in this paper we deal with the existence and nonexistence of nonnegative nontrivial weak solutions for a class of degenerate quasilinear elliptic problems with weights and nonlinearity involving the critical hardy sobolev exponent and a sign changing function some existence results are obtained by splitting the nehari manifold and by exploring some properties of the best hardy sobolev constant together with an approach developed by brezis and nirenberg

on and off diagonal heat kernel behaviors on certain infinite dimensional local dirichlet spaces this paper studies on diagonal and off diagonal bounds for symmetric diffusion semi groups that admit a continuous kernel in the case where the underlying space is typically infinite dimensional for invariant diffusions on the infinite dimensional torus the equivalence between a certain on diagonal estimate and a certain off diagonal behavior is proved this gives a necessary and sufficient condition in terms of the associated infinite symmetric matrix for an elliptic harnack inequality to be satisfied as a consequence elliptic and parabolic harnack inequalities are in fact equivalent properties in this setting in terms of potential theory this gives a necessary and sufficient condition for the associated harmonic sheaf to be a brelot harmonic sheaf a new distance associated to certain dirichlet spaces is also introduced it plays a crucial role in relating on diagonal behavior to off diagonal behavior in cases where the intrinsic distance is infinite almost everywhere

variational approach to semilinear elliptic equation with nonlinear boundary abstract we consider the existence of positive solutions for a certain class of semilinear elliptic equation with nonlinear boundary our approach is based on the variational methods

multiple positive solutions for a quasilinear elliptic equation with critical exponential nonlinearity abstract let be a bounded domain in r n n 2 with c 2 boundary in this work we study the existence of multiple positive solutions of the following problem p div u 2 n 2 1 u 2 n u u 2 n 2 1 u 2 n u u p e u n n 1 u 0 in u 2 n 2 1 u 2 n u u q on where 0 p 2 n 1 q 0 n 1 and is a non negative holder continuous function we show that there exists 0 such that p admits at least two solutions in w 1 n for 0 and no solution for

a variational approach to discontinuous problems with critical sobolev exponents abstract we employ variational techniques to study the existence and multiplicity of positive solutions of semilinear equations of the form u h x h u a uq u2 1 in rn where a 0 are parameters h x is both nonnegative and integrable on rn h is the heaviside function 2 is the critical sobolev exponent and 0 q

problemes de neumann non lineaires sur les varietes riemanniennes nonlinear neumann problems on riemannian manifolds let m g be a c compact riemannian manifold of dimension n 2 whose boundary b is an n 1 dimensional submanifold and let m m b be the interior of m study of neumann problems of the form x 0 in m d dn g y 0 on b where for every t x y r m b t x and g t y are bounded by c 1 t a or c exp t a application to the determination of a conformal metric for which the scalar curvature of m and the mean curvature of b take prescribed values

the existence and multiplicity of classical positive solutions for a singular nonlinear elliptic problem with any growth exponents a b s t r a c t in this paper we consider a class of singular elliptic boundary problems u u p x u u 0 in u 0 on with 1 0 1 and p c 0 0 1 we emphasize that there is no restriction on the boundedness of from above by means of the approximating variational arguments and truncation two classical positive solutions are obtained 2014 the authors published by elsevier ltd this is an open access article under the cc by nc nd license http creativecommons org licenses by nc nd 3 0

nondegeneracy and uniqueness of positive solutions for robin problem of second order ordinary differential equations and its applications abstract this article studies positive solutions of robin problem for semi linear second order ordinary differential equations nondegeneracy and uniqueness results are proven for homogeneous differential equations necessary and sufficient conditions for the existence of one or two positive solutions for inhomogeneous differential equations or differential equations with concave convex nonlinearities are obtained by making use of the nondegeneracy and uniqueness results for positive solutions of homogeneous differential equations

a new proof of moser s parabolic harnack inequality using the old ideas of nash in 1958 nash published his fundamental work on the local holder continuity of solutions of second order parabolic equations with coefficients which need not be smooth 8 the primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution though the work is often cited in the literature about weak solutions of elliptic and parabolic equations one feels that nash s ideas were never fully understood and maybe still are not and that because of this the more understandable and seemingly more fruitful ideas of de giorgi 4 and moser 6 7 were subsequently adopted

radial solutions of dirichlet problems with concave convex nonlinearities abstract we prove the existence of a double infinite sequence of radial solutions for a dirichlet concave convex problem associated with an elliptic equation in a ball of r n we are interested in relaxing the classical positivity condition on the weights by allowing the weights to vanish the idea is to develop a topological method and to use the concept of rotation number the solutions are characterized by their nodal properties

a numerical solution to nonlinear second order three point boundary value problems in the reproducing kernel space abstract in this paper a new numerical algorithm is provided to solve nonlinear three point boundary value problems in a very favorable reproducing kernel space which satisfies all boundary conditions its reproducing kernel function is discussed in detail we also prove that the approximate solution and its first and second order derivatives all converge uniformly the numerical experiments show that the algorithm is quite accurate and efficient for solving nonlinear second order three point boundary value problems

finite difference approximate solutions for a mixed sub superlinear equation in ben mabrouk and ben mohamed 2006 a ben mabrouk m l ben mohamed on some critical and slightly super critical sub superlinear equations far east j appl math 23 1 2006 73 90 special volume of pdes a ben mabrouk m l ben mohamed nodal solutions for some nonlinear elliptic equations appl math comp in press doi 10 1016 j amc 2006 08 003 it has proved some theoretical results dealing with some boundary value problem du f u 0 in b and u 0 on b where b is a domain in r d with smooth boundary in the present paper we perform a difference scheme method to approximate the solution of a nonlinear evolutionary problem associated to the elliptic problem studied in ben mabrouk and ben mohamed 2006 we give at the end some numerical implementations based on a mixed sublinear superlinear term of the form f u u p email protected u q 1u

multiple solutions for nonhomogeneous p laplacian equations with nonlinear boundary conditions on r n abstract in this paper we study the existence of multiple solutions for the nonlinear boundary value problem 0 1 div u p 2 u v x u p 2 u h x x r n u p 2 u h 1 x u q 2 u h 2 x u r 2 u x r n where r n x x n r n 1 r is an upper half space in r n and 1 p n 0 and 1 q p r p p n 1 n p and denotes the unit outward normal to the boundary r n the functions v x h x h 1 x and h 2 x satisfy some suitable conditions using the mountain pass theorem and ekeland s variational principle we prove that there exist 0 m 0 0 such that problem 0 1 admits at least two solutions provided that 0 0 and h p m 0 c 1 p 1 r q where the constant c 1 0 is independent of 0 on the other hand if h 2 0 we prove that problem 0 1 admits at least one solution for any 0 and h l p r n

on the singular second order elliptic equation abstract let m g be a smooth and compact riemannian manifold with dimensions n 3 this paper deals with the existence of nontrivial solutions for a nonlinear second order elliptic equation with a singular term and critical sobolev growth we provide a geometric application of this singular equation

the nehari manifold for a semilinear elliptic problem with the nonlinear boundary condition abstract using the nehari manifold and fibering maps we prove the existence theorem of the nonlinear boundary problem u u u p 2 u for x u n u q 2 u for x on a bounded domain r n we discuss how the nehari manifold changes as changes and show how the existence of solutions depends on the properties of the nehari manifold

on the evolution and qualitative behaviors of bifurcation curves for a boundary value problem abstract we study the evolution and qualitative behaviors of bifurcation curves of positive solutions for u x u 1 sin u u p 1 x 1 u 1 u 1 0 where 0 is a bifurcation parameter and p 1 is an evolution parameter on the u plane we prove that the bifurcation curve has exactly one turning point where the curve turns to the left for p 2 it is a monotone curve for p 2 it has at least two turning points for 1 p p 1 2 cos 1 sin 1 1 2 cos 1 sin 1 1 629 and it has infinitely many turning points for p 1 hence we are able to determine the exact number of positive solutions in particular we give complete descriptions of the structure of bifurcation curves when p 2 we also give some numerical simulations of bifurcation curves for p 1

existence and multiplicity results for quasilinear elliptic exterior problems with nonlinear boundary conditions abstract in this paper we study a class of quasilinear elliptic exterior problems with nonlinear boundary conditions existence of ground states and multiplicity results are obtained via variational methods

existence and nonexistence of solutions for quasilinear elliptic equations abstract in this paper we study the problem p u f x u u in u 0 on where r n is a smooth bounded domain n 2 and p u div u p 2 u defines the p laplacian we provide some necessary and sufficient conditions on f under which the problem admits a weak solution for the case p 2 we obtain more general conditions on f the main ingredients are degree theory and the super subsolution method

on α convex operators in this paper the existence and uniqueness of positive fixed points for a class of convex operators is obtained by means of the properties of cone concave operators and the monotonicity of set valued maps in the end we give a simple application to certain integral equations

compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity we study the branch of semistable and unstable solutions i e those whose morse index is at most 1 of the dirichlet boundary value problem u f x 1 u 2 on a bounded domain n which models among other things a simple electrostatic microelectromechanical system mems device we extend the results of 11 relating to the minimal branch by obtaining compactness along unstable branches for 1 n 7 on any domain and for a large class of permittivity profiles f we also show the remarkable fact that powerlike profiles f x x can push back the critical dimension n 7 of this problem by establishing compactness for the semistable branch on the unit ball also for n 8 and as long as as a byproduct we are able to follow the second branch of the bifurcation diagram and prove the existence of a second solution for in a natural range in all these results the conditions on the space dimension and on the power of the profile are essentially sharp 2007 wiley periodicals inc

three positive solutions for dirichlet problems involving critical sobolev exponent and sign changing weight abstract in this paper we study the decomposition of the nehari manifold via the combination of concave and convex nonlinearities furthermore we use this result and the lusternik schnirelman category to prove that a semilinear elliptic equation involving a sign changing weight function has at least three positive solutions

four positive solutions for a semilinear elliptic equation involving concave and convex nonlinearities abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of positive solutions for a semilinear elliptic equation with the help of the nehari manifold and the center mass function we prove that there are at least four positive solutions for a semilinear elliptic equation in a finite strip with a hole