harnack estimates and extinction profile for weak solutions of certain singular parabolic equations we establish an intrinsic harnack estimate for nonnegative weak solutions of the singular equation 1 1 below for m in the optimal range n 2 n 1 intrinsic means that due to the singularity the space time dimensions in the parabolic geometry must be rescaled by a factor determined by the solution itself consequences are sharp supestimates on the solutions and decay rates as t approaches the extinction time analogous results are shown for p laplacian type equations

the beginning of the fucik spectrum for the p laplacian has a nontrivial solution u here 1 p 2p u div u p 2 u u max u 0 u uu in 1 1 denoting by 1 2 the first two eigenvalues of 2p on w 1 p 0 0 cf the end of this introduction 7p clearly contains in particular 1 1 2 2 and the two lines 1 r and r 1 this generalized notion of spectrum was introduced for p 2 i e in the linear case in the 1970s by fuc ik 21 and dancer 15 in connection article id jdeq 1999 3645 available online at http www idealibrary com on

flatness of bernoulli jets level sets of minimizers of a possibly singular or degenerate elliptic variational problem related with fluid jets are shown to possess a harnack type inequality a few flatness and one dimensional results thence follow

higher holder regularity for the fractional p laplacian in the superquadratic case abstract we prove higher holder regularity for solutions of equations involving the fractional p laplacian of order s when p 2 and 0 s 1 in particular we provide an explicit holder exponent for solutions of the non homogeneous equation with data in l q and q n s p which is almost sharp whenever s p p 1 n q the result is new already for the homogeneous equation

optimal wloc2 2 regularity pohozhaevʼs identity and nonexistence of weak solutions to some quasilinear elliptic equations abstract we begin by establishing a sharp optimal w loc 2 2 regularity result for bounded weak solutions to a nonlinear elliptic equation with the p laplacian p u def div u p 2 u 1 p we develop very precise optimal regularity estimates on the ellipticity of this degenerate for 2 p or singular for 1 p 2 problem we apply this regularity result to prove pohozhaev s identity for a weak solution u w 1 p of the elliptic neumann problem p p u w u f x in u 0 on here is a bounded domain in r n whose boundary is a c 2 manifold x 0 denotes the outer unit normal to at x 0 x x 1 x n is a generic point in and f l w 1 1 the potential w r r is assumed to be of class c 1 and of the typical double well shape of type w s 1 s for s r where 1 are some constants finally we take an advantage of the pohozhaev identity to show that problem p with f 0 in has no phase transition solution u w 1 p 1 p n such that 1 u 1 in with u 1 in 1 and u 1 in 1 where both 1 and 1 are some nonempty subdomains of such a scenario for u is possible only if n 1 and 1 1 are finite unions of suitable subintervals of the open interval r 1

extremal solutions of p laplacian problems in d1 p rn via wolff potential estimates abstract we prove the existence of smallest positive and greatest negative solutions in d 1 p r n of the p laplacian problem of logistic type in r n with 1 p n u d 1 p r n p u a x u p 2 u g u and consider its extended version with a harvesting term h as well unlike papers on the topic here we only assume a superlinear growth for the nonlinearity g r r at zero and at infinity i e even supercritical growth is allowed and moreover no monotonicity for s g s s p 1 is required we prove a vazquez type maximum principle for nonnegative d 1 p r n solutions of some p laplacian problem and establish a sub supersolution method in d 1 p r n which along with wolff potential estimates for superharmonic functions will be used as our main tools

existence and multiplicity of positive solutions for semilinear elliptic systems with sobolev critical exponents abstract in this paper we study the existence and multiplicity of positive solutions to the following system u f u u v e g x v f v u v e h x in u v 0 in and u v 0 on where is a bounded smooth domain in r n f c 1 r 2 r is positively homogeneous of degree g h c 1 0 and e is a positive parameter using sub supersolution method we prove the existence of positive solutions for the above problem by means of the variational approach we prove the multiplicity of positive solutions for the above problem with 2 2

pruning error minimization in least squares support vector machines the support vector machine svm is a method for classification and for function approximation this method commonly makes use of an spl epsi insensitive cost function meaning that errors smaller than spl epsi remain unpunished as an alternative a least squares support vector machine lssvm uses a quadratic cost function when the lssvm method is used for function approximation a nonsparse solution is obtained the sparseness is imposed by pruning i e recursively solving the approximation problem and subsequently omitting data that has a small error in the previous pass however omitting data with a small approximation error in the previous pass does not reliably predict what the error will be after the sample has been omitted in this paper a procedure is introduced that selects from a data set the training sample that will introduce the smallest approximation error when it will be omitted it is shown that this pruning scheme outperforms the standard one

existence of a non trivial solution for the p laplacian equation with fucik type resonance at infinity abstract we prove the existence of a non trivial weak solution for the equation p u a u p 1 b u p 1 f x u in u 0 on by variational methods we consider the case where a 1 or b 1 1 denotes the first eigenvalue of p or the case where a is close to b while the nonlinearity f belongs to a class that contains for example u q 2 u 1 q p

existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations in connection with the maximizing problem for the functional r u u lq u lp in w01 p 0 we consider the equation div u p 2 u x u q 2 u x x 1 p resp q p e has no nontrivial resp nonnegative nontrivial solution belonging to the class p u w01 p lq xi u q 2u lp p 1 i 1 2 n w01 p provided that is star shaped the crucial point of the proof of our result is to obtain an l estimate of weak solutions and to verify a certain pohozaev type inequality for weak solutions belonging to p

weighted eigenvalue problems for quasilinear elliptic operators with mixed robin dirichlet boundary conditions abstract we investigate the existence of principal eigenvalues type problems with weights for the quasilinear operator p v p with mixed weighted robin dirichlet boundary conditions in a bounded regular domain we also give some results on the existence of nonprincipal eigenvalues

heat flow on finsler manifolds this paper studies the heat flow on finsler manifolds a finsler manifold is a smooth manifold m equipped with a minkowski norm f x txm on each tangent space mostly we will require that this norm be strongly convex and smooth and that it depend smoothly on the base point x the particular case of a hilbert norm on each tangent space leads to the important subclasses of riemannian manifolds where the heat flow is widely studied and well understood we present two approaches to the heat flow on a finsler manifold as gradient flow on l2 m m for the energy as gradient flow on the reverse l2 wasserstein space 2 m of probability measures on m for the relative entropy both approaches depend on the choice of a measure m on m and then lead to the same nonlinear evolution semigroup we prove 1 regularity for solutions to the nonlinear heat equation on the finsler space m f m typically solutions to the heat equation will not be 2 moreover we derive pointwise comparison results a la cheeger yau and integrated upper gaussian estimates a la davies 2008 wiley periodicals inc

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

global l gradient estimates for solutions to a certain degenerate elliptic equation abstract in view of the applications in the study of regularity properties of minimizers for a continuous model of transportation which is a kind of divergence constrained optimization problem we prove a global l gradient estimate for solutions of an elliptic equation whose ellipticity constants become degenerate at every point where u with 0 the exposition is as self contained as possible

on a free boundary problem for thep laplacian abstract by a maximum principle we obtain radial symmetry of domain and weak solution to div du p 2du 0 on a starshaped ring 0 1 inrn with the overdetermined boundary conditionsu 1 and du c1uniformly a e asx 1 u 0 and du c0uniformly a e asx 0 c1 c0 0 1

existence of blow up solutions for a non linear equation with gradient term in rn in this paper we study the existence of positive large solutions for the equation pu u p 1 x f u in rn where f is a non negative non decreasing function and is a non negative continuous function we show under some hypotheses detailed below the existence of positive solutions which blow up at infinity

spectral theory for linearized p laplace equations abstract we continue and completely set up the spectral theory initiated in castorina et al 2009 5 for the linearized operator arising from p u f u 0 we establish existence and variational characterization of all the eigenvalues and by a weak harnack inequality we deduce holder continuity for the corresponding eigenfunctions this regularity being sharp the morse index of a positive solution can be now defined in the classical way and we will illustrate some qualitative consequences one should expect to deduce from such information in particular we show that zero morse index or more generally non degenerate solutions on the annulus are radial

bayesian framework for least squares support vector machine classifiers gaussian processes and kernel fisher discriminant analysis the bayesian evidence framework has been successfully applied to the design of multilayer perceptrons mlps in the work of mackay nevertheless the training of mlps suffers from drawbacks like the nonconvex optimization problem and the choice of the number of hidden units in support vector machines svms for classification as introduced by vapnik a nonlinear decision boundary is obtained by mapping the input vector first in a nonlinear way to a high dimensional kernel induced feature space in which a linear large margin classifier is constructed practical expressions are formulated in the dual space in terms of the related kernel function and the solution follows from a convex quadratic programming qp problem in least squares svms ls svms the svm problem formulation is modified by introducing a least squares cost function and equality instead of inequality constraints and the solution follows from a linear system in the dual space implicitly the least squares formulation corresponds to a regression formulation and is also related to kernel fisher discriminant analysis the least squares regression formulation has advantages for deriving analytic expressions in a bayesian evidence framework in contrast to the classification formulations used for example in gaussian processes gps the ls svm formulation has clear primal dual interpretations and without the bias term one explicitly constructs a model that yields the same expressions as have been obtained with gps for regression in this article the bayesian evidence framework is combined with the ls svm classifier formulation starting from the feature space formulation analytic expressions are obtained in the dual space on the different levels of bayesian inference while posterior class probabilities are obtained by marginalizing over the model parameters empirical results obtained on 10 public domain data sets show that the ls svm classifier designed within the bayesian evidence framework consistently yields good generalization performances

on full two scale expansion of the solutions of nonlinear periodic rapidly oscillating problems and higher order homogenised variational problems we consider a scalar quasilinear equation in the divergence form with periodic rapid oscillations which may be a model of e g nonlinear conducting dielectric or deforming in a restricted way hardening elastic plastic composites with outer periodicity conditions of a fixed large period under some natural growth assumptions on the stored energy function we construct for uniformly elliptic problems a full two scale asymptotic expansion which has a precise double series structure separating the slow and the fast variables in all orders so that its slowly varying part solves asymptotically an infinite order homogenised equation cf bakhvalov n s panasenko g p homogenisation averaging processes in periodic media nauka moscow 1984 in russian english translation kluwer 1989 and whose higher order terms depend on the higher gradients of the slowly varying part we prove the error bound i e that the truncated asymptotic expansion is higher order close to the actual solution in appropriate norms the approach is extended to a non uniformly elliptic case for two dimensional power law potentials we prove the non degeneracy using topological index methods examples and explicit formulae for the higher order terms are given in particular we prove that the first term in the higher order homogenised equations is related to the first order corrector to the mean flux and has in general the form of a fully nonlinear operator which is quadratic with respect to its highest second derivative being a linear combination of the second minors of the hessian with coefficients depending on the first gradient and in dimension two is of monge ampere type we show that this term is present at least for some examples three phase power law laminates

generalized minimizer solutions for equations with the p laplacian and a potential term let varomega be a domain in mathbb r n possibly unbounded n geq2 1 and let v in l mathrm loc infty varomega consider the energy functional mathcal q v on c mathrm c infty varomega and its g teaux derivative mathcal q v prime respectively given by mathcal q v u eqdef frac 1 p int varomega nabla u p v u p mathrm d x qquad mathcal q v prime u mathrm div nabla u p 2 nabla u v u p 2 u for u in c mathrm c infty varomega assume that mathcal q v 0 on c mathrm c infty varomega setminus 0 and that mathcal q v does not have a ground state in the sense of a null sequence for mathcal q v that converges in l mathrm loc p varomega to a positive function varphi in c mathrm loc 1 varomega a ground state finally let f in mathcal d prime varomega be such that the functional u mapsto mathcal q v u langle u f rangle c mathrm c infty varomega to mathbb r is bounded from below then the equation mathcal q v prime u f has a solution u 0 in w 1 p mathrm loc varomega in the sense of distributions this solution also minimizes the functional u mapsto mathcal q v u langle u f rangle c mathrm c infty varomega to mathbb r where mathcal q v denotes the bipolar vargamma regularization of mathcal q v and mathcal q v is the largest convex weakly lower semicontinuous functional on c mathrm c infty varomega that satisfies mathcal q v leq mathcal q v the original energy functional mathcal q v is not necessarily convex

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

symmetry properties for nonnegative solutions of non uniformly elliptic equations in the hyperbolic space abstract we are interested in monotonicity and symmetry properties for nonnegative solutions of elliptic equations defined in geodesic balls of the hyperbolic space h n which is the simplest example of manifold with negative curvature more precisely let b be a geodesic ball in h n and let u w 1 p b l b be a sufficiently regular solution of p u f u 0 in b with boundary condition u 0 where p is the p laplace beltrami operator with p 2 then we prove local or global symmetry results for nonnegative solutions according to the assumptions about the zeros of the nonlinearity f s which is merely continuous

bounded solutions to a quasilinear and singular parabolic equation with p laplacian abstract in this paper we study the existence of a positive local in time solution for the following singular nonlinear problem with homogeneous dirichlet boundary conditions t u p u u f x u u in 0 t q t u 0 on 0 t u 0 in q t u 0 x u 0 0 in where stands for a regular bounded domain of r n p u is the p laplacian defined by p u div u p 2 u 2 p 0 and t 0 the nonlinear term f r r n r is a caratheodory function satisfying the growth condition f x s a s q 1 b c p p q for a a x s r and m where a c m 0 and b 0 are some constants and q p p where p p n n p if p n and p if p n we prove for any initial nonnegative data u 0 l r with r 2 large enough the existence of at least one weak solution to p in the case 2 1 p 1 we prove the uniqueness of the solution and further regularity results for that we use some estimates based on logarithmic sobolev inequalities to get ultracontractivity of a closely related semi group of linear operators

proximinality in banach spaces in this paper we study approximatively compact and strongly chebyshev sets where is the norm or the weak topology we show that the metric projection onto strongly chebyshev sets are norm continuous we characterize approximatively compact and strongly chebyshev hyperplanes and use them to characterize factor reflexive proximinal subspaces in almost locally uniformly rotund spaces we also prove some stability results on approximatively compact and strongly chebyshev subspaces

ground states of degenerate quasilinear schrodinger equation with vanishing potentials abstract in this paper we study the existence of nontrivial ground state solutions for the following class of p laplacian type equation div a x u v x x p u p 2 u k x x p f u in r n where 1 p n n 3 n p p e 1 d 1 e p p e n p n d p critical hardy sobolev exponent f has a quasicritical growth v and k are nonnegative potentials the function a satisfies a x u c 0 x p h 0 x u p 1 c 0 1 x p h 1 x u p 1 for any r n a e x r n for any two positive functions h 1 l l o c r n h 0 l p p 1 r n with p p

an end point orlicz type estimate for nonlinear elliptic equations abstract we study nonlinear elliptic equations of p laplacian type in divergence form to establish a natural calderon zygmund type theory of an orlicz space type where the lebesgue space is the special case with t t q p f p l 1 q 2 r d u p l 1 q r in the previous results the estimates obtained were strictly above the natural exponents and the function such as t t log 1 t was ruled out for the candidate of but with our approach can be selected up to the end point case of the estimates and the functions t t and t t log 1 t can be additionally selected for

bounded perturbations of homogeneous quasilinear operators using bifurcations from infinity this paper deals with existence results for the following nonlinear problem with the dirichlet p laplacian p in a bounded domain rn pu u p 2u h x u in u 0on here pu defdiv u p 2 u where p 1 is a fixed number h h x s is a given function from r into r and r stands for a spectral parameter we focus on close to 1 including the resonant case 1 the nonlinearity h is assumed to be of landesman lazer type but we can deal with vanishing nonlinearities as well our asymptotic method substitutes the lyapunov schmidt method in some sense unlike in the semilinear case p 2 our method can treat more general nonlinearities if p 2 vanishing nonlinearities with very fast decay

regularity and existence of solutions of elliptic equations with p q growth conditions for some positive constants m a4 and for exponents q 2 p 2 2 under 1 2 1 3 and some other assumptions by assuming also that the quotient q p is sufficiently close to one in dependence on n precisely if q p c n n 2 then we prove that every weak solution to 1 1 of class 4 4 0 is locally lipschitz continuous in q moreover there are positive constants 8 c and 0 2 1 such that

eigencurves of non definite sturm liouville problems for the p laplacian abstract the eigencurves are studied for a weighted quasi linear sturm liouville type problem of the form p y p 1 r q s sgn y y p 1 on 0 1 with sturmian type boundary conditions p being the p laplacian topics include holder classes c k and asymptotics including existence of oblique asymptotes an example illustrates precision of our holder exponents when p 2

c1 α local regularity of weak solutions of degenerate elliptic equations abstract it is demonstrated the local c 1 alpha nature of weak solutions of elliptic equations of the type 1 1 in the introduction under the degeneracy or singularity assumptions a sub 1 a sub 3

the existence and non existence of positive solutions to a singular quasilinear elliptic problem in rn abstract this paper deals with the existence and nonexistence of entire positive solutions of the quasilinear elliptic equation p u a x g u b x f u in r n 1 p n n 3 with u x 0 as x here either g or f or both of them are singular at 0 in the sense that g t f t as t 0 by using a perturbation method which eliminates the singularity on a ball with radius r and then let r tends to infinity with help of the bounded super solution of the original problem we obtain the existence of a weak solution of the problem the main results of this paper improve the corresponding results of dragos patru covei the existence and asymptotic behavior of a positive solution to a quasilinear elliptic problem in r n nonlinear anal 69 2008 2615 2622 j v goncalves c a santos singular elliptic problems existence non existence and boundary behavior nonlinear anal 66 2007 2078 2090

global boundedness of the gradient for a class of schrodinger equations abstract in this paper via applying the method developed by a cianchi and v maz ya the author obtains the global boundedness of the gradient for solutions to the dirichlet and the neumann problems of a class of schrodinger equations under the minimal assumptions for the integrability on the data and the regularity on the boundary of the domain moreover the case of arbitrary bounded lipschitz domains satisfying a uniform exterior ball condition is also considered

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

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

necessary condition for eigensolutions of the 1 laplace operator by means of inner variations eigenfunctions of the p laplace operator for p 1 are defined to be critical points of an associated variational problem or equivalently to be solutions of the corresponding euler lagrange equation in the highly degenerated limit case of the 1 laplace operator eigenfunctions can also be defined to be critical points of the corresponding variational problem if critical points are understood on the basis of the weak slope however the associated euler lagrange equation has many solutions that are not critical points and thus it cannot be used for an equivalent definition the present paper provides a new necessary condition for eigenfunctions of the 1 laplace operator by means of inner variations of the associated variational problem and it is shown that this condition rules out certain solutions of the euler lagrange equation that are not eigenfunctions

existence uniqueness and qualitative properties of the solution of a degenerate nonlinear parabolic system abstract in this paper we are concerned with the study of the existence of solutions of a nonlinear parabolic system with density dependent diffusion subject to homogeneous third type boundary conditions a comparison result is shown that allows the study of the asymptotic behavior of the solutions via the same methods used for systems with classical diffusion finally an application of the results to an epidemic system is presented sufficient conditions are given for the positivity of the solution of this system

regularity of subelliptic p harmonic systems with subcritical growth in carnot group abstract for a class of subelliptic p harmonic systems with the subcritical growth defined in carnot groups we prove that the weak solutions belong to loc 1 regularity with respect to the carnot caratheodory distance provided that p is close to 2 as a consequence we make it clear that the critical growth of nonlinearity for p harmonic type systems in carnot groups is just a sharp borderline of partial regularity of the small excess energy due to the counterexamples of p harmonic maps with p close to 2

extremal solutions of logistic type equations in exterior domain in r2 abstract let r 2 b 0 1 be the exterior of the closed unit ball we prove the existence of extremal constant sign solutions in d 0 1 2 of the following logistic type boundary value problem u a x u g u in u 0 on b 0 1 under the assumption that the nonnegative coefficient a decays like x 2 with 0 and the growth for the continuous nonlinearity g r r at zero and at infinity is superlinear which includes even exponential growth we are looking for solutions in the space d 0 1 2 which is the completion of c c with respect to the 2 norm for general unbounded domains in r 2 including the whole plane this completion may result in objects that do not belong to any function space this is one of the main reasons to consider the problem in the exterior of the unit ball instead in the whole plane unlike in the situation of r n with n 3 i e n p 2 the behavior of the solutions in the case n p 2 considered here is significantly different in particular it will be seen that the solutions are not decaying to zero at infinity and instead are bounded away from zero to prove our main results new tools have to be developed here such as for example a hopf type lemma and a sub supersolution method in d 0 1 2

regularity for quasilinear equations on degenerate singular sets we prove a new universal gradient continuity estimate for solutions to quasilinear equations with varying coefficients at singular set of degeneracy fancyscript s u x d u x 0 our main theorem reveals that along fancyscript s u u is asymptotically as regular as solutions to constant coefficient equations in particular along the critical set fancyscript s u du enjoys a modulus of continuity much superior than the possibly low continuity feature of the coefficients the results are new even in the context of linear elliptic equations where it is herein shown that h 1 weak solutions to text div left a ij x du right 0 with a ij elliptic and dini continuous are actually c 1 1 along fancyscript s u the results and insights of this work foster a new understanding on smoothness properties of solutions to degenerate or singular equations beyond typical elliptic regularity estimates precisely where the diffusion attributes of the equation collapse

existence uniqueness and qualitative properties of positive solutions of quasilinear elliptic equations we study the following quasilinear elliptic equation p pu x a x up 1 b x g u 0in rn where p 1 a b l rn b g 0 b 0 and lloc rn infrn we provide a sharp criterion in term of generalized principal eigenvalues for existence nonexistence of positive solution of p in suitable classes of functions uniqueness result for p in those classes is also derived under additional conditions on we further show that i either for every 0 nonexistence phenomenon occurs ii or there exists a threshold value 0 in the sense that for every 0 existence and uniqueness phenomenon occurs and for every nonexistence phenomenon occurs in the latter case we study the limits as 0 and of the sequence of positive solutions of p our results are new even in the case p 2

on multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in rn abstract in this work we study the existence multiplicity and concentration of positive solutions for the following class of quasilinear problems n n u v x u n 2 u f u in r n u x 0 x r n where n u is the n laplacian operator is a positive parameter n 2 v r n r is a continuous function and f r r is a c 1 function having critical exponential growth

compactly uniformly convex spaces and property β of rolewicz abstract we study different facets of property of rolewicz we remark that the notions of compact uniform convexity and property are isometrically equivalent and present new examples of spaces with that property an observation is made that the property can be formulated in terms of graphs and an estimate of the modulus is also given

travelling waves in a convection diffusion equation abstract we study existence and stability of travelling waves for nonlinear convection diffusion equations in the 1 d euclidean space the diffusion coefficient depends on the gradient in analogy with the p laplacian and may be degenerate unconditional stability is established with respect to initial data perturbations in l 1 r

linear potentials in nonlinear potential theory pointwise gradient bounds via riesz potentials such as those available for the linear poisson equation actually hold for general quasilinear degenerate equations of p laplacean type the regularity theory of such equations completely reduces to that of the classical poisson equation up to the c1 level

on singular quasi monotone p q laplacian systems we combine the sub and supersolution method and perturbation arguments to obtain positive solutions of singular quasi monotone p q laplacian systems

kernel least squares models using updates of the pseudoinverse sparse nonlinear classification and regression models in reproducing kernel hilbert spaces rkhss are considered the use of mercer kernels and the square loss function gives rise to an overdetermined linear least squares problem in the corresponding rkhs when we apply a greedy forward selection scheme the least squares problem may be solved by an order recursive update of the pseudoinverse in each iteration step the computational time is linear with respect to the number of the selected training samples

on the gibbons conjecture for equations involving the p laplacian in this paper we prove the validity of gibbons conjecture for the quasilinear elliptic equation delta p u f u on mathbb r n the result holds true for 2n 2 n 2 p 2 and for a very general class of nonlinearity f

up to boundary regularity for a singular perturbation problem of p laplacian type abstract in this paper we are interested in establishing up to boundary uniform estimates for the one phase singular perturbation problem involving a nonlinear singular degenerate elliptic operator our main result states if r n is a c 1 domain f c 1 for some 0 1 and u e verifies div a x u e u e e u e in 0 u e 1 in u e f on where e 0 e t 1 e t e and 0 t b 0 t 1 r e t d t m 0 with some positive constants b and m then there exists a constant c 0 independent of e such that u e l c

maximum principle of optimal control for degenerate quasi linear elliptic equations optimal control problems governed by degenerate quasi linear partial differential equations of elliptic type are considered the optimal control systems considered may lack cesari type conditions and therefore the corresponding approximate optimal control problem may have no solution to yield the maximum principle of optimal pairs relaxed controls are used to overcome the difficulties occurring when considering approximate problems the relaxed controls used are defined by finite additive measures so that the case of the control set being noncompact can be treated

some remarks on the fucik spectrum of the p laplacian and critical groups let be a bounded domain in n n 1 for 1 p the fucik spectrum of the p laplacian on w 1 p 0 is defined as the set p of those points a b 2 such that pu a u p 1 b u p 1 u w 1 p 0 1 1 has a nontrivial solution here pu div u p 2 u and u max u 0 it is known that the first eigenvalue 1 of p on w 1 p 0 is positive simple and admits a positive eigenfunction 1 w 1 p 0 c1 see lindqvist 8 so p contains the two lines 1 and 1 a first nontrivial curve c2 in p through 2 2 where 2 is the second eigenvalue of p was recently constructed and variationally characterized by a

quasilinear elliptic problems under strong resonance conditions abstract in this paper we establish the existence and multiplicity of solutions for a quasilinear elliptic problem under strong resonance conditions at infinity in order to control the resonance we consider a new hypothesis on the nonlinear term in all results we use variational methods critical groups and the morse theory