stefan problems with nonlinear diffusion and convection abstract we consider a class of stefan type problems having a convection term and a pseudomonotone nonlinear diffusion operator assuming data in l 1 we prove existence uniqueness and stability in the framework of renormalized solutions existence is established from compactness and monotonicity arguments which yield stability of solutions with respect to l 1 convergence of the data uniqueness is proved through a classical l 1 contraction principle obtained by a refinement of the doubling variable technique which allows us to extend previous results to a more general class of nonlinear possibly degenerate operators

positive and negative solutions for a class of kirchhoff type problems on unbounded domain abstract in this paper we study the existence of multiple positive and negative solutions for a class of kirchhoff type problems on an unbounded domain using the variational methods we show that our problem admits at least two positive and two negative solutions under different conditions a cut off functional is utilized to obtain the bounded palais smale sequences

multiplicity of positive solutions for the kirchhoff type equations with critical exponent in rn abstract in this work we study the existence and multiplicity of solutions to the following kirchhoff type problem with critical nonlinearity in r n a b r n u p d x p u u p 1 f x u x r n u d 1 p r n where n 2 p a b 0 and the nonlinearity f x u satisfies certain subcritical growth conditions by using topological and variational methods infinitely many positive solutions are obtained

nontrivial solutions of kirchhoff type problems via the yang index abstract we obtain nontrivial solutions of a class of nonlocal quasilinear elliptic boundary value problems using the yang index and critical groups

renormalised solutions of nonlinear parabolic problems with l1 data existence and uniqueness in this paper we prove the existence and uniqueness of a renormalised solution of the nonlinear problem where the data f and u 0 belong to l 1 0 t and l 1 and where the function a 0 t n n is monotone but not necessarily strictly monotone and defines a bounded coercive continuous operator from the space into its dual space the renormalised solution is an element of c 0 0 t l 1 such that its truncates t k u belong to with this solution satisfies the equation formally obtained by using in the equation the test function s u where belongs to and where s belongs to c with

adaptive estimation for a class of nonlinearly parameterized dynamical systems the technical note presents a new technique for the adaptive parameter estimation in nonlinear parameterized dynamical systems the technique proposes an uncertainty set update approach that guarantees forward invariance of the true value of the parameters in addition it is shown that in the presence of sufficiently exciting state trajectories the parameter estimates converge to the true values and the uncertainty set vanishes around the true value of the parameters the set based approach is also shown to identify when the parameter estimates are biased two simulation examples are presented that demonstrate the effectiveness of the technique

a pocket guide to nonlinear differential equations in musielak orlicz spaces abstract the musielak orlicz setting unifies variable exponent orlicz weighted sobolev and double phase spaces they inherit technical difficulties resulting from general growth and inhomogeneity in this survey we present an overview of developments of the theory of existence of pdes in the setting including reflexive and non reflexive cases as well as isotropic and anisotropic ones particular attention is paid to problems with data below natural duality in absence of lavrentiev s phenomenon

existence of solution for a class of nonvariational kirchhoff type problem via dynamical methods abstract in this paper we use dynamical methods to establish the existence of nontrivial solution for the following class of kirchhoff type problem p 1 b u 2 d x u f u h u u x u 0 x where r n n 2 3 is a smooth bounded domain b 0 f r r and h r r n 0 are c 1 functions that satisfy some technical conditions

renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and l1 data abstract in this paper we prove the existence and uniqueness of both renormalized solutions and entropy solutions for nonlinear parabolic equations with variable exponents and l 1 data and moreover we obtain the equivalence of renormalized solutions and entropy solutions

estimation of states and parameters for linear systems with nonlinearly parameterized perturbations we consider systems that can be described by a linear part with a nonlinear perturbation where the perturbation is parameterized by a vector of unknown constant parameters under a set of technical assumptions about the perturbation and its relationship to the outputs we present a modular design technique for estimating the system states and the unknown parameters the design consists of a high gain observer that estimates the states of the system together with the full perturbation and a parameter estimator constructed by the designer to invert a nonlinear equation we illustrate the technique on a simulated dc motor with friction

on the existence of weak solutions for quasilinear parabolic initial boundary value problems structure conditions on a strongly nonlinear operator a u are given under whichthe initial dirichlet boundary value problem for has weak solutions

existence and multiplicity of solutions for a quasilinear nonlocal problem abstract in this paper we establish two critical points theorems for functional which is not defined on the whole space but c 1 on a closed subset as applications we study the existence and multiplicity of solutions for a class of kirchhoff type equations with bounded nonlocal term

positive solutions of logistic equations with dependence on gradient and nonhomogeneous kirchhoff term abstract consider the equation m x u p u f x u u g x u u in u 0 on the main aim of this paper is to prove existence results for both non degenerate and degenerate cases of the function m our approach relies on the fixed point index and the cone theoretic argument

existence of solution for a class of quasilinear schrodinger equation in rn with zero mass abstract in this paper we use variational methods to establish a berestycki lions type result for two classes of quasilinear schrodinger equations in r n in the zero mass situation

l1 contraction and uniqueness for quasilinear elliptic parabolic equations abstract we prove the l 1 contraction principle and uniqueness of solutions for quasilinear elliptic parabolic equations of the form formula where b is monotone nondecreasing and continuous we assume only that u is a weak solution of finite energy in particular we do not suppose that the distributional derivative t b u is a bounded borel measure or a locally integrable function

steady flow of non newtonian fluids monotonicity methods in generalized orlicz spaces abstract our purpose is to show the existence of weak solutions to steady flow of non newtonian incompressible fluids with nonstandard growth conditions of the cauchy stress tensor we are motivated by the fluids of strongly inhomogeneous behavior and characterized by rapid shear thickening since l p framework is not sufficient to capture the described model we describe the growth conditions with the help of a general x dependent convex function and formulate our problem in generalized orlicz spaces

nonlinear parabolic equations with measure data in this paper we give summability results for the gradients of solutions of nonlinear parabolic equations whose model isu div u p 2 u on 0 t p with homogeneous cauchy dirichlet boundary conditions wherep 1 and is a bounded measure on 0 t we also study how the summability of the gradient improves if the measure is a function inlm 0 t withm small moreover we give a new proof of the existence of a solution for problem p

on ground states for the kirchhoff type problem with a general critical nonlinearity abstract in this paper we consider the following kirchhoff type problem a b r 3 u 2 u f u in r 3 u h 1 r 3 u 0 in r 3 where a b 0 are constants and f has a critical growth the aim of this paper is to study the existence of ground state solutions for kirchhoff type equations with a general nonlinearity in the critical growth without the assumption of the monotonicity of the function t f t t 3 moreover we will show that the mountain pass value gives the least energy level and also obtain a mountain pass solution

on adaptive control of nonlinearly parameterized nonlinear systems towards a constructive procedure abstract a new framework to design immersion and invariance adaptive controllers for nonlinearly parameterized nonlinear systems was recently proposed by the authors the key step is the construction of a monotone mapping via a suitable selection of a controller tuning function which has to satisfy some integrability conditions this translates into the need to solve a partial differential equation pde in this paper this result is extended providing some answers to the questions of characterization of monotonizable systems and solvability of the pde first adding to the design a nonlinear dynamic scaling we obviate the need to solve the pde second for the case of factorizable nonlinearities the following results are established i it is shown that the monotonicity condition is satisfied if a linear matrix inequality is feasible ii directly verifiable involutivity conditions that ensure the solution of the pde are presented iii an explicit formula for the required tuning function is given provided the regressor matrix satisfies some rank conditions hence adding a dynamic scaling this yields a constructive solution to the problem

well posedness of parabolic equations in the non reflexive and anisotropic musielak orlicz spaces in the class of renormalized solutions abstract we prove existence and uniqueness of renormalized solutions to general nonlinear parabolic equation in musielak orlicz space avoiding growth restrictions namely we consider t u div a x u f l 1 t on a lipschitz bounded domain in r n the growth of the weakly monotone vector field a is controlled by a generalized nonhomogeneous and anisotropic n function m the approach does not require any particular type of growth condition of m or its conjugate m neither 2 nor 2 the condition we impose on m is continuity of log holder type which results in good approximation properties of the space however the requirement of regularity can be skipped in the case of reflexive spaces the proof of the main results uses truncation ideas the young measures methods and monotonicity arguments uniqueness results from the comparison principle

existence and multiplicity of positive solutions for a class of kirchhoff laplacian type problems this paper deals with the existence and multiplicity of positive solutions for a class of kirchhoff laplacian type problems in whole r3 the multiplicity of solutions is established by variational methods and ljusternik schnirelmann theory

existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms the authors study the nonlinear elliptic equation div a x u du div u g x u f x in with the boundary condition u 0 on where is a bounded open subset of rn a u div a x u du is a nonlinear operator of leray lions type from w1 p0 into its dual c0 r n g x t t 0 and f w 1 p no growth hypothesis is assumed on the vector valued function u the term div u may be meaningless for usual weak solutions even as a distribution the authors deal with a weaker form of solutions of which in w1 p0 are called the renormalized solutions of the original problem this weaker form can be formally obtained from by means of pointwise multiplication by h u where h is a c1 function with compact support they prove the existence of renormalized solutions of and give sufficient conditions under which a renormalized solution is a usual weak solution furthermore the authors study ls and l regularity of renormalized solutions

nonlinear parabolic problems in musielak orlicz spaces abstract our studies are directed to the existence of weak solutions to a parabolic problem containing a multi valued term the problem is formulated in the language of maximal monotone graphs we assume that the growth and coercivity conditions of a nonlinear term are prescribed by means of a time and space dependent n function this results in the formulation of the problem in generalized musielak orlicz spaces we are using density arguments hence an important step of the proof is a uniform boundedness of appropriate convolution operators in musielak orlicz spaces for this purpose we shall need to assume a kind of logarithmic holder regularity with respect to t and x

existence and uniqueness of renormalized solutions to nonlinear elliptic equations with variable exponents and l1 data abstract we prove the existence and uniqueness of a renormalized solution to nonlinear elliptic equations with variable exponents and l 1 data the functional setting involves sobolev spaces with variable exponents w 1 p

infinitely many small solutions for the kirchhoff equation with local sublinear nonlinearities abstract we study the following nonlinear kirchhoff equation on the whole space 1 b r 3 u 2 v x u 2 d x u v x u f x u in r 3 where b 0 is a constant and the nonlinear term f x u is only locally defined for u small and satisfies some mild conditions it is proved that the above problem admits infinity many weak solutions provided that some additional assumptions are satisfied moreover we show that these solutions tend to zero in l r 3 our method relies upon a variant of the symmetric mountain pass lemma due to kajikiya and moser iteration method

renormalized solutions of nonlinear elliptic problems in generalized orlicz spaces abstract we study a general class of nonlinear elliptic problems associated with the differential inclusion x u div a x u f u f where f l 1 the vector field a is monotone in the second variable and satisfies a non standard growth condition described by an x dependent convex function that generalizes both l p x and classical orlicz settings using truncation techniques and a generalized minty method in the functional setting of non reflexive spaces we prove existence of renormalized solutions for general l 1 data under an additional strict monotonicity assumption uniqueness of the renormalized solution is established sufficient conditions are specified which guarantee that the renormalized solution is already a weak solution to the problem

positive solutions for a quasilinear elliptic equation of kirchhoff type this paper is concerned with the existence of positive solutions to the class of nonlocal boundary value problems of the type m w u 2dx du f x u email protected u 0 email protected email protected where w is a smooth bounded domain of n m is a positive function and f has subcritical growth

infinitely many radial solutions for kirchhoff type problems in rn abstract in this paper we concern with a class of kirchhoff type problems in r n by using the fountain theorem we obtain three existence results of infinitely many radial solutions for the problem

signed and sign changing solutions for a kirchhoff type equation in bounded domains abstract the main concern of this article is a kirchhoff type equation of the form m u 2 u f u where is a bounded smooth domain in r n with n 3 and is a positive parameter under certain assumptions on m and f the existence results of signed and sign changing solutions are established for large and when converges to infinity the asymptotic behavior of these solutions is also studied the proofs are based on a careful study of the ground state and least energy nodal solutions of an auxiliary problem which is constructed by making a refined truncation on m furthermore we get the ground state and least energy nodal solutions and prove the energy doubling property for all 0 under more restricted assumptions on m and f

on kirchhoff type equations with critical sobolev exponent abstract we study the following brezis nirenberg problem of kirchhoff type a b u 2 d x u u q 2 u u 2 u in u 0 on where r 4 is a bounded domain with the smooth boundary 2 q 4 and a b are positive parameters we obtain some new existence and nonexistence results depending on the values of the above parameters which improve some known results

existence of a least energy nodal solution for a schrodinger kirchhoff equation with potential vanishing at infinity in this paper we study a class of nonlocal schrodinger kirchhoff problems involving only continuous functions using a minimization argument and a quantitative deformation lemma we find a least energy nodal or sign changing solution to this problem moreover when the problem presents symmetry we show that it has infinitely many nontrivial solutions

on a schrodinger kirchhoff type equation involving the p x laplacian abstract in this paper we deal with a schrodinger type equation involving a nonlocal kirchhoff type coefficient and depending on two real parameters working within the framework of variable exponent spaces and using the variational approach we obtain several results of existence and multiplicity of solutions depending on the range of the parameters

classification and geometrical properties of the xθ valued function spaces abstract this paper is devoted to investigate the x valued function spaces based on the notions of bounded topological lattice banach space net continuous modular net and the dual space net we divide x valued function spaces into two classes norm modular spaces and modular modular spaces for the first class we study the separability of l p i x give a representation of the dual space l p i x find sufficient conditions of the equality l p i x l p i x and prove the reflexivity of l p i x under some reasonable conditions and for the second class we prove the completeness and uniform convexity of l i x in suitable situations to show the naturality and rationality of these results some concrete function spaces such as l p t i l p x t l p t i w 1 p x t and l p t i w 0 1 p x t together with l p p t i l p x t are taken into account

sign changing solutions for a fractional kirchhoff equation abstract using a minimization argument and a quantitative deformation lemma we establish the existence of least energy sign changing solutions for the following nonlinear kirchhoff problem a b u 2 s u v x u k x f u in r 3 where a b 0 are constants s 0 1 s is the fractional laplacian v k are continuous positive functions allowed for vanishing behavior at infinity and f is a continuous function satisfying suitable growth assumptions moreover when the nonlinearity f is odd we obtain the existence of infinitely many nontrivial weak solutions not necessarily nodals

a continuous asymptotic tracking control strategy for uncertain nonlinear systems in this note we present a new continuous control mechanism that compensates for uncertainty in a class of high order multiple input multiple output nonlinear systems the control strategy is based on limited assumptions on the structure of the system nonlinearities a new lyapunov based stability argument is employed to prove semiglobal asymptotic tracking

multiple solutions for p kirchhoff equations in rn abstract in this paper we prove the existence of multiple ground state solutions for the nonhomogeneous p kirchhoff elliptic equation 0 1 m r n u p v x u p d x p u v x u p 2 u f x u g x in r n where v x c r n and v x as x the nonlinear function f x u is continuous and satisfies some conditions the solutions will be obtained by the mountain pass theorem ekeland s variational principle and krasnoselskii s genus theory in struwe 2000 1

study of a nonlinear kirchhoff equation with non homogeneous material abstract in this paper we study a non homogeneous elliptic kirchhoff equation with nonlinear reaction term we analyze the existence and uniqueness of positive solution the main novelty is the inclusion of non homogeneous term making the problem without a variational structure we use mainly bifurcation arguments to get the results

remarks on an elliptic equation of kirchhoff type abstract we present a short survey on the nonlocal elliptic boundary value problem m u 2 d x u f x u in u 0 on where is a smooth bounded domain of r n m is a positive function and f has subcritical growth

nonlinear perturbations of a periodic elliptic problem with critical growth abstract using the well known mountain pass theorem due to ambrosetti and rabinowitz we establish conditions for the existence of a positive solution for a class of nonlinear elliptic problems in r n with critical growth

existence of renormalized solutions to elliptic equation in musielak orlicz space abstract we prove existence of renormalized solutions to general nonlinear elliptic equation in musielak orlicz space avoiding growth restrictions namely we consider div a x u f l 1 on a lipschitz bounded domain in r n the growth of the monotone vector field a is controlled by a generalized nonhomogeneous and anisotropic n function m the approach does not require any particular type of growth condition of m or its conjugate m neither 2 nor 2 the condition we impose is log holder continuity of m which results in good approximation properties of the space the proof of the main results uses truncation ideas the young measures methods and monotonicity arguments

existence of renormalized solutions of degenerate elliptic parabolic problems we consider a general class of degenerate elliptic parabolic problems associated with the equation b t div a d f existence of renormalized solutions is established for general l 1 data uniqueness of renormalized solutions has already been shown in a previous work

multiple solutions for kirchhoff type equations in rn this paper is devoted to the existence of infinitely many solutions for a class of kirchhoff type equations setting on rn based on the minimax methods in critical point theory we obtain infinitely many large energy and small energy solutions which unify and sharply improve the recent results of wu existence of nontrivial solutions and high energy solutions for schrodinger kirchhoff type equations in rn nonlinear anal real world appl 12 1278 1287 2011

renormalized solution for a nonlinear parabolic problems with noncoercivity in divergence form in orlicz spaces we prove the existence result in the setting of orlicz sobolev spaces the n function m does not satisfy the 2 condition existence of a renormalized solution for a class of nonlinear parabolic equations the data belongs to l 1 and no continuous assumption is made on divergence form in this paper we prove the existence results for renormalized solution of the following nonlinear parabolic problems in orlicz sobolev spaces 1 p b x u t div a x t u u div x t u f in q t 0 t b u t 0 b u 0 in u 0 on 0 t where b x u is unbounded function of u the term div a x t u u is a leray lions operator defined on a subset of w 0 1 x l m q t where m is a n function without assuming a 2 condition the second term f l 1 q t and is a noncoercive function which satisfies only the growth condition

adaptive control of continuous time systems with convex concave parametrization this paper deals with adaptive control of a class of nonlinear dynamic systems with nonlinear parametrization in this class the state variables are assumed to be accessible and the nonlinearity in the parameters is assumed to be either convex or concave by introducing a tuning function and an adaptive law based on a min max strategy it is shown that such a class of dynamic systems can be adaptively controlled in a stable manner global boundedness of the overall adaptive system and tracking to within a desired precision are established with the new adaptive controller the proposed controller is applied to a precise positioning problem in the presence of nonlinearly parametrized friction dynamics it is shown that the controller leads to position errors and friction compensation that are orders of magnitude better than those based on estimation of linear parametrizations the fact that the new parameter estimation strategy used is distinct from the traditionally used gradient schemes permits the expansion of the scope of adaptive control which has been restricted hitherto in most cases to systems with linear parametrization

local higher integrability of the gradient of a quasiminimizer under generalized orlicz growth conditions abstract we study local quasiminimizers of the dirichlet energy under generalized growth conditions special cases include standard variable exponent and double phase growths we show that the gradient of a local quasiminimizer has local higher integrability

existence and concentration result for the kirchhoff type equations with general nonlinearities in this paper we study the existence and concentration behaviors of positive solutions to the kirchhoff type equations varepsilon 2 m left varepsilon 2 n int mathbf r n nabla u 2 mathrm d x right delta u v x u f u quad rm in mathbf r n quad u in h 1 mathbf r n n geqq 1 where m and v are continuous functions under suitable conditions on m and general conditions on f we construct a family of positive solutions u varepsilon varepsilon in 0 tilde varepsilon which concentrates at a local minimum of v after extracting a subsequence ek

applications of hysteresis switching in parameter adaptive control the hysteresis switching algorithm of r h middleton et al ibid vol 33 no 1 p 50 8 jan 1988 is reexamined in a broader context to demonstrate its utility the algorithm is applied to various families of identifier based parameterized controllers of both the direct and indirect control types application to the direct control type results in a model reference adaptive controller capable of stabilizing without excitation any siso process which can be modeled by a minimum phase linear system whose transfer function has relative and mcmillan degrees not exceeding prescribing integers m and n respectively it is shown that such processes can also be adaptively stabilized with indirect adaptive controllers and hysteresis switching a simple numerical example involving a non minimum phase process model is used to illustrate how hysteresis switching might be applied to implicitly tuned parameterized controllers to realize an adaptive controller with capabilities which might prove very difficult if not impossible to achieve without hysteresis switching or some other form of discontinuous control

the existence of nontrivial solutions for the critical kirchhoff type problem in rn abstract in this paper we study the problem 0 1 a b r n u 2 d x u v x u f u g x u 2 1 in r n where n 3 a b 0 when n 3 and 0 is small under suitable conditions on v f and g we prove 0 1 has two nontrivial solutions when n 4 and g x 1 we also obtain the existence and non existence results

existence and uniqueness for a degenerate parabolic equation with l1 data in this paper we study existence and uniqueness of solutions for the boundary value problem with initial datum in l1 ut div a x du in 0 u a u on 0 u x 0 u0 x in where a is a caratheodory function satisfying the classical leray lions hypothesis a is the neumann boundary operator associated to a du the gradient of u and is a maximal monotone graph in r r with 0 0

renormalized solutions to nonlinear parabolic problems in generalized musielak orlicz spaces abstract we will present the proof of existence of renormalized solutions to a nonlinear parabolic problem t u div a d u f with right hand side f and initial data u 0 in l 1 the growth and coercivity conditions on the monotone vector field a are prescribed by a generalized n function m which is anisotropic and inhomogeneous with respect to the space variable in particular m does not have to satisfy an upper growth bound described by a 2 condition therefore we work with generalized musielak orlicz spaces which are not necessarily reflexive moreover we provide a weak sequential stability result for a more general problem t u div a d u f u f where is a monotone function with respect to the second variable and f is locally lipschitz continuous within the proof we use truncation methods young measure techniques the integration by parts formula and monotonicity arguments which have been adapted to nonreflexive musielak orlicz spaces

renormalized solutions for a nonlinear parabolic equation with variable exponents and l 1 data abstract we prove the well posedness existence and uniqueness of a renormalized solution to nonlinear parabolic equations with variable exponents and l 1 data the functional setting involves lebesgue sobolev spaces with variable exponents

on a kirchhoff type problem in r n abstract in this paper we investigate the following kirchhoff type problem p r n u 2 u 2 d x u u u p 2 u in r n u h 1 r n where n 1 2 p 2 2 2 n n 2 if n 3 2 if n 1 2 and are two positive parameters by studying the decomposition of the nehari manifold to p and using the scaling technique we give a total description on the positive solutions to p we also make an observation on the sign changing solutions to p in the current paper

on some class of nonlinear parabolic inequalities in orlicz spaces abstract the paper deals with the existence of solutions of some unilateral problems in the framework of orlicz spaces

structural stability for variable exponent elliptic problems i the p x laplacian kind problems abstract we study the structural stability i e the continuous dependence on coefficients of solutions of the elliptic problems under the form b u n div a n x u n f n the equation is set in a bounded domain of r n and supplied with the homogeneous dirichlet boundary condition on here b is a non decreasing function on r and a n x n is a family of applications which verifies the classical leray lions hypotheses but with a variable summability exponent p n x 1 p p n p the need for varying p x arises for instance in the numerical analysis of the p x laplacian problem uniqueness and existence for these problems are well understood by now we apply the stability properties to further generalize the existence results the continuous dependence result we prove is valid for weak and for renormalized solutions notice that besides being interesting on its own the renormalized solutions framework also permits us to deduce optimal convergence results for the weak solutions our technique avoids the use of a fixed duality framework like the w 0 1 p x w 1 p x duality and thus it is suitable for the study of problems where the summability exponent p also depends on the unknown solution itself in a local or in a non local way the sequel of this paper will be concerned with well posedness of some p u laplacian kind problems and with existence of solutions to elliptic systems with variable solution dependent growth exponent

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

uniqueness of renormalized solutions of degenerate elliptic parabolic problems we consider a general class of degenerate elliptic paraboic problems associated with the equation b v t div a v dv f using kruzhkov s method of doubling variables both in space and time we prove uniqueness and a comparison principle in l 1 for renormalized solutions

existence of positive ground state solutions for kirchhoff type problems abstract in this paper we study the existence of positive ground state solutions for the nonlinear kirchhoff type problem a b r 3 u 2 u v x u f u in r 3 u h 1 r 3 u 0 in r 3 where a b 0 are constants f c r r is subcritical near infinity and superlinear near zero and satisfies the berestycki lions condition by using an abstract critical point theorem established by jeanjean and a new global compactness lemma we show that the above problem has at least a positive ground state solution our result generalizes the results of li and ye 2014 concerning the nonlinearity f u u p 1 u with p 2 5

positive solutions for the kirchhoff type problem involving general critical growth part i existence theorem involving general critical growth abstract in this paper we consider the following kirchhoff type problem a r 3 u 2 d x b r 3 u 2 d x u b u f u in r 3 u h 1 r 3 u 0 in r 3 where 0 is a parameter a b are positive constants and f reaches the critical growth without the ambrosetti rabinowitz condition we prove the existence of positive solutions for the kirchhoff type problem with a general critical nonlinearity we also study the asymptotics of solutions as 0 numerical solutions for related problems will be discussed in the second part

convergence of a full discretization of quasi linear parabolic equations in isotropic and anisotropic orlicz spaces convergence of a subsequence of approximate solutions arising from a full discretization is shown for a general class of quasi linear parabolic problems the numerical method combines the backward euler method for the time discretization with a generalized internal approximation scheme for the spatial discretization the governing monotone elliptic differential operator is described by a nonlinearity that may have anisotropic and nonpolynomial growth but fulfills a coercivity condition in terms of a generalized mathscr n function if the problem admits a unique solution which is shown in the case of a strictly monotone nonlinearity and in a class of sufficiently smooth solutions then the whole sequence of approximate solutions converges moreover an a priori error estimate for the temporal semidiscretization is provided and a numerical illustration is given

renormalized solutions to parabolic equations in time and space dependent anisotropic musielak orlicz spaces in absence of lavrentiev s phenomenon abstract the paper concerns existence and uniqueness of solutions to a nonlinear parabolic equation with merely integrable data on a lipschitz bounded domain in r n our focal point is to involve the leading nonlinear part of the operator whose growth is described by anisotropic n function m inhomogeneous in the space and the time variables the main goals are proven in absence of lavrentiev s phenomenon to ensure which we impose a certain type of balance of interplay between the behavior of m for large and small changes of time and space variables its instances are log holder continuity of variable exponent inhomogeneous in time and space or optimal closeness condition for powers in double phase spaces possibly changing in time new delicate approximation in time result is proven and applied in the construction of renormalized solutions

stability and robustness properties of a simple adaptive controller a low order adaptive tracking controller is proposed for linear time invariant plants with a relative degree not exceeding two the order of the plant is not required to be known a priori the algorithm is robustly stable with respect to small linear time invariant plant perturbations and bounded disturbances the robustness is achieved by using a projection of the parameter estimates in the control law an additional a priori information needed to design the controller is bounds on the plant parameters in the absence of plant perturbations and disturbances perfect tracking is obtained

nonlinear perturbations of a periodic kirchhoff equation in rn abstract this paper is concerned with the existence of positive solutions for the following class of nonlocal problem m r n u 2 d x r n v x u 2 d x u v x u f u u in r n where 5 for n 3 and 1 for n 1 2 is a positive parameter and 0 1 moreover m v and f are continuous functions satisfying some conditions

the critical problem of kirchhoff type elliptic equations in dimension four abstract we study the following kirchhoff type elliptic problem p a b u 2 d x u u q u 3 u 0 in u 0 on where r 4 is a bounded domain with smooth boundary moreover we assume a 0 b 0 and 1 q 3 in this paper we prove the existence of solutions of p our tools are the variational method and the concentration compactness argument for ps sequences