Study Of Solutions To Some Nonlinear Elliptic Equations In Weighted Sobolev Spaces

In the past few decades,many mathematicians were interested in the study of the existence,uniqueness and regularity of solutions of partial differential equations,especial-ly of the nonlinear elliptic equations.The paper discusses the existence and nonexistence of solutions to some elliptic equations with constant exponents and variable exponents in the framework of the weighted Sobolev spaces.The main topics include the existence and nonexistence of solutions to nonlinear elliptic equation;the renormalized solutions or the entropy solutions to nonlinear elliptic equation with degenerate coercivity;the renormalized solutions and entropy solutions to a nonlinear p(x)-Laplace equation with zero-order term;the entropy solutions and the renormalized solutions to nonlinear elliptic equation.In Chapter 1 we first describe the background of the problems considered in this paper.Then we briefly recall the related works about the elliptic equations obtained by the scholars both in China and aboard.We also state our problems and some methods to be used.At the end,we introduce the main content of the weighted Sobolev space.In Chapter 2,we investigate the existence and nonexistence of weak solutions to nonlinear elliptic equation with Dirichlet boundary conditions.Here,? is a bounded domain in RN(N ? 2).The right-hand side term has the form?=f-divF,where be a vector of measurable weighted function strictly positive a.e.in such thatWe denote by the space of all real-valued functions u?Lp(?,?0)such that the derivatives in the sense of distributions satisfy endowed with the normFurthermore,with regard to a,g,we also assume that(Al)a(x,s,?)={ai(x,s,?)}1?i?N:?×R×RN?RN is a Caratheodory vector-valued function,for almost every x??,s?R,??RN,satisfyingwhere k(x)is a positive function in Lp'(?),1/p+1/p'=1,? is weighted function and theconstants c0,c1 are both positive.(A2)Let g(x,s,?)is a Caratheodory function,for almost every x??,s?R,??RN,there exist h,?>0,such thatwhere b:R+?R+ is a continuous increasing function and d(x)is a nonnegative function in L1(?).In the first part of Chapter 2,we investigate the existence of weak solutions of Problem(9).There are two main differences compared to the other papers.First,we discuss the existence of weak solutions in the framework of Sobolev space,the embedding relationship has changed.Second,our sign assumption on g in(10)is different from the usual sign assumption on g in other paper,namely,It is easy to see that(12)hold for every s ? R while in our case,the requirement for the sign condition when |s|<h is removed.In other word,we only require(10)holds for|s|?h(large values of |s|).Therefore,we obtain the approximate equation corresponding to Problem(9),after dealing with the gradient term and the right-hand side term by truncation approximation.Considering the sign condition(10),we choose appropriate test functions and construct the needed estimates to the approximate equation.At the end,by proceeding a limit process,we embrace the existence of weak solutions of Problem(9):Theorem 1.(the existence of weak solutions)Assume(Al),(A2)hold,let f be in L1(?)and F??i=1NLp'(?,?i*),then there exists at least one solution u of Problem(9).In the second part of Chapter 2,we study the nonexistence of solutions of Problem(9).When the right-hand term of the equation is only a bounded Radon measure ??Mb(?),we decompose it by the previous proposition for ?=?0+?.The result of Theorem 1 then ensures that Problem(9)has a solution with datum ? if and only if?=0,?=?0?f-div F.Suppose now that ?0=0,so that ?=? is singular with respect to the p-capacity,what can we say about u?We need to construct a suitable collection of cut-off functions and take appropriate test functions,then we use the measure theory to show that the solution of Problem(9)may not exist,or at least the nontrivial W01,p solution may not be approximated by the sequence of solutions to the approximate problem,i.e.if un(?)u weakly in W01,p(?,?),where u is the solution of Problem(9),then u ? 0.Our main result is as follows:Theorem 2.(nonexistence result)Let ? be a positive measure in Mb(?),concen-trated on a set E such that capp(E,?)= 0,and let fn be a sequence of nonnegative L?(?)functions such that Suppose that g satisfies(11),(12).Let un be a solution of the equationThen,there exists k>0(depending on g and c0),such thatMoreover,un converges weakly to zero in w01,p(?,?),andIn the first section of Chapter 3,we consider the following nonlinear elliptic equation with degenerate coercivity and lower order term in the setting of the weighted Sobolev space.Here,? is a bounded domain in RN(N? 2),?>0,f?L1(?).We denote by W01,p(?,?)(1<p<?)weighted Sobolev space,is a vector of measurable weighted function,with regard to a and g,we also assume that:(A3)a(x,?)={ai(x,?)1?i<?N:?×RN?RN is a Caratheodory vector-valued function,satisfyingwhere k(x)is a positive function in Lp'(?),1/p,1/p'=1 and the constants ?,? are bothpositive.(A4)Let g(x,s)be a Caratheodory function,for almost every x?? Q,s?R,satisfying g(x,s)sgn(s)?0.We investigate the existence of renormalized solutions in W01,p(?,?)by the truncation method.Notice that the first term of(13)is not coercive,the existence of lower order term and that the right-hand side term f is only in L1(?).As a result,the method we used to estimate the approximating solution sequence un,in Chapter 2 will no longer be effective.With the help of Marcinkiewicz estimate,through some priori estimates for the sequence of solutions of the approximate problem,we prove that un converges in measure.Based on Riesz theorem,there exists a subsequence of un that is almost everywhere convergence.Then we choose suitable test functions to the approximate equation and obtain the needed estimates.At the end,proceeding a limit process,we obtain the existence of renormalized solutions to Problem(13):Theorem 3.(the existence of the renormalized solutions)Assume(A3),(A4)hold,f?L1(?)then there exists at least one renormalized solution u of Problem(13).Following this,we study the entropy solutions and the renormalized solutions of elliptic equation in weighted Sobolev space with variable exponents.The assumptions of the weighted Sobolev space with variable exponents,weighted function ?(x)and variable exponents p(x)are as follows:For any p?C+(?),we denote the weighted variable exponents Lebesgue space Lp(x)(?,?)as a space consist of all measurable functions such thatand endowed with the Luxemburg normDenoteWk,p(x)(?,?)= {u?Lp(x)(?,?):D?u?Lp(x)(?,?),|?|?k},We also assume the following:(A5)(1)??Lloc1(?),?-1/p(x)-1?Lloc 1(?).satisfy 1<p?p+<N,furthermore,p(x)satisfy log—Holder continuity condition.In the second section of Chapter 3,we consider the renormalized solutions to a p(x)?Laplace equation in weighted Sobolev space with degenerate coercivity and zero-order term.Here,? is a bounded domain in RN(N>2)with Lipschitz boundary(?)?,?(x)is a weighted function,?(x)?C(?),?(x)>0,f?L1(?).The assumption of g is as follows:(A7)Let g(x,s)be a Caratheodory function,for almost every x??,s?R,??RNand for any k?R+,satisfyingFirst of all,the first term in is not coercive,one cannot seek for a weak energy solution in the classical sense.Besides,the existence of the zero-order term and the fact that the right-hand side term f is only in L1(?),not in W-1,p'(x)(?),all prevent us from employing the classical duality argument or nonlinear monotone operator theory directly.Therefore we make use of the truncation method,taking appropriate test functions and a accurate limit process to obtain the existence of entropy solutions to Problem(14):Theorem 4.(the existence of entropy solutions)Assume(A5),(A6)and(A7)hold,let f?L1(?)then there exists at least one entropy solution u of Problem(14).In chapter 4,we consider a nonlinear p(x)-Laplace equation in weighted Sobolev space with variable exponent.Here,? is a bounded domain in RN(N ? 2)with Lipschitz boundary(?)?,?(x)is a weighted function,f?L1(?)g satisfies the hypothesis(A7).In this chapter,we also utilize truncation method to get the priori estimates to the approximate equation.However,there are many differences comparing to the constant exponent case.First,it is impossible to utilize De Giorgi iteration and take appropriate test function to obtain the uniform L? bound.In order to overcome these difficulties,with the help of Marcinkiewicz estimate with variable exponent and some embedding of weighted Sobolev space with variable exponent,then we obtain almost everywhere convergence of un,and deal with the lower order term.Then,by choosing suitable test functions and proceeding a limit process,we prove the existence of renormalized solutions and entropy solutions of Problem(15):Theorem 5.(the existence of the entropy solutions and renormalized solu-tions)Assume(A5),(A6)and(A7)hold,let f L1(?),then there exist at least one renormalized solution and one entropy solution u of Problem(15).In chapter 5,we consider the entropy solutions to nonlinear elliptic equation with variable exponent in weighted Sobolev space.Here,? is a bounded domain in RN(N?2)with Lipschitz boundary(?)? and the variable exponent We denote by W0,1p(?,?)weighted Sobolev space,?(x)is weighted function.Besides a,g satisfy the following assumptions:vector-valued function,and for all i=1,...,N,satisfying for almost every x??(s,?)?R×RN,where k(x)is a nonnegative function in Lp'(x)(?),and ?,? are positive number.(A9)Let g:?×R×RN?R be a Caratheodory function satisfying the following assumptions for almost every x??,s?R,??RN.|g(x,s,?)|?b(|s|)(c(x)+?|?|p(x)).where b:R+?R+ is a continuous increasing function,c:?R+ satisfies c?L1(?).We study the existence of entropy solutions and renormalized solutions for a class of nonlinear elliptic equation in the framework of weighted Sobolev space with variable exponent.Notice that,the weighted function has increased the difficulty in solving this problem,especially in the aspect of embedding.Moreover,there is no growth condition on?,that is to say,the term in(16)may not make sense even as a distribution.Nevertheless,the integrability of the right-hand term is not good enough.Consequently,one cannot seek for a weak energy solution in the classical sense.To overcome this obstacle,we investigate(16)in the framework of entropy solutions and renormalized solutions using truncation method.With the help of the embedding relationship in weighted Sobolev space with variable exponent,we take appropriate test solutions and a limit process.Then,we obtain the strong convergence of Tk(un),based upon which we prove the existence of entropy solutions to Problem(16).In the next step,we prove that the entropy solution u is also a renormalized solution of Problem(16),the main results are as follows:Theorem 6.(the existence of entropy solutions)Assume(A5),(A6),(A8),(A9)hold,let f?L1(?),F?W-1,p'(x)(?,?*),then there exists at least one entropy solution u of Problem(16).Theorem 7.(the existence of renormalized solutions)Assume(A5),(A6),(A8),(A9)hold,let f?L1(?),F?W-1,p'(x)(?,?*),then the entropy solution u?W01,p(x)(?,?)is also a renormalized solution of Problem(16).