Study Of Solutions To A Class Of Nonlinear Elliptic Equations With Singular Terms

In this paper we mainly study the existence and regularity of solutions to a class of nonlinear elliptic equations with singular terms. These problems arise from mathemat-ical physics models, such as the non-Newtonian fluid, chemical heterogeneous catalysts, nonlinear heat conduction etc. This paper is divided into four chapters, the contents are as follows:In Chapter one, we first describe the background of the problems considered in this paper. Then we briefly recall some related results about the semilinear or quasilinear elliptic equations obtained by authors both in China and abroad. At the end of this Chapter, we state our problems and sketch the methods and the techniques to be used.In Chapter two, we first investigate the existence of solutions to the following Dirich-let boundary value problem of semilinear elliptic equation involving a singular nonlinear term where Ω is a bounded domain in RN(N≥2) with smooth boundary (?)Ω,α＞1,f is a nonnegative function belonging to some Lebesgue space, and M is a bounded positive definite matrix; i.e., there exists0＜γ≤β such that for every η in RN, for almost every x in Ω.We mainly discuss the effect of the summability of the weighted function f(x) and the singular nonlinear term on the right hand side of the equation. We will use the technique of approximation with the help of the integrability of the solution to the first step of the approximate problem and some a priori estimates to prove the existence of weak solutions for problem (8) when m＞1,1＜α＜2-1/m. Our main result is as follows:Theorem1. Let f be a nonnegative function in Lm(Q)(m＞1), f(?)0. If1＜α＜2-1/m, then there exists a solution u in H01(Ω) for problem (8), in the sense thatIn the second part of Chapter two, we investigate the existence of solutions to the following Dirichlet boundary value problem of semilinear elliptic equation involving a singular nonlinear term and a variable exponent. where Ω is a bounded domain in RN (N≥2) with smooth boundary (?)Ω, α(x) is a continuous function, α(x)＞0,f is a nonnegative function belonging to some Lebesgue space, and M is a bounded positive definite matrix.The equation is not scaling invariance since the appearance of the singular term on the right hand side of the equation, and the variable exponent in the singular term. It is hence difficult for us to prove the existence of solutions to this problem. We utilize the supremum and infimum of α(x) and some a priori estimates to solve the difficulties caused by the variable exponent and use the integrability of solution to the approximate problem with n=1to overcome the difficulties arising from the singular term, then we obtain the existence of the weak solution to the problem with the help of the method of regularization and the Schauder fixed point theorem as well as a necessary compactness argument.Furthermore.we show that the existence and regularity of solutions depend on the summability of f(x)and the range of α(x).The supremum and infimum of α(x)are denoted respectively by α+and α-,that is,In the case0＞α-≤α(x)≤α+＜1,the existence of solutions for problem(10)is obtained if f is more regular than L1(Ω),we have the following three results:Theorem2.Suppose that f is a nonnegativc function in Lm(Ω)(f(?)0),with m=and let0＜α-≤α(x)≤α+＜1.Then problem (10)has a solution u∈H01(Ω)satisfyingTheorem3.Suppose that and0＜α-≤α(x)≤α+＜1.Then problem(10)has a solution u in W01,1(Ω)with Theorem4.Suppose that f∈Lm(Ω)and0＜α-≤α(x)≤α+＜1.Then problem(10)has a solution u in W01,1(Ω) with In the case1＜α-≤α(x)≤α-,we can prove that problem(10)has a solutionu in H01(Ω)if f is more regular than L1(Ω)and α+and α-are close to1,we have the following result: Theorem5.Suppose that f∈Lm(Ω)(m＞1)and1＜α-＜α-+＜2-1/m,f(?)0.Then problem(10)has a solution u in H01(Ω). Moreover,we can prove that problem(10)has a solution u in Hloc1(Ω)if f∈Lm(Ω)(m＞1)and α+is close to α-.The result is: Theorem6.Suppose that f is a nonnegative function in Lm(Ω)(m＞1),f(?)0,1＜α≤α(x)≤α+and α+-α-＜1-1/m.Then problem(10)has a solution u in Hloc1(Ω). Furthermore,belongs to H01(Ω). In the case0＜α-＜1＜α+,we can also prove that problem(10)has a solution uin H01(Ω)if f is more regular than L1(Ω).The result we obtain is: Theorem7.Suppose that f∈Lm(Ω) with and0＜α-＜1＜α+＜2-1/m,f(?)0.Then problem(10)has a solution u in H01(Ω). In Chapter three, we investigate the existence of solutions to the following Dirichlet boundary value problem of quasilinear elliptic equation involving a singular nonlinear term, where Ω is a bounded domain in RN(N≥1) with smooth boundary (?)Ω, f∈Lm(Ω),f≥0,f(?)0, p＞1, α≥1.The difference between the problem discussed in this Chapter and the problem in the first part of Chapter two is that the linear differential operator on the left hand side of the equation is replaced by the quasilinear differential operator, which makes the approximating solution sequence un to the approximate problem we constructed may not have the following weak convergence|â–½un|p-2â–½unâ†'|â–½u|p-2â–½u in (Ω, RN), where u is the solution of problem (12). All of these pose a challenge for us to obtain the existence of the solutions. We overcome the difficulty by choosing suitable test functions and a priori estimate techniques, and prove that the existence of weak solutions for problem (12) when f∈L1(Ω), α=1; then we utilize the measure theory to show that the solution of problem (12) may not exist, at least the solution may not be approximated by the sequence of solutions for the approximate problem if the inhomogeneous function f(x) is not in L1(Ω) but a nonnegative bounded Radon measure μ. Finally, we generalize the result of the first part in Chapter two to obtain the existence of weak solutions for problem (12) when m＞1,1＜α＜2-1/m. Unlike most previous works concerning similar problems, we do not need to discuss the existence of solutions by means of upper-lower solution techniques. Our main results are as follows:Theorem8. Suppose that f is a nonnegative function in L1(Ω),f(?)0and α=1. Then problem (12) has a solution in W01,p(Ω).Theorem9. Suppose that μ is a nonnegative Radon measure concentrated on a Borel set E of zero p-capacity, and that{gn} is a sequence of nonnegative bounded L1(Ω) functions which converges to μ in the narrow topology of measures. Let un be the solution of the approximate problem with the non-homogeneous function fn=gn(x). Then Although Theorem9does not imply that problem (12) has no solution when the inhomogeneous function f(x) is not in L1(Ω) but a nonnegative bounded Radon measure μ, it at least shows that the solution may not be approximated by the sequence of solutions for the approximate problem.As for the existence of solutions to our problem, we have the following theorem.Theorem10. Let f be a nonnegative function in Lm(Ω)(f(?)0)(m＞1). If1α＜2-1/m, then problem (12) has a solution u∈W01,p(Ω) satisfyingIn Chapter four, we investigate the existence of solutions to the following Dirich-let boundary value problem of quasilinear elliptic equation involving singular term and variable exponent where Ω(?)Rn(N≥p) is a bounded domain with a smooth boundary, p＞2, α(x) is a continuous function, α(x)＞0,f is a nonnegative function belonging to some Lebesgue space.In addition to the difficulties in solving problem (12), the variable exponent α(x) increases greatly the complexity of problem (13). With the help of the supremum and infimum of α(x) and some a priori estimates, we overcome the difficulties arising from variable exponent and nonlinear differential operator by choosing some suitable test func-tions and by using Sobolev embedding theorem and Schauder fixed point theorem. The existence of solutions to problem (13) is obtained.The supremum and infimum of α(x) are denoted respectively by α+and α-, that is,In the case0＜α-≤α(x)≤α+＜1, the existence of solutions for problem (13) is obtained if f is more regular than L1(Ω), the theorems are the following:Theorem11. Suppose that f is a nonnegative function in Lm(Ω)(f(?)0), with and let0＜α-≤α(x)≤α+＜1. Then problem (13) has a solution u u∈W01,p(Ω) satisfyingTheorem12. Suppose that and0＜α-≤α(x)≤α+＜1. Then problem (13) has a solution u in W01,1(Ω) withTheorem13. Suppose that and0＜α-≤α(x)≤α+＜. Then problem (13) has a solution u iu W01,1(Ω) withIn the case1＜α-≤α(x)≤α+, we can prove that problem (13) has a solution u in W01,p(Ω) if/is more regular than L1(Ω) and α+and α-are close to1, we have the following result:Theorem14. Suppose that f∈Lm(Ω)(m＞1) and1＜α-＜α+＜2-1/m,f(?)0Then problem (13) has a solution u in W01,p(Ω).Moreover, we can prove that problem (13) has a solution u in Wloc1,p(Ω) if f∈Lm(Ω)(m＞1) and α+is close to α-, we have the following result:Theorem15. Suppose that f is a nonnegative function in Lm(Ω)(m＞1),f(?)0,1＜α-≤α(x)≤α+and α+-α-＜1-1/m. Then problem (13) has a solution u in Wloc1,p(Ω). Furthermore,belongs to W01,p(Ω).In the case0＜α-＜1＜α+, we can prove that problem (13) has a solution u in W01,p(Ω) if/is more regular than L1(Ω), we have the following result:Theorem16. Suppose that f∈Lm(Ω) with and0＜α-1＜α+＜2-1/m,f(?)0. Then problem (13) has a solution u in W01,p(Ω).In this Chapter, α(x) is clarified completely, we obtain the existence and regularity of solutions for problem (13) when α(x) is at the different range which shows the effect of the summability of f(x) and the range of α(x) on the existence and regularity of solutions to this problem.