Existence And Multiplicity Of Solutions For Boundary Value Problems With P-Laplacian

Existence and multiplicity of solutions for boundary value problems of differential equations has been one of the most active subject in modern mathematics. One reason is that the problems arises in some areas of science and technology such as physics, chemistry, biology, economics. On the other hand, some purely mathematical properties of the p-Laplacian seem to be a challenge for nonlinear analysis and their study lead to the development of new methods and approaches. In recent years, due to the close relation with the non-Newton filtration, population dynamics and nonliear elastic, etc., existence and multiplicity of solutions for p-Laplacian boundary value problems has attracted great attention from researchers. However, until now, we know few about the spectrum and solutions of the various boundary value problems involving the p-Laplacian. The purpose of this thesis is to study the existence and multiplicity of solutions for boundary value problems with p-Laplacian using a combination of varia-tional method, upper and lower solutions method, degree theory, critical point theory and homotopy continuation method. In Chapter 2 we consider the periodic boundary value problem of the one dimensional p-Laplacianwhereφ(u) =|u|^p-2u,f∈C([0,2Ï€]×R,R).Circle sports appear everywhere in the nature. Ever since the investigation of three-body problem by Poincare at the end of the nineteenth century, there has been an active research for periodic solutions of ordinary differential equations. One of the most interesting topic in the periodic problems is resonance problems due to the appearance of resonance phenomenon which arises frequently in physics and plenty of results have been obtained by using all kinds of methods and techniques. When p = 2, the operator -(φ_p(u'))' is just the operator -u",which has a sequence of eigenvalues {n²} at the periodic boundary value conditions. In general, the nonlinearity is called nonresonant if the asymptotic behavior at infinity of the ratio (?) stays strictly between two consecutive eigenvalue of -u". If the ratio (?) stays asymptoticlly between two consecutive eigenvalues with touching the eigenvalues allowed, we say it as double resonance. Specially, if the ratio (?) takes some eigenvalue at infinity, we say that resonancehappens.In 1967, Loud studied the solvability of the resonant periodic problem(1) with p=2. Subsequently, many famous mathematicians, such as Ressig, Gossez, Omari and Zanolin, have studied the solvability of problem(l) with p=2 under resonance or nonresonance conditions. In 1976, Dancer[55] and Fu(?)(?)k[68] introduced the Fu(?)(?)k spectrum, which is denoted by∑_p¹.Obviously, (n²,n²)∈E∑_p¹.Hence the Fu(?)(?)k spectrum is a generalized spectrum. Since then, resonance problem has been studied by many people under the ??frameworks of the Fu(?)(?)k spectrum(see [57, 60]).On the other hand, in 1989, Habets and Metzen[75] introduced the property P as follows. Definition We say that the set Q has property P, if(i) there exists v∈L^∞(0,2Ï€) such that (t, v(t))∈Q,(?)t∈[0, 2Ï€],and(ii) if∈L^∞(0,2Ï€) satisfies (i), thenhas only the trivial solution.Subsequently, many authors used this property to investigate resonant problems in ordinary differential equations (see[66, 70, 126]). In 1989, Fonda and Mawhin[67] applied the variational methods to prove that problem(1) with p=2 is solvable if there exist a, 6∈L¹(0,2Ï€) such thatand the sethas property P. Here the property P is different from neither the eigenvalue nor the Fu(?)(?)k and hence can be seen as a generalized spectrum.In the three types of resonance problems stated above, the asymptotic limits of the ratio (?) at infinity stay on the spectrum or outside the spectrum. Naturally, is there a solution if the asymptotic behavior of f at infinity cross the spectrum? In 1993, Fonda obtained the solvability of problem(1) with p =2, f(x,u) = f(u) + h(x) by assuming that the ratio stay(?) stays asymptoticlly between the first eivenvalue and the first curve of the Fu(?)(?)k spectrum, where F(u) =∫₀^u f(s)ds. In 2006, Liu and Li [94] extended this result. They proved that if / satisfies the sign condition and has linear growth at infinity, then problem(l) is solvable under the assumption that (?) stays asymptoticlly at infinity between two consecutive curve of Fu(?)(?)k spectrum. For general p∈(1, +∞), is there some solvability of the problem (1)? In 1992, del Pino, Man(?)sevich and Mur(?)a[?] first studied the periodic problem of the one dimensional p-Laplacian. There they studied the Fu(?)(?)k spectrum of the one dimensional p-Laplacian and obtained the solvablity of problem(1) when the ratio (?) is nonresonant with respect to the Fu(?)(?)k spectrum. Subsequently, many researchers studied the periodic problem of the one dimensional p-Laplacian under the frame of Fu(?)(?)k spectrum, see[81, 127]. In this chapter we obtain the solvability of problem(1) by using the asymptotic limit at infinity of (?.)For p∈(1,+∞),we obtain the solvability of problem(1) when f satisfies p-linear growth at infinity and the ratio (?) stays asymptoticlly at infinity between two consecutive curves of the Fu(?)(?)k spectrum. Note that we do not assume the asymptotic limit of the ratio (?) at infinity to exist, the ratio (?) may be oscillatory and may cross a finite number of consecutive curves of the Fu(?)(?)k spectrum of the one dimensional p-Laplacian. In addition, whether there is a similar result if we utilize the property P to study the solvability of problem(1)? We give an affirmative answer in Chapter 2.In the last of Chapter 2, using the asymptotic behavior of the ratio (?) at infinity and at zero, we apply the mountain pass lemma to prove that problem(1) admits at least one nonnegative solution and onenonpositive solution. In Chapter 3 we consider the following Dirichlet problem with p-Laplacianwhereâ–³_pu = div(|(?)u|^p-2(?)u),Ω(?) Rⁿ(n≥1) is a bounded region,f∈C((?)×R,R).In 1984, Otani[105] first investigated the eigenvalue problems for the p-Laplacian with n-1 under Dirichlet boundary value condtion. In recent years, Dirichlet boundary value problems involving the p-Laplacian has attracted great attention from researchers. It is well known that (-â–³_p,W₀^1,p(Ω)) admits a sequence of variational eigenvalues {λ_k}(see [58]) such that 0 <λ₁ <λ₂ <…<λ_k→+∞.For the cases n =1,1 < p < +∞and n≥2,p= 2,{λ_k} are just the complete eigenvalues, However,when n≥2,1 < p < +∞, it is not clear that if there are any other eigenvalues? If there are some other eigenvalues then how many? In 1986, Boccardo, Dr(?)bek, Giachetti and Ku(?)era[28] obtained the solvability of problem(2) if the nonlinearity f is nonresonant atλ₁.Subsequently, Anane and Gossez[15], Ambrosetti and Arcoya[8], Arcoya and Orsina[17] studied respectively the solvability of problem(2) if the nonlinearity f is resonant atλ₁.In 1999, Dr(?)bek and Rabinson[58] dealt with the resonance problem of the p-Laplacian for generalλ∈R.They obtained the existence of solution of problem(2) by assuming that there exists (?)∈L^?(Ω) such thatand the Landesman-Lazer conditions hold. In 2004, Liu and Li[94] proved that problem(2) always has a solution if nonresonance happens.On the other hand, in 1999, Cuesta,de Figueiredo and Gossez[42] first studied the Fu(?)(?)k spectrum for the p-Laplacian. Denote by∑_p the Fu(?)(?)k spectrum of-Δ_p in W₀^1,p(Ω). They constructed the first nontrivial curve l of∑_p with variational character and obtained the solvability of (2) if the ratio (?) stays asymptoticlly between the first eigenvalue and l.Later on, Perera[108, 109] proved that∑_p contains a sequence of hyperbolic curves and dealt with the resonance and nonresonance problems with respect to the Fu(?)(?)k spectrum. Note that in the previous papers, the asymptotic limit of the ratio (?) at infinity stays on the∑_p or in the set R²\∑_p.If f cross the spectum or the Fu(?)(?)k spectrum, is there any solution? Similar to the discussion in the Chapter 2, for p€(1,+∞), using the interaction of the asymptotic behavior of F at infinity with the set Q satisfying property P and the Fu(?)(?)k spectrum respectively, we establish the existence of solutions by a prior estimate and degree theory.For the multiplicity, because of the fact that W₀^1,p(Ω) is not a Hilbert space, we can't obtain multiplicity of solutions by utilizing the space structure of W₀^1,p(Ω).If f satisfies the superlinear subcritial growth, one usually uses the superquadratic condition (Ambrosetti-Rabinowitz condition) (see [12] if p=2)whereθ> p, to assure the compactness of the corresponding energy functional and obtain the multiplicity of solutions, see[24, 54, 107]. Note that the (AR)_θcondition implies that f is superlinear and it obviously does not hold if the nonlinearities admits p-linear growth. While in the study of guided modes of an electromagnetic field in a nonlinear medium, laser beams in plasma and selection-migration models in population genetics(see[64, 120, 121, 122]), the nonlinearity has p-linear growth. In recent years, the study of multiplicity of solutions for problem (2) when f has p-linear growth has attracted much attention. Carl and Perera[31] obtained three nontrivial solutions by using variational method and upper and lower solution method, Li and his group[128, 129] also obtained three nontrivial solutions by using variational method and flow invariance. In 2004, Jiang obtained three nontrivial solutions using the Morse theory. In 2007, D. Motreanu, V. V. Motreanu and N. S. Papageorgiou[101] got four nontrivial solutions by the degree theory and upper and lower solution method. In Chapter 3 we apply the mountain pass theorem, together with the asymptotic behavior of (?)|u|^p-2u) at infinity and at zero, to obtain two nontrivial solutions when f has p-linear growth. Furthermore, if there exist a positive upper solution and a negative lower solution, we obtain five nontrivial solutions by combining with variational method, upper and lower solution method, degree theory, critical point theory and truction technology. In addition, if f is odd, we obtain k pair of nontrivial solutions by using the symmetric mountain pass theorem.In Chapter 4 we study the following Dirichlet problem involving the p-Laplacian with combined nonlinearitieswhereΩ(?) Rⁿ(n≥1) is a smooth bounded region,1∞(Ω),f∈C((?)×R,R).Since the famous paper of Ambrosetti, Brezis and Cerami[10], there are several works on the elliptic problems with combined nonlinearities, such as [25, 94]. Most of these papers dealt with the superlinear problems. There are few papers which deal with the case that f satisfies p-linear growth.(see [90,108]). In this thesis we will consider three cases:1.h∈L^∞(Ω),h(x)(?)0;2.h(x)≡-λ;3.h(x)≡λ.Using the asymptotic limits at infinity and at zero of the ratio (?) we obtain a sequence of multiplicity results of solutions for problem (4) when the nonlinearity f satisfies p-linear growth.When h(x)≡λ>0,problem (1.8) becomes the following problemFor p = 2 and f(x,u)≡u^r,Ambrosetti, Brezis and Cerami[10] proved that if 1* > 0 such that problem (5) has, at least, two positive solutions forλ<λ^*, one positive solution at least ifλ<λ^*, and no positive solution at all ifλ>λ^*. For p≠2 and f(x,u)≡|u|^r-2u with r∈(p,p^*), where p～*=(?) if p < n and p～* = +∞if p > n,Ambrosetti, Azorero and Peral[9] dealt with the radial solutions using a priori estimates and topological arguments. In 2000, Azorero, Peral and Manfredi[22] considered the nonradial case using variational setting and upper and lower solution method. In [90], by assuming that p = 2, and f is asymptoticlly linear at infinity, Li, Wu and Zhou obtained that there existsλ^* >0 such that problem (5) has at least positive solutions forλ∈(0,λ^*). Note that in [90],λ^* may be small. In this thesis, we show that for the general case 1< p<+∞and f is of asymptoticlly p-linear growth at infinity, there is a similar result to [10].In Chapter 5 we consider the following semilinear elliptic Dirichlet problem whereΩ(?)Rⁿ(n≥2) is a smooth bounded region,f∈C(Ω×R,R).Similar to the discussion in the case of periodic boundary value condition,we study the solvability of problem (6) by using the interaction between the asymptotic behavior at infinity of (?) and the spectrum of -Δ.Denote 0<λ₁<λ₂<λ_k→+∞the distinct eigenvalues of -Δon H₀¹(Ω). In this thesis we obtain the solvability of problem (6) when f admits subcritical growth and there exists k∈Z⁺ such thatHere we don't assume that the asymptotic limits of (?) stays between two consecutive eigenvalues of (-Δ,H₀¹(Ω)),which implies that (?) may be oscillatory and may cross a finite number of consecutive eigenvalues of (-Δ,H₀¹(Ω)) at infinity. Note that here the corresponding functional may don't satisfy that (PS) condition or even the weak (PS) condition, and hence we can't apply the critical point theory. By combining with thehomotopy continuation method and the saddle point reduction method, we overcome this difficulty and obtain the existence of solution of problem(6).In 1980, Amann and Zehnder[6] first investigated the multiplicity of solutions using the interaction of the asymptotic limits of the ratio (?) at infinity and at zero with the operator -Δ.Later, many famous mathematicians,such as Hofer, Dancer, Castro, Lazer, Schechter, Chang, etc., studied this topic extensivelysilly and plenty of classic results are obtained by using of variational method, degree theory and critical point theory, see [7, 33, 34, 45, 77] or see the monographs [35, 115, 119]. In this chapter, combing with the asymptotic limits of (?) and (?) at infinity and at zero, we obtain some multiplicity results of solutions for problem (7) by us- ing the space structure of H₀¹(Ω),Lyapunov-Schmidt reduction, variational method and degree theory.In Chapter 6 we study the following Neumann problem involving the p-LaplacianwhereΩ(?)Rⁿ(n≥1) is a bounded smooth region,f∈C((?)×R,R),v is the outer unit normal to (?)Ω.Due to the close relation with many problems appearing in physics, Neumann boundary value condition has long been one of the most important concerns in nonlinear analysis. By now, it is well known that the operator -Δ_p adimits a sequence of eigenvalues {λ_k} on W^1,p(Ω).If n=1,1 < p < +∞or n≥2,p=2,{λ_k} are just the complete eigenvalues. However, for n≥2,p∈(1,+∞),the spectrum of -Δ_p is not clear. In this chapter, as in the Dirichlet case, we obtain the solvability of problem (7) by using the interaction of asymptotic limits of (?) and (?) at infinity with some set Q satisfying porperty P and the Fu(?)(?)k spectrum respectively.