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MINISTRY OF EDUCATION AND TRAINING HANOI PEDAGOGICAL UNIVERSITY BUI KIM MY ANALYTICAL METHODS FOR STUDYING SOME DEGENERATE ELLIPTIC PROBLEMS SUMMARY OF DOCTORAL THESIS IN MATHEMATICS Major: Mathematical Analysis Code: 46 01 02 HA NOI, 2019 This dissertation has been written at Hanoi Pedagogical University Supervisor: Assoc Prof Dr Cung The Anh Referee 1: Referee 2: Referee 3: The thesis shall be defended at the University level Thesis Assessment Council at Hanoi Pedagogical University on This thesis can be found in: - The National Library of Vietnam; - HPU2 Library Information Centre INTRODUCTION Motivation and history of the problem Many type of elliptic equations are associated with the study of the steady of evolutionary processes in physics, chemistry, mechanics and biology and many important classes of nonlinear elliptic equations are also started from the problems of differential geometry (see, for instance, Ambrosetti and Malchiodi (2007), Evans (1998), Gilbarg and Trudinger (1998), Quittner and Souplet (2007), Willem (1986)) Therefore, studying these classes are meaningful in science and technology In addition, the study of elliptic equations motivates and provides ideas for the development of tools and the results of many analytical fields, such as theory of functional spaces, nonlinear analysis, Especially, the development of these issues lead to progress in the theory of elliptic equations Thus, the theory of elliptic equations has been attracting the attention of many scientists in the world As mentioned above, the research on elliptic equations by using analytical methods has been studing by many domestic and foreign mathematicians In recent decades, a lot of qualitative results for many classes of problems involving both nondegenerate elliptic operators and degenerate elliptic operators are obtained (see, for example, Ambrosetti and Malchiodi (2007), Quitter and Souplet (2007), Willem (1996), Figueiredo (1996) and Kogoj (2018)) Among the degenerate operators, ∆λ -Laplace operator is an important class which is following form N ∂xi (λ2i (x)∂xi u), ∆λ u = i=1 where λi are functions satisfying some suitable conditions This operator is first introduced by Franchi and Lanconelli in 1982, and it is recently reconsidered It was named as ∆λ -Laplacians by Kogoj and Lanconelli in 2012 Especially, it contains many important elliptic operators such as Laplace operator ∆u = N uxi xi , Grushin operator Gs u = ∆x u + |x|2s ∆y u (see Grushin (1971)), and i=1 strongly degenerate operator Pα,β u = ∆x u + |x|2α ∆y u + |y|2β ∆z u, (see N.M Tri et al (2002, 2012)), etc We now recall some recent important results related to the existence and qualitative properties of solutions to elliptic equations and systems which are involving the content of my thesis • Semilinear elliptic equations In the last decades, the boundary value problem for semilinear elliptic equation has form  −∆u = f (x, u), x ∈ Ω, (1)  u = 0, x ∈ ∂Ω has been studied by many authors Many important problems arise in studying progress of the above equation, for instance, existence, regularity, qualitative estimates, the effect of the domain topology on the number of solutions to the equations Many methods are used to study problem (1) such as the method of sub-supersolutions (see Evans (1998)), the topological degree method (see Li (1989)), However, one of the most effective methods in order to study the existence of weak solutions is the variational methods The idea of the method is that we transform the problem (1) into finding the critical of the differentiable functional J associated to problem (1) The following Ambrosetti-Rabinowitz condition introduced by Ambrosetti and Rabinowitz (1973) (AR) ∃R0 > 0, θ > cho < θF (x, s) ≤ sf (x, s), ∀|s| ≥ R0 , ∀x ∈ Ω, plays an important role in their studies This condition not only ensures that the Euler-Lagrange functional associated to problem (1) has a mountain pass geometry, but also guarantees that Palais-Smale sequences of the Euler-Lagrange functional is bounded With this (AR) condition, one can use the classical version of the Mountain Pass Theorem of Ambrosetti and Rabinowitz to study the existence of solutions (see e.g AmbrosettiRabinowitz (1973), Ambrosetti (1986)) Although (AR) condition is quite natural and important but there are many problems where the nonlinear term f (x, u) does not satisfy the (AR) condition, and thus this condition is restrictive to many nonlinearities Because of this reason, in recent years, some authors has studied the problem (1) by trying to drop the (AR) condition, for instance, Schechter and Zou (2004), Liu and Wang (2004), Miyagaki and Souto (2008), Liu (2010), Lam and Lu (2013, 2014), Binlin et al (2015) The existence of nontrivial weak solutions of problem (1) where Laplace operator replaced by degenerate operators has been studied by many authors, e.g., V.V Grushin (1971), N.M Tri (1998), N.T.C Thuy and N.M Tri (2002), P.T Thuy and N.M Tri (2012, 2013) In 2017 many authors concerned with the Dirichlet boundary problem for semilinear elliptic equations involved the strongly degenerate elliptic ∆λ Specifically, it is the following problem  −∆ u + V (x)u = f (x, u), x ∈ Ω, λ (2)  u = 0, x ∈ ∂Ω, where Ω is a bounded domain in RN , N ≥ Some results in existence, multiplicity and regularity of weak solutions to problem (2) has been considered by Kogoj and Lanconelli (2012), D.T Luyen and N.M Tri (2015, 2018), Luyen (2017), Chen and Tang (2018), Rahal (2018), where V (x) is potential function and the nonlinearities allow is discontinuous but (AR) condition is still required (see also the survey paper of Kogoj (2018)) Therefore, we can see that for the degenerate elliptic equations, most of all results are only obtained in the cases where the nonlinearity has standard growth (i.e., has subcritical growth and satisfies (AR) condition) In my knowledge, there are still many open problems in this topic, for instance, studying the existence of weak solutions to problem (4) when the nonlinearity term f (x, u) has subcritical or critical growth, • Semilinear elliptic systems of Hamiltonian type Beside studying the scalar elliptic equations, the elliptic systems are also of interest to many mathematicians, one of typically elliptic systems is class of Hamiltonian as follows:    −∆u = |v|p−1 v,   −∆v = |u|q−1 u,     u = v = 0, x ∈ Ω, x ∈ Ω, (3) x ∈ ∂Ω, where p, q > and Ω is a bounded domain in RN , N ≥ with smooth boundary ∂Ω For system (3), we know that the critical hyperbola is 1 N −2 + = p+1 q+1 N For exponents (p, q) lying on or above this curve, that is, N −2 + ≤ , p+1 q+1 N then nonexistence of positive classical solutions of systems (3) in starshaped bounded domain has been proved in works of Pucci and Serrin (1986), Mitidieri (1993) In the degenerate operator, some results in existence and nonexistence to the Hamiltonian/gradient systems are obtained by N.T Chung (2014) and by N.M Chuong et al (2004, 2005) In case of (p, q) below the critical hyperbola, by using variational method and Fountain theorem of Bartsch and Figueiredo (1996), the existence of weak solutions of (3) has shown (see some works of Peletier and van der Vorst (1992), Hulshof and van der Vorst (1993), de Figueiredo and Felmer (1994) and a survey paper of Figueiredo (1996)) However, the corresponding results for degenerate elliptic systems are still very few; for example, existence, multiplicity and nonexistence to systems (3) when Laplace operator is replaced by the strongly degenerate ∆λ is not considered • Some Liouville type theorems for elliptic equations and systems In recent years, one of the very hot topics is the study Liouville type theorems for elliptic equations and systems The Liouville-type theorem is the nonexistence of solutions in entire space or in half-space The classical Liouville-type theorem stated that a bounded harmonic (or holomorphic) function defined in entire space must be constant This theorem, known as the Liouville Theorem, was first announced by Liouville (1844) Later, Cauchy (1844) published the first proof of the above stated theorem (see also Axler, Bourdon and Ramey (2001)) This classical result has been extended to nonnegative solutions of the semilinear elliptic equations in whole space RN by Gidas and Spruck (1980, 1981), Chen and Li (1991) The Liouville theorem for semilinear elliptic equations or inequalities on a cone Σ in RN was also studied by Dolcetta, Berestycki and Nirenberg (1995) Recently, Liouville type theorems for degenerate elliptic equations have been attracted the interest of many mathematicians The classical Liouville theorem was generalized to p-harmonic functions on the whole space RN and on exterior domains by Serrin and Zhou (2002) The Liouville type theorem for semilinear elliptic inequality involving the Grushin operator has been established by Dolcetta and Cutr`ı in (1997), D’Ambrosio v Lucente (2003), Monticelli (2010), Yu (2014) The Liouville type theorems for elliptic systems and systems of inequalities has also attracted the interest of many mathematicians, for instance, Souto (1995), Serrin and Zou (1996), Mitidieri and Pohozaev (2001), Souplet (2009) In the case elliptic system involving the Grushin operator, some Liouville type for stable solutions established by Hung and Tuan (2017) Therefore, we see that some Liouville type theorems are just obtained for weak degenerate operators and there are still few results The corresponding results for strongly degenerate operators such as ∆λ are still open in many cases Summary, for analysis as above, we would see that, beside the results are known, many problems in elliptic equations and systems involving the strongly degenerate ∆λ still open, for instance: • The existence and multiplicity of weak solutions to semilinear strongly degenerate elliptic has form  −∆ u = f (x, u) x ∈ Ω, λ (4)  u=0 x ∈ ∂Ω, where Ω is a bounded domain in RN , N ≥ and the nonlinear term does not satisfy the Ambrosetti-Rabinowitz condition • Existence and nonexistence of solutions to erate elliptic system    −∆λ u = |v|p−1 v   −∆λ v = |u|q−1 u     u=v=0 a Hamiltonian strongly degen- x ∈ Ω, x ∈ Ω, (5) x ∈ ∂Ω, with p, q > and Ω is a bounded domain in RN , N ≥ • Establishing Liouville type theorems for semilinear elliptic inequality and systems of inequalities involving the strongly degenerate operator ∆λ : − ∆λ u ≥ up x ∈ RN ,  −∆ u ≥ v p λ −∆ v ≥ uq λ x ∈ RN , (N ≥ 2, p > 1), (6) (N ≥ 2, p, q > 0) (7) and systems N x∈R , Therefore, our thesis focus on study existence, nonexistence, multiplicity and establish some Liouville type theorems for some degenerate elliptic problems involving the ∆λ -Laplace operator Purpose of thesis This thesis focus on study some class of elliptic equations and systems involving the ∆λ -Laplace operator Namely, that is the following important issues: • To study the existence of weak solutions; • To study multiplicity of solutions; • To study nonexistence positive classical solutions in starshaped like domain; • To study some Liouville type theorems on the nonexistence of solutions in entire space Object and scope of thesis Research scope: • Content To study existence and multiplicity of solutions in the subcritical case of the semilinear degenerate elliptic equations involving the ∆λ Laplace operator when the nonlinear term does not satisfy the AmbrosettiRabinowitz condition • Content To study existence and nonexistence of solutions to a Hamiltonian elliptic systems involving the ∆λ -Laplace operator • Content To study some Liouville type theorems for semilinear inequality elliptic system involving the ∆λ -Laplace operator Research methods • To study the existence and multiplicity of solutions: Variational methods • To study nonexistence of positive classical solutions: Establishing suitable Pohozaev type identities and exploiting geometry structure of the domain • To study Liouville type theorems: Using the rescaled-test functions methods and establishing suitable integral estimates Results of thesis The thesis achieved the following main results: • Proving the existence of nontrivial weak solutions to problem (4) when the nonlinearity has subcritical polynomial growth and does not satisfy the Ambrosetti-Rabinowitz condition In additions, if the nonlinear term is odd with respect to the second variable, we proved the multiplicity of weak solutions to problem (4) This is the main content of Chapter • Proving the nonexistence of positive classical solutions to Hamiltonian system (5) in the starshaped like domain Proving the multiplicity of weak solutions of systems (5) when (p, q) below critical hyperbola This is the main content of Chapter • Establishing some Liouville type theorems on the nonexistence of nonnegative classical solutions for inequality (6) and elliptic inequalities (7) in entire space This is the main content of Chapter Structures of thesis Beside Introduction, Conclusion, Authors works related to the thesis and References, the thesis includes chapters: • Chapter Preliminaries; • Chapter Existence of solutions to a semilinear degenerate elliptic equation; • Chapter Existence and nonexistence of solutions to a degenerate Hamiltonian system; • Chapter Liouville type theorems for degenerate elliptic inequalities f (x)g(x)dx with f, g ∈ L2 (Ω) where (f, g)L2 = Ω The following result was established by Kogoj and Lanconelli which frequently used in thesis Proposition 1.1 Assume that the functions λi , i = 1, 2, , N satisfy conditions 1) − 4) as in Section 1.1 and Q > Then the embedding ◦ ∗ ∗ pλ W 1,p λ (Ω) → L (Ω), where pλ := pQ , Q−p is continuous Moreover, the embedding ◦ γ W 1,p λ (Ω) → L (Ω) is compact for every γ ∈ [1, p∗λ ) We now prove the following important result Proposition 1.2 Assume that the functions λi , i = 1, 2, , N satisfy conditions 1) − 4) as in Section 1.1 and Q > Then the embedding ◦ γ Wλ2,2 (Ω) ∩ W 1,2 λ (Ω) → L (Ω) is continuous if ≤ γ ≤ 2Q Q−4 We consider the following homogeneous Dirichlet problem:  −∆ u = f (x) inΩ, λ  u = on ∂Ω ◦ (1.1) ◦ 1,2 Proposition 1.3 The operator −∆λ : W 1,2 λ (Ω) → (W λ (Ω)) is surjective, ◦ ◦ 1,2 where (W 1,2 λ (Ω)) is the dual space of W λ (Ω) Corollary 1.1 For each f ∈ L2 (Ω) problem (1.1) has unique weak solution ◦ u ∈ W 1,2 λ (Ω) ◦ ◦ By Proposition 1.3, the inverse operator T = (−∆λ )−1 : (W 1,2 λ (Ω)) → W 1,2 λ (Ω) of the operator −∆λ is well defined Then, we have following proposition 11 Proposition 1.4 The inverse operator T of −∆λ is positive, self-adjoint and compact in L2 (Ω) By Proposition 1.4, there exists a sequence of eigenfunctions ϕj ∈ L2 (Ω) of T which is an orthogonal in L2 (Ω) corresponding to eigenvalues {γj }∞ j=1 with γj → as j → +∞ Since ◦ T : L2 (Ω) → W λ1,2 (Ω) ⊂ L2 (Ω) ◦ this implies ϕj ∈ W 1,2 λ (Ω) for all j = 1, 2, Moreover, since ϕj = T −1 (T ϕj ) = T −1 (γj ϕj ) = γj (−∆λ ϕj ), thus −∆λ ϕj = ϕj , γj ∀j = 1, 2, ◦ 1,2 Therefore, the operator −∆λ has a sequence of eigenfunctions {ϕj }∞ j=1 in W λ (Ω) corresponding to eigenvalues {µj = γ1j }∞ j=1 such that < µ1 ≤ µ2 ≤ · · · ≤ µj ≤ · · · , µj → +∞ as j → +∞ 1.3 Some results of critical points theory We will use the following version of Mountain Pass Theorem Theorem 1.1 Let X be a real Banach space and let J ∈ C (X, R) satisfy the (C)c condition for any c ∈ R, J(0) = 0, and (i) There are constants ρ, α > such that J(u) ≥ α ∀ u = ρ; (ii) There is an u1 ∈ X, u1 > ρ such that J(u1 ) ≤ Then c = inf max J(γ(t)) ≥ α is a critical value of J, where γ∈Γ 0≤t≤1 Γ = {γ ∈ C ([0, 1], X) : γ(0) = 0, γ(1) = u1 } Let X be a reflexive and separable Banach space We know that there exist sequences {ej } ⊂ X, {ϕj } ⊂ X ∗ such that (i) ϕi , ei = δi,j , where δi,j = if i = j and δi,j = otherwise; 12 ∗ w ∞ ∗ (ii) span{ej }∞ j=1 = X and span {ϕj }j=1 = X Xj We define Let Xj = Rej then X = j≥1 k Yk = Xj and Zk = j=1 Xj (1.2) j≥k Because the Mountain Pass Theorem also holds when functionals satisfy the (C)c condition, thus we can establish multiplicity results for problem (2.1) by using the following Fountain Theorem of Bartsch Theorem 1.2 Assume that J ∈ C (X, R) satisfies the (C)c condition for all c ∈ R and J(u) = J(−u) If for every k ∈ N, there exists ρk > rk such that (i) ak = max ϕ(u) ≤ 0; u∈Yk u =ρk (ii) bk = inf ϕ(u) → +∞, k → ∞; u∈Zk u =rk then J has a sequence of critical points {uk } such that J(uk ) → +∞ We next recall some concepts in order to study existence of weak solutions to Hamiltonian system in Chapter Definition 1.5 Let E be a Hilbert space and a functional Φ ∈ C (E, R) Given a sequence F = (En ) of finite dimensional subspaces of E such that En ⊂ En+1 , n = 1, 2, , and ∪∞ n=1 En = E We say that (i) sequence (zk ) ⊂ E with zk ∈ Enk , nk → ∞, is a (P S)Fc -sequence if Φ(zk ) → c and (1 + zk )(Φ |Enk )(zk ) → (ii) Φ satisfies (P S)Fc , at level c ∈ R, if every sequence (P S)Fc -sequence has a subsequence converging to a critical point of Φ We will use the Fountain Theorem established by Bartsch and de Figueiredo to prove the existence of infinitely many of weak solutions to our problem We decompose the Hilbert space E into direct sum E = E + ⊕ E − , denote E1± ⊂ E2± ⊂ · · · be an increasing sequence of finite dimensional subspaces of − ± + ± E ± , respectively, such that ∪∞ n=1 En = E and let En = En ⊕ En , n = 1, 2, 13 Theorem 1.3 Assume Φ : E → R is C (E, R) and satisfies the following conditions: (Φ1) Φ satisfies (P S)Fc , with F = (En ), n = 1, 2, and c > 0; (Φ2) There exists a sequence rk > 0, k = 1, 2, , such that for some k ≥ 2, bk := inf{Φ(z) : z ∈ E + , z⊥Ek−1 , z = rk } → +∞ as k → ∞; (Φ3) There exists a sequence of isomorphisms Tk : E → E, k = 1, 2, , with Tk (En ) = En for all k and n, and there exists a sequence Rk > 0, k = 1, 2, , such that, for z = z + +z − ∈ Ek+ ⊕E − and Rk = max{ z + , z − }, one has Tk z > rk and Φ(Tk z) < 0, where rk is obtained in (Φ2); (Φ4) dk := sup{Φ(Tk (z + + z − )) : z + ∈ Ek+ , z − ∈ E − , z + , z − ≤ Rk } < +∞; (Φ5) Φ is even, i.e., Φ(z) = Φ(−z) Then Φ has an unbounded sequence of critical values We notice that, if Φ mapping bounded sets in E into bounded sets in R then the (Φ4) condition is satisfied 1.4 Some standard conditions on the nonlinearity term In this section, we introduce some standard conditions, for instance, AmbrosettiRabinowitz (AR) condition, subcritical polynomial growth (SCP) and (SCPI), and critical growth conditions on the nonlinearity f (x, s) and give some related results to problem (1) 14 Chapter EXISTENCE OF SOLUTIONS TO A SEMILINEAR DEGENERATE ELLIPTIC EQUATION In this chapter, we study the Dirichlet boundary problem for semilinear degenerate elliptic equations involving the ∆λ -Laplace operator in a bounded domain Ω ⊂ RN , N ≥ 2, where the nonlinearity has subcritical polynomial growth and does not satisfy the Ambrosetti-Rabinowitz condition Firstly, we prove the existence of at least a weak solutions and next we use assumption on the nonlinear term is odd function we get the multiplicity of weak solutions to the problem This chapter is written based on the paper [1] 2.1 Setting of the problem In this paper, we study the existence of nontrivial weak solutions to the following problem  −∆ u = f (x, u), x ∈ Ω, λ (2.1) u = 0, x ∈ ∂Ω, where Ω is a bounded domain in RN , N ≥ Here the nonlinear terms f (x, u) has subcritical polynomial growth and satisfies the following assumptions: (f 1) f : Ω × R → R is continuous and f (x, 0) = for all x ∈ Ω; F (x, u) (f 2) lim = +∞ uniformly on x ∈ Ω, where F (x, u) = u2 |u|→+∞ u f (x, t)dt; 2F (x, u) < µ1 uniformly on x ∈ Ω, where µ1 > is the first |u|2 |u|→0 eigenvalue of the operator −∆λ in Ω with homogeneous Dirichlet boundary conditions; (f 3) lim sup (f 4) There exist C∗ ≥ 0, θ ≥ such that H(x, t) ≤ θH(x, s) + C∗ ∀t, s ∈ R, < |t| < |s|, ∀x ∈ Ω, where H(x, u) = uf (x, u) − F (x, u) 15 (SCP I) f has subcritical polynomial growth on Ω, i.e., 2Q f (x, s) ∗ = 0, = ,Q > ∗ λ Q−2 |s|→+∞ |s|2λ −1 lim where Q denotes the homogeneous dimension of RN with respect to a group of dilations Remark 2.1 In problem (2.1) we not require the (AR) condition imposed on the nonlinear terms We recall the definition of weak solutions of problem (2.1) ◦ Definition 2.1 A function u ∈ W 1,2 λ (Ω) is called a weak solutions of (2.1) if ∇λ u∇λ ϕ dx = ∀ϕ ∈ C0∞ (Ω) f (x, u)ϕ dx, Ω Ω Define the Euler-Lagrange functional associated with problem (2.1) as follows Jλ (u) = |∇λ u|2 dx − Ω F (x, u)dx, Ω u f (x, s)ds By the hypotheses on f , ◦ ◦ 1,2 W λ (Ω) and Jλ ∈ C (W 1,2 λ (Ω), R) with where F (x, u) = well-defined on ◦ ∇λ u∇λ vdx − Jλ (u)v = Ω we can see that Jλ is f (x, u)vdx, ∀ v ∈ W 1,2 λ (Ω) Ω One can also check that the critical points of Jλ are the weak solutions of problem (2.1), and thus we can use a suitable version of the Mountain Pass Theorem to study the existence of weak solutions to problem (2.1) 2.2 Existence of nontrivial weak solutions The main result of this section is the following theorem Theorem 2.1 Assume that f has subcritical polynomial growth on Ω, i.e (SCPI) condition holds, and satisfies (f 1) − (f 4) Then problem (2.1) has a nontrivial weak solution We prove Theorem 2.1 by verifying that all conditions of Lemma 1.1 are satisfied First, we check the condition (i) in this lemma 16 Lemma 2.1 Assume that f satisfies conditions (f 1), (f 3) and (SCP I) Then there exist α, ρ > such that Jλ (u) ≥ α ◦ ∀u ∈ W 1,2 λ , u 1,2 = ρ Next, we check the condition (ii) in Lemma 1.1 Lemma 2.2 Assume that f satisfies (f 2) Then Jλ (tu) → −∞ as t → +∞ ◦ for all functions u ∈ W 1,2 λ (Ω) \ {0} We now show the (C)c conditions is satisfied Lemma 2.3 If f (1) − (f 4) and (SCP I) are satisfied, then Jλ satisfies the (C)c condition for all c ∈ R Proof of Theorem 2.1: Combining Lemmas 2.1-2.3, we deduce that problem (2.1) has a nontrivial weak solution 2.3 The multiplicity of weak solutions Next, when f (x, s) is an odd function in s we obtain the result on the existence of infinitely many weak solutions to problem (2.1) Theorem 2.2 Assume that (f 1) − (f 4) hold and (1) there exist a, b > and q ∈ (2, 2∗λ ) such that (SCP) |f (x, s)| ≤ a + b|s|q−1 (2) f (x, −s) = −f (x, s), ∀(x, s) ∈ Ω × R; ∀(x, s) ∈ Ω × R Then problem (2.1) has a sequence of solutions {un } such that Jλ (un ) → +∞ 17 Chapter EXISTENCE AND NONEXISTENCE OF SOLUTIONS TO A HAMILTONIAN DEGENERATE ELLIPTIC In this chapter, we study nonexistence of classical solutions and existence of infinitely many weak solutions to a semilinear degenerate elliptic system involving ∆λ -Laplace operator in a bounded domain This chapter is written based on the paper [3] 3.1 Setting of the problem We consider the following semilinear degenerate elliptic system of Hamiltonian type    −∆λ u = |v|p−1 v, x ∈ Ω,   (3.1) −∆λ v = |u|q−1 u, x ∈ Ω,     u = v = 0, x ∈ ∂Ω, where p, q > 1, and Ω is a bounded domain in RN , N ≥ with smooth boundary ∂Ω We now define some functional spaces which are used to study problem (3.1) ◦ By definition of the spaces W λ1,2 (Ω) v Wλ2,2 (Ω) as in Chapter 1, we consider the operator ◦ A : Wλ2,2 (Ω) ∩ W λ1,2 (Ω) → L2 (Ω), (3.2) where A = −∆λ with the homogeneous Dirichlet boundary condition We denote E s = D(As ), with s > 0, the space with the inner product As u As v dx, (u, v)E s = u, v ∈ E s , Ω where ∞ s D(A ) = {ϕ = ∞ ∞ µ2s j aj aj ϕj , aj ∈ R| j=1 j=1 s aj µsj ϕj < +∞} and A ϕ = j=1 where ϕj are eigenfunctions of A corresponding to eigenvalues µj , j = 1, 2, We notice that, as a consequence of Proposition 1.2 and interpolation theorem, we obtain following important embeddings 18 Lemma 3.1 Suppose that Q > Then, the following embeddings E s → Lq+1 (Ω) and E t → Lp+1 (Ω) 1 2s 1 2t ≥ − , ≥ − , respectively Moreover, these q+1 Q p+1 Q embeddings are compact if the corresponding inequalities are strict are continuous if For s, t ≥ such that s + t = 1, we consider E = E s × E t , a Hilbert space with the inner product (z, η)E = (u, ϕ)E s + (v, ψ)E t , for z = (u, v), η = (ϕ, ψ) ∈ E We consider the bilinear form (As uAt ψ + As ϕAt v) dx B((u, v), (ϕ, ψ)) = Ω Now we define the functional Φ : E = E s ×E t → R associated to the problem (3.1) by (As uAt ψ + As ϕAt v) dx − Φ(z) = Ω H(u, v)dx, Ω where |v|p+1 |u|q+1 + H(u, v) = p+1 q+1 One can check that Φ is well-defined on E and Φ ∈ C (E, R) with (As u At ψ + At v As φ)dx − Φ (u, v)(φ, ψ) = Ω (uq φ + v p ψ)dx Ω One can also see that the critical points of Φ are the weak solutions of the problem (3.1) in the sense following definition Definition 3.1 We say that z = (u, v) ∈ E = E s × E t is a weak solution of (3.1) if As u At ψ dx − Ω v p ψdx = ∀ψ ∈ E t , Ω At v As φdx − Ω uq φ dx = ∀φ ∈ E s Ω 19 3.2 Nonexistence of positive classical solutions In this section, we prove nonexistence result of our problem when the domain Ω is starshaped in the sense of definition below We first consider the following vector field N T := i xi i=1 ∂ , ∂xi (3.3) and this vector field is the generator of the group of dilation {δt }t>0 Here, a function u is δt -homogeneous of degree m if and only if T u = mu Definition 3.2 A domain Ω is called δt -starshaped with respect to the origin if ∈ Ω and T, ν ≥ at every point of ∂Ω, where ν is the outward normal vector and ·, · denotes the inner product in RN We will denote by Λ2 (Ω) the linear space of the functions u ∈ C(Ω) such that Xj u, Xj2 u, j = 1, , N, exist in the weak sense of distributions in Ω and can be continuously extended ∂ to Ω Here Xj := λj ∂xj We obtain useful following lemma Lemma 3.2 For any u, v ∈ Λ2 (Ω), we have [T (u) ∇λ v, νλ + T (v) ∇λ u, νλ ]dS [T (u)∆λ v + T (v)∆λ u] dx = Ω ∂Ω − ∇λ u, ∇λ v T, ν dS + (Q − 2) ∇λ u, ∇λ v dx, (3.4) Ω ∂Ω where T is the vector field in (3.3), ν is the outward normal to Ω, νλ = (λ1 ν1 , , λN νN ) and ∇λ = (λ1 ∂x1 , , λN ∂xN ) The following theorem is the main result of this section Theorem 3.1 Assume N ≥ 3, and p, q > satisfy Q−2 + ≤ p+1 q+1 Q (3.5) If Ω is bounded and δt -starshaped with respect to the origin, then problem (3.1) has no nontrivial nonnegative solution u ∈ Λ2 (Ω) 20 3.3 Existence of infinitely many of weak solutions In this section we show the existence of infinitely many solutions to the systems (3.1) We have the following result on the existence of infinitely many solutions to the systems (3.1) Theorem 3.2 If p, q > 1, Ω is a smooth and bounded domain in RN and Q−2 + > , p+1 q+1 Q (3.6) then the problem (3.1) has infinitely many weak solutions Here, in order to prove Theorem 3.2 we will check conditions (Φ1) − (Φ4) in Theorem 1.3 in Chapter are satisfied 21 Chapter LIOUVILLE TYPE THEOREMS FOR DEGENERATE ELLIPTIC SYSTEM OF INEQUALITIES In this chapter, we study some Liouville type theorems, that is the nonexistence of positive classical solutions for the elliptic system of inequalities involving the ∆λ -Laplace operator in entire space RN , N ≥ This chapter is written based on the paper [2] 4.1 Setting of the problem We will establish Liouville type theorems for the elliptic system of inequalities  −∆ u ≥ v p , x ∈ RN , λ (4.1) −∆ v ≥ uq , x ∈ RN , λ p, q > We will consider two cases: • Case The exponents p, q > and such that 2(p + 1) 2(q + 1) , } ≥ Q − max{ pq − pq − To this, we use the so-called rescaled test-functions method and exploit the homogeneous properties of the operator ∆λ The idea of this method is the following: we fix a constant R > and consider the equations or systems in bounded domains, next multiply by suitable test functions and use suitable computations, then let R tends to infinity, we obtain new equations or systems in the whole space, hence obtain the results that our solutions are trivial • Case The exponents p, q > such that pq > and satisfy 1 Q−2 + ≥ p+1 q+1 Q−1 To this, we exploit the interesting idea of Souto (1995) for the Laplace operator, namely it is to reduce the problem to a question concerning a scalar equation by introducing a new function w = uv and then use the Liouville type theorem for elliptic inequality 22 4.2 Liouville type theorem for p, q > Our main results are the following theorems Theorem 4.1 Let p, q > Then system (4.1) does not possess positive classical solutions u, v ∈ C (RN ) provided max{a, b} ≥ Q − 2, where a = 2(q + 1) 2(p + 1) ,b= pq − pq − As a direct consequence of Theorem 4.1, when u = v and p = q, we get the following Liouville type theorem for the elliptic inequality − ∆λ u ≥ up RN Corollary 4.1 Assume < p ≤ ity (4.2), then u ≡ Q Q−2 (4.2) If u is a nonnegative solution of inequal- 4.3 Liouville type theorem for p, q > The main result in this section is following Theorem 4.2 Assume p, q > such that pq > 1, and 1 Q−2 + ≥ p+1 q+1 Q−1 Then system (4.1) does not possess any positive classical solutions 23 CONCLUSION Results In this thesis, we obtained the following results: Proved the existence and multiplicity of weak solutions to a class of semilinear degenerate elliptic equation in bounded domain, when the nonlinearity has subcritical polynomial growth and does not satisfy the AmbrosettiRabinowitz condition Proved the nonexistence of positive classical solutions and multiplicity of weak solutions of system to a Hamiltonian strongly degenerate elliptic system in bounded domain Established some Liouville type theorems on the nonexistence of nonnegative classical solutions for the elliptic system of inequalities in entire space RN Recommendation The following research topics can be further developed: 1) To study the existence of solutions to degenerate elliptic equations or systems when the nonlinearities has critical growth To study the regularity of weak solutions; 2) To study some applications of Liouville-type theorems, for instance, universal and singularity estimates, decay estimates, the blow-up rate of solutions, 24 AUTHOR’S WORKS RELATED TO THE THESIS THAT HAVE BEEN PUBLISHED [1] C.T Anh and B.K My, (2016), Existence of solutions to ∆λ -Laplace equations without the Ambrosetti-Rabinowitz condition, Complex Var Elliptic Equ 61 No.1, 137-150 (SCIE) [2] C.T Anh and B.K My, (2016), Liouville type theorems for elliptic inequalities involving the ∆λ -Laplace operator, Complex Var Elliptic Equ 61 No.7, 1002-1013 (SCIE) [3] C.T Anh and B.K My, (2019), Existence and non-existence of solutions to a Hamiltonian strongly degenerate elliptic system, Adv Nonlinear Anal No.1, 661-678 (SCIE) ... in the theory of elliptic equations Thus, the theory of elliptic equations has been attracting the attention of many scientists in the world As mentioned above, the research on elliptic equations... problems involving both nondegenerate elliptic operators and degenerate elliptic operators are obtained (see, for example, Ambrosetti and Malchiodi (2007), Quitter and Souplet (2007), Willem (1996),... • Semilinear elliptic systems of Hamiltonian type Beside studying the scalar elliptic equations, the elliptic systems are also of interest to many mathematicians, one of typically elliptic systems