Special Issue in honor of John W. Neuberger Electronic Journal of Differential Equations, Special Issue 02 (2023), pp. 101–107. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu REMARKS ON COMPACTNESS CONDITIONS AND THEIR APPLICATIONS DAVID G. COSTA In memory of Prof. John W. Neuberger and his legacy to mathematics Abstract. We review typical compactness conditions used in variational tech- niques and some of their properties, and the relationships between them. In particular, we provide some new insights into results related to the Palais- Smale and Cerami conditions, and their comparison. 1. Introduction Let H be a Hilbert space with inner-product 〈·, ·〉 and J : H → R a C1 func- tional defined on H. Researchers in variational techniques and their applications to differential equations (ODEs or PDEs) are familiar with the following compactness conditions, where (un) is a sequence in H: • Palais-Smale condition at level c, (PS)c: If (un) is such that J(un) → c and J ′(un)→ 0, then (un) has a convergent subsequence (see [10]); • Cerami condition at level c, (Ce)c: If (un) is such that J(un) → c and (1 + ‖un‖)J ′(un)→ 0, then (un) has a convergent subsequence (see [5]); • Brézis-Coron-Nirenberg condition at level c, (BCN)c: If (un) is such that J(un)→ c and J ′(un)→ 0, then c ∈ R is a critical value of J (see [4]). Researchers familiar with the above conditions know and it is also easy to show that (PS)c ⇒ (Ce)c ⇒ (BCN)c . Indeed, (PS)c ⇒ (Ce)c as (1 + ‖un‖)J ′(un) ≥ J ′(un), and either (PS)c or (Ce)c implies that c ∈ R is a critical value of J , since the limit ū of the convergence subsequence (still denoted (un)) satisfies J(ū) = c, J ′(ū) = 0. As a side remark, one should notice that (BCN)c simply says that c is a critical value of J , a result that might be applicable in situations where J is periodic with period (say) p > 0. Indeed, one could use ûn with J(ûn) belonging to the closed interval [0, p] and note that J(ûn)→ c and J ′(ûn)→ 0. 2020 Mathematics Subject Classification. 35-04. Key words and phrases. Compactness conditions; Palais-Smale condition; Cerami condition, Brézis-Coron-Nirenberg condition. ©2023 This work is licensed under a CC BY 4.0 license. Published March 27, 2023. 101 102 D. G. COSTA EJDE/SI/02 2. Two results Now we state and prove two new and simple results involving the Palais-Smale as well as the Cerami condition (inspired by results in [6, 7]). Let us start by recalling the notion of Strong Resonant problems, as was introduced by Benci- Bartolo-Fortunato [3] in 1983 for Dirichlet problems in bounded domains Ω ⊂ RN , N ≥ 3 (cf. also [2]). Such problems were also used in the context of unbounded domains (e.g. see [8, 12] and references therein). In [3] the authors considered the “strong” resonant problem below in a bounded domain Ω, with λk denoting the kth eigenvalue of −∆ under Dirichlet condition on ∂Ω, −∆u− λku+ g(u) = 0, u = 0 on H1 0 (Ω) , (2.1) and assumed the conditions (A1) tg(t)→ 0 as |t| → ∞; (A2) G(t) := ∫ t −∞ g(s) ds well-defined and such that G(t)→ 0 as t→∞; (A3) G(t) ≥ 0 for all t ∈ R Then they proved the following three theorems: Theorem 2.1. If (A1)–(A3) hold, then problem (2.1) has at least one solution. Theorem 2.2. If g(0) = 0, g′(0) = sup{g′(t) : t ∈ R} and (A1)–(A3) hold, then problem (2.1) has at least one nontrivial solution. Theorem 2.3. Assume (A2) and (A3) with g odd and G(0) ≥ 0. Moreover, suppose that there exists an eigenvalue λh ≤ λk such that g′(0)+λh−λk > 0. Then problem (2.2) possesses at least m := dimension(Mh ⊕ · · · ⊕Nk) distinct pairs of nontrivial solutions, where Mi denotes the eigenspace corresponding to λi. As pointed out by the authors, the definition of “strong” resonant problem ap- plies to the situation in Theorem 2.1 where the conditions (A1)–(A3) hold (with (A1) weakened to g(t) → 0 as |t| → ∞). In fact, as stated by the authors, con- dition (A1) is simply a technical condition in case g has a “good” behavior at ∞. In addition, in their approach, the authors show that the Cerami condition (Ce)c holds for all c ∈ (0,∞), by making use of “linking” results. In our approach, we plan to show that the stronger (PS)c hods for all c ∈ R except for a finite set of values that can be found explicitly. In particular, given that the authors use linking arguments, another alternative one could have is to use the stronger Palais-Smale condition (PS)c by avoiding the exceptional finite set of values that we shall find in our approach. We may assume, without loss of generality, that the eigenvalues of −∆ under Dirichlet boundary condition are simple; see Remark 2.5. First result. Theorem 2.4. Consider the Dirichlet problem −∆u = λku+ g(u), u = 0 on H1 0 (Ω) , (2.2) and assume the conditions (A4) g(t)→ 0 as |t| → ∞, with g continuous; EJDE-2023/SI/02 COMPACTNESS CONDITIONS AND THEIR APPLICATIONS 103 (A5) G(t) := ∫ t 0 g(s) ds is such that limt→±∞G(t) := G± ∈ (−∞,+∞) , where λk is a given eigenvalue of −∆ under Dirichlet boundary condition. Then there exist a finite set Γk ⊂ R such that the functional J(u) = 1 2 ∫ Ω (|∇u|2 − λku2) dx− ∫ Ω G(u) dx := Q(u)− ∫ Ω G(u) dx , for u ∈ H1 0 (Ω), satisfies (PS)c if and only if c /∈ Γk, where Γk := {−measure([v > 0])G+ −measure([v < 0])G− : v ∈ Nk, ‖v‖ = 1} , and [v > 0] (resp. [v < 0]) denotes the set {x | v(x) > 0} (resp. {x : v(x) < 0}). Proof. Recall we are denoting ‖u‖ = ( ∫ Ω |∇u|2 dx)1/2 the usual norm in H1 0 (Ω), and Nk = Rφk is the one-dimensional eigenspace associated with λk, with ‖φk‖ = 1. Let us also denote by X+, X− the subspaces of H1 0 (Ω) where Q is positive definite, negative definite, respectively, and set X 0 = Nk, so that H1 0 (Ω) = X+ ⊕X− ⊕X 0 . Since g has subcritical growth by (A4), the functional J satisfies (PS)c if and only if any sequence (un) in H1 0 (Ω) satisfying (i) J(un)→ c, and (ii) J ′(un)→ 0, must have a bounded subsequence. So, let us assume that J satisfies (i), (ii), but ‖un‖ → ∞ , and prove that c ∈ Γk. Claim: Assuming (A4) and (A5), the functional J satisfies (PS)c if and only if c /∈ Γk, where we recall that Γk := {−measure([v > 0])G+ −measure([v < 0])G− : v ∈ Nk, ‖v‖ = 1} , and [v > 0] (resp. [v < 0]) denotes the set {x | v(x) > 0} (resp. {x | v(x) < 0}). Proof. Since we are denoting by X+, X− the subspaces of X := H1 0 (Ω) where Q is positive definite, negative definite, respectively, and X 0 = Nk, we shall write u ∈ H1 0 as un = u+ n + u−n + u0 n, where u+ n ∈ X+, u−n ∈ X−, u0 n ∈ X 0 = Nk. And, since g has subcritical growth, the functional J satisfies (PS)c if and only if any sequence (un) in H1 0 verifying (i) J(un)→ c, and (ii) ‖J ′(un)‖H−1 → 0, must have a bounded subsequence. So, by negation, let us then assume that J satisfies (i), (ii), but (iii) ‖un‖ → ∞. and show in this case that c ∈ Γk. Indeed, (ii) implies that |〈∇J(un), u+ n 〉| = |‖u+ n ‖2 − λk‖u+ n ‖2L2 − ∫ Ω g(un)u+ n dx| ≤ C‖u+ n ‖ , (2.3) where C = supn∈N ‖J ′(un)‖H−1 . Also, in view of Holder’s and Sobolev’s inequality, we have that C2‖u+ n ‖L2 ≤ ‖u+ n ‖, where we may replace C2 by a smaller 0 < C0 with 1− C2 0λk > 0 . (2.4) 104 D. G. COSTA EJDE/SI/02 On the other hand, note by (A4) that if q′ ≤ 2N/(N − 2), N ≥ 3 (where q′ = q/(q − 1) denotes the conjugate exponent of q), we can estimate the integral term in (2.3) as ∣∣ ∫ Ω g(un)u+ n dx ∣∣ ≤ ‖g(un)‖Lq‖u+ n ‖Lq′ ≤ C‖g(un)‖Lq‖u+ n ‖ . (2.5) Therefore, using Holder’s inequality and Sobolev’s embedding, it follows from (2.3), (2.4), and (2.5), that (1− C2 0λk)‖u+ n ‖2 ≤ (C‖g(un)‖Lq + Ĉ)‖u+ n ‖ , (2.6) which implies the (u+ n ) is bounded in H1 0 . Similarly, we show that (u−n ) is also bounded. � Thus, by (iii), we must have that ‖u0 n‖ → ∞ and, by setting ûn = un/‖u0 n‖ (and recalling that Nk = Rφk with ‖φk‖ = 1), it follows that ûn → φk ∈ Nk and we may also assume that ûn(x)→ v(x) a.e. in Ω. Hence, un(x)→ +∞ a.e. in [φk > 0], (2.7) un(x)→ −∞ a.e. in [φk < 0] . (2.8) Next, in view of (A4), we apply Lebesgue’s theorem to the sequence G(un(x)) to obtain lim n→∞ ∫ Ω G(un(x)) dx = ∫ [φk>0] G+ dx+ ∫ [φk<0] G− dx , which proves the Claim with v = φk. Therefore, using Holder’s inequality and Sobolev’s embedding as in (2.6), we obtain (with q ≥ 2N/(N + 2), N ≥ 3) | ∫ Ω g(un)u+ n dx| ≤ C( ∫ Ω |g(un(x))|q dx) 1 q ‖u+ n ‖ and, since g(un(x)) → 0 a.e. in Ω in view of (A4), an application of Lebesgue’s theorem once again implies the desired conclusion that c ∈ Γk in case (iii) holds. In other words, assuming (i), (ii) (i.e., that (un) is a Palais-Smale sequence), we have shown through the negation argument (iii) that any Palais-Smale sequence (un) has a convergent subsequence. On the other hand, it is clear that if c ∈ Γk then (PS)c does not hold. � Remark 2.5. Since we are assuming that λk is a simple eigenvalue, the set Γk := {−αk.G+ − βkG− , −βkG+ − αkG−} (where αk := measure([v > 0]), βk := measure([v < 0]) has either one or two elements. When λk is not a simple eigenvalue the set Γk has νk or 2νk elements, where νk is the dimension of the eigenspace (Nk) associated with the eigenvalue λk. Remark 2.6. We should also note that nonlinear resonant problems were origi- nally introduced and studied via different methods by Landesman-Lazer [9] in 1970, and by Ahmad-Lazer-Paul [1] in 1976. Later, in 1986, Rabinowitz [11] published a CBMS monograph (in AMS Conf. Ser. in Math.) introducing Minimax methods in critical point theory with applications to differential equations, where his seminal abstract Saddle-Point Theorem, motivated by the Ahmad-Lazer-Paul paper, pro- vided yet a third different proof for nonlinear resonant problems. It is illustrating EJDE-2023/SI/02 COMPACTNESS CONDITIONS AND THEIR APPLICATIONS 105 to contrast the resonant situations in [9, 1, 11], where G± is infinite with the strong resonant situation in [3] and in the above result, where G± are finite real numbers. We must mention that there is a large literature on both “resonant” and “strong resonant” problems (on bounded and unbounded domains), but we tried to restrict the references to a minimum by only listing those which were related to the very first results on this subject, or that pertain to the results which we wish to address in this short paper. Second result. The next theorem uses the non-quadratic condition at infinity (A8) that was introduced in [7]. Theorem 2.7. Consider the Dirichlet problem −∆u = f(x, u), u = 0 on H1 0 (Ω) , (2.9) where again Ω ⊂ RN , N ≥ 3 is bounded, f is continuous, subcritical, and assume the conditions (A6) λk = lim|s|→∞ 2F (x,s) s2 , uniformly for x ∈ Ω, (A7) λk = lim inf |s|→∞ 2F (x,s) s2 ≤ lim sup|s|→∞ 2F (x,s) s2 = λl, uniformly for x ∈ Ω, (A8) lim|s|→∞[f(x, s)s− 2F (x, s)] = +∞, uniformly for x ∈ Ω, where λk < λl are two eigenvalues of −∆ under Dirichlet boundary condition on ∂Ω. Then the functional J(u) = 1 2 ∫ Ω |∇u|2 dx− ∫ Ω F (x, u) dx := 1 2 ‖u‖2 − ∫ Ω F (x, u) dx satisfies (Ce)c for all c ∈ R. Proof. Recall that the functional J satisfies (Ce)c if any sequence (un) in H1 0 (Ω) such that (i) J(un)→ c, and (ii) ‖J ′(un)‖‖un‖ → 0, has a bounded subsequence. Let us assume by negation that J does not satisfy (Ce)c for some c ∈ R. Then there exists a sequence (un) which satisfies (i) and (ii) above, but ‖un‖ → ∞ . It follows that lim n→∞ ∫ Ω [f(x, un)un − 2F (x, un)] dx = lim n→∞ [2J(un)− J ′(un) · un] = 2c , (2.10) and we shall obtain a contradiction by showing that the left-hand side of (2.10) must go to infinity. Indeed, we make the following claim. Claim: There exists a subset Ω̂ ⊂ Ω with measure(Ω̂) > 0 such that |un(x)| → ∞ a.e. x ∈ Ω̂. Using the Claim, the subcritical growth of f and the assumption (A8), we conclude that the left-hand side of (2.10) goes to infinity. In fact, in this case, the subcritical growth of f and (A8) imply that f(x, un(x))un(x)− 2F (x, un(x))) ≥ −C, for a.e. x ∈ Ω and some C ∈ R , lim n→∞ [f(x, un(x))un(x)− 2F (x, un(x)))] = +∞, for a.e. x ∈ Ω , while Fatou’s lemma with Qn := f(x, un)un − 2F (x, un) gives∫ Ω lim inf n→∞ Qn dx ≥ lim inf n→∞ ∫ Ω̂ Qn dx− C measure(Ω\Ω̂) = +∞ 106 D. G. COSTA EJDE/SI/02 for some C ∈ R, which contradicts (2.10). Now, it remains to prove the claim. To that end, we note that that (A6) and (A7) imply lim sup |n|→∞ 1 ‖un‖2 ∫ Ω [F (x, un)− 1 2 λlu 2 n] dx ≤ 0 . (2.11) And, setting ûn = un/‖un‖, we may assume that ûn converges weakly to some û in H1 0 Ω, and strongly to û in L2(Ω). We shall then define our subset Ω̂ to complete the proof. Indeed, passing to the limit in the equality 1 ‖ûn‖2 J(un) = 1 2 (1− λl‖ûn‖2L2)− 1 ‖un‖2 ∫ Ω [F (x, un)− 1 2 λlu 2 n] dx , and using (2.11), we obtain 0 ≥ 1 2 (1− λl‖û‖2L2) which shows that û 6= 0. The claim is proved by taking Ω̂ = {x ∈ Ω : û(x) 6= 0}. � Remark 2.8. As a final remark, we shall exhibit various possibilities of Γk (indi- cated in Remark 2.5) in terms of the measures of the sets [φk > 0] (denoted αk) and [φk < 0] (denoted βk), as well as the relative signs of the limits G+ and G−. Indeed, let us define the parameters γ ∈ [0, 1] and δ ∈ [−1, 1] , and set βk = γαk, G− = δG+. Then, an easy calculation shows that the finite set Γk can be rewritten as Γk = {−(1 + γδ)αk.G+,−(γ + δ)αkG+} . (2.12) Note that the set Γk has 2 elements (or 1 element, if the above elements coin- cide). Indeed, recall that in Remark 2.5 we assumed λk to be a simple eigenvalue. Clearly, when λk has multiplicity νk (i.e., dimension(Nk) = νk), we’ll get 2νk (or νk) elements in Γk. Finally, we consider some special cases of γ and δ (assuming λk is a simple eigenvalue) where the situation described in Remark 2.8 arises by using Γk in (2.12) . Special cases. Case 1: If γ = 0, then Γk = {−αk.G+,−δαkG+}, and (i) Γk has 1 element if δ = 1, (ii) Γk has 2 elements if δ < 1; Case 2: If γ > 0, then Γk, and (i) Γk has has 1 element if δ = 0, and γ = 1, (ii) Γk has 2 elements if γ < 1 [see (2.12)]; Case 3: If δ < 1, then (i) Γk has (i) 1 element if γ = 1, and (ii) Γk has 2 elements if γ < 1; Indeed, δ < 1, γ = 1⇒ −1− δ = −1− γ, so Γk has 1 element, whereas δ < 1, γ < 1 ⇒ −1 − δγ 6= −γ − δ, so Γk has 2 elements; on the other hand, EJDE-2023/SI/02 COMPACTNESS CONDITIONS AND THEIR APPLICATIONS 107 Case 4: If δ = 1 [i.e. G+ = G−] and γ = 1 [i.e. βk = αk], then βk = αk = measure(Ω)/2, which is equivalent to γ = 1 . References [1] S. Ahmad, A. C. Lazer, J. L. Paul; Elementary critical point theory and perturbations of elliptic boundary value problems at resonance, Indiana Univ. Math. J., 25 (1976), 933–944. [2] D. Arcoya, D. G. Costa; Nontrivial solutions for a strong resonant problem, Diff. Int. Eqs., 8 (1995), 151–159. [3] V. Benci, B. Bartolo, D. Fortunato; Abstract Critical Points Theorems and Applications to some Nonlinear Problems with “Strong” Resonance at Infinity, Nonlinear Analysis, Theory, Methods & Applications 7, (1983), no. 8, 981–1012. [4] H. Brézis, J. M. Coron, L. Nirenberg; Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz, Comm. Pure Appl. Math., 33 (1980), 667–689. [5] G. Cerami; Un criterio de esistenza per i punti critici su variet’a ilimitate, Rx. Ist. Lomb. Sci. Lett., 112 (1978), 332–336. [6] D. G. Costa; An Invitation to Variational Methods in Differential Equations, Birkhauser Boston, MA, 2007. [7] D. G. Costa, C. A. Magalhaes; Variational elliptic problems which are nonquadratic at in- finity, Nonl. Anal., 23 (1994), 1401–1412. [8] D. G. Costa, H. Tehrani; On a class of asymptotically linear elliptic problems in RN , J. Diff. Eqs., 173 (2001), 470–494. [9] E. M. Landesman, A. C. Lazer; Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech., 19 (1970), 609–623. [10] R. S. Palais, S. Smale; A generalized Morse theory, Bull. Amer. Math. Soc. 70 (1964), 165– 171. [11] P. H. Rabinowitz; Minimax Methods in Critical Point Theory with Applications to Differen- tial Equations, CBMS Regional Conf. Ser. in Math., 65, AMS, Providence, RI, 1986. [12] C. A. Stuart, H. S. Zhou; Applying the mountain-pass theorem to an asymptotically linear elliptic equation on RN , Comm. Partial Differential Equations, 24 (1999), 1731–1758. David G. Costa Department of Mathematical Sciences, University of Nevada Las Vegas, Box 454020, Las Vegas, NV 89154-4020, USA Email address: david.costa@unlv.edu 1. Introduction 2. Two results First result Second result Special cases References