Multiplicity of solutions for nonlinear coercive problems

J. Math. Anal. Appl. 528 (2023) 127473 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa Regular Articles Multiplicity of solutions for nonlinear coercive problems J. Diblík a,b, M. Galewski c,∗, V.D. Rădulescu d,e,a, Z. Šmarda a aFaculty of Electrical Engineering and Communication, Department of Mathematics, Brno University of Technology, Technická 3058/10, 616 00 Brno, Czech Republic bFaculty of Civil Engineering, Department of Mathematics and Descriptive Geometry, Veveří 331/95, 602 00 Brno, Czech Republic cInstitute of Mathematics, Lodz University of Technology, al. Politechniki 8, 93-590 Lodz, Poland dFaculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Krakow, Poland eDepartment of Mathematics, University of Craiova, 200585 Craiova, Romania a r t i c l e i n f o a b s t r a c t Article history: Received 9 February 2023 Available online 5 June 2023 Submitted by E. Braverman Keywords: Coercive functional Multiple solutions Nonlinear equations We are concerned in this paper with problems that involve nonlinear potential mappings satisfying condition (S) and whose potentials are coercive. We first provide mild sufficient conditions for the minimizing sequence in the Weierstrass-Tonelli theorem in order to have strongly convergent subsequences. Next, we establish a three critical point theorem which is based on the Pucci-Serrin type mountain pass lemma and which is an infinite dimensional counterpart of the Courant theorem. Ricceri-type three critical point results then follow. Some applications to Dirichlet boundary value problems driven by the perturbed Laplacian are given in the final part of this paper. © 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http:// creativecommons .org /licenses /by -nc -nd /4 .0/). 1. Introduction Let Ebe a separable reflexive real Banach space and let ·,· denote the duality pairing between E∗and E. In this paper we consider nonlinear potential equations of the following type A(u)=0foru∈E, (1) where A :E→E∗is coercive bounded (that is, Ais bounded on bounded sets) and potential operator which satisfies condition (S) and where A :E→Rstands for the potential of A. We examine the solvability of problem (1)by introducing a version of the Weierstrass-Tonelli theorem in which we obtain that the minimizer is the limit of a norm convergent minimizing sequence, as is the case *Corresponding author. E-mail addresses: [email protected] (J. Diblík), [email protected]dz.pl (M. Galewski), [email protected], [email protected] (V.D. Rădulescu), [email protected] (Z. Šmarda). https://doi.org/10.1016/j.jmaa.2023.127473 0022-247X/© 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by -nc -nd /4 .0/). 2J. Diblík et al. / J. Math. Anal. Appl. 528 (2023) 127473 in the finite dimensional case. We examine the multiple solvability for problem (1)in the above introduced setting by introducing some version of the three critical point theorem working in the coercive case. Concerning the multiplicity of solutions to (1)in the coercive case, there are also some results mostly pertaining to the usage of three critical point theorems of Ricceri-type, see for example [17], [18], [19], [20]for the theoretical background. We aim to derive a three critical point result for coercive C1functionals, which can be viewed as an infinite-dimensional version of the Courant theorem, see [12]for the classical version in RN. This classical result asserts that a functional having two distinct local minima must have another critical point which is not a minimizer (i.e., a point which realizes the minimum). Theorem 1 (Courant). [12]Suppose that a C1functional J:RN→Ris coercive and possesses two distinct strict relative minima x1and x2. Then Jpossesses a third critical point x3distinct from x1and x2, which is not a relative minimizer, that is in every neighborhood of x3, there exists a point xsuch that J(x)<J(x3). In the infinite dimensional version of Theorem 1coercivity is replaced by the fact that the Palais-Smale condition is satisfied and then the following is known: Theorem 2. [16, Theorem 2] Let Ebe a Banach space and let J:E→Rbe a C1functional satisfying Palais-Smale condition with 0Eits strict local minimum. If there exists e =0 Esuch that J(e) ⩽J(0E), then there is a critical point ¯xof I, with J(¯x) >J(0E), which is not a local minimum. Our multiplicity result relies on connecting two distinct local minimizers via suitably chosen mountain pass approach cited above. Due to the necessity of obtaining the Palais-Smale compactness condition for the coercive action functional we investigate further relations between this condition and the coercivity. It is well known that a bounded from below C1functional satisfying the Palais-Smale condition is coercive, while the converse holds necessarily in finite dimensional spaces. We wish to obtain the converse in infinite dimensional space under some additional assumption on mapping A. Therefore we first investigate when the minimizing sequence generated by the Weierstrass-Tonelli Theorem is strongly convergent and provide a version of the Weierstrass-Tonelli Theorem which guarantees such a behavior. This is in compliance with what is known about the Weierstrass-Tonelli Theorem in the finite dimensional setting. Such convergence result is next used in providing conditions which guarantee that a coercive functional satisfies the PalaisSmale condition and as a consequence we obtain our three critical point type theorem by demonstrating the suitable mountain geometry. We comment also on the usage of our results about convergence on minimizing sequences in derivation of a Ricceri type multiplicity theorem by providing some version of an already known multiplicity theorem from [4]. In our approach we rely on using tools commonly applied in the monotonicity theory while investigating potential problems as far as the existence and multiplicity are concerned. Applications are provided for the Dirichlet boundary value problems driven by the perturbed p−Laplacian and its various generalizations. Boundary value problems driven by the p−Laplacian attracted a lot of attention from different point of view. Let us mention for example [10] where the Authors determine the structure of the set of the solutions to the Dirichlet problem for the p−Laplacian on the line. Bifurcationtype results describing the set of positive solutions as the parameter varies are considered in [9], while the three critical point theorem due to Ricceri, see for example [17], is employed for the Dirichlet problem with the p−Laplacian in [1]among many sources which exploit its usage. 2. Auxiliary results For the monotonicity we follow [7]and for variational tools [14]. Operator A :E→E∗is called: i) monotone, if for all u, v∈E A(u)−A(v),u−v≥0 J. Diblík et al. / J. Math. Anal. Appl. 528 (2023) 127473 3 and strictly monotone, if the above inequality is strict for u =v; ii) strongly monotone, if for some constant α>0it holds for all u, v∈E A(u)−A(v),u−v≥αu−v2; iii) strongly continuous, if unu 0implies A (un)→A (u0); iv) potential, if there exists a Gâteaux differentiable functional A :E→R, called the potential of Asuch that A=A; v) satisfying condition (S) if, unu 0in Eand A(un)−A(u0),u n−u0→0imply un→u0in E vi) coercive if, lim v→∞ A(v),v v=+∞. Remark 3. There are some related conditions pertaining to condition (S) which make weakly convergent sequences strongly convergent upon some additional convergence condition on the operator involved. These are as follows: i) Condition (S)+: unu 0and lim supn→+∞A(un)−A(u0),u n−u0≤0imply that un→u0in E; ii) Condition (S)0: unu 0, A (un)b, A(un),u n→b, u0imply that un→u0in E. It is known that condition (S)+implies that condition (S)is satisfied and this in turn implies condition (S)0. We decided to apply condition (S) due to the fact that it is satisfied by the perturbed p−Laplacian operator which we consider further on and is much more intuitive than the technical condition (S)0, while being less demanding than (S)+. We mention that adding a strongly continuous perturbation to operator satisfying any of the above mentioned conditions does not violate this condition. A Gâteaux differentiable functional J:E→Rsatisfies the Palais-Smale condition, the (PS) condition, if any sequence (un) ⊂Esuch that i) |J (un)| ≤Mfor all n ∈Nand some M>0, ii) lim n→∞ J(un) =0in E∗ admits a norm convergent subsequence. Theorem 4 (Ekeland Variational Principle - differentiable form). [14]Let J:E→Rbe a Gâteaux differentiable functional which is bounded from below. Then there exists a minimizing sequence (un)consisting of almost critical points, i.e. such that J(un)→inf u∈EJ(u)and J(un)→0(in E∗). We will require the Lagrange Multiplier Rule in the form of Karush-Kuhn-Tucker providing necessary optimality conditions taken after [11]. Let f:E→Rbe a given functional and let g:E→Rbe a constraint functional. Let S={x:g(x)≤0}. 4J. Diblík et al. / J. Math. Anal. Appl. 528 (2023) 127473 Theorem 5. [11] Assume that u0∈Eis such that inf u∈Sf(u)=f(u0). Let fand gbe Fréchet differentiable at u0. Assume that the Slater constraint qualification holds, i.e. if there is some x0that g(x0)<0. Then there is a nonnegative real number μsuch that f(u0)+μg(u0)=0(in E∗). The following theorem about the continuity of the Niemytskij operator is rewritten after [7]: Theorem 6 (Generalized Krasnosel’skii Theorem). Let p1, p2≥1and N≥1be a fixed natural number. Assume that f:[0,1] ×RN→RNis a Carathéodory function. If for any sequence (un)∞ n=1 ⊂Lp1(0,1) convergent to u ∈Lp1(0,1) there exists a function h ∈Lp2(0,1) such that |f(t, un)|≤h(t),for n∈Nand a.e. t ∈[0,1], then the Niemytskij operator induced by f Nf:Lp1(0,1) u(·)−→ f(·,u(·)) ∈Lp2(0,1), is well defined and continuous. 3. On the Weierstrass-Tonelli Theorem In this section we undertake the question about the type of convergence of the minimizing sequence in the Weierstrass-Tonelli Theorem. It is well known that if one minimizes a lower semicontinuous functional Jon a closed bounded set Sin a finite dimensional space or else if the set is unbounded but Jis coercive, then the minimizer is approximated by a convergent minimizing sequence (consisting of points from S). In case we work in an infinite dimensional reflexive Banach space we must require the set Sto be sequentially weakly compact and the functional Jto be sequentially weakly lower semicontinuous (for the latter to hold it suffice to assume continuity and convexity). Then a minimizer is obtained as a limit a weakly convergent minimizing sequence. When a functional is defined on the whole space, we must again assume that J is additionally coercive. However, in the Weierstrass-Tonelli Theorem applied in the infinite dimensional setting the minimizing sequence is strongly convergent as well under the additional condition (S) on the derivative. There is a result by the second author, see [8], answering the question about a convergence of minimizing sequences for a coercive functional with some monotonicity imposed on the derivative. In the proof of this result the Minty Lemma, see Lemma 3.6 in [7], is used. Here we not only drop the assumption about the monotonicity, do not impose a special structure on the action functional but also we simplify the proof methodology. Theorem 7. Assume that A :E→E∗satisfies condition (S) and it is potential with a sequentially weakly l.s.c. and coercive potential A. Let h ∈E∗be fixed. Then there is a solution u0to A(u)=h(2) which minimizes action functional J:E→Rdefined by J(u)=A(u)−h, u J. Diblík et al. / J. Math. Anal. Appl. 528 (2023) 127473 5 over Eand moreover there is a sequence (un)⊂E, un→u0such that J(un)→inf u∈EJ(u)and J(un)→0(in E∗).(3) If in addition Ais strictly monotone, then the solution is unique. Moreover, functional Jsatisfies the (PS) condition. Proof. From the assumptions it follows that functional Jis Gâteaux differentiable, coercive and sequentially weakly l.s.c. Hence it is bounded from below. Moreover, it has at least one minimizer u0which is a critical point, i.e. a solution to (2). By Theorem 4there is a minimizing sequence (un)⊂E, unu 0(the weak convergence follows by coercivity) and which is such that (3) holds. We see from J(un)→0by writing the derivative explicitly that A(un)−A(u0),u n−u0→0.(4) Since Asatisfies condition (S), we obtain that un→u0. Let us take a Palais-Smale sequence, i.e. such a sequence (un)⊂Ethat J(un)is bounded and J(un)→ 0. Due to the assumptions we can assume that unu 0, possibly up to a subsequence which we chose and do not renumber. Hence (4)holds. Since J=A −hsatisfies condition (S) and since unu 0, we see that un→u0, so the remaining assertion follows.  We mention also here that checking condition (S) is technically similar to checking the strong convergence of bounded (PS) sequences which is also why we decided to apply this condition in our reasoning. From the proof of the above result we immediately obtain: Proposition 8. Assume that a Gâteaux differentiable functional J:E→Rhas a derivative J:E→E∗ which satisfies condition (S). Then any bounded (PS) sequence for functional Jadmits a strongly convergent subsequence. Now we are in position to formulate the result about the existence of minimizers for coercive functionals following directly from the above: Corollary 9. Assume that functional J:E→Ris bounded from below, coercive, Gâteaux differentiable and that its derivative J:E→E∗satisfies condition (S). Then there is some u0∈Esuch that J(u0)= inf u∈EJ(u). Proof. From Theorem 7it follows that functional Jsatisfies the (PS) condition. Since it is bounded from below and satisfies the (PS) condition, it necessarily has at least one minimizer. The result now readily follows.  Remark 10. We may replace condition (S) with conditions (S+)or (S)0with retaining the same conclusion in the above, see also Remark 3and remarks in [6]. Now we give some direct applications of our version of the Weierstrass-Tonelli Theorem that are related the known results. We provide a version of Theorem 2.1 from [15]. We recall that a functional whose derivative is monotone and coercive is necessarily bounded from below. In the result that follows we show that for a potential problem which can be tackled by the Banach fixed point theorem the sequence obtained by the method of successive approximations which converges to the unique solution stands also for a minimizing sequence. 6J. Diblík et al. / J. Math. Anal. Appl. 528 (2023) 127473 Proposition 11. Let Ebe a Hilbert space with a scalar product (·,·). Let N:E→Ebe a contraction with the unique fixed point u∗∈E(guaranteed by the Banach contraction theorem). If there exists a C1functional Jsuch that J(u)=u−N(u)for all u∈E, then u∗minimizes the functional E, i.e. J(u∗)= inf u∈EJ(u). Moreover, a sequence (un)⊂E, defined by un+1 =N(un)for any u0∈Eis such that un→u∗and J(un)→inf u∈EJ(u)and J(un)→0.(5) Proof. Since Nis a contraction we see by a direct calculation that Jis strongly monotone. We see that J as a potential of a strongly monotone mapping is coercive and sequentially weakly l.s.c. It also holds that Jis strictly convex. Moreover, since Jis strongly monotone, it satisfies condition (S). Then by Theorem 7 there is minimizer, which is unique by the strict convexity and therefore equal to u∗. Assertion (5) follows by the continuity of N. 4. On a three critical point theorem for a coercive functional In this section we are going to derive the infinite dimensional multiplicity result corresponding to finite dimensional mountain pass theorem due to Courant. For r>0we put Br:= {x:x≤r},S r={x:x=r}. Theorem 12. Assume that I∈C1(E)has a strongly continuous derivative I:E→E∗and that operator A :E→E∗is continuous, bounded, monotone, coercive and satisfies condition (S) and it is potential with the potential A. Denote J:= A+I. Let x∈Eand r>0be fixed. Assume that the following conditions are satisfied: (A.1) lim inf x→∞ I(x) A(x)≥0; (A.2) inf x∈EJ(x) <inf x∈Br J(x); (A.3) x<rand J(x) <inf x∈Sr J(x). Then functional Jhas at least three distinct critical points in E, i.e. equation A(u)+I(u)=0 has at least three distinct solutions two of which are necessarily nontrivial. Proof. By the coercivity and the boundedness of operator Ait follows that its potential is also coercive, see Lemma 5.5 from [7]. From (A.1) it holds for all x ∈Ewith sufficiently large norms that I(x)>−1 2A(x). J. Diblík et al. / J. Math. Anal. Appl. 528 (2023) 127473 7 Indeed, according to [13] lim inf x→∞ I(x) A(x)≥0means that for any α<0 there is some R>0such that for all x>Rwe have I(x) A(x)>α. This implies, for all x ∈Ewith x>R, that J(x)=A(x)+I(x)>1 2A(x) and therefore functional Jis coercive as well. Since Iis strongly continuous, we see that its Ipotential is sequentially weakly continuous. Since Ais monotone, it follows that Ais sequentially weakly l.s.c. and so is functional J. Now from Theorem 7we see that Jsatisfies the (PS) condition. Since Jis sequentially weakly l.s.c. and coercive it has a global minimizer (and a critical point) which due to (A.2) lies outside Br. Moreover, in Brfunctional Jhas a local minimizer which by (A.2) lies inside a ball and therefore it is a critical point as well. By the application of Theorem 2we get the existence of a third critical point which is distinct from these two mentioned.  Remark 13. We can replace (A.1) with the assumption that functional Iis bounded from below. As it can be expected from Theorem 7we see that Theorem 12 has also a more general formulation which does not involve monotonicity. Hence the special structure of Jneed not be assumed. However, due to the fact that we minimize a functional over a closed ball, we have to assume the sequential weak lower semicontinuity of the functional. In order to conclude this section we will work on exploiting the monotonicity theory in derivation of the Ricceri type of a three critical point theorem for coercive functionals following [4]. In the proof the following technical lemma will be utilized and which obtained as a special case of results from [17, Proposition 2.2] and [2, Theorem 1]: Lemma 14. Let D⊆R+be an interval. Assume that Φ ∈C1(E)is such that its derivative Φ :E→E∗is strictly monotone, coercive and satisfies condition (S). Assume that I∈C1(E)is such that I:E→E∗is strongly continuous. Moreover, assume that there exist x1, x2∈Eand σ∈Rsuch that (B.1) Φ(x1) <σ<Φ(x2); (B.2) inf Φ(x)≤σI(x) >(Φ(x2)−σ)I(x1)+(σ−Φ(x1))I(x2) Φ(x2)−Φ(x1); (B.3) lim x→∞ [Φ(x)+λI(x)] = +∞for all λ ∈D. Then there exists a nonempty open set C⊆Dsuch that for all λ ∈Cthe functional Φ +λIhas at least three critical points in E. Lemma 15. Assume that I∈C1(E)is sequentially weakly l.s.c. and that A :E→E∗is continuous, strictly monotone, coercive, satisfies condition (S) and is potential with the potential Asuch that A (0) = 0. Let x∈Eand r>0be fixed. Assume that (A.1) holds and also the following conditions are satisfied: (C.1) inf x∈EI(x) <inf A(x)≤rI(x); (C.2) A(x) <rand I(x) <inf A(x)=rI(x). Then there exists a nonempty open set C⊆(0, +∞)such that for all λ ∈Cthe functional A +λIhas at least three critical points in E, two of which are necessarily non-trivial. Proof. By (A.1) we see that condition (B.3) is satisfied. 8J. Diblík et al. / J. Math. Anal. Appl. 528 (2023) 127473 Since mapping Ais strictly monotone, we see that its potential is necessarily convex. Then set the Lebesgue level set B={x∈E:A(x)≤r}is convex and due to the continuity of Ait is also closed. This means that Bis weakly closed. Since Ais the potential of a coercive mapping it follows that Bis bounded which means that Bis sequentially weakly compact. Now, since functional Jis sequentially weakly l.s.c. it attains its infimum on Bat some x1∈B. We claim that A(x1)<rand I(x1)= inf x∈BI(x).(6) Since A (0) = 0 <rwe observe that the Slater constraint qualification is satisfied. Therefore it follows by the Karush-Kuhn-Tucker Theorem (Theorem 5) that there is a Lagrange multiplier ζ≥0for which it holds I(x1)+ζA(x1)=0andζ(A(x1)−r)=0. By (C.2) it follows that A (x1)<rand therefore we must take ζ=0which implies (6). By (C.1) there exists x2∈Esuch that A(x2)>rand I(x2)<I(x1).(7) Set σ=rand noting that A(x1) <σ<A(x2)we see that (B.1) is satisfied. Moreover by (6)and (7)we have inf A(x)≤rI(x)=I(x1)=I(x1)(A(x2)−A(x1))) A(x2)−A(x1)= I(x1)A(x2)−σI(x1)+σI(x1)−I(x1)A(x1) A(x2)−A(x1)= (A(x2)−σ)I(x1)+(σ−A(x1)) I(x1) A(x2)−A(x1)> (A(x2)−σ)I(x1)+(σ−A(x1)) I(x2) A(x2)−A(x1), so (B.2) is satisfied too. The assertion now follows by Lemma 14. Remark 16. While in the proof of Lemma 15 we use some ideas from [4], we include new arguments, like the usage of the Karush-Kuhn-Tucker Theorem and we also exploit the usage of the monotonicity theory. Condition (C1) can be replaced with the following (C.4). A(x) +λI(x) →+∞as x→∞ with the assertion retained. The assertion is also retained if we assume Ito be bounded from below, see also [5]for some research in this direction. Comparing Lemma 15 with Theorem 12 we see that their applicability coincides when operator Ais strongly monotone, i.e. for the case of the (negative) Laplacian. In contrast when the (negative) p−Laplacian is considered finding a minimizer on a ball is much more convenient than examining the behavior of the Euler action functional. J. Diblík et al. / J. Math. Anal. Appl. 528 (2023) 127473 9 5. Applications 5.1. Applications of the Weierstrass-Tonelli Theorem We note that the application of our version of the Weierstrass-Tonelli Theorem is similar to checking the classical version but it requires verifying that the condition (S) or any related is satisfied instead of checking the sequential weak lower semicontinuity of the action functional. Nevertheless both approaches use similar arguments. Towards the uniqueness we need to determine that the derivative defines the strictly coercive operator. We will need some preparation prior to introduction of the problem under consideration. Let p ≥2and set E:= W1,p 0(0,1). Then Eis a separable, uniformly convex (and thus reflexive) space, see [3]. Recall that for any u ∈Eit holds uC:= max t∈[0,1] |u(t)|≤u:= ˙uLp=⎛ ⎝ 1 0 |˙u(t)|pdt⎞ ⎠ 1/p . Let f:[0,1] ×R →Rbe an L1−Carathéodory function and define F:[0,1] ×R →Rby F(t, u)= u 0 f(t, s)ds for a.e. t ∈[0,1] and all u∈R.(8) We assume that (ϕ1)ϕ :[0,1] ×R+→Ris a Carathéodory function for which there is a constant M>0such that |ϕ(t, x)|≤Mfor a.e. t∈[0,1] and all x∈R+; (ϕ2) there exists a constant γ>0such that ϕ(t, x)x−ϕ(t, y)y≥γ(x−y) for all x ≥y≥0and a.e. t ∈[0,1]; (F.1) f(t, 0) = 0 for a.e. t ∈[0,1], g∈Lp (0,1), g=0; (F.2) for a.e. t ∈[0,1] function x → f(t, x)is nondecreasing. Now we can consider the existence and the uniqueness for the following Dirichlet problem ⎧ ⎪ ⎨ ⎪ ⎩ −d dt ϕt, d dt up−1d dt up−2d dt u+f(t, u (t)) = g(t),a.e. on (0,1) , u(0) = u(1) = 0. (9) The solutions are understood in the weak sense. We say that a function u ∈Eis a weak solution of (9)if for all v∈Eit holds 1 0 ϕt, |˙u(t)|p−1|˙u(t)|p−2˙u(t)˙v(t)dt + 1 0 f(t, u (t))v(t)dt = 1 0 g(t)v(t)dt.