FROM A CAPILLARY PHENOMENA 1. INTRODUCTION

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1 J. Nonlinear Funct. Anal , Article ID htts://doi.org/0.395/jnfa.08. x-laplacian-like PROBLEMS WITH NEUMANN CONDITION ORIGINATED FROM A CAPILLARY PHENOMENA SHAPOUR HEIDARKHANI, AMJAD SALARI Deartment of Mathematics, Faculty of Sciences, Razi University, 6749 Kermanshah, Iran Abstract. In this aer, we study the multilicity results for x-lalacian-like roblems with Neumann condition, originated from a caillary henomena. Using variational methods the critical oint theory, the existence of two three solutions for the roblem is discussed. Keywords. method. Critical oint theory; Multile solutions; Neumann condition; Variable exonent Sobolev sace; Variational 00 Mathematics Subject Classification. 35J0, 35J60.. INTRODUCTION Let R N be an oen bounded domain with smooth boundary. In this aer, we are mainly concerned with existence multilicity results for the x-lalacian-like roblem, originated from a caillary henomena, div + u x + u x u x u + αx u x u = f x,u, in, u ν = 0, on, where R N N is a bounded domain with boundary of class C, ν is the outer unit normal to, > 0, µ 0, α L with essinf α 0, f : R R is an L -Carathéodory function C 0 satisfies the condition N < := inf x x + := su x < +. x During the last fifteen years, differential artial differential equations with variable exonent growth conditions have become increasingly oular. This is artly due to their frequent aearance in alications such as the modeling of electrorheological fluids, image restoration, elastic mechanics Corresonding author. addresses: s.heidarkhani@razi.ac.ir S. Heidarkhani, amjads45@yahoo.com A. Salari. Received October 6, 07; Acceted February 3, 08. P f c 08 Journal of Nonlinear Functional Analysis

2 S. HEIDARKHANI, A. SALARI continuum mechanics, these roblems are also very interesting from a urely mathematical oint of view as well. Recently, Fan Deng [] studied the existence multilicity of ositive solutions for the inhomogeneous Neumann boundary value roblem involving the x-lalacian of the following form div u x u + u x u = f x,u, in, u x u η = ϕ, on, where is a bounded smooth domain in R N, C x > for x, ϕ C 0,γ with γ 0,, ϕ 0 ϕ 0 on. They, under aroriate assumtions on f, obtained that there exists > 0 such that the roblem has at least two ositive solutions if >, has at least one ositive solution if =, has no ositive solution if <. Deng [], based on a local mountain ass theorem without P.S condition Ricceri s variational rincile, obtained the existence multilicity of non-trivial solutions for the following x-lalacian double erturbed Neumann roblem with nonlinear boundary condition x u + ax u x u = f x,u + h x,u, in, u x u γ = gx,u + µh x,u, on, where is a bounded oen domain in R N with smooth boundary, x u = div u x u is the x-lalacian with C, x >,,µ R, a L with essinf x ax = a > 0, γ is the outward unit normal to. Cammaroto Vilasi [3], based on a recent variational rincile due to Ricceri, established the existence of at least three solutions for the Neumann roblem involving the x-lalacian oerator. In [4], the existence of at least three solutions for ellitic roblems driven by a x-lalacian was established based on the variational methods critical-oint theory. The existence of at least one nontrivial solution was also roved. Problem P f is the develoed form of the ellitic Dirichlet roblem u div = f t,u in, + u u = 0 on, where R N N is a bounded oen subset with sufficiently smooth boundary f : R R is a suitable Carathéodory function. It is well-known that a solution u of. defines a Cartesian surface in R N+ whose mean curvature is rescribed by the right h side of the equation this roblem lays an imortant role in differential geometry in the theory of relativity. Existence, non-existence multilicity of ositive solutions of roblem. have been studied recently. We just observe that the one-dimensional roblem has been rather thoroughly discussed, by using different methods; see [5, 6, 7, 8] references therein. In articular, if f x,u = u, where > 0 is given > 0 is a arameter, the existence of ositive solutions of. was established assuming > the convex, or suerlinear, case, with < N+ N if N >, large [9], or = the linear case in a left neighborhood of the rincial eigenvalue of in H 0 [0], or < the concave, or sublinear, case small []. The solutions found in these aers belong at least to H 0 are intended in the usual weak sense. In [], the existence of a second ositive solution has been roved in the concave case when is small, assuming further ]0, N [ if N >. Yet, such a.

3 x-laplacian-like PROBLEMS WITH NEUMANN CONDITION ORIGINATED FROM A CAPILLARY PHENOMENA 3 solution just belongs to BV satisfies the equation only in a generalized sense see also [3] for a similar aroach. We also notice that, with the excetion of [0], where bifurcation theory is used, the roofs of these results are all based on variational methods: a variant of Nehari method is used in [9], a minimization argument in [], non-smooth critical oint theory in []. The action functional to be considered here is + u u +. + The different behavior at zero at infinity of the area term + u, which is quadratic with resect to u near the origin grows linearly at infinity, gives raise to the multilicity henomena devised in the concave case; the lack of coercivity in H however causes several technical difficulties. Also, we refer the reader to aer [4] in which Obersnel Omari, based on variational combines critical oint theory, the lower uer solutions method ellitic regularization, established the existence multilicity of ositive solutions of the rescribed mean curvature roblem. where f C R is suerlinear do not satisfy the Ambrosetti Rabinowitz tye condition, > 0 is a arameter. Caillarity can be briefly exlained by considering the effects of two oosing forces: adhesion, i.e., the attractive or reulsive force between the molecules of the liquid those of the container; cohesion, i.e., the attractive force between the molecules of the liquid. Recently, the study of caillarity henomena has been an interesting toic. In [5], Shokooh used variational methods to obtain existence results for ellitic equations involving the -Lalacian-like. Rodrigues [6], by using Mountain Pass lemma see [7] Fountain theorem see Theorem 3.6 in [8], established the existence of non-trivial solutions for roblem div + u x u = 0, + u x u x u = f x,u, x, x, where R N N is a bounded domain with boundary of class C, is a ositive arameter, C f is a Carathéodory function. Recently, Bin [9] obtained the existence results of nontrivial solutions for every arameter for the nonlinear eigenvalue roblems for x-lalacian-like oerators originated from a caillary henomena of the following form: div + u x u = 0 + u x u x u = f x,u in, on, where R N is a bounded domain with smooth boundary, > 0 is a arameter. Based on the mountain ass theorem a nontrivial solution is constructed for almost every arameter > 0. he considered the continuation of the solutions. In [0], Zhou by emloying variational methods, established the existence of at least one non-trivial solution for the following nonlinear eigenvalue roblem for the x-lalacian-like oerators originated from a caillary henomenon div + u x u = 0 + u x u x u = f x,u in, on,.

4 4 S. HEIDARKHANI, A. SALARI where is a bounded domain in R N with smooth boundary, C, > 0 is a arameter, f C R is suerlinear does not satisfy the Ambrosetti Rabinowitz tye condition. The existence of at least one nontrivial solution was also roved. Our aroach is the variational method the main tools are the local minimum theorem for differentiable functionals due to Bonanno [] Mountain Pass Theorem. Two of the consequences of the local minimum theorem due to Bonanno are here alied see Theorems... Indeed, we investigate the existence of two solutions for roblem P f emloying a consequence of the local minimum theorem due to Bonanno mountain ass theorem under some algebraic conditions with the classical Ambrosetti-Rabinowitz AR condition on the nonlinear term; see []. Moreover, by combining two algebraic conditions on the nonlinear term alying two consequences of the local minimum theorem due to Bonanno, we guarantee the existence of two solutions. Alying the mountain ass theorem given by Pucci Serrin [3], we establish the existence of third solution for roblem P f. The remainder of the aer is organized as follows. In Section, we recall the definitions some roerties of variable exonent Sobolev saces. In Section 3, we state rove the main results of the aer. Here, we state two secial cases of our results, immediately follows from the next Theorems Theorem.. Let > N, diam < N N such that h0 0. Assume that h : R R be a non-negative continuous function hξ lim = + ξ 0 + ξ AR there exist constants ν > R > 0 such that, for all ξ R, Then, for each 0 < ν ξ 0 hζ dζ ξ hξ. 0, + meas su γ γ, γ>0 0 hζ dζ where meas is the Lebesgue measure of, the roblem div + u u u + u u = hux, in, + u.3 u ν = 0, on admits at least two ositive weak solutions. Theorem.. Let > N, diam < N N such that h0 0. Assume that 0 hξ lim = +, ξ 0 + ξ lim h : R R be a non-negative continuous function ξ + hξ ξ = 0 htdt < + meas meas 0 htdt.

5 x-laplacian-like PROBLEMS WITH NEUMANN CONDITION ORIGINATED FROM A CAPILLARY PHENOMENA 5 Then for each + meas 0 htdt, meas + meas, 0 hxdx where meas is the Lebesgue measure of, roblem.3 admits at least three ositive weak solutions.. PRELIMINARIES Our basic instruments include the following theorems which derived from the existence results of a local minimum theorem for differentiable functionals due to Bonanno [, Theorem 3.], which is in turn motivated by Ricceri s variational rincile; see [3, 4] the references therein. For a given non-emty set X, two functionals Φ,Ψ : X R, we define the following functions for all r,r R, r < r, for all r R. βr,r = r,r = r = inf v Φ r,r su v Φ r,r su v Φ r,+ su u Φ r,r Ψu Ψv, r Φv Ψv su u Φ,r ] Ψu Φv r Ψv su u Φ,r] Ψu Φv r Theorem. [, Theorem 5.]. Let X be a real Banach sace let Φ : X R be a sequentially weakly lower semicontinuous, coercive continuously Gâteaux differentiable function whose Gâteaux derivative admits a continuous inverse on X. Let Ψ : X R be a continuously Gâteaux differentiable function whose Gâteaux derivative is comact. Assume that there are r,r R, r < r such that βr,r < r,r. Setting I := Φ Ψ, for each r,r, βr,r, there is u 0, Φ r,r such that I u 0, I u u Φ r,r I u 0, = 0. Theorem. [, Theorem 5.3]. Let X be a real Banach sace let Φ : X R be a continuously Gâteaux differentiable function whose Gâteaux derivative admits a continuous inverse on X. Let Ψ : X R be a continuously Gâteaux differentiable function whose Gâteaux derivative is comact. Fix inf X Φ < r < su X Φ assume that r > 0, for each > r, the functional I := Φ Ψ is coercive. Then for each r,+, there is u 0, Φ r,+ such that I u 0, I u u Φ r,+ I u 0, = 0. We refer readers to the aer [5] in which Theorems.. have been successfully emloyed to obtain the existence of solutions for nonlinear mixed boundary value roblems Here in the sequel, meas denotes the Lebesgue measure of the set, we also assume that C + := { h C : hx >, x } verifies the following condition: N < := inf x x x + := su x < +.. x

6 6 S. HEIDARKHANI, A. SALARI Define the variable exonent Lebesgue sace by { L x := u : R measurable On L x we consider the norms resectively u L x {η = inf > 0 : ux η } ux x dx < +. x } dx. Let X be the generalized Lebesgue-Sobolev sace W,x defined by utting W,x by { } W,x = u L x : u L x endowed with the following norm u W,x := u L x + u L x.. It is well known see [6] that, in view of., both L x W,x, with the resective norms, are searable, reflexive uniformly convex Banach saces. Moreover, since α L, with α := essinf x αx > 0 is assumed, the norm { u α = inf σ > 0 : αx ux x + ux x } dx σ σ on W,x is equivalent to that introduce in.. Since W,x is continuously embedded in W, see [6] or [7] > N, W,x is continuously embedded in C 0 one has u C 0 k u W,. When is convex, an exlicit uer bound for the constant k is { k max D, N meas α L N α α L where D = diam meas is the Lebesgue measure of see [8, Remark ]. On the other h, taking into account that x, [7, Theorem.8] ensures that L x L the constant of such embedding does not exceed + meas. So, one has u W, + meas u W,x + meas α L. }, In conclusion, utting = k + meas, it results u C 0 u α.3 for each u W,x. Put t Fx,t := f t,ξ dξ for all t,x R. 0

7 x-laplacian-like PROBLEMS WITH NEUMANN CONDITION ORIGINATED FROM A CAPILLARY PHENOMENA 7 Definition.3. We mean by a weak solution of roblem P f, any function u X such that [ ux x ux + ux x ux vx + ux x ] + αx ux x uxvx dx f x,uxvxdx = 0 for every v W,x. Proosition.4 [, Proosition.4]. Letting α u = u x + αx u x dx for u W,x, we have u α = u α α u u α +, u α = u + α α u u α. Now for every u X, we define Φu := x ux x + + ux x + αx ux x dx,.4 Ψu = Fx, uxdx..5 Stard arguments show that Φ Ψ are Gâteaux differentiable functionals whose Gâteaux derivatives at the oint u X are given by [ Φ uv = ux x ux + ux x ux vx + ux x ] + αx ux x uxvx Ψ uv = f x,uxvxdx for all u,v X, resectively. Hence, a critical oint of the functional Φ Ψ, gives us a weak solution of P f. We need the following roosition in the roofs of our main results. Proosition.5 [6]. The functional ϕ : X R defined by ϕu := ux x + + ux x x dx is convex the maing ϕ : X X is a strictly monotone bounded homeomorhism. Remark.6. We say that f : R R is an L -Carathéodory function, if a x f x,t is measurable for every t R, b t f x,t is continuous for a.e. x, c for every > 0 there exists a function l L such that dx for a.e. x. su f x,t l x t

8 8 S. HEIDARKHANI, A. SALARI 3. MAIN RESULTS In this section we formulate our main results. For a non-negative constant γ a ositive constant δ with γ δ + α L, we set a γ δ := su t γ Fx,tdx Fx,δdx. γ δ + α L Theorem 3.. Assume that f x,0 0 for all x suose that there exist a non-negative constant γ two ositive constants γ δ, with γ < δ < α + L + γ + α L 3. such that A a γ δ < a γ δ; A there exist ν > + R > 0 such that Then, for each for all ξ R for all x. 0 < νfx,ξ ξ f x,ξ 3. + a γ δ, +, a γ δ roblem P f admits at least two non-trivial weak solutions u u in X, such that γ + u x x + + u x x x + αx u x x dx γ +. Proof. Put I = Φ Ψ, where Φ Ψ are given as in.4.5, resectively. Due to Proosition.5, we have Φu + u α 3.3 for all u X with u >, which follows that Φ is coercive. Of course Φ is sequentially weakly lower semicontinuous continuously Gâteaux differentiable while Proosition.5 gives that its Gâteaux derivative admits a continuous inverse on X. The functional Ψ : X R is well defined is continuously Gâteaux differentiable whose Gâteaux derivative is comact. Choose for δ > define w δ X by Thus Φw δ = = x x r = + γ, r = + γ w δ t = δ. 3.4 w δ x x + + w δ x x + αx w δ x x dx αx δ x dx.

9 x-laplacian-like PROBLEMS WITH NEUMANN CONDITION ORIGINATED FROM A CAPILLARY PHENOMENA 9 According to., we have the following inequalities δ + α L Φw δ δ + Moreover, for all u X with Φu < r, one has α L. u α max{ + r +, + r } = γ. So, due to the embedding X C 0 see.3, one has u u α < γ. It follows that which concludes Φ,r = {u X; Φu < r } {u X; u γ }, Ψu su u Φ,r By the same argument as above we have Therefore, On the other h, one has su Ψu u Φ,r Fx, uxdx su Fx,tdx. t γ su Fx,tdx. t γ βr,r su u Φ,r Ψu Ψw δ r Φw δ + su t γ Fx,tdx Fx,δdx γ δ + α L = + a γ δ. r,r Ψw δ su u Φ,r ] Ψu Φw δ r + su t γ Fx,tdx Fx,δdx γ δ + α L = + a γ δ. Hence, from A, one has βr,r < r,r. Therefore, from Theorem., for each + a γ δ, +, a γ δ the functional I admits at least one non-trivial critical oint u such that r < Φu < r, that is, + γ + x γ u x x + + u x x + αx u x x dx.

10 0 S. HEIDARKHANI, A. SALARI Now, we rove the existence of the second local minimum distinct from the first one. To this aim, we verify the hyotheses of the mountain-ass theorem for the functional I. Clearly, the functional I is of class C I 0 = 0. The first art of roof guarantees that u X is a local nontrivial local minimum for I in X. We can assume that u is a strict local minimum for I in X. Therefore, there is > 0 such that inf u u = I u > I u, so condition [9, I, Theorem.] is verified. By integrating the condition 3., there exist constants a,a > 0 such that Fx,t a t ν a for all x t R. Now, choosing any u X \ {0} taking Proosition.4 into account, one has I τu = Φ Ψτu max{ τu +, τu } Fx, τuxdx max{τ +,τ } max{ u +, u } τ ν a ux ν dx + a as τ +, so condition [9, I, Theorem.] is satisfied. Thus, the functional I satisfies the geometry of mountain ass. Moreover, I satisfies the Palais-Smale condition. Indeed, assume that {u n } n N X such that {I u n } n N is bounded I u n 0 as n +. Then, there exists a ositive constant c 0 such that I u n c 0, I u n c 0 n N. Therefore, we infer to deduce from the definition of I the assumtion A that c 0 + c u n νi u n I u nu n ν + min{ u n α +, u n νfx,u n x f x,u n xu n xdx ν + min{ u n + α, u n α } for some c > 0. Since ν > +, this imlies that u n is bounded. Consequently, since X is a reflexive Banach sace we have, u to a subsequence, u n u in X. By I u n 0 u n u in X, we obtain I u n I u u n u α } From the continuity of f, we have f x,u n x f x,uxu n x uxdx 0, as n So by 3.6 [6, Proosition 3.] the sequence {u n } converges strongly to u in X. Therefore, I satisfies the Palais-Smale condition. Hence, by the classical theorem of Ambrosetti Rabinowitz we establish a critical oint u of I such that I u > I u. Since f x,0 0 for all x, u u are two distinct nontrivial weak solutions of P f the roof is comlete.

11 x-laplacian-like PROBLEMS WITH NEUMANN CONDITION ORIGINATED FROM A CAPILLARY PHENOMENA Remark 3.. In Theorem 3., we guaranteed the existence of at least two nontrivial weak solutions for P f. One of these solutions has been achieved in relation with the classical Ambrosetti-Rabinowitz condition on the data by taking f x,0 0 for all x. If f x,0 0 for all x does not hold, the second solution u of the roblem P f may be trivial, but the roblem has at least a nontrivial solution. Remark 3.3. In Theorem 3., we looked for the critical oints of the functional I naturally associated with roblem P f. We note that, in general, I can be unbounded from the following in X. Indeed, for examle, in the case when f x,ξ = + ξ a + ξ + for x,ξ R with a > +, for any fixed u X\{0} ι R, we obtain I ιu = Φιu Fx, ιuxdx ι + u + ι u L ιa a u a L a as ι +. Hence, we can not use direct minimization to find critical oints of the functional I. Now, we oint out an immediate consequence of Theorem 3.. Theorem 3.4. Assume that f x,0 0 for all x there exist two ositive constants δ γ, with δ < + + γ α L such that the assumtion A in Theorem 3. holds. Furthermore, suose that su t γ Fx,tdx < Fx,δdx. 3.8 γ α L δ + Then, for each + δ + α L Fx,δdx, + γ su t γ Fx,tdx, roblem P f admits at least two non-trivial weak solutions u u in X such that 0 < u x x + + u x x x + αx u x x dx γ +. Proof. The conclusion follows from Theorem 3., by taking γ = 0 γ = γ. Indeed, owing to the inequality 3.8 A, one has a γ δ = su t γ Fx,tdx Fx,δdx γ δ + α L < su t γ Fx,tdx < γ Fx,δdx δ + α L = a 0 δ.

12 S. HEIDARKHANI, A. SALARI In articular, one has which follows + a γ δ < su t γ Fx,tdx γ, γ su t γ Hence, Theorem 3. ensures the conclusion. Fx,tdx < + a γ δ. Now we illustrate Theorem 3.4 by resenting the following examle. Examle 3.5. Let N =, = 0, π R, x = 3 + sinx for all x [0, π ], αx = x9 for all x [0, π ] e x +t 8, t, x 0, π f x,t =, e x 3 t, t <, x 0, π. Thus, = 3, + = 4, meas = π, D = π, α L = 0 π 0, α = π 9 k = k , π so, π + π. Moreover, f is an L -Carathéodory non-negative function, f x,0 0 for all x 0, π e x t + t , t <, x 0, π, Fx,t = e x 3t t3 3, t, x 0, π,. e x t + t , t >, x 0, π. Now, since ξ f x,ξ lim ξ + Fx,ξ = lim ξ + e x t +t 9 = 9e x 9 > 4 = + t + t9 9 for all x 0, π 7, by choosing ν = > 4 = + we can choose R > in a manner that the assumtion 3. is fulfilled. Moreover, by choosing δ = 4 γ = the assumtions are fulfilled. Indeed, δ = 4 < π + 3 su t γ Fx,tdx γ = 8 3 e π Therefore, by aly Theorem 3.4 for each 3 430e π the roblem < π 3 e π α L + α L γ + π 0, πe π Fx,δdx δ +. π 0 + u x u 3+sinx x +sinx u x = f x,t, in 0, π + u 6+4sinx, u 0 = u π = 0

13 x-laplacian-like PROBLEMS WITH NEUMANN CONDITION ORIGINATED FROM A CAPILLARY PHENOMENA 3 admits at least two non-trivial weak solutions u u in the sace W,3+sinx 0, π { = u L 3+sinx 0, π : u L 3+sinx 0, π } such that π 0 < u sinx x 3+sinx + + u x 6+4sinx + x 9 u x 3+sinx dx π 0. 0π + 3 Now, we give an alication of Theorem. which will be used later to obtain multile solutions for roblem P f. Theorem 3.6. Assume that there exist two ositive constants γ δ with δ < + γ + + α L such that Then, for each >, where su t γ Fx,tdx < Fx, δdx Fx,ξ lim su 0 uniformly in R. 3.9 ξ + ξ roblem P f := + δ + α L γ Fx, δdx su t γ Fx,tdx, admits at least one non-trivial weak solution u X such that u x x + + u x x x + αx u x x dx > + γ. Proof. Take the real Banach sace X as defined in Section, ut I = Φ Ψ, where Φ Ψ are given as in.4.5, resectively. Our goal is to aly Theorem. to function I. The functionals Φ Ψ satisfy all assumtions requested in Theorem.. Moreover, for > 0, the functional I is coercive. Indeed, fix 0 < ε < + meas. From 3.9, there is a function ε L such that Fx,t εt + ε x, for every x t R. Taking Proosition.4 into account, it follows that, for each u X with u, Φu Ψu + u ε ux dx ε L + ε meas u α ε L. It follows that lim Φu Ψu = +, u +

14 4 S. HEIDARKHANI, A. SALARI which means the functional I = Φ Ψ is coercive. Put r = + γ w δ t = δ. Arguing as in the roof of Theorem 3., we obtain that r + su t γ Fx,tdx γ δ + α L Fx, δdx. Hence, from our assumtion it follows that r > 0. Therefore, from Theorem. for each >, the functional I admits at least one local minimum u such that u x x + + u x x x + αx u x x dx > γ +. The conclusion is achieved. Remark 3.7. If f is non-negative, then the strong maximum rincile ensures that the weak solutions of roblem P f are non-negative see [30, Lemma.]. Now, we oint out some results in which the function f has searated variables. To be recise, consider the following roblem div + u x u x u + αx u x u + u x P h,θ = θxhu, in, u ν = 0, on, where θ : R is a non-negative non-zero function such that θ L h : R R is a non-negative continuous function. Put t Ht = hξ dξ for all t R. 0 The following existence results are consequences of Theorems , resectively, by setting f x, t = θxht for every x,t R. Theorem 3.8. Assume that h0 0 there exist a non-negative constant γ two ositive constants γ δ, with γ < δ < α + + γ + α L such that Hγ Hδ γ δ + α L < Hγ Hδ. γ δ + α L Furthermore, suose that there exist constants ν > + R > 0 such that for all ξ R the assumtion 0 < νhξ ξ hξ. 3.0

15 x-laplacian-like PROBLEMS WITH NEUMANN CONDITION ORIGINATED FROM A CAPILLARY PHENOMENA 5 Then, for each,, where roblem P h,θ + = + γ δ + α L θ L Hγ Hδ = + γ m δ + α L θ L Hγ Hδ, admits at least two ositive weak solutions u u in X such that γ u x x + + u x x x + αx u x x dx + γ. Theorem 3.9. Assume that h0 0 there exist two ositive constants δ γ, with δ < + + γ +, α L such that Hγ γ < α L Furthermore, suose that the assumtion 3.0 holds. Then, for every + δ + α L θ L Hδ, + γ, θ L Hγ roblem P h,θ 0 < Hδ δ admits at least two ositive weak solutions u u in X such that u x x + + u x x x + αx u x x dx + γ. Theorem 3.0. Assume that there exist two ositive constants γ δ with δ < + γ + +, 3. α L such that Then, for each >, where roblem P h,θ H δ < H γ 3.3 lim ξ + hξ ξ 0. := + δ + α L γ θ L H δ H γ, admits at least one ositive weak solution u X such that u x x + + u x x x + αx u x x dx > + γ.

16 6 S. HEIDARKHANI, A. SALARI Now we illustrate Theorem 3.0 by resenting the following examle. Examle 3.. Let N =, = {x,x R ; x + x < } R, x,x = e +x +x for all x,x, αx,x = for all x +x,x, θx,x = x +x + x for all x,x t, t, ht =, t >. Thus, = e, + = e, meas = π, D =, thus, +π e π ln π α L = r r drdθ = π ln α =, { } k = k e e e π max e, = π ln π ln e πe e π ln θ L = π 0 0 r 3 drdθ = π. Moreover, h is a non-negative continuous function, h0 0 By the exression of h, we have lim ξ + hξ ξ = 0. t 3 t < Ht = t t3 3, t, t + 3, t >. Thus, by choosing δ = e γ = +π we clearly observe that are satisfied. Indeed, δ = e < e e + γ + + α L H δ = e 8 3e 3 < 56 3 = H γ. Therefore, by Theorem 3.0 for every > 6e5 e e e +π e, the roblem πe e e e 3 3π 8 div + ux,x e+ x ux,x e+ x ux,x + ux,x e+ x = x hux,x, in, u v = 0 on, has at least one ositive weak solution u { u L e+ x : u L e+ x }

17 x-laplacian-like PROBLEMS WITH NEUMANN CONDITION ORIGINATED FROM A CAPILLARY PHENOMENA 7 such that where x = x + x. x < e + x + u x,x e+ x + x u x,x e+ x + + u x,x e+ x π ln dx dx > e A further consequence of Theorem 3. is the following existence result. Theorem 3.. Assume that h0 0 hξ lim = ξ 0 + ξ Furthermore, suose that the assumtion 3.0 holds. Then, for every 0, γ, where + γ := θ L su γ>0 roblem P h,θ admits at least two ositive weak solutions in X. Proof. Fix ]0, γ [. Then there is γ > 0 such that γ Hγ, + < θ L From 3.4 there exists a ositive constant δ with δ < + su γ>0 γ Hγ. + γ + α L such that < θ Hδ L. δ + α L Therefore, we can use Theorem 3.4 to comlete the roof. Remark 3.3. Theorem. immediately follows from Theorem 3.. Now we illustrate Theorem 3. by resenting the following examle. Examle 3.4. Let N = 3, = {x,x,x 3 R ; x +x +x 3 < } R3, x,x,x 3 = 4+x +x +x 3 for all x,x,x 3, αx,x,x 3 = x +x +x 3 for all x,x,x 3, θx,x = for all x,x,x 3 ht = { 0 +t 0, t, t, t <. Direct calculations give = 4, + = 5, meas = 4π 3, D =, π π α L = r 4 sinφdrdφdθ = 4π { k = k 4 4 } 4 5 max 4π, 5 4 4π α =, = 0 4 8π

18 8 S. HEIDARKHANI, A. SALARI thus, 0 4 8π + 4π 3. Also θ L = 4π 3, lim γ + γ Hγ = lim γ γ 4 γ γ3 3 = + hξ lim = +. ξ 0 + ξ Moreover, since ξ hξ lim ξ + Hξ = lim 0t +t = > 5 = + ξ + 0t + t by choosing ν = > 5 = + we can choose R > such that the assumtion 3.0 is fulfilled. Hence, by alying Theorem 3., for every > 0 the roblem div + ux,x,x 3 4+ x u x + ux,x,x 3 + ux,x,x 3 8+ x = hux,x,x 3, in, u v = 0 where x = x + x + x 3, has at least two ositive weak solutions. on, Remark 3.5. The non-triviality of the second weak solution ensured by Theorem 3. can be achieved also in the case h0 = 0 requiring the extra condition at zero, that is, there are a non-emty oen set D a set B D of ositive Lebesgue measure such that Indeed, arguing as in [5, 3] let 0 < <, where Then, there exists γ > 0 such that lim su ξ 0 + essinf t B θt.hξ ξ = lim inf ξ 0 + essinf t D θt.hξ ξ > = θ L su γ>0 γ Hγ. θ L γ + < H γ. Let Φ Ψ be as given in.4.5, resectively. Due to Theorem 3., for every 0, there exists a critical oint of I = Φ Ψ such that u Φ,r, where r = γ +. In articular, u is a global minimum of the restriction of I to Φ,r. We will rove that the function u cannot be trivial. Let us show that Ψu lim su = u 0 + Φu Owing to the assumtions , we can consider a sequence {ξ n } R + converging to zero two constants σ,κ with 0 < σ < such that essinf t B θt.hξ n lim = + n + ξ n

19 x-laplacian-like PROBLEMS WITH NEUMANN CONDITION ORIGINATED FROM A CAPILLARY PHENOMENA 9 ess inf θt.hξ κ ξ t D for every ξ [0,σ]. We consider a set G B of ositive measure a function v X such that k vt [0,] for every t, k vt = for every t G, k 3 vt = 0 for every x \ D. Hence, fix N > 0 consider a real ositive number η with Then, there is n 0 N such that ξ n < σ N < + η measg + + κ D\G vt dt. v α ess inf t B θt.hξ n η ξ n for every n > n 0. Now, for every n > n 0, by considering the roerties of the function v that is 0 ξ n vt < σ for n large enough, one has Ψξ n v Φξ n v G θt.hξ ndt + D\G θt.hξ nvtdt Φξ n v > + η measg + + κ D\G vt dt > N. v α Since N could be arbitrarily large, we get Ψξ n v lim n Φξ n v = +, from which 3.7 clearly follows. So, there exists a sequence {ω n } X strongly converging to zero such that, for n large enough, ω n Φ,r I ω n = Φω n Ψω n < 0. Since u is a global minimum of the restriction of I to Φ,r, we obtain so that u is not trivial. I u < 0, 3.8 Next, as a consequence of Theorems , the following theorem of the existence of three solutions is obtained. Theorem 3.6. Suose that h0 0 Hξ lim su ξ + ξ Moreover, assume that there exist four ositive constants γ, δ, γ δ with δ < + + γ + < + γ α L + + < δ α L

20 0 S. HEIDARKHANI, A. SALARI such that hold, are satisfied. Then, for each Λ = max Hγ γ < H δ H γ γ δ + α L {, δ + α L }, + γ, θ L Hδ θ L Hγ 3.0 roblem P h,θ admits at least three ositive weak solutions u, ū u 3 such that u x x + + u x x x + αx u x x dx γ + u x x + + u x x x + αx u x x dx > γ +. Proof. First, in view of 3.0, we have Λ /0. Next, fix Λ. Emloying Theorem 3.9, there is a ositive weak solution u such that u x x + + u x x x + αx u x x dx γ +, which is a local minimum for the associated functional I, while Theorem 3.0 ensures a ositive weak solution u such that u x x + + u x x x + αx u x x dx > γ +, which is a local minimum for I. Arguing as in the roof of Theorem 3.6, from the condition 3.9 we observe that the functional I is coercive. Then it satisfies the P.S condition. Hence, the conclusion follows from the mountain ass theorem as given by Pucci Serrin see [3]. The following existence result is a consequence of Theorem 3.6. Theorem 3.7. Assume that h0 0, lim su ξ 0 + Hξ ξ = + 3. Furthermore, suose that there exist two ositive constants γ δ with + Hξ lim su = ξ + ξ γ + + < δ 3.3 α L such that H γ γ < + α L H δ δ

21 x-laplacian-like PROBLEMS WITH NEUMANN CONDITION ORIGINATED FROM A CAPILLARY PHENOMENA Then, for each α L θ L roblem P h,θ admits at least three ositive weak solutions. δ + H δ, + γ, α L θ L H γ Proof. We easily observe from 3. that condition 3.9 is satisfied. Moreover, by choosing δ small enough γ = γ, one can drive condition 3. from 3. as well as the conditions from 3.4. Hence, the conclusion follows from Theorem 3.6. Remark 3.8. Theorem. immediately follows from Theorem 3.7. Finally, we resent one alication of Theorem 3.7 as follows. Examle 3.9. Let N >, > N R N be an oen bounded domain with smooth boundary such that meas N N. diam Also let + > + meas π 4 e, ht = + t, for all t R +t θ L =. Now, by choosing α in such a way that α L = α = we have < h is a non-negative continuous function, h0 0, Ht = t arctant for allt R, Hξ ξ arctanξ lim = lim = + ξ 0 + ξ ξ 0 + ξ Hξ ξ arctanξ lim = lim = 0. ξ + ξ ξ + ξ Moreover, by choosing γ = e δ =, we see that are satisfied. Hence, by alying Theorem 3.7, for every + meas + e meas π 4,, meas + meas the roblem div + u x u x u + αx u x u + u x = θx + u x, in, +u x u v = 0, on has at least three ositive weak solutions. + meas,

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