Three solutions for elliptic Dirichlet boundary value problem with singular weight.

Transcription

1 arxiv:67.235v [math.ap] 5 Jul 26 Three solutions for ellitic Dirichlet boundary value roblem with singular weight. Kowalski Piotr, Piwnik Joanna December 3, 27

2 Abstract In this aer we rove the existence of two non-trivial weak solutions of Dirichlet boundary value roblem for -Lalacian roblem with a singular art and two disturbances satisfying the roer assumtions. The abstract existence result we use is the famous Ricceri theorem.

3 Introduction Singular ellitic roblems have been intensively and widely studied in recent years. Among others, -Lalacian oerator aears to be mostly investigated ellitic oerator. An imortant subclass of such roblems are roblems involving singular nonlinearities [2]. We recall that denotes the -Lalacian oerator, namely u = div( u 2 u). In this aer we are esecially interested in eigenvalue roblem, derived from the one well exlained by Lindquist in [3]. Some roblems require to find the smallest ositive scalar λ R for which the equation = λ a(x) 2 for a.e. x, () with R n, n 2,beingboundedwithsufficientlysmoothboundary, hasanontrivialsolution in W, (), see [7, 4]. Such λ shall be referred as the rincile eigenvalue. The starting oint is usually the simlest case a(x), but also the weighted version of this equation finds a lot of alications. By the solution we usually understand the weak one. Some researchers are interested in roving the existence of three solutions for small µ,λ > µ > with a semilinear term. For examle or = µa(x) 2 +f(x,) for a.e. x, (2) +µa(x) 2 = f(x,) for a.e. x, (3) where a L () with essinf a(x) >, e.q, [4] or [8]. We should note that the roblem (3) is x essentially different from the roblem in (), since both λ > and µ >. Yet the most interesting cases contain the weight function a: R n R unbounded and having a singularity. Stationary roblem involving such nonlinearities describes some alied economical models and several hysical henomena, for instance conduction in electrically conducting materials. The roblem (2) can be found for examle in [7, 9, ]. In articular, many results on the existence and multilicity of solutions for nonlinear roblems involving the -Lalacian oerator have been obtained by adoting variational methods. We also recall that, several authors have treated the case n, by using comletely different techniques [3, ]. In this aer we consider the following roblem defined on, where R n is bounded, has Lischitz boundary, and 2 < n. By W, () we understand the closure of C () in norm of Sobolev sace W, (). Problem. Find u W, () such that: +µ 2 x = λf()+γg(), for a.e. x, u. The above resented roblem is understand as equivalent to Problem 2. Problem 2. Find u W, () such that for all v W, () 2 v(x)+µ 2 v(x) x = λ f()v(x)dx+γ dx g()v(x) dx.

4 We assume that f: R R satisfies the following conditions f(t) (f) lim =. t t f(t) (f2) lim =. t + t t (f3) suf(t) >, where F(t) = f(s)ds. t R For function g: R R we assume only that there exists c g and < q < n n g(t) c g (+ t q ) for all t R. such that The main tool shall be the Ricceri three critical oint theorem [5], and from it shall follow that there are intervals for λ > and γ > in which the Problem 2 has at least three weak solutions. This aer is organized as follows. In section 2 we recall some results from the theory of uniformly monotone oerators, the roerties of uniformly convex Banach saces and some inequalities related to embedding results. In section 3 we introduce the corresonding energy functional for Problem 2 and we define some auxiliary functionals. We also study roerties of those functionals, such as sequential weak lower semicontinuity, the comactness of derivative and existence of continuous inverse. In section 4 we rove the necessity conditions for a certain three critical oints theorem by Ricceri and we also formulate the main result of this aer. The last section contains an examle. This work is mainly motivated by the study of ellitic roblems with singular and sublinear otentials done in [2]. 2 Preliminaries We recall some of the basic definitions and theorems we require from functional analysis. We start by recalling the definition of comact oerator. Definition 2. ([6, def 4.6] Comact oerator). Assume X,Y are Banach saces, U is oen unit ball contained in X. Oerator A: X Y is called comact if closure of A(U) is comact in Y. We shall rove that a functional satisfies a strong monotonic condition, namely it is an uniformly monotone oerator. Definition 2.2 ([7, def 25.2] Uniformly monotone oerator). Let X be a Banach sace and A: X X. We say that A is uniformly monotone oerator, when a( u v ) u v Au Av;u v for all u,v X, where the continuous function a: R + R +, is strictly monotone increasing with a() = and a(t) = +. lim t + The Ricceri abstract existence result require that derivative of a given oerator admits a continuous inverse. In order to obtain such roerty we shall use the following theorem. 2

5 Theorem 2.3 ([7, th 26.A] Browder-Minty theorem). Let A: X X be a monotone, coercive, and hemicontinuous oerator on the real, searable, reflexive Banach sace X. If A is strictly monotone, then the inverse oerator A : X X exists and is strictly monotone, demicontinuous, and bounded. If A is uniformly monotone, then A is continuous. We denote that if A is uniformly monotone, then A is strictly monotone, coercive and hemicontinuous. The following well known analytic inequality shall be helful. Lemma 2.4 ( [5, age 3]). Let 2. For each x,y R n there occurs ( x 2 x y 2 y) (x y) 2 (2 ) x y. Theorem 2.5 ([, th 6.3] Poincarè inequality). If domain R n is bounded, then there exists a constant C deendent only on and, such that for all u C () u L () C u L (;R n ). SincePoincarèinequalityholds, inthisaer,weshallusew, () given with a norm equivalent to Sobolev s one, i.e. x W, () = x L (). Theorem 2.6 ([, th 6.3] Rellich Kondrachov theorem). Let R n be an oen, bounded Lischitz domain, and let < n. Set = n. Then the Sobolev sace W, n () is continuously embedded in L () and comactly embedded in L q (), where q <. and W, () L (), W, () L q () for q <. By above theorem we immediately obtain the following: W, () L (). The theory of Ricceri is easily alicable to roblems connected to a norm that is an uniformly convex one. Thus we recall some classical results of Clarkson [6]. Definition 2.7 ([6, def ] Uniformly convex sace). A Banach sace B will be said to be uniformly convex if to each ε, < ε 2, there corresonds a δ(ε) > such that the conditions x = y =, x y ε imly x+y 2 δ(ε). In this aer we shall use two uniformly convex saces, W, () and L ( x ; ), simultaneously. Thus the following result concerning the roduct of uniformly convex saces is required. 3

6 Definition 2.8 ([6] Uniformly convex roduct). Let N(a,a 2,,a k ) be a non-negative continuous function of the non-negative variables a i. We say that N is. homogeneous, if for c 2. strictly convex, if N(ca,ca 2,,ca k ) = cn(a,a 2,,a k ); N(a +b,a 2 +b 2,,a k +b k ) < N(a,a 2,,a k )+N(b,b 2,,b k ), unless a i = cb i (i =,2,,k). In the latter case we have equality by condition. 3. strictly increasing, if it is strictly increasing in each variable searately. A familiar examle of a function N satisfying these conditions is N(a,...,a k ) = ( k i= a i ) ( > ); here condition 2. becomes the inequality of Minkowski. Suose now that a finite number of Banach saces B,B 2,,B k are given, and that B is their roduct. We shall call B a uniformly convex roduct of B i if the norm of an element x = (x,x 2,,x k ) of B is defined by x = N( x, x 2,, x k ), where N is a continuous non-negative function satisfying the conditions -3. Theorem 2.9 ([6] Clarkson theorem). The uniformly convex roduct of a finite number of uniformly convex Banach saces is a uniformly convex Banach sace. We shall also need a relation in between Sobolev sace W, () and weighted L ( x, ) sace. Theorem 2. ([2] Hardy inequality). Assume < < n and u W, (), then u L ( x ; ) and x dx C n, dx, ( with C n, =. n ) Finally we shall also require the following toological lemma: Lemma 2.. Let d,d 2 be metrics on metric sace X. Then d(x,y) = d (x,y) +d 2 (x,y) is also a metric on X. Proof. We shall rove metric axioms are satisfied. M) Let x = y. Then d(x,y) = d(x,x) = d (x,x) +d 2 (x,x) = + = =. Conversely d(x;y) = then d (x;y) =. Hence x = y. 4

7 M2) d(x,y) = d (x,y) +d 2 (x,y) = d (y,x) +d 2 (y,x) = d(y,x). Hence d is symmetric. M3) Let x,y,z X. Then Thus d is a metric. d(x,z) = d (x,z) +d 2 (x,z) (d (x,y)+d (y,z)) +(d 2 (x,y)+d 2 (y,z)) = (d (x,y),d 2 (x,y))+(d (y,z),d 2 (y,z)) L (R 2 ) d (x,y),d 2 (x,y) L (R 2 ) + d (y,z),d 2 (y,z) L (R 2 ) = d (x,y) +d 2 (x,y) + d (y,z) +d 2 (y,z) = d(x,y)+d(y,z). 3 Variational Framework In this aer we rove the existence of weak solutions of roblem corresonding to Problem, namely: Problem 2. Find u W, () such that for all v W, () 2 v(x)+µ 2 v(x) x dx = λ f()v(x)dx+γ g()v(x) dx, where R n is bounded, has Lischitz boundary, and 2 < n. We assume that µ (, + ). We also require certain conditions on continuous functions f,g: R R, namely (f) lim t f(t) t =. f(t) (f2) lim =. t + t t (f3) suf(t) >, where F(t) = f(s)ds. t R (g) There exists c g and < q < n n such that g(t) c g (+ t q ) for all t R. 5

8 We define the following functionals, Φ,J,J 2,E: W, () R given by the formulas Φ(u) = +µ x dx, J (u) = F()dx, J 2 (u) = G()dx, where G(t) = t g(s)ds. E(u) = Φ(u) λj (u) γj 2 (u), We start by roving that any critical oint of E is a weak solution of Problem 2. Lemma 3.. Assume, that conditions (f), (f2) and (g) hold. Then functionals Φ, J and J 2 are well defined and they have the following Gâteaux derivatives: Φ (u);v = J (u);v = 2 v(x)+µ 2 v(x) x f()v(x)dx, dx, J 2 (u);v = g()v(x) dx. Proof. All of the functional are well-defined. We calculate the Gâteaux derivative: Φ (u);v =lim λ = = (+λv(x)) +µ +λv(x) x λ µ x +λ v(x)) lim + µ +λv(x) λ λ x lim dx λ λ 2 v(x)+ µ x 2 v(x)dx. The following equality holds due to Lebesgue s dominated convergence theorem. F(+λv(x)) F()dx J F(+λv(x)) F()dx (u);v =lim = lim λ λ λ λ F(+λv(x)) F() = lim dx = f()v(x)dx. v(x)λ λv(x) dx 6

9 Thedominatedconvergencetheoremisalicablesince, wecanestimatethefunction F(+λv(x)) F() λ from above in a following way +λv(x) f λ (x) = F(+λv(x)) F() f(s)ds λ = λ +λv(x) +λv(x) f(s) ds cf s ds = c f λ λ λ c +λv(x) f su t ds c f λ t [,+λv(x)] λ c f λ max{, +v(x) } +λv(x) +λv(x) ds f(s)ds +λv(x) su t [,+v(x)] s ds t ds = c f λ max{, +v(x) }λv(x) = c f max{, +v(x) }v(x). It follows from Hölder inequality that this function is integrable. Calculation for the functional J 2 are very similar: G(+λv(x)) G()dx J 2(u);v G(+λv(x)) G()dx =lim = lim λ λ λ λ G(+λv(x)) G() = lim dx = g()v(x) dx, v(x)λ λv(x) 7

10 G(+λv(x)) G() λ since the following estimate holds: from above: +λv(x) g λ (x) = G(+λv(x)) G() g(s)ds g(s) ds λ = λ +λv(x) +λv(x) ( g(s) ds cg + s q ) ds = c g λ λ λ c +λv(x) g + su t q ds λ t [,+λv(x)] c +λv(x) g λ λv(x) + su t q ds t [,+v(x)] c g ( v(x) +max{ q, +v(x) q }v(x) ). +λv(x) + s q ds We can show, using Theorem 2.6 and Hölder inequality, that the obtained dominant is integrable. Which concludes the roof. Remark 3.2. From the roof of Lemma 3., we can conclude that Φ is bounded on bounded subsets of W, (). We can note that using the functional given above we can rewrite Problem 2 also as: Find u W, () such that E (u);v = Φ (u);v λ J (u);v γ J 2 (u);v =, for all v W, (). Thus, any critical oint to E is a solution of Problem 2. At first we shall concentrate on roerties of functional Φ. We shall rove that it has many similarities to a norm. We define the following functional for every u W, () u u = ( Φ(u)) = +µ x dx. Lemma 3.3. Function W, () u u is a norm on the sace W, (). Proof. We observe that u = ( Φ(u)). We will show u satisfies norm axioms. N) Let u =. Then ( Φ(u)) =, thus Φ(u)=. We note exlicitly +µ x dx =. Since -ower of modulus is non-negative, then and almost everywhere in, hence u = in W, (). The oosite imlication holds instantly. 8

11 N2) Let α R. Then: αu = ( Φ(αu)) = (α) +µ α x dx = α () +µ x dx = α ( Φ(u)) = α u. N3) Define two metrics d,d 2 : W, () W, () R as d (x,y) = x y W, (), d 2 (x,y) = µ x y L ( x ;). Then it follows from by Lemma 2. that d(x,y) = d (x,y) +d 2 (x,y) is also a metric. Thus, for x = u+w,z =,y = w we have: Hence u u is a norm. u+w =( Φ(u+w)) = d (u+w,) +d 2 (u+w,) =d(u+w,) d(u+w,w)+d(w,) = u+w w +µ u+w W, () w L ( x ;) + w W, () +µ w L ( x ;) =( Φ(u)) +( Φ(w)) = u + w. Lemma 3.4. Functional Φ is sequentially weakly lower semicontinuous. Proof. From its definition it follows that Φ(u) = ( ) u W, () +µ u L ( x,). Since W, () and L ( x,) are sequentially weakly lower semicontinuous, then Φ is sequentially weakly lower semicontinuous. Lemma 3.5. Oerator Φ is coercive. Proof. Φ(u) = Thus, Φ is coercive oerator. +µ x dx = u u. W, () 9

12 Now we focus on roerties of J and J 2. Lemma 3.6. Assume (f)-(f2). Then functional J has a comact derivative. Proof. We slit the roof into two arts. At first we shall rove that the image of a bounded set through J is bounded, afterwards we shall rove the existence of a convergent subsequence within this image. We take a bounded sequence (u n ) n N W, (), which means there exists M > such that for all n N u n W, () M. We show that the sequence J (u n) is also bounded, what is equivalent to su v W, ()= J (u n );v < +. By the conditions (f) and (f2) and continuity of f we obtain f(t) c f t, t R. Thus f(u n (x))v(x)dx f(u n (x)) v(x) dx c f u n (x) v(x) dx. From Hölder inequality it follows that: u n (x) v(x) dx ( u n (x) ) dx v(x) dx = v L () u n L () c v W, c M, () u n W, () where c is the constant from Lemma 2.5. It follows that su f(u n (x))v(x)dx v, W c f c M, ()= and this means that for all n N J (u n) W, () c f c M < +. Thus, the image of J is bounded. We know that in a reflexive Banach sace each bounded sequence has a weakly convergent subsequence. So there exists d W, () such that J (u n) d. Suose, that J (u n) d W, () > δ >. Then k (n k) su J (u n ) d;v > δ >. v W, = ()

13 Then we can construct the sequence (v n ) n= W, () such that v n W, () = and k (n k) J (u n) d;v n > δ. Since(v n ) n= isbounded, itadmitsaweaklyconvergentsubsequence. BytheRellich-Kondrachov theorem(theorem2.6)thissequenceadmitsasubsequence convergentstronglyinl (). Without any loss at generality we can assume that v n is weakly convergent in W, () and strongly in L (). The following holds in an obvious way J (u n ) d;v n = J (u n ) d;v + J (u n );v n v d;v n v. The first and the third terms converge to zero. Finally J (u n);v n v = f(u n ) v n v dx c f u n v n v dx c f u n dx v n v dx = c f u n L () v n v L (). Thus, we have a contradiction. The roof for this fact follows the stes of roof of Lemma 3.6 almost identically. We rove thesimilar statement forj 2, aswe alythefollowing comact embedding: W, () L q () instead of W, () L (). Lemma 3.7. Assume (g). Then functional J 2 has a comact derivative.. We will use Theorem 2.3 and Lemma 2.4 to rove the following lemma: Lemma 3.8. The derivative of oerator Φ admits a continuous inverse. Namely for Φ (u);v = 2 v(x)+µ 2 v(x) dx x there exists the continuous inverse (Φ ) : W, () W, (). Proof. We rove that Φ is uniformly monotone. Φ (u ) Φ (u 2 );u u 2 = u (x) 2 u (x) (u (x) u 2 (x))+µ u (x) 2 u (x)(u (x) u 2 (x)) x u 2 (x) 2 u 2 (x) (u (x) u 2 (x)) µ u 2(x) 2 (u (x) u 2 (x)) dx x = ( u (x) 2 u (x) u 2 (x) 2 u 2 (x)) ( u (x) u 2 (x)) +µ ( u (x) 2 u (x) u 2 (x) 2 u 2 (x)) (u (x) u 2 (x)) x dx.

14 By Lemma 2.4 (alied twice) Φ (u ) Φ (u 2 );u u 2 ( a u (x) u 2 (x) + µ ) x u (x) u 2 (x) dx = a ( u u 2 +µ u W, () u 2 L ( x,) a u u 2 u W, () u 2 W, (). One can easily check that this oerator is hemicontinuous. Since it is uniformly monotone oerator it is a monotone and coercive oerator. Thus, by Theorem 2.3 the assertion holds. 4 Existence result In our roblem the role of X sace is layed by W, (). It is searable and reflexive Banach sace. Lemma 4.. A sace ( W, (), ) is uniformly convex Banach sace. ) Proof. We recall that u u = ( Φ(u)) = +µ x dx in a norm on W, (). We will rove that it is a uniformly convex norm. Since we will use two different norms on W, (), it is essential to add that by u W, () = dx we understand the usual norm. In order to rove this, we will use Clarkson s concet of uniformly convex roduct. By Definition 2.9 the following functional W, () L ( x, ) (u,v) (u,v) R +, given by the formula (u,v) = u W, () +µ v L( x,), is uniformly convex roduct; here µ is the same constant as in Φ. By the Clarkson theorem (Theorem 2.9), the sace ( W, () L ( x, ) ), is a uniformly convex Banach sace. We recall that W, () L ( x, ). Thus, we can easily observe that (u,u) = u for u W, (). We will rove now that u u is an uniformly convex norm. Let 2 > ε >, u = v = and u v ε.whence (u,u) = and (v,v) = aswellas (u,u) (v,v) ε. Thus, bytheuniformconvexity of ( W, () L ( x, ) ) ;, wegetthatthereexists δ ε (,) such that ( δ ε (u,u)+(v,v) 2 = u+v, u+v ) = u+v 2 2 2, which roves that ( W, (), ) is uniformly convex Banach sace. 2

15 Definition 4.2. [5] By W X we denote the class of all functionals Φ: X R ossessing the following roerty: if {u n } is a sequence in X converging weakly to u X and lim inf n Φ(u n) Φ(u), then {u n } has a subsequence converging strongly to u. Lemma 4.3. Φ belongs to W W, (). Proof. Assume (u n ) n= W, (), u n u and liminf Φ(u n) Φ(u). From sequentially n weakly lower semicontinuity of functional Φ it follows lim inf Φ(u n) Φ(u). Thus there exists n a subsequence u nk W, () of u n such that lim Φ(u nk ) = Φ(u). Exlicitly: n lim u nk (x) +µ u n k n x dx = +µ x dx = u ; lim n u n k = u ; lim u n n k = u. We roved that W, () is uniformly convex Banach sace, therefore there exists a subsequence (u nk ) of (u n ) such that u nk u. J Lemma 4.4. Assume, that (f)-(f3) holds. Then su (u) >. Φ(u) Φ(u)> Proof. We show equivalently there exists u δ such that J (u δ ) >. By (f3) there exists s Φ(u δ ) R such that F(s ) >. Let δ (,). Then there exists x \ such that there exist R, r such that R > r > with B(x,R) \. We can choose u δ W, () such that. su u δ B(x,R), 2. u δ B(x,r+δ(R r)) s, 3. u δ s. Then J (u δ ) = F(u δ (x))dx F(s ) Vol(N)[r +δ(r r)] N B(x,R) max F(t) ( Vol(N)[R N (r +δ(r r))] N), t [ s,s ] where Vol(N) stands for the measure of N-dimensional unit ball/ As δ, then J (u δ ) C >. Let δ be such that J (u δ) >. Observe, that u δ. So Φ(u δ) > and J (u δ ) >. J Lemma 4.5. Assume, that (f)-(f3) holds. Then lim su (u). Φ(u) u, W () 3

16 ( ) Proof. Let ε > and η, 2. We will use conditions (f) and (f2). For ε n f = (C ) ε there exists by (f) and (f2) δ ε > such that Then for all t R J (u) = f(t) (C ) ε t for t [,δ ε ] [δ ε,+ ). f(t) M ε t +η for t (δ ε,δ ε ). f(t)dt dx f(t) (C ) ε t +M ε t +η. (C ) ε +M ε = (C ) ε dx+m ε +η = (C ) ε u L () +M ε From Theorem 2.6, for η u W, (, f(t) dt dx +η +η dx +η u +η. L +η () 2 n (), so by dividing both sides we obtain J (u) Φ(u) Since u W, J lim su (u). Φ(u) u, W () ) +η dx (C ) ε u L () +M ε +η u +η L +η () u W, ε+m ε u η W, (). () thus limsu u W, () +M (C ) ε t ε t +η dt dx it follows W, () L +η (). We know that Φ(u) () J (u) Φ(u) ε u +M W ε u +η, () u W, () W, () ε. Since ε > was chosen arbitrarily we obtain J Lemma 4.6. Assume, that (f)-(f3) holds. Then lim su (u). Φ(u) u, W () Proof. Let ε > and η (,min{ 2 n ε f = (C ) ε there exists by (f) and (f2) δ ε and there exists η, }). We will use conditions (f) and (f2). For ( ), such that 2 n Then f(t) (C ) ε t for t [,δ ε ] [δ ε,+ ). f(t) M ε t η for t (δ ε,δ ε ). f(t) (C ) ε t +M ε t η,t R. 4

17 J (u) = f(t)dt dx (C ) ε +M ε = (C ) ε dx+m ε η = (C ) ε u L () +M ε f(t) dt dx η η dx η u η. L η () η dx We know that Φ(u) u W, (), so by dividing both sides we obtain J (u) Φ(u) Since u W, (C ) ε u L () +M ε η u η L η () ε+m ε u η W, (). () thus limsu u W, () u W, () J (u) Φ(u) ε. Since ε > chosen arbitrarily thus limsu u W, () J (u) Φ(u). +M (C ) ε t ε t η dt dx ε u +M W ε u η, () u W, () W, () We will use the following Ricceri theorem to show that the Problem 2 has a solution. Theorem 4.7 (Ricceri[5]). Let X be a searable and reflexive real Banach sace; Φ: X R a coercive, s.w.l.s.c. C functional, belonging to W X, bounded on each bounded subset of X and whose derivative admits a continuous inverse on X ; J : X R a C functional with comact derivative. Assume that Φ has a strict local minimum x with Φ(x ) = J (x ) =. Finally, setting α = max β = {,limsu x + J (x) su x Φ ((,+ )) Φ(x), J (x) Φ(x),limsu x x J (x) Φ(x) assume α < β. Then for each comact interval [a,b] (, ), with the conventions β α ( = +, = ), there exists r > with the following roerty: for every λ [a,b] and every + C functional J 2 : X R with comact derivative, there exists δ > such that, for each µ [,δ], the equation Φ (x) = λj (x)+µj 2 (x), has at least three solutions whose norm are less than r. Theorem 4.8 (The existence of three weak solutions of Problem 2). Assume that conditions (f)-(f3) and (g) hold. Then there exists β > such that for each comact interval [a,b] (β,+ ), there exists r > with the following roerty: for every λ [a,b], there exists δ > such that, for each γ [,δ], the Problem 2 has at least three solutions whose norm are less than r. 5 }

18 Proof. W, () is obviously searable and reflexive. By Theorem 4.7 and Lemmas 3., 3.4, 3.5, 3.6, 3.7, 3.8, 4., 4.4, 4.5, 4.6 and since Φ() = J () = and W(, () ) is a strict minimum of Φ. By abstract existence result of Ricceri there exists [a,b], such that for all J 2 C, J 2 has a comact derivative and Problem 2 has 3 solutions. So two of them must be non trivial. 5 Examle Let R n be bounded set with Lischitz boundary such that. Furthermore let 2 < n and µ (,+ ). We consider the following roblem: Problem. Find u W, () such that for all v W, () 2 v(x)+µ 2 v(x) x dx = λ with functions f,g: R R given by f(t) = and g(t) = f()v(x)dx+γ ( π 2r g()v(x) dx ) sin(rt) for t π 2r ( +( π 2r) 2) t +t 2 for t > π 2r + t q for t z (+z 2 )(+z q ) +t 2 for t > z, where r,z > are fixed constants and < q < n. n Both functions f and g are obviously continuous on R. We will show that function f satisfies conditions (f)-(f3). Indeed, ( π 2r) sin(rt) lim t f(t) = lim t t lim t t f(t) = lim t t = lim t ( π 2 ) sin(rt) rt Hence condition (f) holds. Moreover, also condition (f2) is verified, because ( + ( ) ) π 2 t 2r Furthermore suf(t) >, because for t = π 2r t R ( π F = 2r) π 2r f(s)ds = π 2r ( π 2r (+t 2 ) t =. we have ) sin(rs) ( π ds = 2r ) π 2r χ, τ sin(rt) =. sin(rs) ds ( π 2r π 2r ) π 4r ( 2 2 ) ( ds = π ) 2 >. 2 4r 6

19 Also condition (g) is satisfied by function g with constant c g =. For t r we obviously have On the other hand for t r g(t) = + t q. g(t) = (+z2 )(+z q ) +t 2 ( +z q ) ( + t q ). From theorem 4.8, there exists β > such that for each comact interval [a,b] (β,+ ), there exists w > with the following roerty: for every λ [a,b], there exists δ > such that, for each γ [,δ], the Problem 5 has at least three solutions whose norm are less than w. Acknowledgement The authors would like to thank rofessor Giovanni Molica Bisci from Universitá degli Studi Mediterranea in Italy for introducing authors to this tye of roblems and to thank rofessor Aleksander Ćwiszewski from Nicolaus Coernicus University in Poland, for his remarks that allowed to exclude unnecessary assumtions and include the omitted ones. References [] R. A. Adams and J. J. F. Fournier. Sobolev Saces. Elsevier, 23. [2] J. P. García Azorero and I. Peral Alonso. Hardy inequalities and some critical ellitic and arabolic roblems. Journal of Differential Equations, (44):44 476, 998. [3] P. A. Binding, P. Drábek, and Y. X. Huang. Existence of multile solutions of critical quasilinear ellitic Neumann roblems. Nonlinear Analysis, 42:63 629, 2. [4] G. Bonanno and P. Candito. Three solutions to a Neumann roblem for ellitic equations involving the -Lalacian. Archiv der Mathematik, (8): , 23. [5] J. H. Chabrowski. Variational methods for otential oerator equations: with alications to nonlinear ellitic equations, volume 24. Walter de Gruyter, 997. [6] J. A. Clarkson. Uniformly convex saces. Transactions of the American Mathematical Society, (3):396 44, 936. [7] M. Cuesta and H. R. Quoirin. A weighted eigenvalue roblem for the -Lalacian lus a otential. Nonlinear Differential Equations and Alications NoDEA, 6(4):469 49, 29. [8] G. D Aguì and G. Molica Bisci. Existence results for an ellitic Dirichlet roblem. Le Matematiche, (66):33 4, 2. [9] M. Ferrara and G. Molica Bisci. Existence result for ellitic roblems with Hardy otential. Bulletin des Sciences Mathématiques, 38(7): , 24. [] R. Filiucci, P. Pucci, and F. Robert. On a -Lalace equation with multile critical nonlinearities. Journal de mathématiques ures et aliquées, 9(2):56 77, 29. 7

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