Nodal solutions for nonlinear nonhomogeneous Robin problems
Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Dušan D. Repovš
aa r X i v : . [ m a t h . A P ] J a n NODAL SOLUTIONS FOR NONLINEAR NONHOMOGENEOUSROBIN PROBLEMS
NIKOLAOS S. PAPAGEORGIOU, VICENT¸ IU D. R ˘ADULESCU, AND DUˇSAN D. REPOVˇS
Abstract.
We consider the nonlinear Robin problem driven by a nonhomoge-neous differential operator plus an indefinite potential. The reaction term is aCarath´eodory function satisfying certain conditions only near zero. Using suit-able truncation, comparison, and cut-off techniques, we show that the problemhas a sequence of nodal solutions converging to zero in the C (Ω)-norm. Introduction
Let Ω ⊆ R N be a bounded domain with a C -boundary ∂ Ω. We study thefollowing nonlinear nonhomogeneous Robin problem:(1) − div a ( Du ( z )) + ξ ( z ) | u ( z ) | p − u ( z ) = f ( z, u ( z )) in Ω ,∂u∂n a + β ( z ) | u | p − u = 0 on ∂ Ω . In this problem, a : R N → R N is a continuous and strictly monotone map (thus alsomaximal monotone), which satisfies certain regularity and growth conditions listedin hypotheses H ( a ) below. These conditions are general and they incorporate inour framework many differential operators of interest, such as the p -Laplacian andthe ( p, q )-Laplacian. We stress that a ( · ) is not homogeneous and this is a sourceof difficulties in the study of problem (1). The potential function ξ ∈ L ∞ (Ω) isindefinite (that is, sign changing). The reaction term (the right-hand side of (1))is a Carath´eodory function (that is, for all x ∈ R , the function z f ( z, x ) ismeasurable, and for almost all z ∈ Ω, the function x f ( z, x )) is continuous. Weimpose conditions on f ( z, · ) only near zero. In the boundary condition, ∂u∂n a denotesthe conormal derivative corresponding to the differential operator u div a ( Du )and is defined by extension of the map C (Ω) ∋ u ( a ( Du ) , n ) R N , with n ( · ) being the outward unit normal on ∂ Ω.We are looking for nodal (that is, sign-changing) solutions for problem (1). Em-ploying a symmetry condition on f ( z, · ) near zero and using truncation, perturba-tion, comparison, and cut-off techniques, and a result of Kajikiya [7], we generatea whole sequence { u n } n > ⊆ C (Ω) of distinct nodal solutions such that u n → C (Ω).The first result in this direction was produced by Wang [27], who used cut-offtechniques to produce an infinity of solutions converging to zero in H (Ω). In Key words and phrases.
Nodal solutions, indefinite potential, nonhomogeneous differentialoperator, nonlinear regularity theory, truncation and cut-off techniques2010 AMS Subject Classification: 35J20, 35J60.
Wang [27] the problem is semilinear driven by the Dirichlet Laplacian. There isno potential term (that is, ξ ≡ et al. [5], the problem is Neumann (that is, β ≡
0) andthe differential operator is the p -Laplacian (that is, a ( y ) = | y | p − y for all y ∈ R N ,with 1 < p < ∞ ). In Papageorgiou & R˘adulescu [19], the differential operatoris the same as in the present paper, but ξ ≡
0. Also, the hypotheses on f ( z, · )near zero are more restrictive. In the present paper we extend the results of allaforementioned works.2. Preliminaries and Hypotheses
In the study of problem (1) we will use the following spaces: the Sobolev space W ,p (Ω), the Banach space C (Ω), and the boundary Lebesgue spaces L r ( ∂ Ω),1 r ∞ .We denote by || · || the norm on the Sobolev space W ,p (Ω) defined by || u || = (cid:2) || u || pp + || Du || pp (cid:3) p for all u ∈ W ,p (Ω) . The Banach space C (Ω) is an ordered Banach space, with positive (order) cone C + = (cid:8) u ∈ C (Ω) : u ( z ) > z ∈ Ω (cid:9) . This cone has a nonempty interior which contains the open set D + = { u ∈ C + : u ( z ) > z ∈ Ω } . In fact, D + is the interior of C + when furnished with the relative C (Ω)-normtopology.On ∂ Ω we consider the ( N − σ ( · ).Using this measure, we can define in the usual way the Lebesgue spaces L r ( ∂ Ω) , r ∞ . From the theory of Sobolev spaces we know that there exists a uniquecontinuous linear map γ : W ,p (Ω) → L p ( ∂ Ω), known as the “trace map”, suchthat γ ( u ) = u | ∂ Ω for all u ∈ W ,p (Ω) ∩ C (Ω) . So, the trace map assigns “boundary values” to all Sobolev functions. We knowthat the trace map is compact into L r ( ∂ Ω) for all 1 r < ( N − pN − p if p < N , andinto L r ( ∂ Ω) for all 1 r < ∞ if p > N . Furthermore, we have thatker γ = W ,p (Ω) and im γ = W p ′ ,p ( ∂ Ω) (cid:18) p + 1 p ′ = 1 (cid:19) . In what follows, for the sake of notational simplicity, we will drop the use of thetrace map γ ( · ). All restrictions of Sobolev functions on ∂ Ω, are understood in thesense of traces.Let X be a Banach space and ϕ ∈ C ( X, R ). We say that ϕ satisfies the “Palais-Smale condition” (the “PS-condition” for short), if the following property holds:“Every sequence { u n } n > ⊆ X such that { ϕ ( u n ) } n > ⊆ R is bounded and ϕ ′ ( u n ) → X ∗ as n → ∞ ,admits a strongly convergent subsequence.” ONLINEAR NONHOMOGENEOUS ROBIN PROBLEMS 3
We shall need the following result of Kajikya [7].
Theorem 1.
Assume that X is a Banach space, ϕ ∈ C ( X, R ) satisfies the PS-condition, ϕ is even and bounded below, ϕ (0) = 0 , and for every n ∈ N , there existsan n -dimensional subspace V n of X and ρ n > such that sup { ϕ ( u ) : u ∈ V n ∩ ∂B ρ n } < , where ∂B ρ n = { u ∈ X : || u || X = ρ n } . Then there exists a sequence { u n } n > ⊆ X \{ } such that ( i ) ϕ ′ ( u n ) = 0 for all n ∈ N (that is, each u n is a critical point of ϕ ) ;( ii ) ϕ ( u n ) for all n ∈ N ; and ( iii ) u n → in X as n → ∞ . In the sequel, for any ϕ ∈ C ( X, R ), we denote by K ϕ the critical set of ϕ , thatis, K ϕ = { u ∈ X : ϕ ′ ( u ) = 0 } . For X ∈ R , we set x ± = max {± x, } . Then for any u ∈ W ,p (Ω), we define u ± ( · ) = u ( · ) ± . We know that u ± ∈ W ,p (Ω) , u = u + − u − , | u | = u + + u − . Let ϑ ∈ C (0 , ∞ ) be such that ϑ ( t ) > t > < ˆ c ϑ ′ ( t ) tϑ ( t ) c and c t p − ϑ ( t ) c ( t τ − + t p − )for all t > , with c , c > , τ < p. Then the hypotheses on the map a ( · ) are the following: H ( a ) : a ( y ) = a ( | y | ) y for all y ∈ R N with a ( t ) > t > a ∈ C (0 , ∞ ) , t a ( t ) t is strictly increasing on (0 , ∞ ), a ( t ) t → + as t → + and lim t → + a ′ ( t ) ta ( t ) > − |∇ a ( y ) | c ϑ ( | y | ) | y | for all y ∈ R N \{ } , and some c > ∇ a ( y ) ξ, ξ ) R N > ϑ ( | y | ) | y | | ξ | for all y ∈ R N \{ } , ξ ∈ R N ; and(iv) If G ( t ) = Z t a ( s ) sds for all t >
0, then there exists q ∈ (1 , p ] such that t G ( t /q ) is convex and lim sup t → + qG ( t ) t q < + ∞ . Remark 1.
Hypotheses H ( a )( i ) , ( ii ) , ( iii ) are dictated by the nonlinear global reg-ularity theory of Lieberman [10] and the nonlinear maximum principle of Pucci &Serrin [24] . Hypothesis H ( a )( iv ) reflects the particular requirements of our problem.However, H ( a )( iv ) is not restrictive as the examples below illustrate. Hypotheses H ( a ) imply that G ( · ) is strictly convex and strictly increasing. Weset G ( y ) = G ( | y | ) for all y ∈ R N . Evidently, G ( · ) is convex and G (0) = 0. Also,we have ∇ G ( y ) = G ′ ( | y | ) y | y | = a ( | y | ) y = a ( y ) for all y ∈ R N \{ } , ∇ G (0) = 0 . N.S. PAPAGEORGIOU, V.D. R˘ADULESCU, AND D.D. REPOVˇS
So, G ( · ) is the primitive of a ( · ). Moreover, the convexity of G ( · ) implies that(3) G ( y ) ( a ( y ) , y ) R N for all y ∈ R N . The next lemma summarizes the main properties of the map a ( · ) and it is aneasy consequence of hypotheses H ( a ) and condition (2) above. Lemma 2.
If hypotheses H ( a )( i ) , ( ii ) , ( iii ) hold, then (a) a ( · ) is continuous, strictly monotone, hence maximal monotone, too; (b) | a ( y ) | c (1 + | y | p − ) for all y ∈ R N , and some c > ; and (c) ( a ( y ) , y ) R N > c p − | y | p for all y ∈ R N . This lemma and (3) lead to the following growth conditions on G ( · ). Corollary 3.
If hypotheses H ( a )( i ) , ( ii ) , ( iii ) hold, then c p ( p − | y | p G ( y ) c (1 + | y | p ) for all y ∈ R N , and some c > . Example 1.
The following maps a ( y ) satisfy hypotheses H ( a ) : (a) a ( y ) = | y | p − y, < p < ∞ .This map corresponds to the p-Laplace differential operator defined by ∆ p u = div ( | Du | p − Du ) for all u ∈ W ,p (Ω) . (b) a ( y ) = | y | p − y + | y | q − y, < q < p < ∞ . This map corresponds to the ( p, q ) -Laplace differential operator defined by ∆ p u + ∆ q u for all u ∈ W ,p (Ω) . Such operators arise in problems of mathematical physics. Recently ( p, q ) -equations have been studied by Bobkov & Tanaka [1] , Li & Zhang [9] ,Marano & Mosconi [11] , Marano, Mosconi & Papageorgiou [12, 13] , Mug-nai & Papageorgiou [16] , Papageorgiou & R˘adulescu [17] , Sun, Zhang & Su [25] , and Tanaka [26] . (c) a ( y ) = (1 + | y | ) p − y, < p < ∞ . This map corresponds to the generalizedp-mean curvature differential operator defined by div ((1 + | Du | ) p − Du ) for all u ∈ W ,p (Ω) . (d) a ( y ) = | y | p − y (1 + 11 + | y | p ) , < p < ∞ . We denote by h· , ·i the duality brackets for the pair( W ,p (Ω) ∗ , W ,p (Ω)) . Let A : W ,p (Ω) → W ,p (Ω) ∗ be the nonlinear map defined by h A ( u ) , h i = Z Ω ( a ( Du ) , Dh ) R N dz for all u, h ∈ W ,p (Ω) . From Gasinski & Papageorgiou [3], we have:
Proposition 4.
The map A : W ,p (Ω) → W ,p (Ω) ∗ is bounded (maps boundedsets to bounded sets), continuous, monotone (hence maximal monotone, too), andof type ( S ) + , that is, “ u n w −→ u in W ,p (Ω) and lim sup n →∞ h A ( u n ) , u n − u i ⇒ u n → u ” . ONLINEAR NONHOMOGENEOUS ROBIN PROBLEMS 5
The hypotheses on the potential function ξ ( · ) and on the boundary coefficient β ( · ) are the following: H ( ξ ) : ξ ∈ L ∞ (Ω). H ( β ) : β ∈ C ,α ( ∂ Ω) for some α ∈ (0 ,
1) and β ( z ) > z ∈ ∂ Ω . Remark 2. If β ≡ , then we recover the Neumann problem. Finally, we introduce our conditions on the reaction term f ( z, x ): H ( f ): f : Ω × R → R is a Carath´eodory function such that f ( z,
0) = 0 for almostall z ∈ Ω and(i) there exists η > z ∈ Ω , f ( z, · ) | [ − η,η ] is odd;(ii) | f ( z, x ) | a η ( z ) for almost all z ∈ Ω, x ∈ [ − η, η ], with a η ∈ L ∞ (Ω);(iii) with q ∈ (1 , p ] as in hypothesis H ( a )( iv ), we havelim x → f ( z, x ) | x | q − x = + ∞ uniformly for almost all z ∈ Ω; and(iv) there exists ˆ ξ > z ∈ Ω x → f ( z, x ) + ˆ ξ | x | p − x is nondecreasing on [ − η, η ] Remark 3.
We point out that all the above hypotheses concern the behaviour of f ( z, · ) only near zero. Finally, we mention that nonlinear problems with an indefinite potential have re-cently been studied in the context of equations driven by the Neumann p -Laplacianby Gasinski & Papageorgiou [4] (resonant problems) and Fragnelli, Mugnai & Pa-pageorgiou [2] (superlinear problems). Also, nodal solutions for nonlinear Robinproblems with no potential term, were obtained by Papageorgiou & R˘adulescu[21]. 3. Nodal solutions
Let ε ∈ (0 , η ) and consider an even function γ ∈ C ( R ) such that 0 γ γ | [ − ε,ε ] = 1 and supp γ ⊆ [ − η, η ].We set ˆ f ( z, x ) = γ ( x ) f ( z, x ) + (1 − γ ( x )) ξ ( z ) | x | p − x. Evidently, ˆ f ( z, x ) is a Carath´eodory function which is odd in x ∈ R and has thefollowing two additional properties:ˆ f ( z, · ) | [ − ε,ε ] = f ( z, · ) | [ − ε,ε ] for all z ∈ Ω;(4) ˆ f ( z, x ) = ξ ( z ) | x | p − x for all z ∈ Ω , | x | > η. (5)It follows from (5) that(6) ˆ f ( z, η ) − ξ ( z ) η p − = 0 for almost all z ∈ Ω . Since ˆ f ( z, · ) is odd, we have(7) ˆ f ( z, − η ) + ξ ( z ) η p − = 0 for almost all z ∈ Ω . On account of hypothesis H ( f )( iii ), given any µ >
0, we can find δ = δ ( µ ) ∈ (0 , ε ) such that(8) f ( z, x ) x = ˆ f ( z, x ) > µ | x | q for almost all z ∈ Ω , and all | x | δ (see (4)). N.S. PAPAGEORGIOU, V.D. R˘ADULESCU, AND D.D. REPOVˇS
Then (8) combined with hypothesis H ( f )( ii ) implies that given r > p we canfind c > f ( z, x ) x > µ | x | q − c | x | r for almost all z ∈ Ω , and all x ∈ R . We introduce the following function(10) k ( z, x ) = µ | x | q − x − c | x | r − x. This is a Carath´eodory function which is odd in x ∈ R .We consider the following auxiliary nonlinear Robin problem:(11) − div a ( Du ( z )) + | ξ ( z ) || u ( z ) | p − u ( z ) = k ( z, u ( z )) in Ω ,∂u∂n a + β ( z ) | u | p − u = 0 on ∂ Ω . Proposition 5.
If hypotheses H ( a ) , H ( ξ ) , H ( β ) hold, then problem (11) admitsa unique positive solution u ∗ ∈ D + and since k ( z, · ) is odd, v ∗ = − u ∗ ∈ D + is the unique negative solution of (11).Proof. We consider the Carath´eodory function ˆ k ( z, x ) defined by(12) ˆ k ( z, x ) = k ( z, − η ) − η p − if x < − ηk ( z, x ) + | x | p − x if − η x ηk ( z, η ) + η p − if η < x. We set ˆ K ( z, x ) = Z x ˆ k ( z, s ) ds and consider the C -functional ˆ ϕ + : W ,p (Ω) → R defined byˆ ϕ + ( u ) = Z Ω G ( Du ) dz + 1 p Z Ω [ | ξ ( z ) | + 1] | u | p dz + 1 p Z ∂ Ω β ( z ) | u | p dσ − Z Ω ˆ K ( z, u + ) dz for all u ∈ W ,p (Ω) . From (12) and Corollary 3 it is clear thatˆ ϕ + ( · ) is coercive.Also, from the Sobolev embedding theorem and the compactness of the tracemap, we deduce thatˆ ϕ + ( · ) is sequentially weakly lower semicontinuous.So, by the Weierstrass-Tonelli theorem, we can find u ∗ ∈ W ,p (Ω) such that(13) ˆ ϕ + ( u ∗ ) = inf (cid:8) ˆ ϕ + ( u ) : u ∈ W ,p (Ω) (cid:9) . On account of hypothesis H ( a )( iv ), we can find c > G ( y ) c q | y | q for all | y | δ, with δ > u ∈ D + . Then we can find t ∈ (0 ,
1) small such that(15) tu ( z ) ∈ (0 , δ ] and | D ( tu )( z ) | δ for all z ∈ Ω . ONLINEAR NONHOMOGENEOUS ROBIN PROBLEMS 7
Using (10), (12), (14) and (15), we obtainˆ ϕ + ( tu ) t q c q || Du || qq + t q q Z Ω | ξ ( z ) || u | q dz + t q q Z ∂ Ω β ( z ) | u | q dσ + t r r || u || rr − t q q µ || u || qq (since t ∈ (0 , , q p < r ) [ c − µc ] t q for some c , c > u. Choosing µ > c c , we infer thatˆ ϕ + ( tu ) < , ⇒ ˆ ϕ + ( u ∗ ) < ϕ + (0) (see (13)) , ⇒ u ∗ = 0 . From (13) we haveˆ ϕ ′ + ( u ∗ ) = 0 , ⇒ h A ( u ∗ ) , h i + Z Ω [ | ξ ( z ) | + 1] | u ∗ | p − u ∗ hdz + Z ∂ Ω β ( z ) | u ∗ | p − u ∗ hdz = Z Ω ˆ k ( z, ( u ∗ ) + ) hdz (16) for all h ∈ W ,p (Ω) . In (16) we choose h = − ( u ∗ ) − ∈ W ,p (Ω). Using Lemma 2(c), we obtain c p − || D ( u ∗ ) − || pp + || ( u ∗ ) − || pp H ( B )) , ⇒ u ∗ > , u ∗ = 0 . In (16) we choose h = ( u ∗ − η ) + ∈ W ,p (Ω). Then h A ( u ∗ ) , ( u ∗ − η ) + i + Z Ω [ | ξ ( z ) | + 1]( u ∗ ) p − ( u ∗ − η ) + dz + Z ∂ Ω β ( z )( u ∗ ) p − ( u ∗ − η ) + dσ = Z Ω (cid:2) µη q − − c η r − + η p − (cid:3) ( u ∗ − η ) + dz (see (12) and (10)) Z Ω h ˆ f ( z, η ) + η p − i ( u ∗ − η ) + dz (see (9))= Z Ω [ ξ ( z ) + 1] η p − ( u ∗ − η ) + dz (see (6)) h A ( η ) , ( u ∗ − η ) + i + Z Ω [ | ξ ( z ) | + 1] η p − ( u ∗ − η ) + dz + Z ∂ Ω β ( z ) η p − ( u ∗ − η ) + dσ (note that A ( η ) = 0 and see hypothesis H ( β )), ⇒ h A ( u ∗ ) − A ( η ) , ( u ∗ − η ) + i + Z Ω [ | ξ ( z ) | + 1] (( u ∗ ) p − − η p − )( u ∗ − η ) + dz H ( β )) , ⇒ u ∗ η. So, we have proved that(17) u ∗ ∈ [0 , η ] = (cid:8) u ∈ W ,p (Ω) : 0 u ( z ) η for almost all z ∈ Ω (cid:9) . N.S. PAPAGEORGIOU, V.D. R˘ADULESCU, AND D.D. REPOVˇS
From (10), (12), (16) and (17), we infer that u ∗ is a positive solution of problem(11). From Papageorgiou & R˘adulescu [20], we have u ∗ ∈ L ∞ (Ω) . Now the nonlinear regularity theory of Lieberman [10] implies that u ∗ ∈ C + \{ } . From (16) and (17), we have − div a ( Du ∗ ( z )) + | ξ ( z ) | u ∗ ( z ) p − = k ( z, u ∗ ( z )) for almost all z ∈ Ω ,∂u ∗ ∂n a + β ( z ) u ∗ = 0 on ∂ Ω (see Papageorgiou & R˘adulescu [18]) ⇒ − div a ( Du ∗ ( z )) + | ξ ( z ) | u ∗ ( z ) p − > − c u ∗ ( z ) r − for almost all z ∈ Ω (see (10)) , ⇒ div a ( Du ∗ ( z )) (cid:2) c || u ∗ || r − p ∞ + || ξ || ∞ (cid:3) u ∗ ( z ) p − for almost all z ∈ Ω(see hypothesis H ( ξ )) , ⇒ u ∗ ∈ D + (see Pucci & Serrin [24, p. 120]) . Next, we show the uniqueness of this solution. To this end, let ˆ i : L (Ω) → R = R ∪ { + ∞} be the integral functional defined byˆ i ( u ) = Z Ω G ( Du q ) dz + 1 p Z Ω | ξ ( z ) | u pq dz + 1 p Z ∂ Ω β ( z ) u pq dσ if u > , u q ∈ W ,p (Ω)+ ∞ otherwise . From Papageorgiou & Winkert [23] (see the proof of Proposition 3.3), we knowthat ˆ i ( · ) is convex and if u ∗ , v ∗ ∈ D + are two positive solutions of (11), thenˆ i ′ (( u ∗ ) q )( h ) = 1 q Z Ω − div a ( Du ∗ ) + | ξ ( z ) | ( u ∗ ) p − ( u ∗ ) q − hdz ˆ i ′ (( v ∗ ) q )( h ) = 1 q Z Ω − div a ( Dv ∗ ) + | ξ ( z ) | ( v ∗ ) p − ( v ∗ ) q − hdz for all h ∈ C (Ω) . The convexity of ˆ i ( · ) implies the monotonicity of ˆ i ′ ( · ). Hence0 Z Ω (cid:20) − div a ( Du ∗ ) + | ξ ( z ) | ( u ∗ ) p − ( u ∗ ) q − − div a ( Dv ∗ ) + | ξ ( z ) | ( v ∗ ) p − ( v ∗ ) q − (cid:21) (( u ∗ ) q − ( v ∗ ) q ) dz = Z Ω c (cid:2) ( v ∗ ) r − q − ( v ∗ ) r − q (cid:3) (( u ∗ ) q − ( v ∗ ) q ) dz (see (10)) , ⇒ u ∗ = v ∗ (since q p < r ) . This proves the uniqueness of the positive solution u ∗ ∈ D + of (11). Sinceproblem (11) is odd, it follows that v ∗ = − u ∗ ∈ − D + is the unique negativesolution of problem (11). (cid:3) Consider the following Robin problem:(18) − div a ( Du ( z )) + ξ ( z ) | u ( z ) | p − u ( z ) = ˆ f ( z, u ( z )) in Ω ,∂u∂n a + β ( z ) | u | p − u = 0 on ∂ Ω . We denote by S + (respectively S − ) the set of positive (respectively negative)solutions of problem (18) which are in the order interval [0 , η ] = { u ∈ W ,p (Ω) :0 u ( z ) η for almost all z ∈ Ω } (respectively in [ − η,
0] = { v ∈ W ,p (Ω) : − η v ( z ) z ∈ Ω } ). From Papageorgiou, R˘adulescu & Repovˇs [22], weknow that ONLINEAR NONHOMOGENEOUS ROBIN PROBLEMS 9 • S + is downward directed (that is, if u , u ∈ S + , then we can find u ∈ S + such that u u , u u ). • S − is upward directed (that is, if v , v ∈ S − , then we can find v ∈ S − such that v v, v v ).Moreover, reasoning as in the proof of Proposition 5 (with k ( z, x ) replaced byˆ f ( z, x )), we show that ∅ 6 = S + ⊆ D + and ∅ 6 = S − ⊆ − D + . Proposition 6.
If hypotheses H ( a ) , H ( ξ ) , H ( β ) , H ( f ) hold, then u ∗ u for all u ∈ S + and v v ∗ for all v ∈ S − .Proof. Let u ∈ S + and let ˆ k ( z, x ) be given by (12). We introduce the followingtruncation of ˆ k ( z, · ):(19) e + ( z, x ) = x < k ( z, x ) if 0 x u ( z )ˆ k ( z, u ( z )) if u ( z ) < x. This is a Carath´eodory function. We set E + ( z, x ) = Z x e + ( z, s ) ds and considerthe C -functional Ψ + : W ,p (Ω) → R defined byΨ + ( u ) = Z Ω G ( Du ) dz + 1 p Z Ω [ | ξ ( z ) | + 1] | u | p dz + 1 p Z ∂ Ω β ( z ) | u | p dσ − Z Ω E + ( z, u ) dz for all u ∈ W ,p (Ω) . Evidently, Ψ + ( · ) is coercive (see (19)) and sequentially weakly lower semicontin-uous. So, we can find ˆ u ∗ ∈ W ,p (Ω) such that(20) Ψ + (ˆ u ∗ ) = inf (cid:8) Ψ + ( u ) : u ∈ W ,p (Ω) (cid:9) . As in the proof of Proposition 5, using hypotheses H ( a )( iv ) and H ( f )( iii ), weshow that Ψ + (ˆ u ∗ ) < + (0) , ⇒ ˆ u ∗ = 0 . From (20) we haveΨ ′ + (ˆ u ∗ ) = 0 , ⇒ h A (ˆ u ∗ ) , h i + Z Ω [ | ξ ( z ) | + 1] | ˆ u ∗ | p − ˆ u ∗ hdz + Z ∂ Ω β ( z ) | ˆ u ∗ | p − ˆ u ∗ hdσ = Z Ω e + ( z, ˆ u ∗ ) hdz for all h ∈ W ,p (Ω) . (21)In (21), we choose h = − ( u ∗ ) − ∈ W ,p (Ω). Then using Lemma 2(c), we have c p − || D (ˆ u ∗ ) − || pp + Z Ω [ | ξ ( z ) + 1 | ]((ˆ u ∗ ) − ) p dz H ( β ) and (19)) ⇒ ˆ u ∗ > , ˆ u ∗ = 0 . Next, in (21) we choose h = (ˆ u ∗ − u ) + ∈ W ,p (Ω). We have h A (ˆ u ∗ ) , (ˆ u ∗ − u ) + i + Z Ω [ | ξ ( z ) + 1 | ](ˆ u ∗ ) p − (ˆ u ∗ − u ) + dz + Z ∂ Ω β ( z )( u ∗ ) p − (ˆ u ∗ − u ) + dσ = Z Ω [ µu q − − c u r − + u p − ](ˆ u ∗ − u ) + dz (see (19), (12), (10) and recall that u ∈ S + ) Z Ω [ ˆ f ( z, u ) + u p − ](ˆ u ∗ − u ) + dz (see (9))= h A ( u ) , (ˆ u ∗ − u ) + i + Z Ω [ | ξ ( z ) + 1 | ] u p − (ˆ u ∗ − u ) + dz + Z ∂ Ω β ( z ) u p − (ˆ u ∗ − u ) + dσ (since u ∈ S + ) ⇒ ˆ u ∗ u. So, we have proved thatˆ u ∗ ∈ [0 , u ] = { y ∈ W ,p (Ω) : 0 y ( z ) u ( z ) for almost all z ∈ Ω } . This fact, together with (10), (12), (19), (21), imply that − div a ( D ˆ u ∗ z ) + | ξ ( z ) | ˆ u ∗ ( z ) p − = k ( z, ˆ u ∗ ( z )) for almost all z ∈ Ω ,∂ ˆ u ∗ ∂n a + β ( z )(ˆ u ∗ ) p − = 0 on ∂ Ω (see Papageorgiou & R˘adulescu [18]) , ⇒ ˆ u ∗ = u ∗ (see Proposition 5) , ⇒ u ∗ u for all u ∈ S + . Similarly, we show that v v ∗ for all v ∈ S − . This completes the proof. (cid:3)
Now we can establish the existence of extremal constant sign solutions for prob-lem (18), that is, we show that problem (18) has a smallest positive solution and abiggest negative solution.
Proposition 7.
If hypotheses H ( a ) , H ( β ) , H ( ξ ) , H ( f ) hold, then there exists asmallest positive solution u + ∈ S + ⊆ D + and a biggest negative solution v + ∈ S − ⊆ − D + .Proof. Invoking Lemma 3.10 of Hu & Papageorgiou [6, p. 178], we can find adecreasing sequence { u n } n > ⊆ S + such thatinf S + = inf n > u n . Evidently, { u n } n > ⊆ W ,p (Ω) is bounded. So, we may assume that(22) u n w −→ u + in W ,p (Ω) and u n → u + in L p (Ω) and L p ( ∂ Ω) . We have(23) h A ( u n ) , h i + Z Ω ξ ( z ) u p − n hdz + Z ∂ Ω β ( z ) u p − n hdσ = Z Ω ˆ f ( z, u n ) hdx for all h ∈ W ,p (Ω) , n ∈ N . In (23) we choose h = u n − u + ∈ W ,p (Ω), pass to the limit as n → ∞ and use(22). Then(24) lim n →∞ h A ( u n ) , u n − u + i = 0 , ⇒ u n → u + in W ,p (Ω) (see Proposition 4) . ONLINEAR NONHOMOGENEOUS ROBIN PROBLEMS 11
In (23) we pass to the limit as n → ∞ and use (24). Then(25) h A ( u + ) , h i + Z Ω ξ ( z ) u p − hdz + Z ∂ Ω β ( z ) u p − hdσ = Z Ω ˆ f ( z, u + ) hdz for all h ∈ W ,p (Ω) . From Proposition 6, we have(26) u ∗ u n for all n ∈ N , ⇒ u ∗ u + (see (24)), hence u + = 0 . It follows from (25) and (26) that u + ∈ S + ⊆ D + , u + = inf S + . Similarly, we produce v − ∈ S − ⊆ − D + , v − = sup S − . (cid:3) Let τ > || ξ || ∞ and consider the following truncation-perturbation of ˆ f ( z, · ):(27) f ( z, x ) = ˆ f ( z, v − ( z )) + τ | v − ( z ) | p − v − ( z ) if x < v − ( z )ˆ f ( z, x ) + τ | x | p − x if v − ( z ) x u + ( z )ˆ f ( z, u + ( z )) + τ u + ( z ) p − if u + ( z ) < x. We set F ( z, x ) = Z x f ( z, s ) ds and consider the C -functional ϕ : W ,p (Ω) → R defined by ϕ ( u ) = Z Ω G ( Du ) dz + 1 p Z Ω [ ξ ( z ) + τ ] | u | p dz + 1 p Z Ω β ( z ) | u | p dσ − Z Ω F ( z, u ) dz for all u ∈ W ,p (Ω) . Evidently, ϕ ( · ) is coercive (see (27) and recall that τ > || ξ || ∞ ). So, ϕ ( · ) isbounded below and satisfies the PS-condition (see Marano & Papageorgiou [14, 15]). Proposition 8.
If hypotheses H ( a ) , H ( ξ ) , H ( β ) , H ( f ) hold and V ⊆ W ,p (Ω) is afinite dimensional linear subspace, then there exists ρ V > such that sup { ϕ ( u ) : u ∈ V, || u || = ρ V } < . Proof.
Recall that u + ∈ D + and v − ∈ − D + . So, m = min { min Ω u + , − max Ω v − } >
0. We set ǫ = min { ǫ, m } (where ǫ > H ( f )( iii ), given any µ >
0, we can find δ = δ ( µ ) > ∈ (0 , ǫ ) such that(28) F ( z, x ) = ˆ F ( z, x ) + τp | x | p = F ( z, x ) + τp | x | p > µq | x | q + τp | x | p (for almost all z ∈ Ω, and all | x | δ , see (4) and (27)) . Moreover, on account of hypothesis H ( a )( iv ) and Corollary 3, we have(29) G ( y ) c [ | y | q + | y | p ] for some c > , and all y ∈ R N . Since the subspace V ⊆ W ,p (Ω) is finite dimensional, all norms are equivalent.So, we can find ρ V ∈ (0 ,
1] such that(30) u ∈ V, || u || ρ V ⇒ | u ( z ) | δ for all z ∈ Ω . Then for every u ∈ V with || u || ρ V , we have ϕ ( u ) c || u || q − µc || u || q for some c , c > ρ V , q p )Since µ > µ > c c and conclude that ϕ ( u ) < u ∈ V with || u || = ρ V . The proof is now complete. (cid:3)
We now obtain the following multiplicity theorem for the nodal solutions ofproblem (1).
Theorem 9.
Assume that hypotheses H ( a ) , H ( ξ ) , H ( β ) , H ( f ) hold. Then thereexists a sequence { u n } n > ⊆ C (Ω) of nodal solutions of problem (1) such that u n → in C (Ω) . Proof.
We know that ϕ ( · ) is even, bounded below, satisfies the PS-condition, and ϕ (0) = 0. Moreover, using (27) as before, we can check that(31) K ϕ ⊆ [ v − , u + ] ∩ C (Ω) . The aforementioned properties of ϕ ( · ) and Proposition 8 permit us to applyTheorem 1. So, we can find a sequence { u n } n > ⊆ W ,p (Ω) such that(32) u n ∈ K ϕ ⊆ [ v − , u + ] ∩ C (Ω) (see (31)) and u n → W ,p (Ω) . The nonlinear regularity theory of Lieberman [10] implies that we can find γ ∈ (0 ,
1) and c > u n ∈ C ,γ (Ω) , || u n || C ,γ (Ω) c for all n ∈ N . We know that C ,γ (Ω) is compactly embedded in C (Ω). So, it follows from (32)and (33) that u n → C (Ω) , ⇒ − ǫ u n ( z ) ǫ for all z ∈ Ω , and all n > n (recall that ǫ = min { ǫ, m } >
0, see the proof of Proposition 8) . From (4), (32) and the extremality of u + , v − , we get that { u n } n > ⊆ C (Ω) arenodal solutions of (1) and we have u n → C (Ω). (cid:3) Acknowledgements.
This research was supported by the Slovenian ResearchAgency grants P1-0292, J1-8131, J1-7025, N1-0064, and N1-0083. V.D. R˘adulescuacknowledges the support through a grant of the Romanian Ministry of Researchand Innovation, CNCS–UEFISCDI, project number PN-III-P4-ID-PCE-2016-0130,within PNCDI III.
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NoDEA Nonlinear Differential Equations Appl. (2001), 15-33. (N.S. Papageorgiou) National Technical University, Department of Mathematics, Zo-grafou Campus, Athens 15780, Greece & Institute of Mathematics, Physics and Mechan-ics, Jadranska 19, 1000 Ljubljana, Slovenia
E-mail address : [email protected] (V.D. R˘adulescu) Institute of Mathematics, Physics and Mechanics, Jadranska 19,1000 Ljubljana, Slovenia & Faculty of Applied Mathematics, AGH University of Scienceand Technology, 30-059 Krak´ow, Poland & Department of Mathematics, University ofCraiova, 200585 Craiova, Romania
E-mail address : [email protected] (D.D. Repovˇs) Faculty of Education and Faculty of Mathematics and Physics, Uni-versity of Ljubljana & Institute of Mathematics, Physics and Mechanics, Jadranska19, 1000 Ljubljana, Slovenia
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