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1 Calc. Var. (218 57:131 Calculus of Variations Asymmetric critical p-laplacian problems Kanishka Perera 1 Yang Yang 2 Zhitao Zhang 3,4 Received: 14 September 217 / Accepted: 28 July 218 Springer-Verlag GmbH Germany, part of Springer ature 218 Abstract We obtain nontrivial solutions for two types of asymmetric critical p-laplacian problems with Ambrosetti Prodi type nonlinearities in a smooth bounded domain in R, 2. For 1 < p <, we consider an asymmetric problem involving the critical Sobolev exponent p = p/( p. In the borderline case p =, we consider an asymmetric critical exponential nonlinearity of the Trudinger Moser type. In the absence of a suitable direct sum decomposition, we use a linking theorem based on the Z 2 -cohomological index to prove existence of solutions. Mathematics Subject Classification Primary 35B33; Secondary 35J92 35J2 1 Introduction Beginning with the seminal paper of Ambrosetti and Prodi [3], elliptic boundary value problems with asymmetric nonlinearities have been extensively studied (see, e.g., Berger and Podolak [7], Kazdan and Warner [2], Dancer [1], Amann and Hess [2], and the references therein. More recently, Deng [15], de Figueiredo and Yang [12], Aubin and Wang Communicated by P. Rabinowitz. Supported by SFC ( , , , , and the atural Science Foundation of Jiangsu Province of China for Young Scholars (o. BK B Zhitao Zhang zzt@math.ac.cn Kanishka Perera kperera@fit.edu Yang Yang yynjnu@126.com 1 Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 3291, USA 2 School of Science, Jiangnan University, Wuxi , Jiangsu, People s Republic of China 3 HLM, CEMS, HCMS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 119, People s Republic of China 4 School of Mathematical Sciences, University of the Chinese Academy of Sciences, Beijing 149, People s Republic of China (.: V,-vol

2 131 Page 2 of 18 K. Perera et al. [6], Calanchi and Ruf [8],and Zhanget al. [32] have obtained interesting existence and multiplicity results for semilinear Ambrosetti Prodi type problems with critical nonlinearities using variational methods. In the present paper, first we consider the Ambrosetti Prodi type asymmetric critical p-laplacian problem p u = λ u p 2 u + u p 1 + in (1.1 u = on, where is a smooth bounded domain in R, 2, 1 < p <, p = p/( p is the critical Sobolev exponent, λ>isaconstant, and u + (x = max {u(x, } is the positive part of u(x. We recall that λ R is a Dirichlet eigenvalue of p in if the problem p u = λ u p 2 u in (1.2 u = on has a nontrivial solution. The first eigenvalue λ 1 (p is positive, simple, and has an associated eigenfunction ϕ 1 that is positive in. Problem (1.1 has a positive solution when p 2 and <λ<λ 1 (p (see Guedda and Véron [19]. When λ = λ 1 (p, tϕ 1 is clearly a negative solution for any t <. Here we focus on the case λ>λ 1 (p. Our first result is the following. Theorem 1.1 If p 2 and λ>λ 1 (p is not an eigenvalue of p, then problem (1.1 has a nontrivial solution. In the borderline case p = 2, critical growth is of exponential type and is governed by the Trudinger Moser inequality e α u dx <, (1.3 sup u W 1, (, u 1 where α = ω 1/( 1 1, ω 1 is the area of the unit sphere in R,and = /( 1 (see Trudinger [3]andMoser[25]. A natural analog of problem (1.1 for this case is u = λ u 2 ue u + in (1.4 u = on. A result of Adimurthi [1] implies that this problem has a nonnegative and nontrivial solution when <λ<λ 1 ( (see also do Ó [16]. When λ = λ 1 (, tϕ 1 is again a negative solution for any t <. Our second result here is the following. Theorem 1.2 If 2 and λ>λ 1 ( is not an eigenvalue of, then problem (1.4 has a nontrivial solution. Problems (1.1and(1.4 belong to the Ambrosetti Prodi class of problems p u = g(x, u in u = on,

3 Asymmetric critical p-laplacian problems Page 3 of where the nonlinearity g is asymmetric in that the limits g(x, t g = lim t t p 2 t, g g(x, t + = lim t + t p 2 t are different (see Zhang et al. [32]. These problems are of the following three types: (I g <λ 1 (p <g +,whereg = and g + =+ are admissible; (II g and g + are finite and the interval (g, g + contains eigenvalues of p, i.e., g is a jumping nonlinearity (see [5,21] and their references, and the problem is related to the Dancer Fučik spectrum (see [9,33]; (III g is between two consecutive eigenvalues of p and g + =+, i.e., g is asymptotically p-linear at, superlinear at +, and crosses all but a finite number of eigenvalues (see [4,24] and their references. Our problems are of type (III with critical growth at + and g >λ 1,sotheyare different from the critical problems considered in [14,31], and our results complement those in [6,8,12,15,32] concerning the semilinear case p = 2. However, the linking arguments based on eigenspaces of used in those papers do not apply to the quasilinear case p = 2 since the nonlinear operator p does not have linear eigenspaces. Therefore we will use more general constructions based on sublevel sets as in Perera and Szulkin [28]. Moreover, the standard sequence of eigenvalues of p based on the genus does not provide sufficient information about the structure of the sublevel sets to carry out these linking constructions, so we will use a different sequence of eigenvalues introduced in Perera [26] that is based on a cohomological index. The Z 2 -cohomological index of Fadell and Rabinowitz [17] is defined as follows. Let W be a Banach space and let A denote the class of symmetric subsets of W \ {}.ForA A,let A = A/Z 2 be the quotient space of A with each u and u identified, let f : A RP be the classifying map of A,andlet f : H (RP H (A be the induced homomorphism of the Alexander-Spanier cohomology rings. The cohomological index of A is defined by if A =, i(a = sup { m 1 : f (ω m 1 = } if A =, where ω H 1 (RP is the generator of the polynomial ring H (RP = Z 2 [ω]. Example 1.3 The classifying map of the unit sphere S m 1 in R m, m 1 is the inclusion RP m 1 RP, which induces isomorphisms on the cohomology groups H q for q m 1, so i(s m 1 = m. The following proposition summarizes the basic properties of this index. Proposition 1.4 (Fadell Rabinowitz [17]. The indexi : A {, } has the following properties: (i 1 Definiteness: i(a = if and only if A =. (i 2 Monotonicity: If there is an odd continuous map from A to B (in particular, if A B, then i(a i(b. Thus, equality holds when the map is an odd homeomorphism. (i 3 Dimension: i(a dim W. (i 4 Continuity: If A is closed, then there is a closed neighborhood A of A such that i( = i(a. When A is compact, may be chosen to be a δ-neighborhood δ (A = {u W : dist (u, A δ}.

4 131 Page 4 of 18 K. Perera et al. (i 5 Subadditivity: If A and B are closed, then i(a B i(a + i(b. (i 6 Stability: If S A is the suspension of A =, obtained as the quotient space of A [ 1, 1] with A {1} and A { 1} collapsed to different points, then i(sa = i(a + 1. (i 7 Piercing property: If A, A and A 1 are closed, and ϕ : A [, 1] A A 1 is a continuous map such that ϕ( u, t = ϕ(u, t for all (u, t A [, 1], ϕ(a [, 1] is closed, ϕ(a {} A and ϕ(a {1} A 1,theni(ϕ(A [, 1] A A 1 i(a. (i 8 eighborhood of zero: If U is a bounded closed symmetric neighborhood of, then i( U = dim W. For 1 < p <, eigenvalues of problem (1.2 coincide with critical values of the functional { } 1 (u =, u M = u W 1,p u p ( : u p dx = 1. dx Let F denote the class of symmetric subsets of M and set λ k (p := inf sup (u, k. M F, i(m k u M Then <λ 1 (p <λ 2 (p λ 3 (p is a sequence of eigenvalues of (1.2and λ k (p <λ k+1 (p i( λ k(p = i(m\ λk+1 (p = k, (1.5 where a = {u M : (u a} and a = {u M : (u a} for a R (see Perera et al. [27, Propositions 3.52 and 3.53]. As we will see, problems (1.1 and(1.4 have nontrivial solutions as long as λ is not an eigenvalue from the sequence (λ k (p. This leaves open the question of existence of nontrivial solutions when λ belongs to this sequence. We will prove Theorems 1.1 and 1.2 using the following abstract critical point theorem proved in Yang andperera[31], which generalizes the well-known linking theorem of Rabinowitz [29]. Theorem 1.5 Let be a C 1 -functional defined on a Banach space W and let A and B be disjoint nonempty closed symmetric subsets of the unit sphere S = {u W : u = 1} such that i(a = i(s\b <. Assume that there exist R > r > and v S\A such that where sup (A inf (B, sup (X <, (1.6 A = {tu : u A, t R} {R π((1 t u + tv : u A, t 1}, B = {ru : u B }, X = {tu : u A, u = R, t 1}, and π : W \ {} S, u u/ u is the radial projection onto S. Let Ɣ ={γ C(X, W : γ(x is closed and γ A = id A } and set c := inf sup (u. γ Ɣ u γ(x

5 Asymmetric critical p-laplacian problems Page 5 of Then inf (B c sup (X, (1.7 in particular, c is finite. If, in addition, satisfies the (C c condition, then c is a critical value of. This theorem was stated and proved under the Palais-Smale compactness condition in [31], but the proof goes through unchanged since the first deformation lemma also holds under the Cerami condition (see, e.g., Perera et al. [27, Lemma 3.7]. The linking construction used in the proof has also been used in Perera and Szulkin [28] to obtain nontrivial solutions of p- Laplacian problems with nonlinearities that cross an eigenvalue. A similar construction based on the notion of cohomological linking was given in Degiovanni and Lancelotti [13](see also Perera et al. [27, Proposition 3.23]. However, the application of Theorem 1.5 to problems (1.1 and(1.4 is nontrivial. Indeed, verification of the Cerami compactness condition below a suitable threshold level, based on the concentration compactness principle and the fact that only the first eigenfunction of p has constant sign, is more involved (see Lemma 2.1. Moreover, establishing the inequalities in (1.6 under the Ambrosetti Prodi term u + requires more careful estimates [see (2.13and(3.15]. 2 Proof of Theorem 1.1 Weak solutions of problem (1.1 coincide with critical points of the C 1 -functional [ 1 ( (u = u p λ u p 1 ] p p u p + dx, u W 1,p (. We recall that satisfies the Cerami compactness condition at the level c R, orthe (C c condition for short, if every sequence ( 1,p ( u j W ( such that (u j c and 1 + u j (u j, called a (C c sequence, has a convergent subsequence. Let S = inf u W 1,p (\{} be the best constant in the Sobolev inequality. ( u p dx p/p (2.1 u p dx Lemma 2.1 If λ = λ 1 (p, then satisfies the (C c condition for all c < 1 S /p. Proof Let c < 1 S /p and let ( u j be a (Cc sequence. First we show that ( u j is bounded. We have [ 1 ( u j p λ u j p 1 ] p p u p j+ dx = c + o(1 (2.2 and ( u j p 2 u j v λ u j p 2 u j v u p 1 j+ v o(1 v dx = 1 +, u j v W 1,p (. (2.3

6 131 Page 6 of 18 K. Perera et al. Taking v = u j in (2.3 and combining with (2.2gives u p j+ dx = c + o(1, (2.4 and taking v = u j+ in (2.3gives u j+ p dx = (λ u p j+ + u p j+ dx + o(1, so ( 1,p u j+ is bounded in W (. Suppose ρ j := u j for a renamed subsequence. Then ũ j := u j /ρ j converges to some ũ weakly in W 1,p (, strongly in L q ( for 1 q < p, and a.e. in for a further subsequence. Since the sequence ( u j+ is bounded, dividing (2.2byρ p j and (2.3byρ p 1, and passing to the limit then gives 1 = λ ũ p dx, j ũ p 2 ũ vdx = λ ũ p 1 ũ v dx v W 1,p (, respectively. Moreover, since ũ j+ = u j+ /ρ j, ũ a.e. Hence ũ = tϕ 1 for some t < andλ = λ 1 (p, contrary to assumption. Since ( u j is bounded, so is ( u j+, a renamed subsequence of which then converges to some v weakly in W 1,p (, strongly in L q ( for 1 q < p and a.e. in, and u j+ p dx w μ, u p w j+ dx ν (2.5 in the sense of measures, where μ and ν are bounded nonnegative measures on (see, e.g., Folland [18]. By the concentration compactness principle of Lions [22,23], then there exist an at most countable index set I and points x i, i I such that μ v p dx + μ i δ xi, i I ν = v p dx + i I ν i δ xi, (2.6 where μ i,ν i > andν p/p i μ i /S. Letϕ : R [, 1] be a smooth function such that ϕ(x = 1for x 1andϕ(x = for x 2. Then set ( x xi ϕ i,ρ (x = ϕ, x R ρ for i I and ρ >, and note that ϕ i,ρ : R [, 1] is a smooth function such that ϕ i,ρ (x = 1for x x i ρ and ϕ i,ρ (x = for x x i 2ρ. The sequence ( ϕ i,ρ u j+ is bounded in W 1,p ( and hence taking v = ϕ i,ρ u j+ in (2.3gives (ϕ i,ρ u j+ p + u j+ u j+ p 2 u j+ ϕ i,ρ λϕ i,ρ u p j+ ϕ i,ρ u p j+ By (2.5, dx = o(1. ϕ i,ρ u j+ p dx ϕ i,ρ dμ, ϕ i,ρ u p j+ dx Denoting by C a generic positive constant independent of j and ρ, ( u j+ u j+ p 2 u j+ ϕ i,ρ λϕ i,ρ u p j+ dx C ( I 1/p j ρ ϕ i,ρ dν. + I j, (2.7

7 Asymmetric critical p-laplacian problems Page 7 of where I j := B 2ρ (x i So passing to the limit in (2.7gives ϕ i,ρ dμ ( u p j+ dx v p dx Cρ p v p dx B 2ρ (x i B 2ρ (x i ( ϕ i,ρ dν C v p dx B 2ρ (x i 1/p + B 2ρ (x i p/p. v p dx. Letting ρ and using (2.6 nowgivesμ i ν i, which together with ν p/p i μ i /S then gives ν i = orν i S /p. Passing to the limit in (2.4 and using (2.5 and(2.6 gives ν i c < S /p,soν i =. Hence I = and u p j+ dx v p dx. (2.8 Passing to a further subsequence, u j converges to some u weakly in W 1,p (, strongly in L q ( for 1 q < p, and a.e. in. Since ( u p 1 j+ (u j u u p j+ + u p 1 j+ u 2 1 p u p j+ + 1 p u p by Young s inequality, u p 1 j+ (u j u dx by (2.8 and the dominated convergence theorem. Then u j u in W 1,p ( by a standard argument. We recall that the infimum in (2.1 is attained by the family of functions C,p ε ( p/p u ε (x = [ ( ] x p/(p 1 ( p/p, ε > 1 + ε when = R,whereC,p > is chosen so that u ε p dx = u p R R ε dx = S /p. Take a smooth function η :[, [, 1] such that η(s = 1fors 1/4 andη(s = for s 1/2, and set ( x u ε,δ (x = η u ε (x, ε, δ >. δ

8 131 Page 8 of 18 K. Perera et al. We have the well-known estimates ( ε ( p/(p 1 u ε,δ p dx S /p + C, (2.9 R δ ( ε /(p 1 u p R ε,δ dx S /p C, (2.1 δ ε p ( u p R ε,δ dx C Cδ p ε ( p/(p 1 if > p 2 δ ε p ( (2.11 δ C log Cε p if = p 2, ε where C = C(, p > is a constant (see, e.g., Degiovanni and Lancelotti [14]. Let i, M,, andλ k (p be as in the introduction, and suppose that λ k (p <λ k+1 (p. Then the sublevel set λk(p has a compact symmetric subset E of index k that is bounded in L ( C 1,α loc ( (see [14, Theorem 2.3]. We may assume without loss of generality that.letδ = dist (,, take a smooth function θ :[, [, 1] such that θ(s = for s 3/4 andθ(s = 1fors 1, set ( x v δ (x = θ δ v(x, v E, <δ δ 2, and let E δ = {π(v δ : v E}, whereπ : W 1,p (\ {} M, u u/ u is the radial projection onto M. Lemma 2.2 There exists a constant C = C(, p,,k > such that for all sufficiently small δ>, (i (w λ k (p + Cδ p w E δ, (ii E δ λk+1 (p =, (iii i(e δ = k, (iv supp w supp π(u ε,δ = w E δ, (v π(u ε,δ / E δ. Proof Let v E and let w = π(v δ.wehave v δ p dx v p dx + C \B δ ( B δ ( ( v p + v p δ p dx 1 + Cδ p since E M is bounded in C 1 (B δ /2(, and v δ p dx v p dx = v p dx v p dx 1 \B δ ( B δ ( λ k (p Cδ since E λ k(p,so v δ p dx (w = λ k (p + Cδ p v δ p dx if δ> is sufficiently small. Taking δ so small that λ k (p + Cδ p <λ k+1 (p then gives (ii. Since E δ M\ λk+1 (p by (ii, i(e δ i(m\ λk+1 (p = k

9 Asymmetric critical p-laplacian problems Page 9 of by the monotonicity of the index and (1.5. On the other hand, since E E δ,v π(v δ is an odd continuous map, i(e δ i(e = k. So i(e δ = k. Since supp w = supp v δ \B 3δ/4 ( and supp π(u ε,δ = supp u ε,δ B δ/2 (, (ivis clear, and (v is immediate from (iv. We are now ready to prove Theorem 1.1. Proof of Theorem 1.1 We have λ k (p <λ<λ k+1 (p for some k. Fixλ k (p <λ <λ and δ> so small that the conclusions of Lemma 2.2 hold with λ k (p + Cδ p λ,in particular, (w λ w E δ. (2.12 Then take A = E δ and B = λk+1 (p, and note that A and B are disjoint nonempty closed symmetric subsets of M such that i(a = i(m\b = k by Lemma 2.2 (iii and (1.5. ow let R > r >, let v = π(u ε,δ, which is in M\E δ by Lemma 2.2 (v, and let A, B, andx be as in Theorem 1.5. For u λk+1 (p, (ru 1 p ( 1 λ r p λ k+1 (p 1 p S p /p r p by (2.1. Since λ<λ k+1 (p, it follows that inf (B >ifr is sufficiently small. ext we show that ona if R is sufficiently large. For w E δ and t, (tw t p ( 1 λ t p ( λ p (w p λ 1 by (2.12. ow let t 1andsetu = π((1 tw+ tv.since (1 tw+ tv (1 t w + t v = 1 and since the supports of w and v are disjoint by Lemma 2.2 (iv, u p p = (1 tw+ tv p p (1 tw+ tv p (1 tp w p p + t p v p (1 tp p (w by (2.12, and u + p p = [(1 tw+ tv ] + p p (1 tw+ tv p (1 tp λ uε,δ p (1 t p w + p p + t p v p p t p p uε,δ p by (2.9and(2.1ifε is sufficiently small, where C = C(, p,,k >. Then (Ru = R p p u p λr p p u p p R [ ] p λ p u + p p 1 p λ (1 tp 1 R p t p t p C C R p. (2.13

10 131 Page 1 of 18 K. Perera et al. The last expression is clearly nonpositive if t 1 (λ /λ 1/p =: t.fort > t,itis nonpositive if R is sufficiently large. ow we show that sup (X < 1 S /p if ε is sufficiently small. oting that X = {ρπ((1 tw+ tv : w E δ, t 1, ρ R}, let w E δ,let t 1, and set u = π((1 tw+ tv.then [ ] ρ p ( p ρ p sup (ρu sup 1 λ u p ρ R ρ p p u + p p = 1 S u(λ /p when 1 λ u p p >, where S u (λ = 1 λ u p p u + p = (1 tw+ tv p p λ (1 tw+ tv p p [(1 tw+ tv ] + p p = (1 ( tp w p λ w p p + t p ( v p λ v p p ] p/p [(1 t p w + p p + t p v p. p Since w p λ w p p = 1 λ/(w by(2.12, S u (λ 1 λ v p uε,δ p p λ uε,δ p S ε p p C + Cε( p/(p 1 if > p 2 v p = p uε,δ p p S ε p C log ε +Cεp if = p 2 by (2.9 (2.11. In both cases the last expression is strictly less than S if ε is sufficiently small. The inequalities (1.7 now imply that < c < 1 S /p.thensatisfies the (C c condition by Lemma 2.1 and hence c is a critical value of by Theorem Proof of Theorem 1.2 Weak solutions of problem (1.4 coincide with critical points of the C 1 -functional [ ] 1 (u = u λ F(u dx, u W 1, (, where F(t = t First we obtain some estimates for the primitive F. Lemma 3.1 For all t R, s 2 se s + ds. F(t t + 2 e t + + t + C, (3.1 F(t t 1 e t + + t + C, (3.2

11 Asymmetric critical p-laplacian problems Page 11 of where C denotes a generic positive constant and t = max { t, }. Proof For t, F(t = t /. Fort >, integrating by parts gives t F(t = s 1 e s ds = t e t t s + 1 e s ds. For t (/ 1/, the last term is greater than or equal to t s + 1 e s ds (/ 1/ and hence t (/ 1/ s 1 e s ds = F(t F((/ 1/ 2F(t t e t + F((/ 1/. Since F is bounded on bounded sets, (3.1 follows. As for (3.2, F(t = (e t2 1/2 for t > if = 2, and F(t = t e t t s 1 e s ds t 1 e t for t 1/( 1/( 1 if 3. Proof of Theorem 1.2 will be based on the following lemma. Lemma 3.2 If λ = λ 1 ( and = c <α 1 /, then every (C c sequence has a subsequence that converges weakly to a nontrivial critical point of. Proof Let λ = λ 1 (,let = c <α 1 /,andlet ( u j be a (Cc sequence. First we show that ( u j is bounded. We have [ ] 1 u j λ F(u j dx = c + o(1 (3.3 and ( u j 2 u j v λ u j 2 u j e u j+ o(1 v v dx = 1 + v W u 1, j (, (3.4 in particular, ( u j λ u j e u j+ dx = o(1. (3.5 Combining (3.5 with (3.3and(3.1gives u j+ e u j+ dx C, (3.6 and taking v = u j+ in (3.4gives u j+ dx = λ u j+ e u j+ dx + o(1,

12 131 Page 12 of 18 K. Perera et al. so the sequence ( u j+ is bounded in W 1, (. Passing to a subsequence, u j+ then converges to some û weakly in W 1, (, strongly in L q ( for 1 q <, and a.e. in. Then for any v C (, u 1 j+ e u j+ v dx û 1 e û v dx (3.7 by de Figueiredo et al. [11, Lemma 2.1] and (3.6. ow suppose ρ j := u j. Then ũ j := u j /ρ j converges to some ũ weakly in W 1, (, strongly in L q ( for 1 q <, and a.e. in for a further subsequence. Taking v = u j in (3.4, dividing by ρ j, and passing to the limit then gives 1 = λ ũ dx, so ũ =. Since the sequence ( u j+ is bounded, dividing (3.4byρ 1 j gives ũ j 2 ũ j vdx = λ ũ 1 j v dx λ ρ 1 u 1 j+ e u j+ v dx + o(1, j and passing to the limit using (3.7gives ũ 2 ũ vdx = λ ũ 1 v dx v C (. This then holds for all v W 1, ( by density, so ũ = tϕ 1 for some t > andλ = λ 1 (, contrary to assumption. Since the sequence ( u j is bounded, a renamed subsequence converges to some u weakly in W 1, (, strongly in L q ( for 1 q <, and a.e. in. Since u j e u j+ dx is bounded by (3.5, then for any v C (, u j 2 u j e u j+ v dx u 2 ue u + v dx by de Figueiredo et al. [11, Lemma 2.1]. So passing to the limit in (3.4gives ( u 2 u v λ u 2 ue u + v dx =. This then holds for all v W 1, ( by density, so u is a critical point of. Suppose u =. Then u j 1 e u j+ dx by de Figueiredo et al. [11, Lemma 2.1] as above, and hence F(u j dx by (3.2 and the dominated convergence theorem, so u j dx c

13 Asymmetric critical p-laplacian problems Page 13 of by (3.3. Since c <α 1 /, then lim sup u j <α 1/, so there exists β>1/α 1/ such that β u j 1 for all sufficiently large j.for1<γ < givenby1/α β + 1/γ = 1, then ( 1/γ ( 1/α u j e u j+ dx u j γ dx e α β (βu j+ dx since u j inl γ ( and the last integral is bounded by (1.3. Then u j inw 1, ( by (3.5, so (u j, contradicting c =. Let i, M,, andλ k ( be as in the introduction, and suppose that λ k ( <λ k+1 (. Then the sublevel set λk( has a compact symmetric subset E of index k that is bounded in L ( C 1,α loc ( (see Degiovanni and Lancelotti [14, Theorem 2.3]. We may assume without loss of generality that. Forallm so large that B 2/m (, let if x 1/2 m m+1 2 m m 1 ( x 2 m η m (x = m+1 if 1/2 m m+1 < x 1/m m+1 (m x 1/m if 1/m m+1 < x 1/m set 1 if x > 1/m, v m (x = η m (xv(x, v E, and let E m = {π(v m : v E}, whereπ : W 1, (\ {} M, u u/ u is the radial projection onto M. Lemma 3.3 There exists a constant C = C(,,k > such that for all sufficiently large m, (i (w λ k ( + C m 1 w E m, (ii E m λk+1 ( =, (iii i(e m = k. Proof Let v E and let w = π(v m.wehave ( v m dx v dx + ηm j v j v j η m j dx. \B 1/m ( j B 1/m ( j= Since E is bounded in C 1 (B 1/m (, v and v are bounded in B 1/m (. Clearly, η m 1, and a direct calculation shows that Since E m M, it follows that η m j dx B 1/m ( C m 1, v m dx 1 + j =,...,. C m 1.

14 131 Page 14 of 18 K. Perera et al. ext v m dx \B 1/m ( v dx = v dx v dx 1 B 1/m ( λ k ( C m since E λk(.so v m dx (w = λ k ( + C v m m dx 1 if m is sufficiently large. Taking m so large that λ k ( + C/m 1 <λ k+1 ( then gives (ii. SinceE m M\ λk+1 ( by (ii, i(e m i(m\ λk+1 ( = k by the monotonicity of the index and (1.5. On the other hand, since E E m,v π(v m is an odd continuous map, i(e m i(e = k. So i(e m = k. We are now ready to prove Theorem 1.2. Proof of Theorem 1.2 We have λ k ( <λ<λ k+1 ( for some k.fixλ k ( <λ <λ and m so large that the conclusions of Lemma 3.3 hold with λ k ( + C/m 1 λ,in particular, (w λ w E m. (3.8 Then take A = E m and B = λk+1 (, and note that A and B are disjoint nonempty closed symmetric subsets of M such that i(a = i(m\b = k by Lemma 3.3 (iii and (1.5. ow let R > r > andleta and B be as in Theorem 1.5. First we show that inf (B >ifr is sufficiently small. Since e t 1+te t for all t >, F(t t + tμ + e t + t R, where μ = + >.Soforu λk+1 (, (ru r [ r u λr u λr μ u μ + e r u + ] dx ( ( λ 1/2 1 λr μ e 2r u+ dx u + μ 2μ λ k+1 (. If 2 r α,then e 2r u+ dx e α u+ dx,

15 Asymmetric critical p-laplacian problems Page 15 of which is bounded by (1.3. Since W 1, ( L 2μ ( and λ<λ k+1 (, it follows that inf (B >ifr is sufficiently small. Since e t 1 + t for all t >, F(t t + tμ + t R, (3.9 μ so for all w E m and t, [ t (tw w λt ] w dx = t ( 1 λ (w t ( λ λ 1 (3.1 by (3.8. ext we show that sup w E m, s, t (sw + tv < α 1 for a suitably chosen v M\E m.let (log j ( 1/ if x 1/ j v j (x = 1 log x 1 ω 1/ 1 (log j 1/ if 1/ j < x 1 if x > 1. Then v j W 1, (R, v j = 1, and v j = O(1/ log j as j.wetake ( x v (x = ṽ j (x := v j r m with r m = 1/2 m m+1 and j sufficiently large. Since B rm (, ṽ j W 1, ( and ṽ j = 1. For sufficiently large j, 1 (ṽ j = λ rm v j and hence ṽ j / E m by (3.8. For w E m and s, t, (sw + tṽ j = (sw + (tṽ j since w = onb rm ( and ṽ j = on\b rm (. Since(sw by(3.1, it suffices to show that sup t (tṽ j < α 1 for arbitrarily large j. Since(tṽ j as t by (3.9, there exists t j such that (t j ṽ j = t j λ F(t j ṽ j dx = sup (tṽ j (3.11 B rm ( t

16 131 Page 16 of 18 K. Perera et al. and ( (t j ṽ j ṽ j = t j 1 1 λ ṽ j e t B rm ( j ṽ j dx =. (3.12 Suppose (t j ṽ j α 1 / for all sufficiently large j. SinceF(t forallt R, then (3.11givest j α,andthen(3.12gives 1 λ = = r m ṽ j e t B rm ( B 1 ( j ṽ j dx B rm ( v j e α v j dx rm B 1/ j ( ṽ j e α ṽ j dx v j e α v j dx = r m (log j 1, which is impossible for large j. owweshowthat ona if R is sufficiently large. In view of (3.1, it only remains to show that (Ru foru = π((1 tw+ tv, w E m, t 1. Since (1 tw+ tv (1 t w + t v = 1 and w and v are supported on disjoint sets, we have by (3.8, and u = (1 tw+ tv (1 tw+ tv (1 t w + t v (1 t (w (1 t λ (3.13 u + μ μ = [(1 tw+ tv ] + μ μ (1 tw+ tv μ (1 tμ w + μ μ + tμ v μ μ tμ v μ μ. (3.14 By (3.9, (3.13, and (3.14, (Ru R u λr [ ] u Rμ λ μ u + μ μ 1 λ (1 t 1 R 1 μ v μ μ tμ R μ. (3.15 The last expression is clearly nonpositive if t 1 (λ /λ 1/ =: t.fort > t,itis nonpositive if R is sufficiently large. The inequalities (1.7 now imply that < c < α 1.If has no (C c sequences, then satisfies the (C c condition trivially and hence c is a critical value of by Theorem 1.5. If has a (C c sequence, then a subsequence converges weakly to a nontrivial critical point of by Lemma 3.2. Acknowledgements This work was completed while the first- and second-named authors were visiting the Academy of Mathematics and Systems Science at the Chinese Academy of Sciences in 215, and they are grateful for the kind hospitality of the host institution.

17 Asymmetric critical p-laplacian problems Page 17 of References 1. Adimurthi, : Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the n-laplacian. Ann. Scuola orm. Sup. Pisa Cl. Sci. 17(3, ( Amann, H., Hess, P.: A multiplicity result for a class of elliptic boundary value problems. Proc. R. Soc. Edinb. Sect. A 84(1 2, ( Ambrosetti, A., Prodi, G.: On the inversion of some differentiable mappings with singularities between Banach spaces. Ann. Mat. Pura Appl. 93(4, ( Arias, M., Cuesta, M.: A one side superlinear Ambrosetti Prodi problem for the Dirichlet p-laplacian. J. Math. Anal. Appl. 367(2, (21 5. Arcoya, D., Ruiz, D.: The Ambrosetti Prodi problem for the p-laplacian. Commun. Partial Differ. Equ. 31(4 6, (26 6. Aubin, T., Wang, W.: Positive solutions of Ambrosetti Prodi problems involving the critical Sobolev exponent. Bull. Sci. Math. 125(4, (21 7. Berger, M.S., Podolak, E.: On the solutions of a nonlinear Dirichlet problem. Indiana Univ. Math. J. 24, (1974/75 8. Calanchi, M., Ruf, B.: Elliptic equations with one-sided critical growth. Electron. J. Differ. Equ. 89, 21 (22 9. Cuesta, M., de Figueiredo, D., Gossez, J.P.: The beginning of the Fučik spectrum for the p-laplacian. J. Dffer. Equ. 159, ( Dancer, E..: On the ranges of certain weakly nonlinear elliptic partial differential equations. J. Math. Pures Appl. 57(4, ( de Figueiredo, D.G., Miyagaki, O.H., Ruf, B.: Elliptic equations in R 2 with nonlinearities in the critical growth range. Calc. Var. Partial Differ. Equ. 3(2, ( de Figueiredo, D.G., Yang, J.: Critical superlinear Ambrosetti Prodi problems. Topol. Methods onlinear Anal. 14(1, 59 8 ( Degiovanni, M., Lancelotti, S.: Linking over cones and nontrivial solutions for p-laplace equations with p-superlinear nonlinearity. Ann. Inst. H. Poincaré Anal. onlinéaire 24(6, ( Degiovanni, M., Lancelotti, S.: Linking solutions for p-laplace equations with nonlinearity at critical growth. J. Funct. Anal. 256(11, ( Deng, Y.: On the superlinear Ambrosetti Prodi problem involving critical Sobolev exponents. onlinear Anal. 17(12, ( Do Ó, J.M.B.: Semilinear Dirichlet problems for the -Laplacian in R with nonlinearities in the critical growth range. Differ. Integral Equ. 9(5, ( Fadell, E.R., Rabinowitz, P.H.: Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems. Invent. Math. 45(2, ( Folland, G.B.: Real Analysis. Pure and Applied Mathematics. Modern Techniques and Their Applications, 2nd edn. Wiley, ew York ( Guedda, M., Véron, L.: Quasilinear elliptic equations involving critical Sobolev exponents. onlinear Anal. 13(8, ( Kazdan, J.L., Warner, F.W.: Remarks on some quasilinear elliptic equations. Commun. Pure Appl. Math. 28(5, ( Koizumi, E., Schmitt, K.: Ambrosetti Prodi-type problems for quasilinear elliptic problems. Differ. Integral Equ. 18(3, ( Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The limit case. I. Rev. Mat. Iberoam. 1(1, ( Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The limit case. II. Rev. Mat. Iberoam. 1(2, ( Miotto, T.J.: Superlinear Ambrosetti Prodi problem for the p-laplacian operator. onlinear Differ. Equ. Appl. 17, ( Moser, J.: A sharp form of an inequality by. Trudinger. Indiana Univ. Math. J. 2, (197/ Perera, K.: ontrivial critical groups in p-laplacian problems via the Yang index. Topol. Methods onlinear Anal. 21(2, ( Perera, K., Agarwal, R.P., O Regan, D.: Morse Theoretic Aspects of p-laplacian Type Operators, Mathematical Surveys and Monographs, vol American Mathematical Society, Providence ( Perera, K., Szulkin, A.: p-laplacian problems where the nonlinearity crosses an eigenvalue. Discrete Contin. Dyn. Syst. 13(3, ( Rabinowitz, P.H.: Some critical point theorems and applications to semilinear elliptic partial differential equations. Ann. Scuola orm. Sup. Pisa Cl. Sci. 5(1, (1978

18 131 Page 18 of 18 K. Perera et al. 3. Trudinger,.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, ( Yang, Y., Perera, K.: -Laplacian problems with critical Trudinger Moser nonlinearities. Ann. Sc. orm. Super. Pisa Cl. Sci. 16(4, ( Zhang, Z., Calanchi, M., Ruf, B.: Elliptic equations in R 2 with one-sided exponential growth. Commun. Contemp. Math. 6(6, ( Zhang, Z.: Variational, Topological, and Partial Order Methods with Their Applications. Springer, Berlin (213