p-laplacian problems involving critical Hardy Sobolev exponents

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1 Nonlinear Differ. Equ. Appl. 2018) 25:25 c 2018 Springer International Publishing AG, part of Springer Nature /18/ published online June 4, Nonlinear Differential Equations and Applications NoDEA p-laplacian problems involving critical Hardy Sobolev exponents Kanishka Perera and Wenming Zou Abstract. We prove existence, multiplicity, and bifurcation results for p- Laplacian problems involving critical Hardy Sobolev exponents. Our results are mainly for the case λ λ 1 and extend results in the literature for 0 <λ<λ 1. In the absence of a direct sum decomposition, we use critical point theorems based on a cohomological index and a related pseudo-index. Mathematics Subject Classification. Primary 35J92, 35B33; Secondary 35J20. Keywords. p-laplacian problems, Critical Hardy Sobolev exponents, Existence, Multiplicity, Bifurcation, Critical point theory, Cohomological index, Pseudo-index. 1. Introduction Consider the critical p-laplacian problem { Δ p u = λ u p 2 u + u p s) 2 x u in s u =0 on, 1.1) where is a bounded domain in R N containing the origin, 1 <p<n, λ>0 is a parameter, 0 <s<p,andp s) =N s) p/n p) is the critical Hardy Sobolev exponent. Ghoussoub and Yuan [6] showed, among other things, that this problem has a positive solution when N p 2 and 0 <λ<λ 1, where λ 1 > 0 is the first eigenvalue of the eigenvalue problem { Δp u = λ u p 2 u in 1.2) u =0 on. In the present paper we mainly consider the case λ λ 1. Our existence results are the following.

2 25 Page 2 of 16 K. Perera and W. Zou NoDEA Theorem 1.1. If N p 2 and 0 <λ<λ 1, then problem 1.1) has a positive ground state solution. Theorem 1.2. If N p 2 and λ>λ 1 is not an eigenvalue of problem 1.2), then problem 1.1) has a nontrivial solution. Theorem 1.3. If N p 2 )N s) > p s) p 1.3) and λ λ 1, then problem 1.1) has a nontrivial solution. Remark 1.4. We note that 1.3) implies N>p 2. Remark 1.5. In the nonsingular case s = 0, related results can be found in Degiovanni and Lancelotti [4] for the p-laplacian and in Mosconi et al. [7] for the fractional p-laplacian. Weak solutions of problem 1.1) coincide with critical points of the C 1 - functional [ 1 I λ u) = u p λ u p) 1 ] u p s) p p s), u W 1,p 0 ). Recall that I λ satisfies the Palais-Smale compactness condition at the level c R, ortheps) c condition for short, if every sequence u j ) W 1,p 0 ) such that I λ u j ) c and I λ u j) 0 has a convergent subsequence. Let u p μ s = inf 1.4) u W 1,p 0 )\{0} u p s) )p/p s) be the best constant in the Hardy Sobolev inequality, which is independent of see [6, Theorem 3.1.1)]). It was shown in [6, Theorem 4.1.2)] that I λ satisfies the PS) c condition for all c< p s N s) p μn s)/p s) s for any λ>0. We will prove Theorems by constructing suitable minimax levels below this threshold for compactness. When 0 <λ<λ 1,we will show that the infimum of I λ on the Nehari manifold is below this level. When λ λ 1, I λ no longer has the mountain pass geometry and a linking type argument is needed. However, the classical linking theorem cannot be used here since the nonlinear operator Δ p does not have linear eigenspaces. We will use a nonstandard linking construction based on sublevel sets as in Perera and Szulkin [11] see also Perera et al. [9, Proposition 3.23]). Moreover, the standard sequence of eigenvalues of Δ p based on the genus does not give enough information about the structure of the sublevel sets to carry out this construction. Therefore, we will use a different sequence of eigenvalues introduced in Perera [8] that is based on a cohomological index.

3 NoDEA p-laplacian problems Page 3 of For 1 <p<, eigenvalues of problem 1.2) coincide with critical values of the functional { } 1 Ψu) =, u M= u W 1,p u p 0 ) : u p =1. Let F denote the class of symmetric subsets of M, letim) denote the Z 2 - cohomological index of M F see Sect. 2.1), and set λ k := inf sup Ψu), k N. M F, im) k u M Then 0 <λ 1 <λ 2 λ 3 is a sequence of eigenvalues of 1.2) and λ k <λ k+1 = iψ λ k )=im\ψ λk+1 )=k, 1.5) where Ψ a = {u M:Ψu) a} and Ψ a = {u M:Ψu) a} for a R see Perera et al. [9, Propositions 3.52 and 3.53]). We also prove the following bifurcation and multiplicity results for problem 1.1) that do not require N p 2. Set V s ) = x N p) s/p s), and note that u p V s ) p s)/n s) by the Hölder inequality. Theorem 1.6. If λ 1 u p s) )p/p s) μ s V s ) p s)/n s) <λ<λ 1, u W 1,p 0 ) 1.6) then problem 1.1) has a pair of nontrivial solutions ± u λ such that u λ p λ 1 λ 1 λ) N p)/p s) V s ). Theorem 1.7. If λ k λ<λ k+1 = = λ k+m <λ k+m+1 for some k, m N and μ s λ>λ k+1, 1.7) V s )p s)/n s) then problem 1.1) has m distinct pairs of nontrivial solutions ± u λ j,j = 1,...,m such that u λ j p λ k+1 λ k+1 λ) N p)/p s) V s ). 1.8) In particular, we have the following existence result that is new when N<p 2. Corollary 1.8. If λ k μ s V s ) p s)/n s) <λ<λ k for some k N, then problem 1.1) has a nontrivial solution.

4 25 Page 4 of 16 K. Perera and W. Zou NoDEA Remark 1.9. We note that λ 1 μ s /V s ) p s)/n s). Indeed, let ϕ 1 > 0be an eigenfunction associated with λ 1. Then ϕ 1 p μ s λ 1 = ϕ p 1 by 1.4) and 1.6). ϕ p s) 1 ϕ p 1 ) p/p s) μ s V s ) p s)/n s) Remark Since V 0 ) is the volume of, in the nonsingular case s =0, Theorems 1.6 & 1.7 and Corollary 1.8 reduce to Perera et al. [10, Theorem 1.1 and Corollary 1.2], respectively. 2. Preliminaries 2.1. Cohomological index The Z 2 -cohomological index of Fadell and Rabinowitz [5] is defined as follows. Let W be a Banach space and let A denote the class of symmetric subsets of W \{0}. ForA A,letA = 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, and let f : H RP ) H A) be the induced homomorphism of the Alexander- Spanier cohomology rings. The cohomological index of A is defined by { 0 if A = ia) = sup { m 1:f ω m 1 ) 0 } if A, where ω H 1 RP ) is the generator of the polynomial ring H RP )= Z 2 [ω]. Example 2.1. The classifying map of the unit sphere S m 1 in R m,m 1is the inclusion RP m 1 RP, which induces isomorphisms on the cohomology groups H q for q m 1, so is m 1 )=m. The following proposition summarizes the basic properties of this index. Proposition 2.2. Fadell Rabinowitz [5]) The index i : A N {0, } has the following properties: i 1 ) Definiteness: ia) =0if and only if A =. i 2 ) Monotonicity: If there is an odd continuous map from A to B in particular, if A B), then ia) ib). Thus, equality holds when the map is an odd homeomorphism. i 3 ) Dimension: ia) dim W. i 4 ) Continuity: If A is closed, then there is a closed neighborhood N Aof A such that in) =ia). WhenA is compact, N may be chosen to be a δ-neighborhood N δ A) ={u W : dist u, A) δ}. i 5 ) Subadditivity: If A and B are closed, then ia B) ia)+ib).

5 NoDEA p-laplacian problems Page 5 of i 6 ) Stability: If SA 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 isa)=ia)+1. i 7 ) Piercing property: If A, A 0 and A 1 are closed, and ϕ : A [0, 1] A 0 A 1 is a continuous map such that ϕ u, t) = ϕu, t) for all u, t) A [0, 1], ϕa [0, 1]) is closed, ϕa {0}) A 0 and ϕa {1}) A 1, then iϕa [0, 1]) A 0 A 1 ) ia). i 8 ) Neighborhood of zero: If U is a bounded closed symmetric neighborhood of the origin, then i U) = dim W Abstract critical point theorems We will prove Theorems 1.2 and 1.3 using the following abstract critical point theorem proved in Yang and Perera [13], which generalizes the well-known linking theorem of Rabinowitz [12]. Theorem 2.3. Let I be a C 1 -functional defined on a Banach space W,andlet A 0 and B 0 be disjoint nonempty closed symmetric subsets of the unit sphere S = {u W : u =1} such that ia 0 )=is \ B 0 ) <. Assume that there exist R>r>0 and v S \ A 0 such that sup IA) inf IB), sup IX) <, where A = {tu : u A 0, 0 t R} {Rπ1 t) u + tv) :u A 0, 0 t 1}, B = {ru : u B 0 }, X = {tu : u A, u = R, 0 t 1}, and π : W \{0} S, u u/ u is the radial projection onto S. LetΓ= {γ CX, W) :γx) is closed and γ A = id A },andset c := inf sup Iu). γ Γ u γx) Then inf IB) c sup IX), 2.1) in particular, c is finite. If, in addition, I satisfies the PS) c condition, then c is a critical value of I. Remark 2.4. The linking construction used in the proof of Theorem 2.3 in [13] has also been used in Perera and Szulkin [11] 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 [3]. See also Perera et al. [9, Proposition 3.23]. Now let I be an even C 1 -functional defined on a Banach space W,and let A denote the class of symmetric subsets of W.Letr > 0, let S r = {u W : u = r}, let 0 < b +, and let Γ denote the group of odd

6 25 Page 6 of 16 K. Perera and W. Zou NoDEA homeomorphisms of W that are the identity outside I 1 0,b). The pseudoindex of M A related to i, S r, and Γ is defined by i M) = min iγm) S r) γ Γ see Benci [2]). We will prove Theorems 1.6 and 1.7 using the following critical point theorem proved in Yang and Perera [13], which generalizes Bartolo et al. [1, Theorem 2.4]. Theorem 2.5. Let A 0 and B 0 be symmetric subsets of S such that A 0 is compact, B 0 is closed, and ia 0 ) k + m, is \ B 0 ) k for some integers k 0 and m 1. Assume that there exists R>rsuch that sup IA) 0 < inf IB), sup IX) <b, where A={Ru : u A 0 }, B = {ru : u B 0 },andx ={tu : u A, 0 t 1}. For j = k +1,...,k+ m, let and set Then A j = {M A : M is compact and i M) j}, c j := inf M A j max Iu). u M inf IB) c k+1 c k+m sup IX), in particular, 0 <c j <b. If, in addition, I satisfies the PS) c condition for all c 0,b), then each c j is a critical value of I and there are m distinct pairs of associated critical points. Remark 2.6. Constructions similar to the one used in the proof of Theorem 2.5 in [13] have also been used in Fadell and Rabinowitz [5] to prove bifurcation results for Hamiltonian systems and in Perera and Szulkin [11] to prove multiplicity results for p-laplacian problems. See also Perera et al. [9, Proposition 3.44] Some estimates It was shown in [6, Theorem 3.1.2)] that the infimum in 1.4) is attained by the family of functions u ε x) = C N,p,s ε N p)/p s) p [ ε + x p s)/p 1) ] N p)/p s), ε > 0 when = R N, where C N,p,s > 0 is chosen so that p u ε p u s) ε = R N R = μ N s)/p s) s. N

7 NoDEA p-laplacian problems Page 7 of Take a smooth function η :[0, ) [0, 1] such that ηs) =1fors 1/4 and ηs) =0fors 1/2, and set ) x u ε,δ x) u ε,δ x) =η u ε x), v ε,δ x) = δ, ε,δ > 0, p u s) )1/p s) ε,δ R N so that p v s) ε,δ R =1. 2.2) N The following estimates were obtained in [6, Lemma ),3),4)]: v ε,δ p μ s + Cε N p)/p s), 2.3) R N 1 v p ε,δ C εp 1) p/p s) if N>p 2 2.4) R 1 N C εp 1) p/p s) log ε if N = p 2, where C = CN, p, s, δ) > 0 is a constant. While these estimates are sufficient for the proof of Theorem 1.2, we will need the following finer estimates in order to prove Theorem 1.3. Lemma 2.7. There exists a constant C = CN,p,s) > 0 such that v ε,δ p μ s + CΘ N p)/p s) ε,δ, 2.5) R N 1 v p ε,δ C εp 1) p/p s) if N>p 2 2.6) R 1 N C εp 1) p/p s) log Θ ε,δ if N = p 2, where Θ ε,δ = εδ p s)/p 1). Proof. We have and So and hence u ε,δ δx) =δ N p)/p u Θε,δ,1x) R N p u s) ε,δ = R N p u s) Θ ε,δ,1. v ε,δ δx) =δ N p)/p v Θε,δ,1x) v ε,δ δx) =δ N/p v Θε,δ,1x). Then v ε,δ x) p = δ N R N v ε,δ δx) p = R N v Θε,δ,1x) p R N

8 25 Page 8 of 16 K. Perera and W. Zou NoDEA and x) = δn R N v p ε,δ R N v p ε,δ δx) = δp R N v p Θ ε,δ,1 so 2.5) and 2.6) follow from 2.3) and 2.4), respectively. x), Let i, M, Ψ,andλ k be as in the introduction, and suppose that λ k < λ k+1. Then the sublevel set Ψ λ k has a compact symmetric subset E of index k that is bounded in L ) C 1,α loc ) see Degiovanni and Lancelotti [4, Theorem 2.3]). Let δ 0 =dist0, ), take a smooth function θ :[0, ) [0, 1] such that θs) =0fors 3/4 andθs) =1fors 1, and set ) x v δ x) =θ vx), v E, 0 <δ δ 0 δ 2. Since E Ψ λ k is bounded in C 1 B δ0/20)), ) v δ p v p + C v p + v p δ p 1+Cδ N p and v δ p \B δ 0) \B δ 0) v p = B δ 0) 2.7) v p v p 1 Cδ N, 2.8) B δ 0) λ k where C = CN, p, s,,k) > 0 is a constant. By 1.6) and 2.8), v δ p s) 1 2.9) C if δ>0 is sufficiently small. Now let π : W 1,p 0 )\{0} M,u u/ u be the radial projection onto M, and set w = πv δ ), v E. If δ>0 is sufficiently small, v δ p Ψw) = λ k + Cδ N p <λ k ) v δ p by 2.7) and 2.8), and w p s) = v δ p s) ) 1 p s)/p C v δ p 2.11) by 2.7) and 2.9). Since supp w = supp v δ \ B 3δ/4 0) and supp πv ε,δ )= supp v ε,δ B δ/2 0), supp w supp πv ε,δ )=. 2.12)

9 NoDEA p-laplacian problems Page 9 of Set E δ = {w : v E}. Lemma 2.8. For all sufficiently small δ>0, i) E δ Ψ λk+1 =, ii) ie δ )=k, iii) πv ε,δ ) / E δ. Proof. i) follows from 2.10). By i), E δ M\Ψ λk+1 and hence ie δ ) im\ψ λk+1 )=k by the monotonicity of the index and 1.5). On the other hand, since E E δ,v πv δ ) is an odd continuous map, ie δ ) ie) =k. ii) follows. iii) is immediate from 2.12). 3. Proofs 3.1. Proof of Theorem 1.1 All nontrivial critical points of I λ lie on the Nehari manifold { } N = u W 1,p 0 ) \{0} : I λu) u =0. We will show that I λ attains the ground state energy c := inf I λu) u N at a positive critical point. Since 0 <λ<λ 1, N is closed, bounded away from the origin, and for u W 1,p 0 ) \{0} and t>0, tu N if and only if t = t u, where Moreover, where t u = u p λ u p) u p s) N p)/p s) p I λ t u u)=supi λ tu) = p s t>0 N s) p ψ λu) N s)/p s), ψ λ u) = u p λ u p) u p s) )p/p s)..

10 25 Page 10 of 16 K. Perera and W. Zou NoDEA By 2.2) 2.4), μ s ψ λ v ε,δ ) μ s εp 1) p/p s) C εp 1) p/p s) C + Cε N p)/p s) if N>p 2 log ε + Cε p 1) p/p s) if N = p 2, and in both cases the last expression is strictly less than μ s if ε>0 is sufficiently small, so c I λ t vε,δ v ε,δ ) < p s N s) p μn s)/p s) s. Then I λ satisfies the PS) c condition by [6, Theorem 4.1.2)], and hence I λ N has a minimizer u 0 by a standard argument. Then u 0 is also a minimizer, which is positive by the strong maximum principle Proof of Theorem 1.2 We will show that problem 1.1) has a nontrivial solution as long as λ>λ 1 is not an eigenvalue from the sequence λ k ). Then we have λ k <λ<λ k+1 for some k N. Fixδ>0 so small that the first inequality in 2.10) implies Ψw) λ w E δ 3.1) and the conclusions of Lemma 2.8 hold. Then let A 0 = E δ and B 0 =Ψ λk+1, and note that A 0 and B 0 are disjoint nonempty closed symmetric subsets of M such that ia 0 )=im\b 0 )=k 3.2) by Lemma 2.8 i), ii) and 1.5). Now let R>r>0, let v 0 = πv ε,δ ), which is in M\A 0 by Lemma 2.8 iii), and let A, B and X be as in Theorem 2.3. For u B 0, I λ ru) 1 p 1 λ ) r p λ k+1 r p s) p s) μ p s)/p s Since λ<λ k+1,ands<pimplies p s) >p, it follows that inf I λ B) > 0ifr is sufficiently small. Next we show that I λ 0onA if R is sufficiently large. For w A 0 and t 0, I λ tw) tp 1 λ ) 0 p Ψw) by 3.1). Now let w A 0 and 0 t 1, and set u = π1 t) w + tv 0 ). Clearly, 1 t) w + tv 0 1, and since the supports of w and v 0 are disjoint by 2.12), 1 t) w + tv 0 p s) =1 t) p s) w p s). + t p s) v p s) 0.

11 NoDEA p-laplacian problems Page 11 of In view of 2.11), and since v p s) 0 = v p s) ε,δ v ε,δ p ) p s)/p 1 C by 2.2) and 2.3) ifε>0 is sufficiently small, it follows that 1 t) w + tv 0 p s) u p s) x = s 1 1 t) w + tv 0 p s) C. Then I λ Ru) Rp p Rp s) u p s) p s) 0 if R is sufficiently large. Now we show that sup I λ X) < p s N s) p μn s)/p s) s 3.3) if ε>0 is sufficiently small. Noting that X = {ρπ1 t) w + tv 0 ):w E δ, 0 t 1, 0 ρ R}, let w E δ and 0 t 1, and set u = π1 t) w + tv 0 ). Then [ ρ p ) sup I λ ρu) sup 1 λ u p s) ] u ρp p s) 0 ρ R ρ 0 p p s) = p s N s) p ψ λu) N s)/p s), 3.4) where ) + 1 λ u p ψ λ u) = u p s) )p/p s) [ 1 t) w + t v 0 p λ 1 t) w + tv 0 p] ) + = 1 t) w + tv 0 p s) )p/p s) ) + ) + 1 t) 1 p λ w p + t p 1 λ v p 0 1 t) p s) w p s) + t p s) v p s) 0 ) p/p s) 3.5)

12 25 Page 12 of 16 K. Perera and W. Zou NoDEA since the supports of w and v 0 are disjoint. Since 1 λ w p =1 λ Ψw) 0 by 3.1), ψ λ u) ψ λ v 0 ) = [ v ε,δ p λv p ε,δ μ s μ s v p s) ε,δ εp 1) p/p s) C εp 1) p/p s) C )p/p s) ] ) + + Cε N p)/p s) if N>p 2 log ε + Cε p 1) p/p s) if N = p 2 by 2.2) 2.4). In both cases the last expression is strictly less than μ s if ε>0 is sufficiently small, so 3.3) follows from 3.4). The inequalities 2.1) now imply that 0 <c< p s N s) p μn s)/p s) s. Then I λ satisfies the PS) c condition by [6, Theorem 4.1.2)], and hence c is a positive critical value of I λ by Theorem Proof of Theorem 1.3 The case where λ>λ 1 is an eigenvalue, but not from the sequence λ k ), was covered in the proof of Theorem 1.2, so we may assume that λ = λ k <λ k+1 for some k N. Takeδ>0sosmall that 2.10) and the conclusions of Lemma 2.8 hold, let A 0, B 0 and v 0 be as in the proof of Theorem 1.2, and let A, B and X be as in Theorem 2.3. As before, inf I λ B) > 0ifr is sufficiently small, and I λ Rπ1 t) w + tv 0 )) 0 w A 0, 0 t 1 if Θ ε,δ is sufficiently small and R is sufficiently large. On the other hand, I λ tw) tp 1 λ ) k CR p δ N p w A 0, 0 t R p Ψw) by 2.10). It follows that sup I λ A) < inf I λ B) ifδ is also sufficiently small. It only remains to verify 3.3) for suitable choice of δε) and small ε. Maximizing the last expression in 3.5) over0 t 1gives [ ψ λ u) ψ λ v 0 ) N s)/p s) + ψ λ w) N s)/p s)] p s)/n s). 3.6)

13 NoDEA p-laplacian problems Page 13 of By 2.2), 2.5), and 2.6), [ ] v ε,δ p λ k v p ε,δ ψ λ v 0 )= v p s) )p/p s) ε,δ and by 2.10) and 2.11), ψ λ w) = ) + 1 λ k Ψw) w p s) μ s ) + εp 1) p/p s) C + CΘ N p)/p s) ε,δ, 3.7) )p/p s) CδN p. 3.8) Recalling that Θ ε,δ = εδ p s)/p 1), if there exist α 0, p 1)/p s)) and a sequence ε j 0 such that, for ε = ε j and δ = ε α j, ψ λv 0 ) <μ s /3, then ψ λ u) 2μ s /3 for sufficiently large j by 3.6) and 3.8), which together with 3.4) gives the desired result. So we may assume that for all α 0, p 1)/p s)), ψ λ v 0 ) μ s /3 for all sufficiently small ε and δ = ε α. Since p s)/n s) < 1, then 3.6) 3.8) with δ = ε α yield [ ) ] N s)/p s) ψλ w) ψ λ u) ψ λ v 0 ) 1+ ψ λ v 0 ) where ψ λ v 0 )+Cψ λ w) N s)/p s) p 1) p/p s) μ s ε [ 1 C CεN p)n s)α α1)/p s) Cε N p)α2 α)/p 1) p 1) p 0 <α 1 := N p)n s) < N p2 )p 1) N p)p s) =: α 2 < p 1 p s by 1.3). Taking α α 1,α 2 ) now gives the desired conclusion Proofs of Theorems 1.6 and 1.7 We only give the proof of Theorem 1.7. Proof of Theorem 1.6 is similar and simpler. By [6, Theorem 4.1.2)], I λ satisfies the PS) c condition for all c< p s N s) p μn s)/p s) s, so we apply Theorem 2.5 with b equal to the right-hand side. By Degiovanni and Lancelotti [4, Theorem 2.3], the sublevel set Ψ λ k+m has a compact symmetric subset A 0 with ia 0 )=k + m. We take B 0 =Ψ λk+1, so that im\b 0 )=k ],

14 25 Page 14 of 16 K. Perera and W. Zou NoDEA by 1.5). Let R > r > 0 and let A, B and X be as in Theorem 2.5. For u Ψ λk+1, I λ ru) rp 1 λ ) r p s) p λ k+1 p s) μ p s)/p s by 1.4). Since λ < λ k+1, and s < p implies p s) > p, it follows that inf I λ B) > 0ifr is sufficiently small. For u A 0 Ψ λ k+1, I λ Ru) Rp 1 λ ) R p s) p λ k+1 p s) λ p s)/p k+1 V s ) p s)/n p) by 1.6), so there exists R>rsuch that I λ 0onA. Foru X, I λ u) λ ) p k+1 λ u p 1 s)/p u p p p s) V s ) p s)/n p) [ ] λk+1 λ) ρ ρ p s)/p sup ρ 0 p p s) V s ) p s)/n p) = p s N s) p λ k+1 λ) N s)/p s) V s ). So sup I λ X) p s N s) p λ k+1 λ) N s)/p s) V s ) < p s N s) p μn s)/p s) s by 1.7). Theorem 2.5 now gives m distinct pairs of nontrivial) critical points ± u λ j,j=1,...,m of I λ such that 0 <I λ u λ j ) p s N s) p λ k+1 λ) N s)/p s) V s ). Since u λ j p = pi λ u λ j )+λ u λ j p + p p s) u λ j p V s ) p s)/n s) u λ j p s) u λ j p s) ) p/p s) by 1.6), and u λ s) [ j p N s) p = I λ u λ j ) 1 ] p s p I λu λ j ) u λ N s) p j = I λ u λ j ), p s 1.8) follows. This completes the proof of Theorem 1.7., References [1] Bartolo, P., Benci, V., Fortunato, D.: Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity. Nonlinear Anal. 79), )

15 NoDEA p-laplacian problems Page 15 of [2] Benci, Vieri: On critical point theory for indefinite functionals in the presence of symmetries. Trans. Am. Math. Soc. 2742), ) [3] Degiovanni, Marco, Lancelotti, Sergio: Linking over cones and nontrivial solutions for p-laplace equations with p-superlinear nonlinearity. Ann. Inst. H. Poincaré Anal. Non Linéaire 246), ) [4] Degiovanni, Marco, Lancelotti, Sergio: Linking solutions for p-laplace equations with nonlinearity at critical growth. J. Funct. Anal ), ) [5] Fadell, Edward R., Rabinowitz, Paul H.: Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems. Invent. Math. 452), ) [6] Ghoussoub, N., Yuan, C.: Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents. Trans. Am. Math. Soc ), ) [7] Mosconi, S., Perera, K., Squassina, M., Yang, Y.: The Brezis Nirenberg problem for the fractional p-laplacian. Calc. Var. Partial Differ. Equ. 554), ) [8] Perera, Kanishka: Nontrivial critical groups in p-laplacian problems via the Yang index. Topol. Methods Nonlinear Anal. 212), ) [9] 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, RI 2010) [10] Perera, Kanishka, Squassina, Marco, Yang, Yang: Bifurcation and multiplicity results for critical p-laplacian problems. Topol. Methods Nonlinear Anal. 471), ) [11] Perera, Kanishka, Szulkin, Andrzej: p-laplacian problems where the nonlinearity crosses an eigenvalue. Discrete Contin. Dyn. Syst. 133), ) [12] Rabinowitz, Paul H: Some critical point theorems and applications to semilinear elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4) 51), ) [13] Yang, Yang, Perera, Kanishka: N-Laplacian problems with critical Trudinger- Moser nonlinearities. Ann. Sc. Norm. Super. Pisa Cl. Sci. 5) 164), ) Kanishka Perera Department of Mathematical Sciences Florida Institute of Technology Melbourne FL32901 USA kperera@fit.edu

16 25 Page 16 of 16 K. Perera and W. Zou NoDEA Wenming Zou Department of Mathematical Sciences Tsinghua University Beijing China Received: 2 April Accepted: 28 May 2018.