$(p,2)$-equations asymmetric at both zero and infinity

We consider a $(p,2)$-equation, that is, a nonlinear nonhomogeneous elliptic equation driven by the sum of a $p$-Laplacian and a Laplacian with $p>2$. The reaction term is $(p-1)$-linear but exhibits asymmetric behaviour at $\pm\infty$ and at $0^{\pm}$. Using variational tools, together with truncation and comparison techniques and Morse theory, we prove two multiplicity theorems, one of them providing sign information for all the solutions (positive, negative, nodal).

If p = 2, then ∆ 2 = ∆ the Laplacian. In problem (1.1), the reaction term f(z, x) is a Carathéodory function such that f(z, 0) = 0. We assume that f(z, ⋅ ) exhibits (p − 1)-linear growth near ±∞. However, the growth of f(z, ⋅ ) is asymmetric near ±∞. More precisely, the quotient f(z, x) |x| p−2 x crosses at least the principal eigenvalueλ 1 (p) > 0 of (−∆ p , W 1,p 0 (Ω)) as we move from −∞ to +∞ (crossing or jumping nonlinearity). In the negative direction we allow resonance with respect toλ 1 (p) > 0, while in the positive direction resonance can occur with respect to any nonprincipal eigenvalue of (−∆ p , W 1,p 0 (Ω)). We have a similar asymmetric behavior when x → 0 ± . This time the quotient f(z,x) x crossesλ 1 (2) > 0. Under this double asymmetric setting, we prove a multiplicity theorem producing three nontrivial smooth solutions and provide sign information for all of them. A second multiplicity theorem is also proved without sign information for the third solution.
Equations involving the sum of a Laplacian and a p-Laplacian arise in problems of mathematical physics; see Cherfils and Ilyasov [9] (plasma physics) and Benci, D'Avenia, Fortunato and Pisani [6] (quantum physics). Recently, there have been existence and multiplicity results for different classes of such equations. We mention the works of Aizicovici, Papageorgiou and Staicu [3], Cingolani and Degiovanni [10], Gasinski and Papageorgiou [13,15], Papageorgiou and Rădulescu [22,23], Papageorgiou, Rădulescu and Repovš [25], Sun [30], Sun, Zhang and Su [31] and Yang and Bai [32]. In the aforementioned works, only Papageorgiou and Rădulescu [23] deal with an asymmetric p-sublinear reaction term. They consider a reaction term f (z, x) such that the quotient f(z, x) |x| p−2 x crosses only the first eigenvalueλ 1 (p) as we move from −∞ to +∞, and resonance is allowed at −∞. At zero, the behavior of the quotient f(z,x) x is symmetric. Finally, in [23] the multiplicity result does not produce nodal solutions. Concerning asymmetric sublinear problems, we should also mention the semilinear works of D'Agui, Marano and Papageorgiou [11] (Robin problems with an indefinite and unbounded potential) and Recova and Rumbos [28] (Dirichlet problems with zero potential).
Our approach is variational, based on the critical point theory combined with suitable truncation and comparison techniques and Morse theory (critical groups).

Mathematical background
Let X be a Banach space and X * its topological dual. By ⟨ ⋅ , ⋅ ⟩ we denote the duality brackets for the pair (X * , X). Given φ ∈ C 1 (X, ℝ), we say that φ satisfies the "Cerami condition" (the "C-condition" for short) if the following holds: Every sequence {u n } n⩾1 ⊆ X such that {φ(u n )} n⩾1 ⊆ ℝ is bounded and (1 + ‖u n ‖)φ (u n ) → 0 in X * as n → ∞ admits a strongly convergent subsequence.
This is a compactness-type condition on the functional. It leads to a deformation theorem from which one can derive the minimax theory of the critical values of φ. One of the main results in this theory is the so-called "mountain pass theorem" of Ambrosetti and Rabinowitz [5], stated here in a slightly more general form (see Gasinski and Papageorgiou [12]). Theorem 2.1. Assume that φ ∈ C 1 (X, ℝ) satisfies the C-condition, u 0 , u 1 ∈ X, ‖u 1 − u 0 ‖ > r > 0, Then c ⩾ m r and c is a critical value of φ (that is, there exists u ∈ X such that φ(u) = c, φ (u) = 0).
In the study of (1.1), we will use the Sobolev spaces W  The Sobolev space H 1 0 (Ω) is a Hilbert space and, as above, the Poincaré inequality implies that we can choose as inner product The corresponding norm is ‖u‖ H 1 0 (Ω) = ‖Du‖ 2 for all u ∈ H 1 0 (Ω).
The space C 1 0 (Ω) is an ordered Banach space with positive (order) cone given by This cone has a nonempty interior given by Here, by ∂u ∂n we denote the normal derivative of u, with n( ⋅ ) being the outward unit normal on ∂Ω. Recall that C 1 0 (Ω) is dense in both W 1,p 0 (Ω) and H 1 0 (Ω). We consider a function f 0 : Ω × ℝ → ℝ which is Carathéodory function, that is, for all x ∈ ℝ the mapping z → f 0 (z, x) is measurable and for almost all z ∈ Ω the function x → f 0 (z, x) is continuous. We assume that with a 0 ∈ L ∞ (Ω) and (the critical Sobolev exponent for p). We set and consider the C 1 -functional φ 0 : W The next proposition is a special case of a more general result of Aizicovici, Papageorgiou and Staicu [2, Proposition 2]. See also Papageorgiou and Rădulescu [21,24] for corresponding results for the Neumann and Robin problems. The result is essentially a byproduct of the regularity theory of Lieberman [18,Theorem 1].
We will also encounter a weighted version of the eigenvalue problem (2.1). So, let m ∈ L ∞ (Ω), m(z) ⩾ 0 for almost all z ∈ Ω, m ̸ ≡ 0. We consider the following nonlinear eigenvalue problem: Againλ ∈ ℝ is an eigenvalue of (−∆ r , W 1,r 0 (Ω), m) if problem (2.3) admits a nontrivial solution. We have a smallest eigenvalueλ 1 (r, m) > 0 which is isolated, simple and satisfies As before, the infimum is realized on the corresponding one-dimensional eigenspace, the elements of which do not change sign. This fact and (2.4) lead to the following monotonicity property of m →λ 1 (r, m).
Next, we recall some basic definitions and facts from Morse theory (critical groups). So, as before, X is a Banach space, φ ∈ C 1 (X, ℝ) and c ∈ ℝ. We introduce the following sets: we denote the kth relative singular homology group with integer coefficients for the pair (Y 1 , Y 2 ). Suppose that u ∈ K c φ is isolated. The critical groups of φ at u are defined by The excision property of singular homology implies that the above definition of critical groups is independent of the choice of the isolating neighborhood U. Suppose that φ satisfies the C-condition and that inf φ The critical groups of φ at infinity are defined by . Now suppose that φ ∈ C 1 (X, ℝ) satisfies the C-condition and that K φ is finite. We define The Morse relation says that where is a formal series in t ∈ ℝ with nonnegative integer coefficients β k . We conclude this section by fixing our notation and introducing the hypotheses on the reaction term We know that u ± ∈ W 1,p 0 (Ω), and we have u = u + − u − and |u| = u + + u − . By | ⋅ | N we denote the Lebesgue measure on ℝ N , and given f(z, x) a measurable function (for example, a Carathéodory function), we denote by N f ( ⋅ ) the Nemitsky (superposition) map corresponding to f( ⋅ , ⋅ ) and defined by The hypotheses on f(z, x) are the following. Hypotheses H(f ). f : Ω × ℝ → ℝ is a Carathéodory function such that f(z, 0) = 0 for almost all z ∈ Ω and the following conditions hold: (i) For every r > 0, there exists a r ∈ L ∞ (Ω) + such that |f(z, x)| ⩽ a r (z) for almost all z ∈ Ω and all |x| ⩽ r.
uniformly for almost all z ∈ Ω, and for every ρ > 0 there existsξ ρ > 0 such that for almost all z ∈ Ω the mapping

Remark 2.
Hypothesis H(f ) (ii) implies that f(z, ⋅ ) is a crossing nonlinearity. In fact, we can cross any finite number of variational eigenvalues, starting withλ 1 (p) > 0. Note that in the negative direction we can have resonance with respect toλ 1 (p) > 0, while in the positive direction resonance is possible with respect to any nonprincipal eigenvalue of (−∆ p , W 1,p 0 (Ω)). As we will see in the proof of Proposition 3.3, Hypothesis H(f ) (iii) guarantees that at −∞ the resonance with respect toλ 1 (p) > 0 is from the left of the principal eigenvalue in the sense thatλ This makes the negative truncation of the energy functional of (1.1) coercive. So, we can use the direct method of the calculus of variations. Hypothesis H(f ) (iv) implies that at zero, too, we have an asymmetric behavior of the quotient f(z,x) x .

Solutions of constant sign
In this section, using variational tools, we show that problem (1.1) admits two nontrivial smooth solutions of constant sign (one positive and the other one negative). So, let φ : W 1,p 0 (Ω) → ℝ be the energy functional for problem (1.1) defined by . Also, we consider the positive and negative truncations of f(z, ⋅ ), that is, the Carathéodory function We set F ± (z, x) = ∫ x 0 f ± (z, s) ds and consider the C 1 -functionals φ ± : W Also, from (3.1) we have We add (3.4) and (3.5). Recalling that p > 2, we obtain (see Hypotheses H(f ) (i) and (iii)).
We argue by contradiction. So, suppose that Claim 1 is not true. By passing to a subsequence if necessary, we may assume that ‖u − n ‖ → ∞ as n → ∞.
Again we argue indirectly. So, suppose that Claim 2 is not true. Then at least for a subsequence we have ‖u + n ‖ → +∞ as n → ∞.
for some M 4 > 0, all n ∈ ℕ and all h ∈ W 1,p 0 (Ω) (see Hypothesis H(f ) (i)), which implies (3.18) Using the growth condition from (3.10), we see that (Ω), pass to the limit as n → ∞ and use (3.13), (3.16), (3.17) and the fact that p > 2. Then lim From (3.19) and Hypothesis H(f ) (ii) we see that at least for a subsequence we have So, if in (3.18) we pass to the limit as n → ∞ and use (3.16), (3.20), (3.21) and the fact that p > 2, then Therefore, the energy functional φ satisfies the C-condition.
Next, we show that φ + satisfies the C-condition, too.

From (3.26) and (3.27) it follows that
(Ω) is unbounded. So, we may assume that ‖u + n ‖ → ∞. We set v n = u + n ‖u + n ‖ , n ∈ ℕ, and have ‖v n ‖ = 1 and v n ⩾ 0 for all n ∈ ℕ. Hence we can say (at least for a subsequence) that In (3.26), we choose h = u n − u ∈ W 1,p 0 (Ω). Passing to the limit as n → ∞, using (3.29) and following the argument in the last part of the proof of Proposition 3.1 (see the part of the proof after (3.24)), we obtain u n → u in W 1,p 0 (Ω). We conclude that φ + satisfies the C-condition.
For the functional φ − , we have the following result.
Remark 3. From (3.33) we see that the resonance with respect toλ 1 (p) > 0 at −∞ is from the left of the principal eigenvalue.
From the above proposition we infer the following fact about the functional φ − (see [19]).

Corollary 3.4. If Hypotheses H(f ) hold, then φ − satisfies the C-condition.
Next, we determine the nature of the critical point u = 0 for φ + .

Proposition 3.5. If
Hypotheses H(f ) hold, then u = 0 is a local minimizer for φ + .
Then a ∈ C 1 (ℝ N , ℝ N ) and div a(Du) = ∆ p u + ∆u for all u ∈ W 1,p 0 (Ω), which implies with id N being the identity map on ℝ N . For all ξ ∈ ℝ N , we have

Nodal solutions -multiplicity theorems
In this section, using tools from Morse theory (critical groups), we show the existence of a nodal (sign changing) smooth solution and formulate our multiplicity theorems.
To produce a nodal sign, changing solution, we will need one more hypothesis which is the following one.
Hypothesis H 0 . Problem (1.1) has a finite number of solutions of constant sign.

Remark 4.
This condition is equivalent to saying that K φ + and K φ − are finite sets.
We start by computing the critical groups of φ at infinity.
We consider the homotopy Claim. There exist γ ∈ ℝ and τ > 0 such that We argue by contradiction. Since h( ⋅ , ⋅ ) maps bounded sets to bounded sets, if the claim is not true, then we can find two sequences {t n } n⩾1 ⊆ [0, 1] and {u n } n⩾1 ⊆ W 1,p 0 (Ω) such that t n → t, ‖u n ‖ → ∞, h(t n , u n ) → −∞, (1 + ‖u n ‖)h u (t n , u n ) → 0. (4.1) From the last convergence in (4.1) we have for all h ∈ W 1,p 0 (Ω), with ϵ n → 0 + . From the third convergence in (4.1) we see that we can find n 0 ∈ ℕ such that (recall that p > 2 and ϵ n → 0 + as n → +∞). We claim that t < 1. If t n → 1, then let y n = u n ‖u n ‖ , n ∈ ℕ. We have ‖y n ‖ = 1 for all n ∈ ℕ, and so we may assume that y n w → y in W 1,p 0 (Ω) and y n → y in L p (Ω). From (4.2) we have for all n ∈ ℕ.
Let v ∈ int C + and consider the function From the nonlinear Picone's identity of Allegretto and Huang [4] we have where the first equation uses the nonlinear Green's identity (see [12, p. 211]), the second equation follows from (4.14) and the last inequality holds since u, v ∈ int C + .
Next, we compute the critical groups at infinity for the functional φ ± . Claim. There exist γ ∈ ℝ and τ > 0 such that for all t ∈ [0, 1], As in the proof of Proposition 4.1, we argue by contradiction. So, we can find two sequences From the last convergence in (4.19) we have From (4.20) and (4.21) we infer that for all h ∈ W 1,p 0 (Ω), with ϵ n → 0 + . Suppose that ‖u + n ‖ → +∞. We set v n = u + n ‖u + n ‖ , n ∈ ℕ. Then ‖v n ‖ = 1 and v n ⩾ 0 for all n ∈ ℕ, and so we may Reasoning as in the proof of Proposition 3.1 (see the proof of Claim 2), we reach a contradiction, and so we infer that But this contradicts (4.19). Hence the claim holds and, as before (see the proof of Proposition 4.1), we have which then implies C k (φ + , ∞) = 0 for all k ∈ ℕ 0 (see the end of the proof of Proposition 4.1). The proof is complete.
Next, we compute the critical groups of φ at u = 0.
As in the proof of Proposition 4.4, via the homotopy invariance of critical groups we have Recalling that the nonprincipal eigenfunctions of (−∆, H 1 0 (Ω)) are nodal and since β >λ 1 (2), we infer that K Ψ − = {0}. Moreover, as in the last part of the proof of Proposition 4.1, using Picone's identity, we have This completes the proof. Now we are ready for our first multiplicity theorem. Recall that at the beginning of this section we have introduced an extra Hypothesis H 0 , which says that the constant sign solutions of (1.1) are finite. This is equivalent to saying that From Proposition 3.6 we know that m, n ∈ ℕ.
Then we have the following multiplicity theorem.
Proof. From Proposition 3.6 we already have two nontrivial constant sign smooth solutions u 0 ∈ int C + and v 0 ∈ − int C + .
By Hypothesis H 0 , we have We set Since v i ∈ − int C + and u l ∈ int C + , we have This means that problem (1.1) admits at least one nodal solution y 0 ∈ C 1 0 (Ω).
We can drop Hypothesis H 0 at the expense of strengthening the regularity of f(z, ⋅ ). Then we can still have a three nontrivial solutions multiplicity theorem, but without providing sign information about the third nontrivial smooth solution.
The new hypotheses on f(z, x) are the following.
Proof. Again, from Proposition 3.6 we already have two nontrivial constant sign smooth solutions u 0 ∈ int C + and v 0 ∈ − int C + .