On some classes of generalized Schrödinger equations


               <jats:p>Some classes of generalized Schrödinger stationary problems are studied. Under appropriated conditions is proved the existence of at least 1 + <jats:inline-formula>
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                  <jats:italic>dim V</jats:italic>
                  <jats:sub>λ<jats:sub>
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                  </jats:sub> pairs of nontrivial solutions if a parameter involved in the equation is large enough, where <jats:italic>V</jats:italic>
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                  </jats:sub> denotes the eigenspace associated to the i-th eigenvalue λ<jats:sub>
                     <jats:italic>i</jats:italic>
                  </jats:sub> of laplacian operator with homogeneous Dirichlet boundary condition.</jats:p>

Motivated by ideas in [5] and [14], the authors in [19] make use of the following approach: Despite the energy functional associated to (P λ,q ) is not well de ned in H (Ω), Proposition 1.1 allows them to consider the change of variable v = f (u) in the semilinear problem (P λ,q ) in order to obtain the problem −∆v = λg (v) in Ω, v = on ∂Ω, (P λ,q ) where g(s) := f (s)|f (s)| q− f (s), which has the advantage of possessing a well de ned C -energy functional in H (Ω), given by where v := Ω |∇v| dr and G(s) = s g(r)dx = ( /q)|f (s)| q . They also prove that critical points of (1.1) in C (Ω) are weak solutions of (P λ,q ). In this way, by working with (P λ,q ), among other things, they were able to prove that: (a) If q = , λ > ϑ( )λ and (ϑ ) − (ϑ ) holds, then (P λ,q ) has a unique positive solution; (b) If q = , λ > (α / )λ and (ϑ ) − (ϑ ) holds, then (P λ,q ) has at least one positive solution.
Having in mind the previous results, the present paper has as its main goal to improve the results in [19] when one considers the cases q = or q = in problem (P λ,q ). Indeed, since in these two particular cases we prove in Lemma 2.1 that g is asymptotically linear at zero and at in nity, respectively, by using genus theory combined with arguments involving the Nehari manifold, it is possible to show that the number of solutions increases with λ. To be more precise, if dimV λ i denotes the dimension of the eigenspace V λ i associated to i-th eigenvalue λ i of laplacian operator under homogeneous Dirichlet boundary condition, we prove the following multiplicity result: (i) If q = and λ > ϑ( )λm, then problem (P λ,q ) possesses at least + m i= dimV λ i pairs of nontrivial solutions u i with I λ, (f − (u i )) > ; (ii) If q = and λ > (α / )λm, then problem (P λ,q ) has at least + m i= dimV λ i pairs of nontrivial solutions u i with I λ, (f − (u i )) > .
The paper is organized in a unique section where we study both cases, q = and q = .

Multiplicity of solutions
Since, by [19], it does not exist any nontrivial solution when λ ≤ , along of this section we are just considering positive values of λ. Moreover, before proving the main results of this section we need to study the properties of function g. Such properties play an important role throughout the paper. Proof. (i) The monotonicity is a straightforward consequence of Proposition 1.1(ii) and (ϑ ). On the other hand, by Proposition 1.
Moreover, by (ϑ ) (ii) Since f is odd (because ϑ is even), it is su cient to prove this item for s > . Observe that where t := f (s) and Υ(t) := t ϑ(r) / dr. It follows from (ϑ ) that t/ϑ(t) / (and consequently t /Υ(t)) is increasing in ( , ∞). This proves that g(s)/s is increasing in ( , ∞). Moreover, by item (iii) and (vi) of Proposition 1.1, we have (iii) The monotonicity follows immediately from (ii). To prove the second part, note that By (ϑ ) we know that t / ϑ(t) / goes to in nity as t goes to in nity. On the other hand, by (ϑ ), Υ(t)−tϑ(t) / is nonnegative and increasing in ( , ∞). Indeed, by de ning The result follows.
From now on {e j } stands for a Hilbertian basis of H (Ω) composed by eigenfunctions of the laplacian operator with homogeneous Dirichlet boundary condition, V λ j is the eigenspace associated to λ j , S and S d(m) are, respectively, the unit sphere of H (Ω) and the unit sphere of Wm := ⊕ m j= V λ j .

Proof of Theorem 1.2(i):
By Lemma 2.1(i) and Sobolev embedding Therefore I λ, is coercive. Since I λ, is weakly lower semicontinuous, we conclude that I λ, is bounded from below. On the other hand, since We conclude from Lemma 2.1(i), L'Hospital and Lebesgue Dominated Convergence Theorem that Since I λ, is coercive, it is standard to prove that it satis es the (PS)c condition. Finally, as I λ, is an even Cfunctional, it follows from Theorem 9.1 in [18] (see also [4]) that I λ, has at least d(m) pairs of critical points.
Before we are ready to prove Theorem 1.2(ii), we will make a careful study about some topological and geometrical aspects involving the Nehari Manifold. Let be the Nehari manifold associated to I λ, , S the unit sphere in H (Ω) and Lemma 2.2. If ϑ satis es (ϑ ) − (ϑ ) and λ > (α / )λ , the following claims hold: (v) It is su cient to choose a normalized (in H (Ω)) eigenfunction associated to λ . Proof. (i) Observe that hv( ) = . Moreover, for each v ∈ F, we have Thus, in view of Lemma 2.1(ii), L'Hôspital rule and Lebesgue's dominated convergence theorem, it follows that Showing that lims→∞ hv(s) = −∞. Moreover, hv(s) is positive for s small enough. Indeed, reasoning as in the previous limit, we get Hence, there exists a global maximum point sv > of hv which, by Lemma 2.1(ii), is the unique critical point of hv.
Consequently, h v (s) > for all s ∈ ( , ∞). By passing to the limit as n → ∞ in the last inequality, we get a contradiction.
(A ) Let {vn} ⊂ W be a sequence such that sn := sv n → ∞. Since W is compact, up to a subsequence, we get v ∈ W such that vn → v in H (Ω). Hence, passing to the lower limit as n → ∞ in = vn ≥ λ [v≠ ] g(sn vn) sn vn v n χ [vn≠ ] dx, it follows from Lemma 2.1(ii) that showing that v ∈ F c . Since v ∈ W ⊂ F, we have a contradiction.
(A ) We are going to prove that m is continuous. Let {vn} ⊂ F and v ∈ F be such that vn → v in H (Ω). Since m(sw) = m(w) for all w ∈ F and s > , we can assume that {vn} ⊂ S F . Hence, where sn := sv n . By (A ) and (A ), it follows that, passing to a subsequence, sn → s > . Thence, passing to the limit as n → ∞ in (2.3), we have showing that m(vn) = sn vn → sv = m(v). The second part of (A ) is immediate.

4)
Therefore I λ, is bounded from below in N.
Previous functions have important properties which will be stated in the next lemma. The proof is a direct consequence of Lemmas 2.3 and 2.4, see [24].
Passing to the limit as n → ∞, it follows from Lemma 2.1(ii) that Now we are going to consider two cases: (i) If ( λ/α ) ≠ λ j , whatever j > , it follows from (2.11) that v = . But this is a contradiction. Therefore {wn} is bounded in H (Ω).
(ii) If ( λ/α ) = λ j , for some j > , then (2.11) implies that v is an eigenfunction associated to λ j . From (2.11), it follows also that v = ( λ/α ) Ω v dx, i.e., v ∈ ∂F. On the other hand, (2.12) In this case, since wn = sv n vn = sv n , (2.13) passing to the limit as n → ∞ in The main result of this section will be proved through Krasnoselski's genus theory. For this, we start de ning some preliminaries notations: It is important to note that, since S F = −S F , γ j is well de ned. Below we state some standard properties of the genus which can be found, for instance, in [18].
Proof. First inequality follows from Lemma 2.5. On the other hand, the monotonicity of the levels c j is a consequence of Lemma 2.10(ii).
Next proposition is crucial to ensure the existence of multiple solutions. Proof. Suppose that γ(Kc) ≤ p. By Proposition 2.8 and Lemma 2.11, Kc is a compact set. Thus, by Lemma 2.9(iv), there exists a compact neighborhood K (in H (Ω)) of Kc such that γ(K) ≤ p. De ning M := K ∩ S F , we derive from Lemma 2.9(ii) that γ(M) ≤ p. Despite the noncompleteness of S F we can still use Theorem 3.11 in [23] (see also Remark 3.12 in [23]) to ensure the existence of an odd homeomorphisms family η(., t) of S F such that, for each u ∈ S F , the map t → Ψ λ, (η(u, t)) is non-increasing. (2.20) Observe that, although S F is non-complete, from Proposition 2.7 and (2.20), for all u ∈ S F , maps t → η(u, t) are well de ned in t ∈ [ , ∞). Consequently, it makes sense the third claim of Theorem 3.11 in [23], namely, η((Ψ λ, )c+ε\M, ) ⊂ (Ψ λ, )c−ε . (2.21) Let us choose B ∈ γ j+p such that sup B Ψ λ, ≤ c + ε. From Lemma 2.10(iv), B\M ∈ γ j . It follows again from Lemma 2.10(iii) that η(B\M, ) ∈ γ j . Therefore, from (2.21) and the de nition of c, we have that is a contradiction. Then γ(Kc) ≥ p + .
We are now ready to prove the following multiplicity result:

Proof of Theorem 1.2(ii):
Note that ≤ c j < ∞ are critical levels of Ψ λ, . In fact, suppose by contradiction that c j is regular for some j. Invoking Theorem 3.11 in [23], with β = c j , ε = , N = ∅, there exist ε > and a family of odd homeomorphisms η(., t) satisfying the properties of referred theorem. Choosing B ∈ γ j such that sup B ψ < c j + ε and arguing as in the proof of Proposition 2.12 we get a contradiction.
Finally, if the levels c j , ≤ j ≤ d(m), are di erent from each other, by Proposition 2.6(iv) the result is proved. On the other hand, if c j = c j+ ≡ c for some ≤ j ≤ d(m), it follows from Proposition 2.12 that γ(Kc) ≥ . Combining last inequality with Lemma 2.9(vi) and Proposition 2.6(iv), we conclude that (P λ,q ) has in nitely many pairs of nontrivial solutions.