Solutions for a quasilinear elliptic problem with indefinite nonlinearity with critical growth

. We are interested in nonhomogeneous problems with a nonlinearity that changes sign and may possess a critical growth as follows


Introduction
The goal of this paper is to find nontrivial solutions for the problem − div a(|∇u| p )|∇u| p−2 ∇u = λ|u| q−2 u + W(x)|u| r−2 u in Ω, u = 0 on ∂Ω, (P λ ) where Ω ⊂ R N is a bounded domain with smooth boundary ∂Ω, N ≥ 2, 1 < p ≤ q < N, q < r ≤ q * and λ ∈ R, where q * = Nq N−q is the critical Sobolev exponent.We introduce the hypotheses on the function a in the sequel.
As a direct consequence of (a 3 ), we obtain that the function a and its derivative a ′ satisfy a ′ (t)t ≤ (q − p) p a(t) for all t > 0. (1.1) Now, if we define the function h(t) = a(t)t − q p A(t), using (1.1) we can prove that function h is nonincreasing.Then, 1 q a(t)t ≤ 1 p A(t), for all t ≥ 0. (1.2) To illustrate the degree of generality of the kind of problems studied here, and with adequate hypotheses on the functions a, which will be made clear shortly, we present some examples of problems that are also interesting from a mathematical point of view and have a wide range of applications in physics and related sciences.
Problem 1: Let a(t) = t q−p p .In this case we are studying problem as −∆ q u = λ|u| q−2 u + W(x)|u| r−2 u in Ω, u = 0 on ∂Ω, (P λ ) and it is related to the main result showed in [6].See also the work [7].
Such class of problems arise from applications in physics and related sciences, such as biophysics, plasma physics and chemical reactions (for instance, see [16,17,24]).
The interest in studying nonlinear partial differential equations with p&q operator has increased because many applications arising in mathematical physics may be stated with an operator in this form.We refer the reader to the works [9][10][11]15], where the authors have considered nonhomogeneous elliptic problems involving several type of function a.
Problems involving indefinite nonlinearities, that is, signal changing nonlinearities, have attracted the attention of many researchers over the past few decades, either because of their application in population dynamics describing the stationary behavior of a population in a heterogeneous environment (see [1,19,22,23]) or because of their mathematical relevance.Researchers have studied this type of problem using: variational methods (see [2,3,8,12,20,21]), sub-supersolution method (see [12,13,20]) and Morse theory (see [2,19]).
This paper deals with the class of problem (P λ ) that brings important characteristics, which are the nonlinearities that change signal (see the hypotheses on W below) with subcritical or critical growth and the generality of the operator that includes, for instance, p−Laplacian and p&q−Laplacian operators.These characteristics provoke some behaviors in the geometry of the energy functional associated to problem (P λ ) which make it difficult to find nontrivial solutions.As far as we know, this is the only work that proves existence and multiplicity of ground state solutions of problem (P λ ) under our assumptions.
Let us consider a weight function W : Ω → R which changes sign in Ω.More specifically, function W satisfies We are going to require another important hypothesis on W. For this, let λ 1 be the first eigenvalue of the operator (−∆ q ) on Ω, with zero Dirichlet boundary condition, and let φ 1 be the first eigenfunction associated to λ 1 .The weight function W satisfies only one of the following two hypotheses: By the variational characterization of λ 1 , we have i) If the weight function W satisfies (W 1 ) and (W + 2 ), then λ * = λ 1 .
We are now ready to state our first main result concerning the subcritical case.Item (i) of Theorem 1.1 provides some interesting qualitative properties on nontrivial solutions of problem (P λ ).For example: ⊂ Ω is a domain with smooth boundary and u is a nontrivial solution of (P λ ), then u ̸ = 0 a.e in Ω \ Ω 0 ; 2) If Ω + := { x ∈ Ω : W(x) > 0} and Ω − := { x ∈ Ω : W(x) < 0} have positive measure, and u is a nontrivial solution of (P λ ), then u must "belong" more to Ω + than Ω − , that is, 3) If Ω is a symmetric set and W ∈ C(Ω) is an odd function, then a nontrivial solution u of (P λ ) can be neither an even nor an odd function.In fact, otherwise To illustrate this, consider Ω = x ∈ R N : |x| < 2π and W : R N → R given by W(x) = cos(|x|).
To show the existence of solutions to the problem in the critical case, we will need to add a new hypothesis on the weight function W. The new hypothesis is as follows.
The above hypothesis is fundamental to overcome the lack of compactness generated by the critical exponent r = q * .It is important to highlight that, up to our knowledge, (W 3 ) is a new hypothesis in the literature, which makes it one of the relevant points of this work.
To provide an example of a function that satisfies hypothesis (W 3 ), just consider and W : Ω → R, given by Now, let S > 0 be the best constant of the Sobolev embedding W 1,q 0 (Ω) → L q * (Ω).Our second main result, concerning the critical case, is the following.Theorem 1.2.Consider r = q * and λ < λ 1 .Let a satisfy (a 1 )-(a 4 ) and the weight function W satisfy (W 1 ), (W 3 ) and .
Then, there are two nontrivial solutions for problem (P λ ).
The paper is organized as follows: in Section 2, we will prove technical results and the first part of Theorem 1.1.In Section 3, we will demonstrate the second part of Theorem 1.1, namely, the existence of least energy solutions that do not change sign.Finally, in Section 4, we will establish the last part of Theorem 1.1, that is, the existence of least energy nodal solutions that change sign exactly once.

Variational framework and preliminary results
The natural space to look for weak solutions to problem (P λ ) is the Sobolev space W Since the approach is variational, consider the energy functional associated We know that J λ is differentiable on W 1,q 0 (Ω) and, for all u, v ∈ W 1,q 0 (Ω), Thus, u ∈ W 1,q 0 (Ω) is a critical point of J λ if, and only if, u is a weak solution of problem (P λ ).Moreover, let us define the Nehari manifold and the nodal Nehari set where u + (x) := max {u(x), 0} and u − (x) := min {u(x), 0} .
Notice that u = u + + u − and N ± λ ⊂ N λ .Now we introduce some important subsets of N λ .Consider Since we want to use the method of minimization, we begin to study the behavior of the functional J λ on N λ .Proposition 2.1.Assume that the function a satisfies (a 1 )-(a 3 ).Then, there exist positive constants K 1 , K 2 and K 3 such that the following properties hold: Proof.For every u ∈ N λ , by (1.2), we have Hence, by (a 1 ) and the Poincaré inequality, Then item (i) follows.
We now prove item (ii).Taking u ∈ N λ , by (a 1 ) and the Poincaré inequality, one has Hence, Finally, using that W ∈ L ∞ (Ω), the Sobolev embeddings and (2.6), there exists a positive constant C 1 such that This inequality proves item (ii).Item (iii) follows directly from inequality contained in item (ii) and by (2.6).In fact, The next result is a direct consequence of Proposition (2.1).
The other inclusions follow from item (iii) of previous proposition.
By the same arguments of Proposition 2.1, but using the definition of λ * instead of Poincaré inequality, the next result follows.
Proposition 2.3.Assume that function a satisfies (a 1 )-(a 3 ).Then, there exist positive constants K 1 , K 2 and K 3 such that the following properties hold: Therefore, from Proposition 2.1 and Proposition 2.3, the following real numbers are well defined: Then, there exists a unique t u > 0 satisfying J λ (t u u) := max t≥0 J λ (tu) > 0.
Moreover, if J ′ λ (u)u < 0, then t u ∈ (0, 1].Proof.Let u ∈ W 1,q 0 (Ω)\{0} and t ∈ (0, +∞).So, by (a 1 ), we obtain and Therefore, lim sup (2.10) Thus, since Ω W(x)|u| r dx > 0, we ensure the existence of t u ∈ (0, +∞) such that To guarantee that the value t u > 0 is unique, let us prove that the equation J ′ λ (su)su = 0 is satisfied only for s = t u .Indeed, this equation is equivalent to By (a 3 ), the right-hand side of the equation above is a nonincreasing function on s > 0, while the left side, an increasing function on s > 0 provided r > q and Ω W(x)|u| r dx > 0. This shows the uniqueness of the value t u > 0. With the same arguments, we obtain that t u > 1 implies J ′ λ (u)u ≥ 0, and the proof of the lemma follows.
Lemma 2.5.If q < r < q * and W : Proof.From (W 1 ), we may consider two open balls B 1 and B 2 contained in Ω such that Arguing as in [6, Lemma 2.3], we have two negative solutions Then, by Lemma 2.4, there are t 1 , Proof of item (i) of Theorem 1.1 Proof.The proof follows directly from item (iii) of Proposition 2.1.

Existence of two least energy solutions which do not change sign
In this section, we are going to show that cλ is attained by some function which is a solution of problem (P λ ).For our purposes, we write where the functionals Φ λ , I ∈ C 1 (W 1,q 0 (Ω), R) are given by

Let us consider the set
We now present some properties of the functionals Φ λ and I when λ < λ * .Lemma 3.1.If λ < λ * , then the following properties hold: (i) Φ λ and u → Φ ′ λ (u)u are weakly lower semicontinuous and
To prove (ii), arguing as Proposition 2.3, we have On the other hand, by (W 1 ), , and then, by Sobolev embeddings, From (3.4) and (3.5) the item (ii) holds.Since ∂Y = {0}, this shows that the item (iii) holds.Now let us prove item (iv).Since q < r and u ∈ Y, we obtain which implies that t → I ′ (tu)u t q−1 is increasing in (0, +∞) and for every u ∈ Y.Moreover, lim sup t→+∞ I(tu) t q = lim sup On the other hand, note that is a nonincreasing function by (a 3 ).Moreover, we also have Then, by (3.6) and (3.7), we conclude the proof of item (iv).
Using the previous lemma and [14, Corollary 3.1], we obtain the next result.
We now show that problem (P λ ) has two least energy solutions when λ < λ * .
Proposition 3.3.If λ < λ * , then there exists a nontrivial function v λ which is a least energy solution of (P λ ), and ṽλ := −v λ is also a least energy solution of (P λ ).Moreover, if λ < λ 1 these solutions are ground state solutions.
Proof.Let v λ be the solution found in Corollary 3.2 and let us assume by contradiction that v ± λ ̸ = 0. Since v λ is a critical point of functional J λ and the intersection of the support of the functions v ± λ is empty, we have that Since Proposition 2.3 holds, then either Without loss of generality, we can assume that Therefore, by (3.8) and (3.9), This contradiction proves that the least energy solution does not change sign.
We may assume that v λ is nonnegative.Then, setting ṽλ = −v λ , we have that Moreover, using that v λ is a critical point of J λ , we have for all φ ∈ W Thus, ṽλ is a critical point of J λ .Therefore, problem (P λ ) has a nonpositive solution and a nonnegative solution.Furthermore, when λ < λ 1 , by Corollary 2.2, M λ = N λ .Thus, these solutions are ground state solutions of (P λ ).

Proof of item (ii) of Theorem 1.1
Proof.The proof follows directly from Corollary 3.2 and Proposition 3.3.

Existence of two nodal solutions
We begin this section by showing that dλ is attained by some function which is a least energy nodal solution of problem (P λ ).
Proof.Let (w n ) ⊂ M ± λ be a minimizing sequence, that is, a sequence satisfying By item (i) of Lemma 2.1, we obtain that functional J λ is coercive on M ± λ , and hence (w n ) is bounded in W 1,q 0 (Ω).Then, by Sobolev embeddings and the continuity of the maps w → w + and w → w − continuous from L r (R N ) in L r (R N ) (for details, see [4,Lemma 2.3] with suitable adaptations), there exists w λ ∈ W 1,q 0 (Ω) such that, up to a subsequence, we have We claim that w ± λ ̸ = 0 and Ω W(x)|w ± λ | r dx > 0. Indeed, using that W ∈ L ∞ (Ω) and item (iii) of Lemma 2.1, we obtain and that proves our claim.Therefore, by Lemma 2.4, there exists We claim that t ± λ ∈ (0, 1).In fact, by Fatou's Lemma and (4.2), we have Hence, by Lemma 2.4, the claim follows.Similarly, with the same arguments of Proposition 3.2, we obtain .
Corollary 4.2.Let wλ be a minimizer found in Propositions 4.1.Then, wλ is a critical point of J λ and has exactly two nodal domains.
Proof.The proof that wλ ∈ M ± λ is a critical point of J λ is done using a suitable quantitative deformation lemma and Brouwer's topological degree properties.It is done, with suitable modifications, as in [5,Lemma 4.3] and [5,Theorem 1.1].To show that the nodal solution wλ has exactly two nodal domains, or in other words it changes sign exactly once, see for instance [5, pages 1230-1232] .
Using the same arguments as in Proposition 3.3 one can immediately prove the following result.
Corollary 4.3.If λ < λ * , then there exists a function wλ which is a nodal least energy solution of (P λ ), and w λ := − wλ is also a nodal least energy solution of (P λ ).Moreover, if λ < λ 1 , then these solutions are ground state solutions of (P λ ).

Proof of item (iii) of Theorem 1.1.
Proof.It follows directly from Corollaries 4.2 and 4.3.

A nontrivial solution for the indefinite critical problem
In this section we consider the following critical problem where Ω ⊂ R N is a bounded domain with smooth boundary ∂Ω, N ≥ 2, 1 < p ≤ q < N and λ ∈ R, where q * = Nq N−q is the critical Sobolev exponent.Here, we consider the associated functional Let us show that the associated functional to the indefinite critical problem has a mountain pass geometry.
Recall that, if E is a Banach space, Φ ∈ C 1 (E, R) and c ∈ R we say that Φ satisfies the Palais-Smale condition at level c (shortly: Φ satisfies (PS) c ) if every sequence (u n ) ∈ E such that Φ(u n ) → c in R and Φ ′ (u n ) → 0 in E ′ , as n → ∞, admits a subsequence that converges for a critical point of Φ.This sequence is called a (PS) c sequence for Φ.
where W 1 , W 2 are positive constants given by (W 3 ), then I λ has a nontrivial critical point.

Proof of Theorem 1.2.
Proof.If λ < λ 1 , by Proposition 5.1 and Lemma 5.2, we have that there exists a critical point u λ of I λ .Thus, if , then, by Proposition 5.4, u λ is nontrivial solution.Moreover, using the same arguments as in Proposition 3.3, one can immediately shows that −u λ is also a nontrivial solution.