Multiplicity of solutions for a class of elliptic problem of $p$-Laplacian type with a $p$-Gradient term

We consider the following problem $$(P) \begin{cases} -\Delta_{p}u= c(x)|u|^{q-1}u+\mu |\nabla u|^{p}+h(x)&\ \ \mbox{ in }\Omega, u=0&\ \ \mbox{ on } \partial\Omega, \end{cases}$$ where $\Omega$ is a bounded set in $\mathbb{R}^{N}$ ($N\geq 3$) with a smooth boundary, $10$, $\mu \in \mathbb{R}^{*}$, and $c$ and $ h$ belong to $L^{k}(\Omega)$ for some $k>\frac{N}{p}$. In this paper, we assume that $c\gneqq 0$ a.e. in $\Omega$ and $h$ without sign condition, then we prove the existence of at least two bounded solutions under the condition that $\|c\|_{k}$ and $\|h\|_{k}$ are suitably small. For this purpose, we use the Mountain Pass theorem, on an equivalent problem to $(P)$ with variational structure. Here, the main difficulty is that the nonlinearity term considered does not satisfy Ambrosetti and Rabinowitz condition. The key idea is to replace the former condition by the \textbf{nonquadraticity condition at infinity}.


Introduction and main result
Let Ω be a bounded set in R N (N ≥ 3) with a smooth boundary ∂Ω. In this paper, we are concerned with the following problem where ∆ p u := div(|∇u| p−2 ∇u) is the p-Laplacian operator, 1 < p < N , q > 0, µ ∈ R * , c and h belong to L k (Ω) for some k > N p . In the literature there are many results concerning the existence, the uniqueness or the multiplicity of solutions for models like (P ) under various assumptions on c and h. At first, it is important to mention that the sign of c plays a crucial role in the problem (P ) regarding uniqueness, as well as existence, of bounded solutions. In this setting, we refer to ( [21]) for more details. In the case c(x) ≤ −α 0 a.e. in Ω for some α 0 > 0, which is referred to as the coercive case, Boccardo,Murat and Puel ([7,9,8]), proved the existence of bounded solution for a general divergence form equation with quadratic growth in the gradient by using the sub and supersolution method, as well as the uniqueness has been shown by Barles and Murat ([6]) and by Barles et at. ([5]). When we just take c(x) ≤ 0 a.e. in Ω, Ferone and Murat ( [14], [15]) observed that the solvability of (P ) becomes rather complex and it is necessary to impose some strong regularity conditions on the data . For the particular case, c ≡ 0 there had been many contributions ( [1,22,24]), but the general case, c ≤ 0 may vanish only on some parts of Ω, the existence of a unique solution was left open until the recent paper authored by Arcoya et at. ( [4]). This result is proved for p = 2, q = 1 and under the following condition              c, h belong to L k (Ω) for some k > N 2 , µ ∈ L ∞ (Ω) and meas(Ω\Supp c) > 0, inf where W c := {w ∈ H 1 0 (Ω) : c(x)w(x) = 0, a.e. in Ω}. For a related uniqueness result for some large class of problem see also Arcoya et at. ([3]).
The case where c(x) 0 a.e. in Ω, the question of non-uniqueness was an open problem given by Sirakov ([26]) and it has received considerable attention by many authors. Moreover, it should be pointed out that the sign of h and whether µ is a function or a constant, generate additional difficulties to the problem. In this setting, Jeanjean and Sirakov ([21]) showed the existence of two bounded solutions assuming that µ ∈ R * , c and h are in L k (Ω) for some k > N 2 and satisfying and C N is the optimal constant in Sobolev's inequality. We note that, here h is allowed to change sign. After, this result was extended by Coster and Jeanjean ([11]) where µ is a bounded function such that µ(x) ≥ µ 1 > 0. They have used the degree topological method to show their result.
Finally, in the case where c is allowed to change sign and assuming c(x) 0 a.e. in Ω, Jenajean and Quoirin ([20, Theorem 1.1]) showed the existence of two bounded positive solutions when h 0, µ is a positive constant and when c + , µh are suitably small.
We would like also to mention that all the above quoted multiplicity results were restricted to the Laplacian operator with quadratic growth in the gradient (i.e. p = 2) with q = 1. Moreover, it is also interesting to mention that when c is allowed to change sign the solutions obtained are positive.
In this work we prove the multiplicity of bounded solutions for the problem (P ) by assuming the following assumption (H) c, h belongs to L k (Ω) for some k > N p , h is allowed to change sign , c a.e. in Ω and µ ∈ R * Now, we give a brief exposition of how we show our multiplicity result. At first, without loss of generality, we resolve our problem (P ) by restricting it to the case µ is a positive constant. For µ is a negative constant, we replace u by −u in (P ), then we conclude. Next, we observe that the type of problem (P ) do not have a variational formulation due to the presence of the p-gradient term. In order to solve this difficulty, we perform the Kazdan-Kramer change of variable, namely v = (e µu p−1 − 1)/µ. Thus, we obtain the following new problem (P ′ ) We mean by bounded weak solutions of (P ′ ), the functions v ∈ W 1,p 0 (Ω)∩L ∞ (Ω) satisfying Obviously, if v > −1 µ is a solution of (P ′ ), then u = p−1 µ ln(1 + µv) is a solution of (P ). Moreover, the solutions obtained here are not necessarily positive (compare with ([20])).
One of the most fruitful ways to deal with (P ′ ) is the variational method, which take into account that the weak solutions of (P ′ ) are critical points in W 1,p 0 (Ω) of the C 1 -functional with G(s) = s 0 g(t)dt, and F (s) = s 0 f (t)dt. Here, to obtain our two critical points for I we use the Mountain Pass Theorem and the standard lower semicontinuity argument respectively. For the first one, according to the famous paper by Ambrosetti and Rabinowitz ( [2]), the most important step is to show that I satisfies the Palais-Smale condition at the level c (see Definition 2.2). The fulfillment of this condition relies on the well-known Ambrosetti-Rabinowitz condition ((AR c ) for short), namely there exist θ > p and s 0 > 0 such that 0 < θG(s) ≤ sg(s), as |s| > s 0 .
Unfortunately, this condition is somewhat restrictive and not being satisfied by many nonlinearities g. However, many researches have been made to drop the (AR c ) condition. We refer, for instance, to [10,27,23,16,19]. We observe that our nonlinearity g does not satisfy (AR c ). Moreover, since we assumed that the sign of h is allowed to change, the Palais-Smale condition turns out more delicate (see eg. [17,20]). To the best of our knowledge, only Jenajean and Quoirin ( [20]), very recently, proved the Palais-Smale condition under the assumption that c change the sign and h is positive without assuming (AR c ). Their arguments (in the particular cases p = 2 and q = 1) are based on the explicit formulation of the function H(s) = g(s)s − 2G(s), and the requirement of the positivity of h. In our situation, as we can not obtain the explicit formulation of H together with the fact that h is allowed to change sign, the arguments used in ( [20]) are not be adapted.
In this work, the key point to show the Palais-Smale condition is to prove that g, among other conditions, satisfies the following condition (see Lemma 3.1), This condition is a variant of the well known nonquadraticity condition at infinity which was introduced by Costa and Malgalhães ( [10]) as follows We observe that, since ν > 0 then (N Q) is weaker than (CM ). Moreover, it should be noted that (N Q) was supposed by Furtado and Silva in their recent paper ( [16]). Our result follows by using similar arguments.
Concerning the existence of the second critical point, we use the standard lower semicontinuity argument. Namely, we look for such point as a local minimum in W 1,p 0 (Ω) for the functional I. Indeed, by the geometrical structure of I (see Proposition 2.1 below), we observe that I takes positive values in a large sphere (i.e. I remains sufficiently large) and in view that I(0) = 0.

Now we state our main result
Theorem 1.1. Assume that c k and h k are sufficiently small. Then, I has at least two critical points. Hence, the problem (P ) has at least two bounded weak solutions.
The paper is organized as follows. In Section 2 we recall some preliminary results and we show that the functional I has a geometrical structure (see Proposition 2.1). Finally in Section 3 we prove our main result, Theorem 1.1.

1) The Lebesgue norm
2) The spaces W 1,p 0 (Ω) and W −1,p ′ (Ω) are equipped with Poincaré norm and the dual norm, u := ( Ω |∇u| p ) 1 p and · * := · W −1,p ′ (Ω) respectively. 3) We denote by B(0, R) the ball of radius R centered at 0 in W 1,p 0 (Ω), ∂B(0, R) its boundary. 4) We denote by C i , c i > 0 any positive constants which are not essential in the arguments and which may vary from one line to another.

Preliminaries and geometry of the functional I
In this section, we recall the standard definition of Palais-Smale sequence at the levelc, Palais-Smale condition at the levelc, and we prove that our functional I defined as in (1.3) has a geometrical structure.
Let us define the level atc as follows Definition 2.2. Let (u n ) be a Palais-Smale sequence at the levelc of E. We say that (u n ) satisfies the Palais-Smale condition at the levelc if (u n ) possesses a convergent subsequence.
Before to prove that I has a geometrical structure, we need some properties of g, which we gather in the following lemma without proof , and for all r ∈ (p − 1, p).
Moreover, by simple calculation we get By combining all cases, (1) holds. To prove the propriety (2) we use (2) of the Lemma 2.1 and by the same previous argument, (2) is satisfied.
Proposition 2.1. Assume that c k and h k are sufficiently small. Then, the functional I has a geometrical structure, that is, I satisfies the following properties i) there exists ρ > 0, such that for all v in ∂B(0, ρ), I(v) ≥ β, where β > 0.
Proof. i) To prove this lemma we distinguish two cases on q . Firstly, if 0 < q < p−1, by using (2) of the Lemma 2.2 and Hölder's inequality, we get We choose r > p − 1, close to p − 1 such that (r + 1)k ′ < pN N −p , which exists due to the assumption k > N p . Obviously, (q + 1)k ′ < pN N −p , thus by using Sobolev's embedding we get Moreover, from the definition (1.2) of the function f , we have and by using Sobolev's embedding we get By the definition (1.3) of I, we deduce that Now, let v in ∂B(0, ρ), then we have We take ρ sufficiently large and such that c k ≤ ρ −r−2+p and h k ≤ ρ −1 (which are sufficiently small by hypothesis), then Secondly, if q ≥ p − 1 we choose again r as above and due to pk ′ < (r + 1)k ′ < pN N −p , then by using (1) of the Lemma 2.2 and Sobolev's embedding we get Now, we procedure as the first case, we get Then, we summarize the two cases we get 2) To prove the second property we show that I(tv) → −∞ when t → +∞. For this, let v ∈ C ∞ 0 (Ω) be a positive function such that cv 0. By the definition (1.3) of I, we have From inequality (2.1), we get Thus we deduce the desired result.
Finally, we stress that since I has a geometrical structure, then the existence of a Palais-Smale sequence at the levelc for I is assured. This can be observed directly from the proof given in ( [2]), or alternatively using Ekeland's variational principle ( [13]).

Proof of Theorem 1.1
In this section we give the proof of our main result which is divided into two steps as follows. In the first step, we show the existence of the first critical point (equivalently bounded weak solution for (P )) for the C 1 -functional I by using the Mountain Pass Theorem due to Ambrosetti-Rabinowitz ( [2]). For this, we show that the functional I satisfies the Palais-Smale condition at the levelc. In the second step, we show the existence of the second critical point of I (which is a local minimum) by using the lower semicontinuity argument in B(0, ρ). Moreover, we are going to see that this second critical point is different to the first one. Finally, we show that any solution of problem (P ) are bounded.

Palais-Smale condition.
In this subsection, we prove that I satisfies the Palais-Smale condition at the levelc. For this end, the main step is to show that the Palais-Smale condition at the levelc is bounded in W 1,p 0 (Ω). Then, we show that any Palais-Smale sequence of I at the levelc has a strongly convergent subsequence.
Before to prove the boundedness of the Palais-Smale sequence at the levelc in W 1,p 0 (Ω), we stress that, as we mentioned in the introduction, the key point is to show that g verifies the nonquadraticity condition at infinity (N Q). Hence, we have the following lemma Proof. To prove (N Q) we show that H is increasing and unbounded for s sufficiently large. We recall that H(s) = g(s)s − pG(s), and by simple calculation we get Now, we tend s → +∞ we obtain H(s) s δ → +∞ and M s δ → 0. Hence, we have a contradiction.
Proof. Let (u n ) be a Palais-Smale sequence at the levelc of W 1,p 0 (Ω). We prove by contradiction that (u n ) is bounded in W 1,p 0 (Ω). We suppose that (u n ) is unbounded in W 1,p 0 (Ω), then u n → +∞. For all integer n ≥ 0, we define We are going to prove that I(z n ) → +∞ and also (I(z n )) is bounded respectively, which is the desired contradiction.

Part 1: I(z n ) → +∞
We set v n := un un , then (v n ) is bounded in W 1,p 0 (Ω). Hence, there exists a subsequence denoted again (v n ) such that v n converges weakly and strongly to v in W 1,p 0 (Ω) and in L s (Ω) for some 1 ≤ s < p * respectively, where p * = N p N −p , is Sobolev conjugate, moreover, v n also converges to v almost everywhere in Ω. Now, we claim by contradiction that v ≡ 0 a.e. in Ω.
Since (u n ) is Palais-Smale type sequence, then we have for all ϕ ∈ W 1,p 0 (Ω) and for some ǫ n → 0 as n → +∞. We divide (3.2) by u n p−1 , then to obtain On the one hand, since v n converges weakly to v in W 1,p 0 (Ω) and by inequality (2.1), then for n large enough the second and the third terms of the righthand side of (3.3) are bounded. On the other hand, if v ≡ 0 in Ω, then cv ≡ 0 in Ω. Now, we choose ϕ ∈ W 1,p 0 (Ω) such that cvϕ > 0 in Ω ϕ and cvϕ ≡ 0 in Ω\Ω ϕ , with |Ω ϕ | > 0. Since v n u n = u n in Ω, then by using (3) of Lemma 2.1, we obtain Hence, by using the Fatou's lemma in (3.3) we obtain the unbounded term of the left-hand side of (3.3). Hence the claim. Since u n → +∞, then for some M > 0, we have u n > M , for some n large enough. Moreover, we have In what follows, we treat only the case 0 < q < p−1. By the same arguments we treat other case. Then, from inequality (2) of Lemma 2.2 we have |G(s)| ≤ c 1 |s| r+1 + c 2 |s| q+1 , where p − 1 < r < p. Since c ∈ L k (Ω), for some k > N p and v n converges strongly to v in L s (Ω) with 1 ≤ s < p * , then we obtain Ω c(x)G(M v n ) → 0, as n → +∞, due to v ≡ 0 a.e. in Ω.

Part 2: (I(z n )) is bounded
In this part, we prove that (I(z n )) is bounded. Now, we distinguish two cases: t n ≤ 2 un and t n > 2 un . For the first case we treat only the case 0 < q < p − 1 and other case we follow as the proof of Proposition 2.1 to obtain Sobolev embedding. For the second case we distinguish the cases on q.
Case t n ≤ 2 un : By the definition of (z n ) and I ∈ C 1 (W 1,p 0 (Ω), R), we have I ′ (t n u n ), t n u n = 0, which means that t p n u n p = Ω c(x)g(t n u n )t n u n + Ω h(x)f (t n u n )t n u n .
By the definition (1.3) of I, we have where the function H is defined as in (N Q) and K(s) := f (s)s − pF (s). Moreover, from inequality (2) we have By choosing r and q as in the proof of Proposition 2.1, we get By inequality (2.1) and Sobolev's embedding, we get then by (3.5), (3.6) and (3.4), we obtain where C is independent of n. Thus, (I(z n )) is bounded. Hence, this contradicts the fact that (I(z n )) is unbounded (see the Part 1 of the this proof).
Case t n > 2 un : Now, we are proceeding the technique inspired by ( [16]). For this, we need the following technical lemma for any positive function z in Ω, and p > 1.
Proof. Obviously we have i). To prove ii), we follow the same approach given in ( [16]) for the case p = 2 and z(x) = 1. We conclude the result for any positive function z and p > 1.
Now, we complete the proof of the Lemma 3.2. From the Lemma 3.1, we have H(s) ≥ σ, for s large enough and for some σ > 0 (which will be chosen later). Moreover, if 0 < q < p − 1, then from (2) of Lemma 2.1 we have for s sufficiently small, Then, by continuity of H we have for all s > − 1 Let 0 < s < t, we have We handle the terms A and B respectively, then we have By using (3.7), we get Moreover, we have In this case, the term integral A (see (3.8)) becomes as follows Moreover, since (p − 1)k ′ < pk ′ < N p N −p then by using Sobolev embedding, the rest of the proof is similar to the first case. Hence, we have also the contradiction.
We finish the proof of the Palais-Smale condition by the following lemma Lemma 3.4. Any Palais-Smale sequence at the levelc of W 1,p 0 (Ω) has a strongly convergent subsequence.

Second critical point.
In this step, by using the first property of the geometrical structure of I and the standard lower semicontinuity argument, we show the existence of the second critical point. We state the result as follows Proof. Since I(0) = 0, then inf v∈B(0,ρ) I(v) ≤ 0. If h ≡ 0, then we obtain that inf v∈B(0,ρ) I(v) < 0. Indeed, we choose v ∈ C ∞ 0 (Ω) a positive function satisfies cv > 0 and hv > 0. From the definition of I we have for t > 0 If q ≥ p − 1, from (2) of the Lemma 2.1, we have G(s)/s p → c < +∞ as s → 0 + . If 0 < q < p − 1, we use again Lemma 2.1, we get G(s)/s p → +∞ as s → 0 + . Moreover F (s) s p → +∞ as s → 0 + in both cases. Hence, by using this limits in (3.11) we get that I(tv) < 0 for t > 0 small enough. Now, we set m := inf v∈B(0,ρ) I(v) and by Proposition 2.1 we have I(v) ≥ β > 0 for v = ρ. Then, there exists a sequence (v n ) ⊂ B(0, ρ) such that I(v n ) converges to m. Since (v n ) is bounded in W 1,p 0 (Ω), then there exists a subsequence denoted again (v n ) such that v n converges weakly and strongly to v in W 1,p 0 (Ω) and L s (Ω) for some 1 ≤ s < p * respectively. Then, we get Moreover, we have v p ≤ lim inf n→∞ v n p , then we obtain that I(v) ≤ m = inf v∈B(0,ρ) I(v). Hence, we conclude that v is local minimum of I in B(0, ρ).
Remark 3.2. By the subsection 3.1 , I has a critical point at the levelc, that is, there exists w in W 1,p 0 (Ω) such that I(w) =c and I ′ (w) = 0. Since I(w) =c > 0 ≥ I(v), with v ∈ B(0, ρ) is the second critical point given in the previous theorem, then w is different to v. Hence, we have two distinct solutions of problem (P ).

Boundedness of solutions.
Now, to finish the proof of our main result, it remains to show the boundedness of the solutions. Then, we have the following result Proof. If |u| ≤ 1, it is over. Otherwise, since the problem (P ′ ) may be written as −∆ p u = a(x)(1 + |u| p−1 ), where a(x) = c(x)g(u) + h(x)f (u) 1 + |u| p−1 .
Let m > 1 and m ′ it's conjugate, by using use Hölder's inequality in (3.12), we obtain