Multiplicity Results for Weak Solutions of a Semilinear Dirichlet Elliptic Problem with a Parametric Nonlinearity

Tis paper deals with the existence of weak solutions to a Dirichlet problem for a semilinear elliptic equation involving the diference of two main nonlinearities functions that depends on a real parameter λ . According to the values of λ , we give both nonexistence and multiplicity results by using variational methods. In particular, we frst exhibit a critical positive value such that the problem admits at least a nontrivial non-negative weak solution if and only if λ is greater than or equal to this critical value. Furthermore, for λ greater than a second critical positive value, we show the existence of two independent nontrivial non-negative weak solutions to the problem.


Introduction
In the last years, most works studied the existence, nonexistence, and multiplicity of nontrivial weak solutions of a semilinear Dirichlet problem of the form as follows: − Δu � f λ (x, u), x ∈ Ω, where Ω is a bounded domain in R N , λ is a real parameter, and f λ : Ω × R ⟶ R is a nonlinear function taking different forms. According to the values of λ, Ambrosetti et al. studied in [1], the existence and multiplicity of non-negative weak solutions of the problem (1) when f λ (x, u) � λu q + u p with 0 < q < 1 < p. For example, by using variational method, they show the existence of infnitely many solutions of the problem as follows: − Δu � λ|u| q− 1 u + |u| p− 1 u, x ∈ Ω, for λ > 0 and small. Later, Alama and Tarantello in [2] studied the semilinear Dirichlet problem (1) by searching non-negative solutions with where λ ∈ R, Ω ⊂ R N (N ≥ 3) is a bounded domain with smooth boundary, ω and h are suitable functions, and 1 < q < r. In this case also, the authors show the infuence of values of λ on the existence and multiplicity of weak solutions of the problem. Tese diferent studies on nonexistence, existence, and multiplicity results for nontrivial weak solutions depending on a parameter for a Dirichlet problem for a semilinear elliptic equation were extensively investigated in the literature (see, for e.g., [3][4][5][6] and the references therein). Similar results, depending on a real parameter, are obtained in the case of quasilinear elliptic equations in bounded domains or in entire space R N . For example, we can mention the papers [7][8][9], which are devoted to the unbounded case. In [7], the authors deal with the nonexistence and existence of nontrivial weak solutions of the quasilinear problem: − div a(x)|∇u| p− 2 ∇u + |u| q− 2 u � λg(x)|u| r− 2 u in Ω, a(x)|∇u| p− 2 z ] u + b(x)|∇u| p− 2 u � 0, on zΩ, where Ω is a smooth exterior domain in R N , ] is the unit vector of the outward normal on zΩ, 1 < p < N, (p < r < q < p * � (Np/(N − p))) or p < q < r < p * , and a, b, and g are suitable functions. Tey showed in diferent cases that the existence of weak solutions of the problem depends on the values of λ relative to the value of some critical value.
In [8], Pucci and Radulescu studied the following problem in whole space: where h > 0 satisfes λ > 0 is a parameter and 2 ≤ p < q < min r, p * with p * � (Np/(N − p)) if N > p and p * � ∞ if N ≤ p. Tey obtained that the nonexistence and multiplicity of nontrivial weak solutions of this quasilinear elliptic equation are corresponding to the smallness and the largeness of λ, respectively. In [10], Autuori and Pucci extended the results in [8] by solving a more general quasilinear elliptic equation with the same variational method. Motivated by these previous results, we are concerned in this paper with the existence, nonexistence, and multiplicity of nontrivial weak solutions of the following Dirichlet problem for a semilinear elliptic equation: where Ω ⊂ R N , N ≥ 3, is a bounded domain with smooth boundary, a, b are suitable non-negative functions, and λ is a real parameter. By taking inspiration on the method developed in [8,10], we use variational arguments to study the existence and the multiplicity of nontrivial weak solutions of problem (ε λ ) according to the values of the parameter λ. To obtain our results in this work, we require in(ε λ ) the following assumptions: (A 1 )a is a function continuous on R + and of C 1 ((0, +∞), R + ) such that lim t⟶+∞ a(t) � +∞.
Let us set Let us set (B 2 ) Tere exists r ∈ ]2, N[ such that q < r and Some examples of functions a and b in (ε λ )satisfying the previous assumptions (A 1 ), (A 2 )(B 1 ), and (B 2 ): (1) For u ≥ 0, we can have a(u) � u p− 2 and b(u) � u r− 2 in (ε λ ) with 2 < p < r < N < 2 * . (2) Another example of functions a and b is the following: , for u > 0, 0, for u � 0, , where t 0 is the unique solution of the equation σ ln(σ + t) − t � 0 on(0, +∞). Hence, for all t > 0, we have Tus, it is clear that the functions a and b of the Dirichlet problem (ε λ ) of our present work generalize the functions |u| q− 2 and |u| r− 2 (2 < q < r) which appear in the main equation studied in [7,8] or [10].
Te main goal of this paper is the proof of the following two theorems: Theorem 1. By the fulfllment of assumptions (A 1 ), (A 2 )(B 1 ), and (B 2 ), there exists a critical value λ > 0 such that the Dirichlet problem (ε λ ) admits at least a nontrivial non-negative weak solution if and only if λ ≥ λ. Theorem 2. Suppose that the assumptions (A 1 ), (A 2 )(B 1 ), and (B 2 ) are fulflled. Ten, there exists a critical value λ satisfying λ ≥ λ such that for all > λλ, the Dirichlet problem (ε λ ) admits at least two nontrivial non-negative weak solutions.
In Section 2, we talk about Orlicz spaces, which we will use in our work. In Section 3, we give diferent imbeddings between the working spaces of this paper and prove the nonexistence of a nontrivial weak solution when λ in (ε λ ) is least than a positive number. Te conditions for existence of weak solutions of (ε λ ) are established in Section 4. Section 5 has devoted to prove Teorem 1, and Section 6 deals with the proof of Teorem 2.

Notions on Orlicz Spaces (See Chapter 8 in [11])
Defnition 1. (defnition of a N-function). Let ψ be a realvalued function defned on [0, ∞) and having the following properties: Ten, the real-valued function Ψ defned on [0, +∞) by is called an N-function. Any such N-function Ψ has the following properties: For any N-function Ψ � Ψ(t) and an open set Ω ⊂ R N , the Orlicz space L Ψ (Ω) is defned. When Ψ satisfes Δ 2 -condition, i.e., for some constant k > 0, then Endowed with the norm which is called the Luxembourg norm, the Orlicz space Te complement of Ψ is given by the Legendre transformation as follows: We say that Ψ and Ψ are complementary N-functions of each other.
For all t > 0, we have the inequality Ψ(Ψ(t)/t) ≤ Ψ(t). From Young's inequality a generalized version of Hölder's inequality is obtained as follows: International Journal of Mathematics and Mathematical Sciences

Preliminaries and Nonexistence of Nontrivial Solution for λ Small
By condition (A 1 ) and the defnition of A and by condition (B 1 ) and the defnition of B, the functions A and B are N-functions, with ψ(s) � sa(s) or ψ(s) � sb(s), respectively.
Proof. Tis proof is based on the proof of Teorem 8.12 in [11]. Fix k, with k ≥ 1, then By the proof of (Teorem 8.12 [11]), where ω � max 1, International Journal of Mathematics and Mathematical Sciences It follows that ‖u‖ A ≤ 2ωk‖u‖ B and consequently L B (Ω) is imbedded continuously in L A (Ω). □ Lemma 6. Te N − functionA satisfes the following: Proof. Te result is followed from property (34). In fact, by (34), for all k ≥ 0, for all t ≥ 1, we have Proof. Te N-function A increases more slowly than t ⟼ t 2 * near infnity and by (34) it follows that for all By the proof of (Teorem 8.12 [11]), It follows that ‖u‖ A ≤ 2ω(A(1)) (1/2 * ) ‖u‖ 2 * and consequently (48) Proof. Te continuity of the imbedding H 1 0 (Ω) ⟶ L A (Ω) is followed from Lemmas 3 and 7. By Lemma 6 and Teorem 8.36 in [11], it follows that the imbedding then λ > 0 and there exists two positive constants k 1 and k 2 independent of u such that where f 1 and f 2 are two functions of λ.
It is clear that λ ≥ λ 0 > 0. In Section 5, we will prove that λ is the required critical value of the Teorem 1.

Basic Results for Existence of Nontrivial Solution
Te results in the previous section require us to work from now on with λ > 0.
Lemma 11. Te energy functional Φ λ in coercive on X.
Proof. Let u ∈ X.
International Journal of Mathematics and Mathematical Sciences By using Young's inequality, as we use it in (55), we have where c λ � max λ (r/(r− p)) (A(1)/(B(1)) (p/r) ) (r/(r− p)) , In conclusion, Φ λ is coercive in X. A and B respectively be the complements of N-functions A and B. Ten, we have

Lemma 12. Let
where A ′ (s) and B ′ (s) are respectively the derivatives of A(s) and B(s).
Proof. Te proof comes from the coercivity of Φ λ in X and the refexivity of the space X. □ Lemma 14. Te functional Φ Δ (u) � (1/2) Ω |∇u| 2 dx is convex, of class C 1 and is particularly sequentially weakly lower semicontinuous in X.
Let us show the continuity of Φ Δ (u) on X. Let (u n ) n be a sequence of which elements are in X and let u ∈ X such that u n ⟶ u in X. Let (u n k ) k be an arbitrary subsequence of (u n ) n . Te subsequence u n k ⟶ u in X and hence, by Lemma 3, ∇u n k ⟶ ∇u in (L 2 (Ω)) N . By Teorem 4.9 in 8 International Journal of Mathematics and Mathematical Sciences [15], there exists a subsequence (u n k j ) j of u n k and a function ψ ∈ L 1 (Ω) such that ∇u n k j ⟶ ∇u a.e in Ω as j ⟶ ∞ and |∇u n k j | ≤ ψ a.e in Ω for all j ∈ N. So |∇u n k j | 2 ⟶ |∇u| 2 a.e in Ω as j ⟶ ∞ and |∇u n k j | 2 ≤ ψ 2 ∈ L 1 (Ω) a.e in Ω for all j ∈ N . Te dominated convergence theorem implies that |∇u n k j | 2 ⟶ |∇u| 2 in L 1 (Ω) as j ⟶ ∞. Te subsequence (u n k ) k being arbitrary, we deduce that |∇u n | 2 ⟶ |∇u| 2 in L 1 (Ω) as n ⟶ ∞. Ten, we get the continuity of Φ Δ on X and also, Φ Δ is sequentially weakly lower semicontinuous in X by Corollary 3.9 in [15]. Moreover, Φ Δ is Gateaux-differentiable in X and for all u, φ ∈ X, we have Let us fnish the proof by showing that Φ Δ ′ is continuous. Let (u n ) n be a sequence of functions in X and u ∈ X such that u n ⟶ u in X when n ⟶ ∞. By a simple calculus we have Tis inequality yields the continuity of Φ Δ ′ . □ Lemma 15. Te functional Φ 2 (u) � (1/2) Ω |u| 2 dx is convex, of class C 1 and sequentially weakly lower semicontinuous in X. Furthermore, if (u n ) is a sequence of elements of X and u belongs to X such that u n converge weakly to u in X, then Proof. Te convexity of the functional Φ 2 is obvious. Te continuity of Φ 2 in X follows from Lemma 3. Tus, Φ 2 is sequentially weakly lower semicontinuous in X by Corollary 3.9 in [15]. To complete the proof of the theorem, it sufces to show the last part of the theorem. Terefore, let us take (u n ) ⊂ X and u ∈ X such that u n ⇀ u weakly in X. Since X ⟶ ⟶ L 2 (Ω) is compact by lemma 3, we have ‖Φ 2 ′ (u n )−

Proof
(i) Suppose that u n converge strongly to u in L A (Ω). Let (v n k ) k be a fxed subsequence of the sequence n⟼a(|u n |)u n . For all k , we have v n k � a(|u n k |)u n k , where (u n k ) k is a subsequence of the sequence (u n ), which converge to u in L A (Ω). Tus, Ω A(|u n k − u|)dx ⟶ 0, as k ⟶ ∞. It follows that, there exists a subsequence (u n k j ) j of (u n k ) k and a positive function ψ ∈ L 1 (Ω) such that A(|u n k j − u|) ⟶ 0 a.e in Ω and 0 ≤ A(|u n k j − u|) ≤ ψ(x) a.e in Ω, for all j ∈ N. As A is continuous and strictly increasing in R + , we have u n k j ⟶ ua.e in Ω. Hence
(ii) Te proof is analogous to (i). where Let (u n ) n be a sequence of X and u an element of X such that u n ⟶ u in X. u belongs to L A (Ω) by Lemma 8 and we have Ω A(|u|)dx < ∞.u n ⟶ u in X implies u n ⟶ u in H 1 0 (Ω). So, u n ⟶ u a.e in Ω. By (35), properties of function ζ 1 and Lemma 8, we have As u n ⟶ u in X, there exists M > 0 such that for all integer n, Since u belongs to L A (Ω), A(|θu|) ∈ L 1 (Ω) for all θ ∈ R and thus, 0 ≤ Ω ψ ε (u)dx < ∞, for all ε > 0. (78) Hence, by Teorem 2 in [16], it follows that Φ A (u n ) ⟶ Φ A (u), as n ⟶ ∞. Tis assures the continuity of the functional Φ A . It then follows that Φ A is sequentially weakly lower semicontinuous in X by Corollary 3.9 in [15]. Furthermore, Φ A is Gateaux-diferentiable in X and for all u, φ ∈ X, Let (u n ) ⊂ X and u ∈ X such that u n ⇀u in X as n ⟶ ∞. Let us show that Φ A ′ (u n ) ⟶ Φ A ′ (u) in X ′ . By Hölder's inequality, we have Since u n ⇀u in X, by Lemma 8 and Proposition 3, it follows that ‖a(|u n |)u n − a(|u|)u‖ A ⟶ 0 and consequently, Φ A ′ (u n ) ⟶ Φ A ′ (u) in X ′ . In particular, this shows that Φ A is class C 1 in X and the proof of the Lemma is fnished. □ Lemma 17. Te functional Φ B (u) � Ω B(|u|)dx is convex, of class C 1 and sequentially weakly lower semicontinuous in X . Moreover, if (u n ) n is a sequence of elements in X and u ∈ X such that u n ⇀u ∈ X as n ⟶ ∞, Proof. Te convexity of Φ B follows from the convexity of the positive function t⟼B(|t|) defned on R. Te proof of continuity of Φ B in X is analogous to the proof of continuity of Φ A in X in the previous Lemma. In fact, by Lemma 3 in [16], for all k > 1, for ε > 0, such that, 0 < ε < (1/k), we have where Let (u n ) n ⊂ X and u ∈ X such that u n ⟶ u in X. By the defnition of the space X, we have X ⟶ L B (Ω), Ω B(|u|)dx < ∞, and u n ⟶ u in L B (Ω). u n ⟶ u in X implies u n ⟶ u in H 1 0 (Ω). So, u n ⟶ ua.e in Ω. By (35) and properties of function ζ 2 , we have As u n ⟶ u in X, there exists M > 0 such that for all integer n, 0 ≤ Ω φ ε (u n − u)dx ≤ k r M. Since u belongs to L B (Ω), B(|θu|) ∈ L 1 (Ω) for all θ ∈ R and thus, Hence, by Teorem 2 in [16], it follows that Φ B (u n ) ⟶ Φ B (u) as n ⟶ ∞. Tis assures the continuity of the functional Φ B in X. It then follows that Φ B is sequentially weakly lower semicontinuous in X by Corollary 3.9 of [15]. Furthermore, Φ B is Gateaux-diferentiable in X and for all u, φ ∈ X, Let (u n ) ⊂ X and u ∈ X such that u n ⟶ u in X as n ⟶ ∞. Let us show that Φ B ′ (u n ) ⟶ Φ B ′ (u) in X ′ . By Hölder's inequality, we have According to Proposition 3 (ii), it follows that ‖b(|u n |)u n − b(|u|)u‖ B ⟶ 0 and consequently, Let us show the last part of this Lemma. Let (u n ) ⊂ X and u ∈ X such that u n ⇀u in X. We shall proof that For any integer n, let us set v n � b(|u n |)u n and consider an arbitrary subsequence (v n k ) k of the sequence (v n ). We have v n k � b(|n k |)u n k , where (u n k ) k is a subsequence of (u n ). Ten, u n k ⇀u weakly in X when k ⟶ ∞. By Lemma 3, it follows that u n k ⟶ u in L 2 (Ω). Tus, there exists a subsequence (u n k j ) j of (u n k ) k such that u n k j ⟶ u a.e in Ω and |u n k j | ≤ g a.e in Ω, where g ∈ L 1 (Ω). In consequence, for all φ ∈ X, v n k j φ ⟶ b(|u|)uφ a.e in Ω and there exists h ∈ L 1 (Ω) such that |b(|u n k j |)u n k j φ − b(|u|)uφ| ≤ h a.e in Ω. In fact, by (30) and (36) we have In conclusion, Φ B ′ (u n )⇀ * Φ B ′ (u) in X ′ , as n ⟶ ∞, since (v n k ) k is an arbitrary subsequence of the sequence (v n ).
Proof. Fix λ > λ. By defnition of λ, there exists h ∈ (λ, λ) such that Φ h has a nontrivial critical point u h ∈ X. Without loss of generality, we assume that u h ≥ 0a.e in Ω, since |u h | is also a solution of (ε h ). We can easily see that u h is a subsolution for (ε λ ). Let us consider the following minimization problem: We remark easily that C is a closed and convex set. Tus, C is weakly closed. Moreover, being Φ λ is coercive in X by Lemma 11, it follows that it is coercive in C. Finally, Φ λ is sequentially weakly lower semicontinuous in X and so in C.