Multiplicity Results for Weak Solutions of a Semilinear Dirichlet Elliptic Problem with a Parametric Nonlinearity
This paper deals with the existence of weak solutions to a Dirichlet problem for a semilinear elliptic equation involving the difference of two main nonlinearities functions that depends on a real parameter . According to the values of , we give both nonexistence and multiplicity results by using variational methods. In particular, we first exhibit a critical positive value such that the problem admits at least a nontrivial non-negative weak solution if and only if is greater than or equal to this critical value. Furthermore, for greater than a second critical positive value, we show the existence of two independent nontrivial non-negative weak solutions to the problem.
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:where is a bounded domain in , is a real parameter, and is a nonlinear function taking different forms. According to the values of , Ambrosetti et al. studied in , the existence and multiplicity of non-negative weak solutions of the problem (1) when with . For example, by using variational method, they show the existence of infinitely many solutions of the problem as follows:for and small. Later, Alama and Tarantello in  studied the semilinear Dirichlet problem (1) by searching non-negative solutions withwhere is a bounded domain with smooth boundary, and are suitable functions, and . In this case also, the authors show the influence of values of on the existence and multiplicity of weak solutions of the problem. These different 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., [3–6] 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 . For example, we can mention the papers [7–9], which are devoted to the unbounded case. In , the authors deal with the nonexistence and existence of nontrivial weak solutions of the quasilinear problem:where is a smooth exterior domain in , is the unit vector of the outward normal on , or , and , and are suitable functions. They showed in different cases that the existence of weak solutions of the problem depends on the values of relative to the value of some critical value.
In , Pucci and Radulescu studied the following problem in whole space:where satisfies is a parameter and with if and if . They obtained that the nonexistence and multiplicity of nontrivial weak solutions of this quasilinear elliptic equation are corresponding to the smallness and the largeness of , respectively. In , Autuori and Pucci extended the results in  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 , , is a bounded domain with smooth boundary, , are suitable non-negative functions, and 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 according to the values of the parameter . To obtain our results in this work, we require in the following assumptions:
is a function continuous on and of such that
Let us set
There exist such that and
is a function continuous on and of such that
Let us set
There exists such that and
Some examples of functions and in satisfying the previous assumptions , , and :(1)For , we can have and in () with .(2)Another example of functions and is the following: In this case, for all t > 0, we have(3)Let fixed in . For all ,. Hence, for all t > 0, we have
The main goal of this paper is the proof of the following two theorems:
Theorem 1. By the fulfillment of assumptions , , and , there exists a critical value such that the Dirichlet problem admits at least a nontrivial non-negative weak solution if and only if .
Theorem 2. Suppose that the assumptions , , and are fulfilled. Then, there exists a critical value satisfying such that for all , the Dirichlet problem 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 different imbeddings between the working spaces of this paper and prove the nonexistence of a nontrivial weak solution when in is least than a positive number. The conditions for existence of weak solutions of are established in Section 4. Section 5 has devoted to prove Theorem 1, and Section 6 deals with the proof of Theorem 2.
2. Notions on Orlicz Spaces (See Chapter 8 in )
Definition 1. (definition of a N-function). Let be a real-valued function defined on and having the following properties:(a)(b) is nondecreasing, that is, implies (c) is right continuous, that is, if , then Then, the real-valued function defined on byis called an N-function.
Any such N-function has the following properties:(i) is continuous on (ii) is strictly increasing(iii) is convex(iv) and (iv)The function is strictly increasing on For any N-function and an open set , the Orlicz space is defined. When satisfies -condition, i.e.,for some constant , thenEndowed with the normwhich is called the Luxembourg norm, the Orlicz space is a Banach space. It is known that if , then with .
The complement of is given by the Legendre transformation as follows:We say that and are complementary N-functions of each other.
For all , we have the inequality .
From Young’s inequalitya generalized version of Hölder’s inequality is obtained as follows:
3. Preliminaries and Nonexistence of Nontrivial Solution for Small
By condition and the definition of and by condition and the definition of , the functions and are N-functions, with or , respectively.
Lemma 1. The -function satisfies the -condition.
Lemma 2. The -function satisfies the -condition.
Proof. In fact, by (17)and satisfies the -condition by Theorem 4.1 in .
Let us consider the Sobolev space , which is the completion of with the normLet denote the completion of with respect to the normwhereis the Luxembourg norm in the Orlicz space . The space is the space in which we will find our nontrivial weak solutions.
Lemma 3. The embeddings are continuous with and .
Moreover, and are compact for all such that .
Proof. The first imbedding that i.e., is followed from the definition of the norm in . The second imbedding is followed from Talenti’s work in ; is the Talenti constant. The imbedding compact for is obtained by Rellich’s Theorem. It follows that the mapping is compact for .
Lemma 4. Let , , , and . Then,
Proof. The proof is given in  (see Lemma 2.1 of ). In fact by integrating the inequalities (29) and (30) respectively, we get inequalities (34) and (35). From (34) and (35) and the definition of Luxembourg norm we get respectively (35) and (37).
Lemma 5. The space is imbedded continuously in with , wherewith and .
Proof. This proof is based on the proof of Theorem 8.12 in . Fix , with , thenby (34) and (36). Thus, by taking , we haveBy the proof of (Theorem 8.12 ),Let such that . Let us set and It follows that and consequently is imbedded continuously in .
Lemma 6. The satisfies the following:
Lemma 7. The imbedding is continuous withwhere, .
Proof. The -function increases more slowly than near infinity and by (34) it follows that for all . By the proof of (Theorem 8.12 ),Let such that . Let us set and .It follows that and consequently is imbedded continuously in .
Lemma 8. The imbedding is continuous with
Moreover, the imbeddings and are compacts.
Proof. The continuity of the imbedding is followed from Lemmas 3 and 7. By Lemma 6 and Theorem 8.36 in , it follows that the imbedding is compact. It follows also that is compact because is continuous by Lemma 3.
Lemma 9. If is an element of and is a real such thatthen and there exists two positive constants and independent of such thatwhere and are two functions of .
Proof. Let us take and such that (49) hold. Thus, we have . As is a non-negative function in , it follows that . Let us show the second part of the Lemma. Firstly, by using respectively (29), (35), (48), and (49), we getThus, this last inequality yieldsSecondly, by (29), (30), (48), and (49) we haveLet us set and . Since , (34) and (36) implyLet or . By applying Young’s inequality for and and , we havewhere , , , and . Inequality (55) yieldsbeing . Hence, (54) yieldsAs , it follows, from (58), thatThus, by (53) and (59), we getFinally, the last inequality and (52) yield the result (50).
Definition 2. An element of is a weak solution of ifConsequently, the weak solutions of are exactly the critical points of the energy functional defined by
Lemma 10. If has a nontrivial weak solution , then , wherewith
Proof. If admits a nontrivial weak solution , then equality (49) is satisfied and by Lemma 9. Let us now show that . By inequalities (50) and (60) of the proof of Lemma 9, we get the result.
We claim that the set is not empty and bounded above. Indeed, by Lemma 9, for all , admit only trivial solution. Thus, . Now suppose that for all , there exists such that . Therefore, there exists a sequence of elements of such that For all , for all such that , admits only trivial solution. By hypothesis, the sequence tends to . Hence, for all , admits only trivial solution. This contradicts the Lemma 9.
Let us defineIt is clear that . In Section 5, we will prove that is the required critical value of the Theorem 1.
4. Basic Results for Existence of Nontrivial Solution
The results in the previous section require us to work from now on with .
Lemma 11. The energy functional in coercive on .
Proof. Let .By using Young’s inequality, as we use it in (55), we havewhere . Therefore, we haveIn conclusion, is coercive in .
Lemma 12. Let and respectively be the complements of -functions and . Then, we havewhere and are respectively the derivatives of and .
Proposition 1. The spaces and are reflexives and separables Banach spaces.
Proof. The N-functions and satisfy -condition respectively by Lemmas 1 and 2. Moreover, the inequalities (69) and (70) of Lemma 12 imply that the N-functions and satisfy the -condition. Thus, by (Theorem 8.20 and Remark 8.22 in ), the spaces and are reflexives and separables Banach spaces.
Proposition 2. is a separable reflexive Banach space.
Proof. Let , which we endow with the norm . Since and are reflexives Banach spaces, then is a separable and reflexive Banach space by Theorem 1.23 in . Let us consider the operator defined by . is well defined, linear, and isometric. Therefore, is closed subspace of and so is separable and reflexive by Theorem 1.22 in . Consequently, is a separable reflexive Banach space, being isomorphic to a separable, reflexive space. Finally, we conclude that is a separable reflexive Banach space because reflexivity and separability are preserved under equivalent norms.
Lemma 13. Let be a sequence in such that is bounded. Then, admits a weakly convergent subsequent in .
Proof. The proof comes from the coercivity of in and the reflexivity of the space .
Lemma 14. The functional is convex, of class and is particularly sequentially weakly lower semicontinuous in .
Proof. The convexity of functional is followed from the convexity of function in .
Let us show the continuity of on . Let be a sequence of which elements are in and let such that in . Let be an arbitrary subsequence of . The subsequence in and hence, by Lemma 3, in . By Theorem 4.9 in , there exists a subsequence of and a function such that a.e in as and a.e in for all . So a.e in as and a.e in for all . The dominated convergence theorem implies that in as . The subsequence being arbitrary, we deduce that in as . Then, we get the continuity of on and also, is sequentially weakly lower semicontinuous in by Corollary 3.9 in . Moreover, is Gateaux-differentiable in and for all , we haveLet us finish the proof by showing that is continuous. Let be a sequence of functions in and such that in when . By a simple calculus we haveThis inequality yields the continuity of .
Lemma 15. The functional is convex, of class and sequentially weakly lower semicontinuous in . Furthermore, if is a sequence of elements of and belongs to such that converge weakly to in , then in .
Proof. The convexity of the functional is obvious. The continuity of in follows from Lemma 3. Thus, is sequentially weakly lower semicontinuous in by Corollary 3.9 in . To complete the proof of the theorem, it suffices to show the last part of the theorem. Therefore, let us take and such that weakly in . Since is compact by lemma 3, we have and then in .
Proposition 3. (i) If converge strongly to in , then converge to in . (ii) If converge strongly to in , then converge to in .
Proof. (i) Suppose that converge strongly to in . Let be a fixed subsequence of the sequence . For all , we have , where is a subsequence of the sequence , which converge to in . Thus, , as . It follows that, there exists a subsequence of and a positive function such that a.e in and a.e in . As is continuous and strictly increasing in , we have a.e in . Hence, a.e in and a.e in . On other hand, there exists such thata.e in . Indeed, by the fact that a.e in and the function is continuous and increases strictly in , we haveBy (13) and since satisfy -condition, we have