Positive solutions for a semipositone anisotropic p -Laplacian problem

In this paper, a semipositone anisotropic p -Laplacian problem –


Introduction
Mathematically, a positione is a particular kind of eigenvalue problem involving a nonlinear function on the reals that is continuous, positive, and monotone.A semipositone is an eigenvalue problem that would be a positone eigenvalue problem except that the nonlinear function is not positive when its argument is zero.
Semipositone problems naturally arise in various studies.For example, consider the Rozenwig-McArthur equations in the analysis of competing species where "harvesting" takes place.The study of positive solutions to these problems, unlike the positone case, turns into a nontrivial question as 0 is not a subsolution, making the method of subsupersolutions difficult to apply.Semipositone problems, again unlike positone problems, give rise to the interesting phenomenon of symmetry breaking (see [8]).
Consider the nonlinear eigenvalue problems of the form When f is positive and monotone, it is referred to in the literature as a positone problem.The case where f satisfies, f (0) < 0, f is monotone and eventually positive, is referred to in the literature as a semipositone problem.The study of positive solutions to semipostone problems is considerably more challenging, since the range of a solution must include regions where f is negative as well as where f is positive.The study of semipositone problems was first formally introduced by Castro et al. in 1988 (see [7]) in the case of Dirichlet boundary conditions, where several challenging differences were noted in their study when compared to the study of positone problems.Perera et al. [16] consider the p-superlinear semipositone p-Laplacian problem and proved the the existence of ground-state positive solutions (see [4][5][6]9] for other cases).Alves et al. [2] prove the existence of a solution for the class of the semipositone problem via the variational method together with estimates that involve the Riesz potential (see also [1,10,11,21]).Fu et al. [14] prove the existence of positive solutions for a class of semipositone problems with singular Trudinger-Moser nonlinearities.The proof is based on compactness and regularity arguments.
Castro et al. [6] study the existence of positive weak solutions to the problem (1.1).Here, we refer to [6] and study the existence of positive weak solutions to the problem where -− → p is the anisotropic p-Laplace operator, is an open smooth bounded domain in R N , N ≥ 2 and the function f : R → R is a differentiable function with f (0) < 0 (semipositone), which implies that u = 0 is not a subsolution to (1.2), making the finding of positive solutions rather challenging (see [15]).We set − → p := (p 1 , . . ., p N ), where Here and after, we assume p + < p .Thus, p ∞ = p : (H 1 ) Suppose there exist q ∈ (p + -1, p -1), A > 0, B > 0 such that (1.3) (H 2 ) Assume an Ambrosetti-Rabinowitz-type condition, i.e., that there exist θ > p + and M ∈ R such that where 3) implies that there exist positive real numbers A 1 , B 1 such that and With respect to the above, the main result of this paper is Theorem 1.2.Our result extends the result of [5, Theorem 1.1] and [6, Theorem 1.1].Theorem 1.2There exists λ * > 0 such that if λ ∈ (0, λ * ), then the problem (1.2) has a positive weak solution u λ ∈ L ∞ ( ).
The rest of the paper is organized as follows.In Sect.2, the suitable function space that is the anisotropic Sobolev space is recalled and necessary facts are also recalled.In Sect.3, we study the Mountain-Pass Theorem and Palais-Smale condition for the problem.In Sect.4, we present the proof of the main result, Theorem 1.2, which shows the existence of a positive solution of the problem (1.2).

Function spaces
Here, we define the anisotropic Sobolev spaces (see [18][19][20] and references therein), to which the solutions for our problems naturally belong, by with the norm We recall the following theorem [13, Theorem 1].
Theorem 2.1 Let ⊂ R N be an open bounded domain with a Lipschitz boundary.If (2.3) Remark 2.3 J λ is a functional of class C 1 and the critical points of the functional J λ are the weak solutions of (1.2) (see [17] for a similar argument).
By the Mountain-Pass Theorem we can prove the existence of one solution of (1.2) and then we show for the proper value of λ that the solution is positive.

Mountain-Pass Theorem and Palais-Smale condition
The next two lemmas prove that J λ satisfies the geometric hypotheses of the Mountain-Pass Theorem. .
Proof By the Sobolev embedding Theorem 2.1, there exists where we have used that τ ≤ (2p + K q+1 1+rp+ , the lemma is proven. Next, using the Mountain-Pass Theorem we prove that (1.2) has a solution There exists c 2 > 0 such that, for each λ ∈ (0, λ 3 ), the functional J λ has a critical point u λ of mountain-pass type that satisfies J λ (u λ ) ≤ c 2 λ -p + r .
Proof First, we show that J λ satisfies the Palais-Smale condition.

Assume that {u
Let K be a constant such that |J λ (u n )| ≤ K for all n = 1, 2, . . . .From (1.4), we obtain From the last two inequalities we have Now, we consider two cases.Case (i): This shows (3.5) can be written as This proves that {u n } is a bounded sequence.Thus, without loss of generality, we may assume that {u n } converges weakly.Let u ∈ W 1, − → p 0 ( ) be its weak limit.Since q < Np + (N-p + ) , by the Sobolev embedding theorem we may assume that {u n } converges to u in L q ( ).These assumptions and Hölder's inequality imply From (3.6) and lim n→+∞ J λ (u n ) = 0, we have Using again that u is the weak limit of {u n } in W The relations (3.7)-(3.9)imply that This shows that for each i = 1, . . ., N This proves that J λ satisfies the Palais-Smale condition.
Proof By (1.4) and the definition of u λ , where we have used 0 < λ < 1.Now, the result follows from (3.14) and the fact that u λ is a weak solution of (1.2).

Existence of a positive solution
Now, we can prove Theorem 1.2 as follows.
Proof Suppose there exists a sequence {λ j } j , 1 > λ j > 0 for all j, converging to 0 such that the measure m({x ∈ ; u λ j (x) ≤ 0}) > 0. Letting From Lemmas 3.5 and 3.6 there is a constant C 3 such that This shows that for each i = 1, . . ., N and therefore ( ) be the solution of The solution v j of the equation is given by v j = (-K j ) 1 ≥ K j , it follows by the comparison principle in [12,Lemma 2.5] that w j ≥ v j .Then, the fact that v j (x) → 0 as j → 0 implies that w(x) ≥ 0 for all x ∈ .Since, by hypothesis, q > p + -1, we have s = Np + r (N-p + ) > 1.This result, together with the Sobolev embedding Theorem, (1.3) and Lemma 3. where C > 0 is a constant independent of j and, without loss of generality, we have assumed u λ j ∞ ≥ 1.From (4.7) and the fact that rNp + (sN-sp + ) = 1 we see that {λ j f (u λ j ) u λ j (1-p + ) ∞ } is bounded in L s ( ), so we may assume that it converges weakly.Let z ∈ L s ( ) be the weak limit of such a sequence.Since λ j u λ j (1-p + ) ∞ → 0 as j → +∞ and f is bounded from below,  Therefore, since {w j } j converges w in L ∞ ( ), for sufficiently large j, w j (x) > 0 for all x ∈ .Hence, u λ j (x) > 0 for all x ∈ , which contradicts the assumption that m x; u λ j (x) < 0 > 0. This contradiction proves Theorem 1.2.