Positive Solutions for Slightly Subcritical Elliptic Problems Via Orlicz Spaces

This paper concerns semilinear elliptic equations involving sign-changing weight function and a nonlinearity of subcritical nature understood in a generalized sense. Using an Orlicz–Sobolev space setting, we consider superlinear nonlinearities which do not have a polynomial growth, and state sufficient conditions guaranteeing the Palais–Smale condition. We study the existence of a bifurcated branch of classical positive solutions, containing a turning point, and providing multiplicity of solutions.


Introduction
In this paper we study the classical positive solutions to the Dirichlet problem for a class of semilinear elliptic equations whose nonlinear term is of subcritical nature in a generalized sense and involves indefinite nonlinearities. More precisely, given Ω ⊂ R N , N > 2, a bounded, connected open subset, with C 2 boundary ∂Ω, we look for positive solutions to: where λ ∈ R is a real parameter, a ∈ C 1 (Ω) changes sign in Ω, |s| q−2 s = L 2 , for some L 2 ≥ 0, and q ∈ 2, 2N N −2 (H) g |g (s)| ≤ C(1 + |s| q−2 ), for s ∈ R.
We will say that f satisfies hypothesis (H) whenever ( When λ = 0, a(x) ≡ 1 and g(s) ≡ 0, this kind of nonlinearity has been studied in [5][6][7]16], and in [11] for the case of the p−laplacian operator, with α > p N −p . It is known the existence of uniform L ∞ a-priori bounds for any positive classical solution, and as a consequence, the existence of positive solutions. When α → 0, there is a positive solution blowing up at a non-degenerate point of the Robin function as α → 0, see [9] for details.
In order to prove (PS) condition, Alama and Tarantello ( [1]) assume that the zero set Ω 0 has a non empty interior. This is also a common hypothesis for other authors when dealing with changing sign superlinear nonlinearities [8,20,23]. But this is a technical hypothesis. (PS)-condition will be proved in Proposition 3.1 without assuming that hypothesis. We neither use Ambrosetti-Rabinowitz condition.
Assume that u is a non-negative nontrivial solution. It is easy to see that the solution is strictly positive. Indeed, adding ±C 0 a(x)u to the r.h.s. of the equation, splitting a = a + − a − , taking into account (1.4) and (1.7), and letting in each side the nonnegative terms, we can write Now, the strong Maximum Principle implies that u > 0 in Ω, and ∂u ∂ν < 0 on ∂Ω. Our main result is the following theorem. Theorem 1.1. Assume that g ∈ C 1 (R) satisfies hypothesis (H). Let C 0 > 0 be defined by (1.6). If a changes sign in Ω, and (1.5) holds, then there exists a Λ ∈ R, The paper is organized in the following way. Section 2 contains a regularity result and a non existence result. (PS)-condition and an existence of solutions result for λ < λ 1 based in the Mountain Pass Theorem will be proved in Sect. 3. A bifurcation result for λ > λ 1 is developed in Sect. 4. The main result is proved in Sect. 5. Appendix A contains some useful estimates. Orlicz spaces, and a Orlicz-Sobolev embeddings theorems, will be treated in Appendix B.

A Regularity Result and a Non Existence Result
Next, we recall a regularity Lemma stating that any weak solution is in fact a classical solution.

Proposition 2.2.
Let f satisfy hypothesis (H) and let C 0 be defined in (1.6). Assume that a changes sign in Ω.
Letφ be the positive eigenfunction of − Δ, H 1 0 (int Ω + ∪ Ω 0 of L 2 -norm equal to 1. For simplicity, we will also denote byφ the extension by 0 ofφ in all Ω. By Hopf's maximum principle, we have ∂φ Again, if we multiply the equation (1.1) byφ and integrate along int Ω + ∪ Ω 0 we find, after integrating by parts, 2. Let λ ≥ λ 1 int (Ω 0 ) and, by contradiction, assume the existence of a positive solution u ∈ H 1 0 (Ω) of problem (1.1) for the parameter λ. Letφ be a positive eigenfunction associated to λ 1 int (Ω 0 ) < +∞. For simplicity, we will also denote byφ the extension by 0 in all Ω. If we multiply equation (1.1) byφ and integrate along Ω 0 we find, after integrating by parts, On the other hand

An Existence Result for λ < λ 1
In this section, we prove the existence of a nontrivial solution to equation (1.1) for λ < λ 1 , through the Mountain Pass Theorem.

On Palais-Smale Sequences
In this subsection, we define the framework for the functional J λ associated to the problem (1.1) λ . Hereafter, we denote by · the usual norm of H 1 0 (Ω): Consider the functional J λ : H 1 0 (Ω) → R given by Take note that for all v ∈ H 1 . The functional J λ is well defined and belongs to the class C 1 with for all ψ ∈ H 1 0 (Ω). As a result, non-negative critical points of the functional J λ correspond to non-negative weak solutions to (1.1).
Then any (PS) sequence, that is, a sequence satisfying the conditions Proof. 1. Let {u n } n∈N be a (PS) sequence in H 1 0 (Ω) and, in contradiction, assume that u n → +∞. Let us first prove the following claim: Claim. Let v ∈ H 1 0 (Ω) be the weak limit of v n = u n u n and assume that v n → v, strongly in L 2 * −1 (Ω) and a.e. Then v = 0 a.e. in Ω.
Assume that v ≡ 0 and write γ n = u n . Let ω n := {x ∈ Ω : v + n (x) > 1}, then for any ψ ∈ C 1 0 (Ω), Let x ∈ Ω \ ω n , based on the estimates (A.1), Besides, by the reverse of the Lebesgue dominated convergence theorem, see for instance [2, Theorem 4.9, p. 94] , there exists h i ∈ L 1 (Ω), 1 ≤ i ≤ 3 such that, up to a subsequence, e. x ∈ Ω, for all n ∈ N, and therefore By Lebesgue's dominated convergent theorem, we have We have used here that if v + (x) = 0, then lim n→+∞ ln(e + γ n ) ln(e + γ n v + n (x)) = 1, and if v + (x) = 0, then lim n→+∞ ln(e + γ n ) ln(e + γ n v + n (x)) On the other hand ln(e + γ n ) α Hence, using (J 2 ) for an arbitrary test function ψ, multiplying by ln(e+γ n ) α γ 2 * −1 n and passing to the limit we find In Dividing by u n and passing to the limit we have On the other hand, taking u − n as a test function in the condition (J 2 ), n → 0 and then v − ≡ 0, and we conclude the proof of the claim. 2. In order to achieve a contradiction, we use a Hölder inequality, and properties on convergence into an Orlicz space, cf. Appendix B.
To this end, the analysis of Lemma A.2 gives us the existence of α * > 0 such that the function s → In this case, we will denote and define the non-decreasing function It follows that Since v n 0 in H 1 0 (Ω) and strongly in L 2 (Ω), it follows from (J 2 ) applied to Since the Hölder inequality into Orlicz spaces, see Proposition B.11.(ii), By using definition B.8 of M * and identities of Proposition B.9 we have Observe that |f (s)| ≤ C(1 + m(s)), so then see Proposition B.11.(iii) and (i), concluding that the l.h.s. is bounded for each n. [15], Theorem 14.2).
By testing (J 2 ) against ψ = u n − u and using (3.10), and (3.11) we get In order to prove this claim, we use, as in the above proposition, a Hölder inequality and a compact embedding into some Orlicz space, c.f. Appendix B.
By Theorem B.3 and Theorem B.12 we have where m, and M are defined by (3.2)-(3.4), as in the above proposition. On the other hand, because there exists Now, using Holder's inequality (B.6) and that and it follows from (3.13) that u n − u → 0.

An Existence Result for λ < λ 1
The next theorem provides a solution to (1.1) for λ < λ 1 based on the Mountain Pass Theorem.
Assume that the nonlinearity f defined by (1.2) satisfies (H), and that the weight a ∈ C 1 (Ω). Then, the boundary value problem (1.1) λ has at least one classical positive solution for any λ < λ 1 .
Proof. We verify the hypothesis of the Mountain Pass Theorem, see [14, Theorem 2, Section 8.5]. Observe that the derivative of the functional J λ : In view of (3.14), and as a result of the Poincaré inequality, we get taking |λ| < λ 1 , r > 0 small enough, and using that p, q, then f (tu 0 )/t → +∞ as t → +∞ in Ω + . From definition, and integrating by parts, It can be easily seen that lim s→+∞ G(s) sf (s) = 0. Therefore, using l'Hôpital's rule we can write Let C 0 ≥ 0 be such that F (s) + 1 2 C 0 s 2 ≥ 0 for all s ≥ 0 (see (1.7)), and let Ω + δ := {x ∈ Ω + : a(x) = a + (x) > δ}. (3.18) By definition, u 0 ≡ 0 in Ω − , so, introducing ± 1 2 C 0 (tu 0 ) 2 , splitting the integral, and using (3.17)-(3.18) we obtain Hence, there exists a positive constant C > 0 such that Step 3. We have at last checked that all the hypothesis of the Mountain Pass Theorem are accomplished. Let is a critical value of J λ , that is, the set K c : (3.19) and thereby, u is a nontrivial weak solution to (3.19). By Lemma 2.1, u is a classical solution, and by (1.8), u > 0 in Ω.

Proof of Theorem 1.1
First we prove an auxiliary result.
Moreover, for every δ > 0 satisfying the function u = δϕ 1 is a subsolution for (1.1) λ whenever λ > λ 1 . Let δ > 0 satisfying (5.1) and such that g(s) ≥ 0 for any s ∈ [0, δ ϕ 1 L ∞ (Ω) ]. For any ψ ∈ H 1 0 (Ω), ψ > 0 with in Ω we deduce This allows us to take u = δϕ 1 as a subsolution for (1.1) λ with u < u 0 . The sub-and supersolution method now guarantees a positive solution u to (1. Step 2. Existence of a minimal positive solution u λ for any λ ∈ (λ 1 , Λ). To show that there is in fact a minimal solution, for each x ∈ Ω we define Firstly, we claim that u λ ≥ 0, u λ ≡ 0. Assume that u λ ≡ 0 by contradiction. This would yield a sequence u n of positive solutions to (1.1) λ such that u n C(Ω) → 0 as n → ∞, or in other words, (λ, 0) is a bifurcation point from the trivial solution set to positive solutions. Set v n := u n u n C (Ω) . Observe that v n is a weak solution to the problem → 0 in C(Ω) as n → ∞. Therefore, the right-hand side of (5.2) is bounded in C(Ω). Hence, by the elliptic regularity, v n ∈ W 2,r (Ω) for any r > 1, in particular for r > N. Then, the Sobolev embedding theorem implies that ||v n || C 1,α (Ω) is bounded by a constant C that is independent of n. Then, the compact embedding of C 1,μ (Ω) into C 1,β (Ω) for 0 < β < μ yields, up to a subsequence, v n → Φ ≥ 0 in C 1,β (Ω).
Using the weak formulation of equation (5.2), passing to the limit, and taking into account that λ is fixed and v n → Φ, we obtain that Φ ≥ 0, Φ ≡ 0, is a weak solution to the equation Then, by the maximum principle, it follows that Φ = ϕ 1 > 0, the first eigenfunction, and λ = λ 1 is its corresponding eigenvalue, which contradicts that λ > λ 1 . Secondly, we show that u λ solves (1.1) λ . We argue on the contrary. Observe that the minimum of any two positive solutions to (1.1) λ furnishes a supersolution to (1.1) λ . Assume that there are a finite number of solutions to (1.1) λ , then u λ (x) := min u(x) : u > 0 solves (1.1) λ and u λ is a supersolution. Choosing ε 0 small enough so that ε 0 ϕ 1 < u λ , the sub-supersolution method provides a solution ε 0 ϕ 1 ≤ v ≤ u λ . Since v is a solution and u λ is not, then v ≤ u λ , v = u, contradicting the definition of u λ , and achieving this part of the proof.
Let u n := min{v n−1 , u n+1 }. Choosing ε n small enough so that ε n ϕ 1 < u n , the sub-supersolution method provides a solution ε n ϕ 1 ≤ v n ≤ u n ≤ v n−1 . With this induction procedure, we build a monotone sequence of solutions v n , such that Since monotonicity and Lemma 2.1, v n C(Ω) ≤ v 1 C(Ω) , by elliptic regularity, v n C 1,μ (Ω) ≤ C for any μ < 1, and by compact embedding v n → v in C 1,β (Ω) for any β < α. Using the weak formulation of equation (1.1) λ , passing to the limit, and taking into account that λ is fixed, we obtain that v is a weak solution to the . Also, and due to (5.3), u n (x) ↓ v(x) pointwise for x ∈ Ω, and inf u n (x) = v(x).
On the other hand, by construction u n ≤ u n+1 , so, for each x ∈ Ω, v(x) = inf u n (x) ≤ inf u n (x) = u λ (x). Therefore, and by definition of u λ , necessarily v = u λ , proving that u λ solves (1.1) λ , and achieving the proof of step 2.
(iii) Since [4, Theorem 2] if there exists an ordered pair of L ∞ bounded sub and supersolution u ≤ u to (1.1) λ , and neither u nor u is a solution to (1.1) λ , then there exist a solution u < u < u to (1.1) λ such that u is a local minimum of J λ at H 1 0 (Ω). Reasoning as in (i), u := u μ with μ > λ is a strict supersolution to (1.1) λ , and u := δϕ 1 is a strict sub-solution for δ > 0 small enough, such that u(x) < u(x) for each x ∈ Ω. This achieves the proof. First, reasoning as in Proposition 5.1(iii), we get a local minimumũ λ > 0 of J λ . Ifũ λ = u λ , thenũ λ is the second positive solution, ending the proof. Assume thatũ λ = u λ . Now we reason as in [12,Theorem 5.10] on the nature of local minima. Thus, either words,ũ λ is a strict local minimum, or (ii) for each ε > 0, there exists u ε ∈ H 1 0 (Ω) such that J λ has a local minimum at a point u ε with u ε −ũ λ = ε and J λ (u ε ) = J λ (ũ λ ).
Let us assume that (i) holds, since otherwise case (ii) implies the existence of a second solution.
Consider now the functional I λ : .
Let v = tv 0 for a certain t = t 0 > 0 to be selected a posteriori, and evaluate Reasoning as in the proof of Theorem 3.3 for large positive t, since F (t)/t 2 → ∞ as t → ∞, and using also (3.1) we obtain that for t = t 0 big enough, and where Ω + δ is defined by (3.18 is a critical value of I λ , and thereby K c : Since for any g ∈ Γ we have ] for all t ∈ [0, 1], it follows that g + ∈ Γ, and we derive the existence of a sequence v n such that On the other hand, w n := u λ + v n is a (PS) sequence for the original functional We next see that u Λ < +∞, reasoning on the contrary. Assume that there exists a sequence of solutions u n := u λ n such that u λ n → +∞ as λ n → Λ. Set v n := u n / u n , then there exists a subsequence, again denoted by v n such that v n v in H 1 0 (Ω), and v n → v in L p (Ω) for any p < 2 * and a.e. Arguing as in the claim of On the other hand, from the weak formulation, for all ψ ∈ C ∞ c (Ω), Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

A. Some Estimates
First, we prove an useful estimate of ln(e+s) ln(e+as) . Notice that we have (s 0 ) ≤ 1 a . In order to find a better upper bound of ln( e+as 0 e+s 0 ) let us denote for all s ≥ 0 θ(s) = (e + as) ln(e + as) − a(e + s) ln(e + s).

B. A Compact Embedding Using Orlicz Spaces
For references on Orlicz spaces see [15,21]. Throughout Ω ⊂ R N is an bounded open set. We will denote The proof of the following property is trivial, we just quoted it for the sake of completeness. Assume also that M satisfies the Δ 2 -condition, that is, Let Ω n = {x ∈ Ω : |u n (x) − u(x)| > M −1 (ε)}. As a consequence of Egoroff's theorem, the sequence (u n ) n∈N converge in measure to u so there exists n 0 ∈ N such that |Ω n | < δ. In order to prove that, for the sequence of our theorem, the set M |u n | : n ∈ N has equi-absolutely continuous integrals we are going to use the following lemma : Remark B.7. Whenever (B.2) is satisfied we say that the sequence {u n k } k∈N converges in M -mean to u.
One can formulate Theorem B.3 as a compact embedding of H 1 0 (Ω) in some vector space endowed of the Luxembourg norm associate to M (see [15,21]). Instead, we are going to use the Orlicz-norm which is more suitable to our purposes. We will see later in Theorem B.12 that the convergence in M -mean implies the convergence with respect to the Orlicz-norm, provided that the Δ 2 -condition is satisfied. Next, let us introduce the Orlicz norm associated to M :