Electronic Journal of Qualitative Theory of Differential Equations

We examine the semilinear resonant problem −∆u = λ1u + λg(u) in Ω, u ≥ 0 in Ω, u|∂Ω = 0, where Ω ⊂ RN is a smooth, bounded domain, λ1 is the first eigenvalue of −∆ in Ω, λ > 0. Inspired by a previous result in literature involving power-type nonlinearities, we consider here a generic sublinear term g and single out conditions to ensure: the existence of solutions for all λ > 0; the validity of the strong maximum principle for sufficiently small λ. The proof rests upon variational arguments.


Introduction
Let Ω ⊂ R N , N ≥ 1, be a bounded domain of class C 2 , and let λ 1 be the first eigenvalue of −∆ in Ω with Dirichlet boundary conditions. The issue of the existence of solutions of the problem s ∈ (1, 2), r ∈ (1, s), and µ > 0, has been the subject of study of the recent [3]. As a distinctive feature, the right-hand side term f (t) := λ 1 t + t s−1 − µt r−1 in (1.1) is not locally Lipschitz near 0, and moreover satisfies the sign property f −1 ((−∞, 0]) ⊇ (0, a], for some a > 0. As a result, from the celebrated paper [13] (see also [8]), it is known that the strong maximum principle may fail to be valid in this context. By adopting minimax and perturbation techniques, the author of [3] showed instead that such a principle does hold as long as the perturbation parameter is chosen sufficiently large. More precisely, the main results in [3] state that problem (1.1) has non-zero solutions for the entire positive range of µ; positive solutions for µ large enough. The fact that, after a rescaling, (1.1) can be turned into the problem for a suitable λ > 0, raises the natural question whether, as explicitly expressed in [3,Remark 2.4], the same results mentioned above continue to hold when the powers in (1.2) are replaced by a generic nonlinear term g. And, if it is so, it would be interesting of course to identify some "minimal" structure conditions on g for the validity of such results. In the present paper we address these questions and consider the problem where g : [0, +∞) → R is continuous, g(0) = 0, and obeys the following conditions: where, as usual, Problems like (P λ ) are being investigated since Landesman and Lazer's pioneering work [9], in which sufficient conditions, based on the interaction between the nonlinearity and the spectrum of the linear operator, were given for them to have a solution. Noteworthy contributions following that work can be found in [2,5,12] and also in [6,7,10,11,14] (see the related references as well) in which several classes of elliptic problems at resonance are investigated via variational and topological methods.
Coming back to (P λ ), our approach develops along the same line of reasoning as [3]. We prove initially that (P λ ) has at least a non-zero solution for all λ > 0. This is accomplished by considering a sequence of problems near resonance whose solutions are shown to converge to a solution of the original problem. In this regard, assumption (g 4 ) comes into play to prove the boundedness of the sequence of approximating solutions. Then, by exploiting the classical decomposition of H 1 0 (Ω) into the first eigenspace and its orthogonal complement, we show that, for sufficiently small λ, the set of solutions to (P λ ) is contained in the interior of the positive cone of C 1 0 (Ω). It still remains an open question to investigate the uniqueness of positive solutions to (P λ ) (in the one-dimensional case and for power-nonlinearities it has instead been established in [4]), as well as the existence of non-zero solutions compactly supported in Ω, in the spirit of [8].
Our main results, Theorems 2.3 and 2.4, are stated and proved in the coming section. Before going on, we arrange some notation and the variational framework for (P λ ). We set u := Ω |∇u| 2 dx 1 2 , for all u ∈ H 1 0 (Ω), and denote by · p , p ∈ [1, +∞], the classical L p -norm on Ω. We also set and denote by φ 1 the positive eigenfunction associated with λ 1 and normalized with respect to · ∞ . We recall that the first two eigenvalues λ 1 , λ 2 of −∆ in Ω admit the variational characterization Given a set E ⊂ R N , its Lebesgue measure will be denoted by the symbol |E|. Throughout this paper, the symbols C, C 1 , C 2 , . . . represent generic positive constants whose exact value may change from occurence to occurrence. For all λ > 0, we denote by I λ : H 1 0 (Ω) → R the energy functional associated with (P λ ), where u + = max{u, 0}. By a weak solution to (P λ ) we mean any u ∈ C 0 (Ω)

Results
As already mentioned, we start by considering a sequence of approximating problems.
admits a non-zero weak solution u n , with positive energy, for all n ≥n.
Proof. Fix λ > 0 and let n ∈ N with n > 1 λ 1 . Let us first show that the energy functional I n : H 1 0 (Ω) → R corresponding to (P n ), for all u ∈ H 1 0 (Ω), has the mountain pass geometry for sufficiently large n ∈ N. Fix k ∈ (2, 2 * ) and set By (g 1 ) and (g 2 ) one has 0 < M < +∞ and λ for any n ∈ N, where S R := {u ∈ H 1 0 (Ω) : u = R}. Now, let us show that there exist u 1 ∈ H 1 0 (Ω), with u 1 > R, andn ∈ N, such that I n (u 1 ) < 0 for all n ≥n. Owing to (g 3 ), there exist L, b > 0 such that If we denote by Since the function ψ(t) := qγt +γ q t q is continuous in (0, +∞) and With the aid of (g 1 ) and (2.4) we then obtain As a result, there existsn ∈ N, withn > 1 λ 1 , such that for all n ≥n. Therefore, the functional I n satisfies the geometric conditions required by the mountain pass theorem for all n ≥n. Moreover, by (g 1 ) and Sobolev embeddings, one has and thus I n (u) → +∞ as u → +∞. This fact, in addition to standard arguments (see for instance Example 38.25 of [15]), ensures that I n satisfies the Palais-Smale condition. Then, by invoking the classical mountain pass theorem, I n admits a critical point u n ∈ H 1 0 (Ω) \ {0} for all n ≥n, and, by (2.2), one also has where Γ := {γ ∈ C 0 ([0, 1], H 1 0 (Ω)) : γ(0) = 0, γ(1) = u 1 }. This concludes the proof.

Lemma 2.2.
Let λ > 0,n ∈ N and let u n , with n ≥n, be as in Lemma 2.1. Then, the sequence {u n } n≥n is bounded in H 1 0 (Ω).
It is straightforward to verify that w n ∈ C 1,α (Ω) is a weak solution to and therefore, also by (g 1 ), one has (2.7) From (2.7), it follows that for some C > 0. We claim that the sequence {t n } n≥n is bounded in R. Arguing by contradiction, assume that, up to a subsequence, t n → +∞ as n → +∞. Without loss of generality, we can assume that t n ≥ 1 for all n ≥n and, since and then w n ≤ C 2 t q−1 n . Therefore, fixing p > max N 2 , q q−1 , we obtain w n ∞ ≤ C 3 w n p + g(t n φ 1 + w n ) p + t n n φ 1 p Dividing the first and the last side of the previous inequality by t n and bearing in mind that y m ≤ 1 + y, for all m ∈ [0, 1] and y > 0, we get It follows that and, as a consequence, lim n→+∞ w n t n ∞ = 0, i.e., u n t n → φ 1 uniformly in Ω.
At this point, set and lett > 0 such that and n * ≥ñ such that t n ≥¯t γ for all n ≥ n * . Then, for all n ≥ n * , taking also (2.5) into account, we obtain a contradiction. Therefore, the sequence {t n } n≥n is bounded in R and (2.8) yields the boundedness of {w n } n≥n in H 1 0 (Ω), as well. As a consequence, we get the boundedness of {u n } n≥n in H 1 0 (Ω), as desired.
Collecting the results of the previous lemmas, it is now easy to derive our first existence result.
Proof. Let {u n } be the sequence of solutions to (P n ) in Lemma 2.1. By Lemma 2.2 there exists u * ∈ H 1 0 (Ω) such that, up to a subsequence, Fixing v ∈ H 1 0 (Ω) and taking the limit as n → +∞ in the identity I n (u n )(v) = 0, we get I λ (u * )(v) = 0, i.e. u * is a weak solution to (P λ ). To justify that u * = 0, observe that, by (2.5) one has ≤ λk 1 u n 1 + u n q q + 2λk 1 u n 1 + 1 q u n q q , and so, letting n → +∞, the conclusion is achieved.
We now show that, when λ approaches zero, every non-zero solution to (P λ ) is actually positive. To this aim, for all λ > 0, set and denote by P the interior of the positive cone of C 1 0 (Ω), i.e.
Proof. We first observe that, by the regularity theory of elliptic equations, for all λ > 0 and u λ ∈ S λ , one has u λ ∈ C 1,α (Ω), for some α ∈ (0, 1). If u λ ∈ S λ , it is straightforward to check that v λ := λ −1 u λ is a solution to the problem clearly equivalent to (P λ ). Note that (g 2 ) ensures the existence of some a > 0 such that g(t) < 0 for all t ∈ (0, a), and moreover it must hold v λ ∞ ≥ a λ , (2.9) otherwise we would get g(u λ ) < 0 in Ω \ u −1 λ (0), and so against the definition of λ 1 . From now on, we will then focus on (P λ ). We split the proof in several steps.