a priori estimates for subcritical semilinear elliptic equations with a Carath´eodory non-linearity

PublishedonlineFebruary6,2023c


Introduction
Let us consider the following semilinear boundary value problem: − Δu = f (x, u), in Ω, u= 0, on ∂Ω, (1.1) where Ω ⊂ R N , N > 2, is a bounded, connected, open subset with C 2 boundary ∂Ω, and the non-linearity f : Ω×R → R is a Carathéodory function (that is, the mapping f (·, s) is measurable for all s ∈ R, and the mapping f (x, ·) is continuous for almost all x ∈ Ω), that is subcritical (see Definition 1.1).
We analyze the effect of the smoothness of the non-linearity f = f (x, ·) on the L ∞ (Ω) a priori estimates of weak solutions to (1.1). Degree theory combined with a priori bounds in the sup-norm of solutions of parametrized versions of (1.1), is a very classical topic in elliptic equations, posed by Leray and Schauder in [18]. It provides a great deal of information about existence of solutions and the structure of the solution set. This study is usually focused on positive classical solutions, see the classical references of de Figueiredo-Lions-Nussbaum, and of Gidas-Spruck [11,14], see also [7,8].
A natural question concerning the class of solutions is the following one: (Q1) can those L ∞ (Ω) estimates be extended to a bigger class of solutions, in particular to weak solutions (with possibly sign changing solutions)?.
Another question concerning the class of non-linearities, can be stated as follows: (Q2) can those estimates be extended to a bigger class of non-linearities, in particular to non-smooth non-linearities (with possibly sign changing weights)?.
In this paper, we provide sufficient conditions guarantying uniform L ∞ (Ω) a priori estimates for any u ∈ H 1 0 (Ω) weak solution to (1.1), in terms of their L 2 * (Ω) bounds, in the class of Carathéodory generalized subcritical problems. In this class, we state that any set of weak solutions uniformly L 2 * (Ω) a priori bounded is universally L ∞ (Ω) a priori bounded. Our theorems allow sign changing weights, and singular weights, and also apply to sign changing solutions.
Usually the term subcritical non-linearity is reserved for power like nonlinearities. We expand this concept in this paper below. Let where r is the conjugate exponent of r, 1/r + 1/r = 1.
is obviously satisfied with c 0 = 1. 2. Thanks to Sobolev embeddings, for any u ∈ H 1 0 (Ω), 3. Again, by Sobolev embeddings, for any u ∈ H 1 0 (Ω), If a(x) = |x| −μ , then a ∈ L p (Ω) for any p < N/μ, hence f (·, u) ∈ L p (Ω) for any p < 2N N +2 . From the sharp Caffarelli-Kohn-Nirenberg interpolation inequality for singular weights, in the particular case where α = β = 0, p = 2, q = 2 * (see [4], see also Theorem A.1 and Corollary A.2), there exists a constant C > 0 such that Throughout the paper, by a solution we will refer to this weak solution. This definition of solution is tied to question (Q1). By an estimate of Brezis-Kato [3], based on Moser's iteration technique [21], and elliptic regularity, we will state sufficient conditions guarantying that any weak solution to (1.1) with a Carathéodory subcritical non-linearity is a continuous function, and in fact it is a strong solution, see Lemma 2.1 and Lemma 3.1.
Our definition of a subcritical non-linearity includes functions such as for any α > 0, and either any a ∈ L r (Ω), with N/2 < r ≤ ∞, or any μ ∈ (0, 2). These non-linearities exemplify question (Q2). One of the main results, Theorem 1.5, applied to f (x, s) = f (1) (x, s) with a ∈ L r (Ω) for r ∈ (N/2, N], implies that for any ε > 0 there exists a constant C > 0 depending only on ε, Ω, r and N such that for any u ∈ H 1 0 (Ω) solution to (1.1), the following holds: where C is independent of the solution u.
Related results concerning f (1) (x, s) with r = ∞ can be found in [10] for the p-Laplacian case, in [9] analyzing what happen when α → 0, in [19] for systems, in [25] for the radial case, and in [23,24] for a summary.
To state our main results, for a non-linearity f satisfying (H0), define Let u be a solution to (1.1). We estimate h u ∞ , in terms of its L 2 * (Ω)norm. This result is robust, and holds for solutions and non-linearities without any sign restriction. Our first main result is the following theorem. Then, for any u ∈ H 1 0 (Ω) weak solution to (1.1), the following holds: where h is defined by (1.9), (1.12) and C depends only on ε, c 0 (defined in (1.8)), r, N , and Ω, and it is independent of the solution u.
Our second main result is the following theorem. Theorem 1.6. Assume that f : Ω × R → R is a Carathéodory function satisfying (H0)' and (H1). Assume also that one of the following two conditions hold (a) Either μ ≤ 4/N ; Then, for any u ∈ H 1 0 (Ω) solution to (1.1), the following holds: independent of the solution u, (ii) either, for any ε > 0 there exists a constant C > 0 such that where h is defined by (1.10), 13) and C depends only on ε, c 0 (defined in (1.8)), μ, N , and Ω, and it is independent of the solution u. As an immediate consequence, as soon as we have a universal a priori L 2 * (Ω)-norm for weak solutions in H 1 0 (Ω), then solutions are a priori universally bounded in the L ∞ (Ω)-norm. Our third main result is the following theorem. Theorem 1.7. (L ∞ uniform a priori bound) Assume that f : Ω × R → R is a Carathéodory function satisfying either hypothesis of Theorem 1.5, either hypothesis of Theorem 1.6. Assume also that there exists constants K i > 0, i = 1, 2, and q > 2 such that Then, there exists a constant C > 0 such that for every non-negative weak solution u of (1.1), where C depends only on N and Ω, but it is independent of the solution u.
As far as we know, our definition of weak solution is the optimal one for the purpose of L ∞ (Ω) a priori bounds. There are more singular solutions which are unbounded in L ∞ (Ω), which we briefly discuss below. We will say that a function u is an is the distance function with respect to the boundary, and Joseph and Lundgren in [17] shows that those L ∞ (Ω) a priori estimates are not applicable for L 1 (Ω)-weak solutions, or for super-critical nonlinearities.
They posed the study of singular solutions. Working on non-linearities such as f (s) := e s or f (s) := (1 + s) p , they consider the following BVP depending on a multiplicative parameter λ ∈ R, and look for classical radial positive solutions in the unit ball B 1 . They obtain singular solutions as limit of classical solutions. In particular, they obtain the explicit weak solution to (1.14), when N > 2, λ = 2(N − 2), and f (s) := e s , see [17, p. 262]. They also found the explicit L 1 (Ω)-weak solution to (1.14), where f (s) := (1+s) p , and λ = 2 Let us focus on BVP with radial singular weights, (1.15) with N > 2, μ < 2 and p > 1. It can be checked that . Above examples of radially symmetric singular solutions to BVP's on spherical domains, solve either super-critical problems (u * 1 ) or are L 1 (Ω)-w eak solutions not in H 1 0 (Ω) (u * 2 and u * 3 ). Consequently, we restrict our study for u ∈ H 1 0 (Ω) weak solutions to (1.1), in the class of generalized subcritical problems. It is natural to ask for uniform L ∞ (Ω) a priori estimates over non power non-linearities in non-spherical domains.
This paper is organized in the following way. In Sect. 2, using Gagliardo -Nirenberg inequality, we prove Theorem 1.5. In Sect. 3, we prove Theorem 1.6. It needs the Caffarelli-Kohn-Nirenberg inequality, which is written in Appendix A, by the sake of completeness. In Sect. 4, we prove Theorem 1.7.

Estimates of the L ∞ (Ω)-norm of the solutions with Carathéodory non-linearities
In this section, assuming that f satisfy the subcritical growth condition (H0), we prove Theorem 1.5. We first collect a regularity Lemma for any weak solution to (1.1) with a non-linearity of polynomial critical growth.

Proof of Theorem 1.5
The arguments of the proof use Gagliardo-Nirenberg interpolation inequality (see [22]), and are inspired in the equivalence between uniform L 2 * (Ω) a priori bounds and uniform L ∞ (Ω) a priori bounds for solutions to subcritical elliptic equations, see [ We first use elliptic regularity and Sobolev embeddings, and next, we invoke the Gagliardo-Nirenberg interpolation inequality (see [22]).
From now on, C denotes several constants that may change from line to line, and are independent of u.

L ∞ (Ω) a priori bounds of the solutions
As as immediate corollary of Theorem 1.5, we prove that any sequence of solutions in H 1 0 (Ω), uniformly bounded in the L 2 * (Ω)-norm, is also uniformly bounded in the L ∞ (Ω)-norm.

Corollary 2.2. Let f : Ω × R → R be a Carathéodory function satisfying (H0)-(H1).
Let {u k } ⊂ H 1 0 (Ω) be any sequence of solutions to (1.1) such that there exists a constant C 0 > 0 satisfying Then, there exists a constant C > 0 such that (2.12) Proof. We reason by contradiction, assuming that (2.12) does not hold. So, at least for a subsequence again denoted as u k , u k ∞ → ∞ as k → ∞. Now, part (ii) of the Theorem 1.5 implies that From hypothesis (H0) (see in particular (1.11)), for any ε > 0 there exists s 1 > 0 such that h(s) ≥ 1/ε for any s ≥ s 1 , and so h u k ∞ ≥ 1/ε for any k big enough. This contradicts (2.13), ending the proof.
We next state a straightforward corollary, assuming that the non-linearitỹ f : R → (0, +∞) satisfies also the following hypothesis: where h is defined by (1.9), A is defined by (1.12), C = C(c 0 , r, N, ε, |Ω|), and C is independent of the solution u.
Since hypothesis (H1)', for any sequence {u k } ⊂ H 1 0 (Ω) of weak solutions to (1.1), The proof can be achieved just reproducing Step 1 and Step 2 of the proof of Theorem 1.5, which now hold for any any sequence of weak solutions to (1.1).

Estimates of the L ∞ (Ω)-norm of the solutions with radial singular weights
In this section, assuming that 0 ∈ Ω and that f satisfies (H0)' and (H1), we prove Theorem 1.6. First, we also collect a regularity Lemma for any weak solution to (1.1) withf (s) of polynomial critical growth, according to Caffarelli-Kohn-Nirenberg inequality.

Proof of Theorem 1.6
Since Lemma 3.1, assuming either (a) or (b), a solution u ∈ H 1 0 (Ω) to (1.1) is in L ∞ (Ω). In the proof of Theorem 1.6, we will not distinguish if we are assuming condition (a) or (b).
Since the infimum is not attained, for any ε > 0, there exists a constant C = C(ε, c 0 , μ, N, Ω) such that which ends the proof.

Uniform L ∞ a priori estimates
Proof of Theorem 1.7. We will show u H 1 0 (Ω) ≤ C, where C is independent of u, once achieved, either Theorem 1.5, either Theorem 1.6 will finished the proof. We prove it by contradiction. Suppose there exists a sequence {u k } of nonnegative weak solutions of (1.1) such that . Then, by the reflexivity of H 1 0 (Ω), U k U in H 1 0 (Ω) up to a subsequence. By the compactness of the trace operator, U k → U in L p+1 (Ω).
Step 1: U = 0 a.e. on Ω. Since u k is a weak solution of (1.1), we have Then dividing both sides of (4.1) by u k H 1 0 (Ω) , we have Taking ψ = U k as a test function, we have which implies since q > 2. Therefore, so U k L q (∂Ω) → 0. Since U k → U in L s (Ω) for all s < 2 * , we have that U = 0 a.e. on Ω.