Correction to: $L^\infty (\Omega )$ a priori estimates for subcritical semilinear elliptic equations with a Carathéodory non-linearity

Correction to: J. Fixed Point Theory Appl. (2023) 25:44https://doi.org/10.1007/s11784-023-01048-w

The author would like to correct an error in the original paper [1], which was kindly pointed out by Prof. Alfonso Castro. In the proof of Theorem 1.7, lines 4–5 in p. 18 of 22 were wrong, and so the conclusions. Let $\Omega \subset \mathbb {R}^N,$ $N > 2,$ be a bounded, connected open subset, with $C^2$ boundary $\partial \Omega ,$ and $\{g_k\}_{k\in \mathbb {N}}\subset L^\infty (\Omega )$ such that $\int _\Omega g_k \psi \rightarrow 0$ for any $\psi \in H_0^1(\Omega ).$ Density alone is not enough and the above do not imply that $\int _\Omega g_k \psi \rightarrow 0$ for any $\psi \in L^2(\Omega ),$ it will hold if $\{g_k\}_{k\in \mathbb {N}}$ is uniformly bounded.

We present here a new statement of Theorem 1.7 and a new proof based on different arguments.

FormalPara Definition 0.1

We will say that a sequence $\{u_n\}\subset H_0^1(\Omega )$ of positive weak solutions to (1.1) has uniformly bounded energy if there exists a constant $C_1>0,$ such that $J[u_n]\le C_1,$ where

$$\begin{aligned} J[u] := \frac{1}{2}\int _{\Omega } |\nabla u|^2\, dx - \int _{\Omega } F(x,u)\, dx, \end{aligned}$$

and where $F(x,t):=\int _{0}^{t}f(x,s)\, ds$.

The next Theorem states that a sequence of solutions to (1.1) is uniformly $L^\infty$ a priori bounded if and only if it has uniformly bounded energy, for subcritical nonlinearities, under Ambrosetti–Rabinowitz condition.

FormalPara Theorem 1.7

$(L^\infty$ uniform a-priori bound) Assume that $f:\overline{\Omega }\times \mathbb {R}\rightarrow \mathbb {R}$ is a subcritical Carathéodory function satisfying either hypotheses of Theorem 1.5,  either hypotheses of Theorem 1.6. Assume also that there exists two constants $\theta >2,$ and $s_0>0$ such that

$$\begin{aligned} \mathrm{(AR)}\qquad \qquad \qquad \theta F(x,s)\le sf(x,s),\qquad \forall x\in \Omega ,\quad \forall s>s_0. \end{aligned}$$

Let $\{u_n\}\subset H_0^1(\Omega )$ be a sequence of positive weak solutions to (1.1).

Then,  there exists a uniform constant $C>0$ (depending only on $\Omega ,$ N and f) such that the following holds

$$\begin{aligned} \Vert u_n\Vert _{L^\infty (\Omega )}\le C, \end{aligned}$$

if and only if $\{u_n\}\subset H_0^1(\Omega )$ has uniformly bounded energy.

FormalPara Proof of Theorem 1.7

Let $\{u_n\}\subset H_0^1(\Omega )$ be a sequence of positive weak solutions to (1.1). Obviously, and since $\Omega$ is bounded, a uniform $L^\infty$ a priori bound implies a uniform bound for the energy. Now, we shall prove the reciprocal.

Since (AR), there exists a constant $C>0$ such that for any positive weak solution to (1.1),

$$\begin{aligned} \int _{\Omega } F(x,u_n)\, dx\le \frac{1}{\theta }\int _{\Omega } u_nf(x,u_n)\, dx +C. \end{aligned}$$

By hypothesis, there exists a constant $C_1>0$ such that

$$\begin{aligned} \frac{1}{2}\int _{\Omega } |\nabla u_n| ^2\, dx - \int _{\Omega } F(x,u_n)\, dx\le C_1. \end{aligned}$$

Hence,

$$\begin{aligned} \frac{1}{2}\int _{\Omega } |\nabla u_n| ^2\, dx - \frac{1}{\theta } \int _{\Omega } u_nf(x,u_n)\, dx\le C. \end{aligned}$$

On the other hand, taking $u_n$ as a test function in the definition of a weak solution,

$$\begin{aligned} \left( \frac{1}{2} - \frac{1}{\theta } \right) \int _{\Omega } |\nabla u_n|^2\, dx \le C, \end{aligned}$$

and applying either Theorem 1.5, or Theorem 1.6, and Sobolev embeddings, the proof is complete.$\square$