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.\(\square \)