Correction to: Positive Solutions for Slightly Subcritical Elliptic Problems Via Orlicz Spaces

1 Correction to: Milan J. Math. Vol. 90 (2022) 229–255 https://doi.org/10.1007/s00032-022-00354-1

The authors would like to correct an error in the original paper [1], which was kindly pointed out to us by Prof. Alfonso Castro. In the proof of Proposition 3.1, the two last lines in p. 8, and in the proof of Theorem 1.1, the two last lines in p. 18 were wrong, and so the conclusions. Indeed, 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 proof based on a different argument. The consideration presented below should modify part of the proof of Proposition 3.1, and part of the proof of Theorem 1.1.

Replace line 3 in page 3, by:

Assume that

(A1):

$\quad \Omega ^+,$ and $\ \Omega ^-$ are non-empty sets,

(A2):

$\quad {\overline{\Omega }}^{\,+} \cap {\overline{\Omega }}^{\,-} =\emptyset .$

We add hypothesis (A1)–(A2) to the statement of Theorem 1.1.

Theorem 1.1

Assume that $g \in C^1({\mathbb {R}})$ satisfies hypothesis (H). Let $C_0>0$ be defined by (1.6). Assume that a changes sign in $\Omega$, that (A1)–(A2), and that (1.5) hold, then there exists a $\Lambda \in {\mathbb {R}},$

$$\begin{aligned} \lambda _1< \Lambda \le \min \Big \{\lambda _1\big (\text { int}\,(\Omega ^0)\big ),\quad \lambda _1\big (\text { int}\,\big (\Omega ^+\cup \Omega ^0\big )\big ) +C_0\sup a^+ \Big \} \end{aligned}$$

such that (1.1) has a classical positive solution if $\lambda < \Lambda$ and there is no positive solutions if $\lambda >\Lambda$.

Moreover, there exists a continuum (a closed and connected set) ${\mathscr {C}}$ of classical positive solutions to (1.1) emanating from the trivial solution set at the bifurcation point $(\lambda ,u)=(\lambda _1,0)$ which is unbounded. Besides,

• (a) For every, $\lambda \in \big (\lambda _1, \Lambda )$, (1.1) admits at least two classical ordered positive solutions.

• (b) Assume $\Lambda <\lambda _1\big (\text { int}\,(\Omega ^0)\big )$, for $\lambda =\Lambda$ problem (1.1) admits at least one classical positive solution.

• (c) For every $\lambda \le \lambda _1$, problem (1.1) admits at least one classical positive solution.

We also add hypothesis (A1)–(A2) to the statement of Proposition 3.1. We present here a more general result, with a new $(J_2)$ condition, more general than usual.

Proposition 3.1

Assume that $f \in C^1({\mathbb {R}})$ fulfills hypothesis (H) and that $\lambda<\lambda _1 (\text { int}\,\Omega ^0)<+\infty$. Assume that $a \in C^1({\overline{\Omega }})$, and $\Omega ^\pm ,\ \Omega ^0$ satisfy hypothesis (A1)–(A2).

Then any (PS) sequence, that is, a sequence $\{u_n\}$ satisfying the conditions

$(J_1)$:

$J_\lambda [u_n] \le C$,

$(J_2)$:

$\big |J_\lambda '[u_n]\, \psi \big | \le \varepsilon _n\, \Vert u_n\Vert \,\Vert \psi \Vert$, where $\varepsilon _n \rightarrow 0$ as $n \rightarrow +\infty$

is a bounded sequence.

Proof

A careful reading of Step 1 show that the conclusion holds under new hypothesis $(J_2)$ assumed here, just changing l. 15–16, p. 7 by the following:

$$\begin{aligned} \left| -\int _\Omega |\nabla u_n^-|^2 dx - \int _\Omega a(x)f (u_n^+)u_n^- dx \right| = \int _\Omega |\nabla u_n^-|^2 dx \le \epsilon _n \Vert u_n\Vert \Vert u_n^-\Vert \end{aligned}$$

so $\Vert v_n^-\Vert \rightarrow 0$ and then $v^-\equiv 0,$ and we conclude the proof of the claim.

2. Claim: $\lim _{n\rightarrow \infty }\int _\Omega a(x)\frac{f(u_n^+)u_n^+}{\Vert u_n\Vert ^2}\, dx =1$.

Taking into account that $v\equiv 0$ a.e. in $\Omega$, it follows from $(J_2)$ applied to $\psi =u_n$ that

$$\begin{aligned} \int _\Omega a(x)\frac{f(u_n^+)u_n^+}{\Vert u_n\Vert ^2}\, dx =1 +o(1). \end{aligned}$$

3. Claim: $0\le \limsup _{n\rightarrow \infty }\int _\Omega a^-(x)\frac{f(u_n^+)u_n^+}{\Vert u_n\Vert ^2}\, dx <+\infty .$

To prove this claim we will make use of hypothesis. It comes from this hypothesis that there exists $0\le \psi \in C^1(\Omega )$ such that $\psi =1$ on $\Omega ^-$ and $\psi =0$ on $\Omega ^+$. Taking $\phi =u_n \psi$ as a test function in $(J_2)$ we get

$$\begin{aligned} \int _\Omega \psi |\nabla v_n|^2\, dx +\int _\Omega v_n \nabla v_n\cdot \nabla \psi \, dx + \int _\Omega a^-(x)\frac{f(u_n^+)u_n^+}{\Vert u_n\Vert ^2}\, dx =\lambda \int _\Omega (v_n^+)^2 \psi \, dx +o(1) \end{aligned}$$

from which it comes directly that the sequence $\left\{ \int _\Omega a^-(x)\frac{f(u_n^+)u_n^+}{\Vert u_n\Vert ^2}\, dx \right\} _{n\in {\mathbb {N}}}$ is bounded and non-negative, so the claim 3 is proved.

In order to achieve a contradiction, we use $(J_1)$ and $(J_2)$. Multiplying by 1/2 the inequality $(J_2)$ applied to $u_n$ and adding $(J_1)$

$$\begin{aligned} \int _\Omega a(x)\bigg [\frac{1}{2}f(u_n^+)u_n^+- F(u_n^+)\bigg ]\, dx&\le C+ \varepsilon _n\Vert u_n\Vert ^2. \end{aligned}$$

Consequently

$$\begin{aligned} \limsup _{n\rightarrow \infty }\int _\Omega a(x)\bigg [\frac{\frac{1}{2}f(u_n^+)u_n^+- F(u_n^+)}{\Vert u_n\Vert ^2}\bigg ]\, dx\le 0. \end{aligned}$$

(0.1)

Since hypothesis (H)$_{g'}$, and using l’Hôpital’s rule, we can write

$$\begin{aligned} \lim _{s\rightarrow \infty }\, \frac{\frac{1}{2}sf(s)-F(s)}{sf(s)}=\frac{1}{2}-\frac{1}{2^*}=:\frac{1}{N}>0. \end{aligned}$$

(0.2)

Fixing $\varepsilon >0$ and using (0.2), there exists $C_\varepsilon$ such that

$$\begin{aligned} -C_\varepsilon +\left( \frac{1}{N} -\varepsilon \right) \int _\Omega a^\pm (x)\,f(u_n^+)u_n^+\, dx&\le \int _\Omega a^\pm (x)\left[ \frac{1}{2}f(u_n^+)u_n^+- F(u_n^+)\right] \, dx\\&\le C_\varepsilon + \left( \frac{1}{N} +\varepsilon \right) \int _\Omega a^\pm (x)\,f(u_n^+)u_n^+\, dx . \end{aligned}$$

Consequently,

$$\begin{aligned}&\int _\Omega \big (a^+(x)-a^-(x)\big )\left[ \frac{1}{2}f(u_n^+)u_n^+- F(u_n^+)\right] \, dx\\&\quad \ge -2C_\varepsilon +\left( \frac{1}{N} -\varepsilon \right) \int _\Omega \big (a^+(x)-a^-(x)\big )\,f(u_n^+)u_n^+\, dx\\&\qquad -2\varepsilon \int _\Omega a^-(x)\,f(u_n^+)u_n^+\, dx, \end{aligned}$$

and then

$$\begin{aligned} \left( \frac{1}{N}-\varepsilon \right) \int _\Omega a(x)\,f(u_n^+)u_n^+\, dx&\le 2C_\varepsilon + \int _\Omega a(x)\,\left[ \frac{1}{2}f(u_n^+)u_n^+- F(u_n^+)\right] \, dx \\&\quad +2\varepsilon \int _\Omega a^-(x)\,f(u_n^+)u_n^+\, dx, \end{aligned}$$

so, using (0.1)

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _\Omega a(x)\,\frac{f(u_n^+)u_n^+}{\Vert u_n\Vert ^2}\, dx \le \frac{2\varepsilon }{\frac{1}{N}-\varepsilon }\,\limsup _{n\rightarrow \infty } \int _\Omega a^-(x)\frac{f(u_n^+)u_n^+}{\Vert u_n\Vert ^2}\, dx. \end{aligned}$$

Letting $\varepsilon \rightarrow 0$, we get a contradiction with claim 2 and claim 3. We have just proved that any Palais–Smale sequence (PS) is bounded. $\square$

We observe that Theorem 3.2 still holds under the more general $(J_2)$ condition.

Proof of Theorem 1.1

(b) Step 2. Existence of a classical positive solution for $\lambda =\Lambda$. We prove the existence of a solution for $\lambda =\Lambda$. First notice that for any $\lambda \in (\lambda _1,\Lambda )$,

$$\begin{aligned} \lim _{t\rightarrow 0^+} J_\lambda [t\varphi _1]/t^2=(\lambda _1-\lambda )\int _\Omega \varphi _1^2 dx <0. \end{aligned}$$

Consequently there exists $t_0>0$ such that $J_\lambda [t\varphi _1]\le 0$ for all $0<t\le t_0$. Besides, we recall that $t\varphi _1$ is a subsolution of problem (1.1) for all $t>0$ small enough. It follows that for each $\lambda \in (\lambda _1,\Lambda )$, problem $(1.1)_\lambda$ admits a positive weak solution ${\overline{u}}_\lambda$ minimising $J_\lambda$ in some interval of the form $[t\varphi _1, {\overline{u}}]$ with $t<t_0$ and ${\overline{u}}$ a positive solution of (1.1) for a $\mu \in (\lambda , \Lambda )$. Hence $J_\lambda [{\overline{u}}_\lambda ]\le J_\lambda [t\varphi _1]$ and

$$\begin{aligned} J_\Lambda [{\overline{u}}_\lambda ]=J_\lambda [{\overline{u}}_\lambda ]+\frac{1}{2}(\lambda -\Lambda )\int _\Omega {\overline{u}}_\lambda ^2 dx \le J_\lambda [t\varphi _1]\le 0. \end{aligned}$$

Moreover

$$\begin{aligned} J_\Lambda '[{\overline{u}}_\lambda ]\psi =J_\lambda '[{\overline{u}}_\lambda ]\psi + (\lambda -\Lambda )\int _\Omega {\overline{u}}_\lambda \psi dx=(\lambda -\Lambda )\int _\Omega {\overline{u}}_\lambda \psi dx \end{aligned}$$

so, for any sequence $\lambda _n\rightarrow \Lambda$, the sequence $({\overline{u}}_{\lambda _n})_n$ satisfies $(J_1)$ and $(J_2)$ for the functional $J_\Lambda$. Since $\Lambda < \lambda _1 \left( int \, ( \Omega ^0\right) )$, by Proposition 3.1 the sequence $({\overline{u}}_{\lambda _n})_n$ is bounded. Hence, using Theorem 3.2 of [1], there exists a function ${\overline{u}}_\Lambda$ such that, up to a subsequence, ${\overline{u}}_{\lambda _n} \rightarrow u_\Lambda$ strongly in $H_0^1(\Omega )$. The possibility of $u_\Lambda =0$ is ruled out by considering the sequence $v_n=\frac{{\overline{u}}_{\lambda _n}}{ \Vert {\overline{u}}_{\lambda _n}\Vert }$ which will converge weakly to some $0\le v\not \equiv 0$ since $1=\Lambda \Vert v\Vert ^2$, and satisfying, weakly in $H_0^1(\Omega )$, $-\Delta v=\Lambda v$, contradicting that $\Lambda >\lambda _1$. We conclude that ${\overline{u}}_\Lambda$ is a positive solution of (1.1) for $\lambda =\Lambda$.

(c) Case $\lambda =\lambda _1$. A solution can be obtained by a Mountain pass as follows. As above, any sequence of solutions $\{u\}_n$ converging to 0 in $H_0^1(\Omega )$ satisfies, up to a sub-sequence, $\frac{u_n}{\Vert u_n\Vert }\rightharpoonup c\varphi _1$ for some $c>0$. Then, using $\mathrm{(H)}_0$ and l’Hôpital rule we have,

$$\begin{aligned} \lim _{\Vert u_n\Vert \rightarrow 0} \frac{J_{\lambda _1}[u_n]}{\Vert u_n\Vert ^p}=-L_1 c^p \int _\Omega a(x)\varphi _1^p dx >0 \end{aligned}$$

so 0 is a strict local minimum of $J_{\lambda _1}$. Besides, by choosing suitably $u_0\in C_0^1(\Omega ^+)$, ${\lim \nolimits _{t\rightarrow +\infty }J_{\lambda _1}[tu_0]=-\infty }$. Since $\lambda _1<\lambda _1 \big (\textrm{int}\, (\Omega ^0)\big )$, $J_{\lambda _1}$ satisfies the (PS) condition and we then inferee the existence of a nontrivial solution of (1.1) for $\lambda =\lambda _1$. By the strong maximum principle, this solution is $>0$.$\square$