On the Brezis–Nirenberg problem for a Kirchhoff type equation in high dimension

Abstract

The present paper deals with a parametrized Kirchhoff type problem involving a critical nonlinearity in high dimension. Existence, non existence and multiplicity of solutions are obtained under the effect of a subcritical perturbation by combining variational properties with a careful analysis of the fiber maps of the energy functional associated to the problem. The particular case of a pure power perturbation is also addressed. Through the study of the Nehari manifolds we extend the general case to a wider range of the parameters.

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Acknowledgements

The authors would like to thank the anonymous Referee for his/her remarkable suggestions which helped improving the quality of the manuscript. F. Faraci has been supported by the Università degli Studi di Catania, “Piano della Ricerca 2016/2018 Linea di intervento 2”. She is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). K. Silva has been supported by CNPq-Grant 408604/2018-2.

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Appendices

Appendix A: Some topological properties of the Nehari manifolds

We collect some topological properties concerning the Nehari manifold \({\mathcal {N}}^-\). Since the dependency on each parameter will be considered, we will write the full notation \(\Phi _{a,b,\lambda }\), \(t_{a,b,\lambda }^-(u)\), \({\mathcal {N}}_{a,b,\lambda }^-\) and so on.

Similarly to Proposition 4.2 we can prove:

Lemma A.1

For each \(a>0\), \(b\in {\mathbb {R}}\), \(\lambda \in {\mathbb {R}}\) and \(u\in H^1_0(\Omega ){\setminus }\{0\},\) only one of the next \(i)-iv)\) occurs.

  1. (i)

    The function \( \psi _{a,b,\lambda ,u}\) is increasing and has no critical points.

  2. (ii)

    The function \( \psi _{a,b,\lambda ,u}\) has only one critical point in \(]0,+\infty [\) at the value \(t_{a,b,\lambda }(u)\). Moreover, \( \psi ''_{a,b,\lambda ,u}(t_{a,b,\lambda }(u))=0\) and \( \psi _{a,b,\lambda ,u}\) is increasing.

  3. (iii)

    The function \( \psi _{a,b,\lambda ,u}\) has only two critical points, \(0< t^-_{a,b,\lambda } (u) < t^+_{a,b,\lambda } (u)\). Moreover, \(t^-_{a,b,\lambda }(u)\) is a local maximum and \(t^+_{a,b,\lambda }(u)\) is a local minimum with \( \psi _{a,b,\lambda ,u}''(t^-_{a,b,\lambda }(u))<0<\psi _{a,b,\lambda ,u}''(t^+_{a,b,\lambda }(u))\).

  4. (iv)

    The function \( \psi _{a,b,\lambda ,u}\) has only one critical point in \(]0,+\infty [\) at the value \(t^-_{a,b,\lambda }(u)\). Moreover, \(t^-_{a,b,\lambda }(u)\) is a local maximum and \( \psi _{a,b,\lambda ,u}''(t^-_{a,b,\lambda }(u))<0\).

Remark A.1

If \(b\le 0\) and \(\lambda \ge 0\), then only item iv) of Lemma A.1 occurs. Moreover, if \(b>0\), then only one of the items \(i)-iii)\) occurs.

Lemma A.2

Fix \(u\in H_0^1(\Omega ){\setminus }\{0\}\) and \(a>0\). Let \(V\subset {\mathbb {R}}^2\) be an open set and assume that \(t_{a,b,\lambda }^-(u)\) is defined for all \((b,\lambda )\in V\). Then the function \(V\ni (b,\lambda )\mapsto t_{a,b,\lambda }^-(u)\) is \(C^1\). Moreover the following holds true.

  1. (i)

    The functions \(t_{a,b,\lambda }^-(u)\) and \(\psi _{a,b,\lambda ,u}(t_{a,b,\lambda }^-(u))\) are increasing with respect to b;

  2. (ii)

    The functions \(t_{a,b,\lambda }^-(u)\) and \(\psi _{a,b,\lambda ,u}(t_{a,b,\lambda }^-(u))\) are decreasing with respect to \(\lambda \).

Proof

Denote \(t_{b,\lambda }=t_{a,b,\lambda }^-(u)\) and note from the implicit function theorem that \(\psi _{a,b,\lambda ,u}'(t_{b,\lambda })=0\) and \(\psi _{a,b,\lambda ,u}''(t_{b,\lambda })<0\) implies that \(t_{b,\lambda }\) is \(C^1\) as a function of \((b,\mu ,\lambda )\in V\). Since

$$\begin{aligned} at_{b,\lambda }^2\Vert u\Vert ^2+bt_{b,\lambda }^4\Vert u\Vert ^4-t_{b,\lambda }^{2^*}\mu \Vert u\Vert _{2^*}^{2^*}-\lambda t_{b,\lambda }^p\Vert u\Vert _p^p=0, \end{aligned}$$

we conclude by differentiating both sides, with respect to b, that

$$\begin{aligned} \frac{\partial {t_{b,\lambda }}}{\partial b}=-\frac{t_{b,\lambda }^4\Vert u\Vert ^4}{\psi _{a,b,\lambda ,u}''(t_{b,\lambda })}>0, \end{aligned}$$

and hence \(t_{b,\lambda }\) is increasing in b. Now let \(f(b)=\psi _{a,b,\lambda ,u}(t_{a,b,\lambda }^-(u))\) and observe that

$$\begin{aligned} f'(b)=\frac{\partial {t_{b,\lambda }}}{\partial b}\psi _{a,b,\lambda ,u}'(t_{b,\lambda })+\frac{t_{b,\lambda }^4\Vert u\Vert ^4}{4}>0, \end{aligned}$$

which implies that f is increasing and hence (i) is proved. The proof of (ii) is similar. \(\square \)

Remark A.2

Note that a similar result can also be proved with respect to the functions \(t_{a,b,\lambda }^+(u)\) and \(\psi _{a,b,\lambda ,u}(t_{a,b,\lambda }^+(u))\).

Denote

$$\begin{aligned} {\mathcal {M}}_{a,b,\lambda }=\left\{ \frac{u}{\Vert u\Vert }:u\in {\mathcal {N}}_{a,b,\lambda }^-\right\} . \end{aligned}$$

Lemma A.3

There holds:

  1. (i)

    If \(b_1< b_2\), then \({\mathcal {M}}_{b_2}\subset {\mathcal {M}}_{b_1}\).

  2. (ii)

    If \(\lambda _1< \lambda _2\), then \({\mathcal {M}}_{\lambda _1}\subset {\mathcal {M}}_{\lambda _2}\).

Proof

(i) Take \(u\in {\mathcal {M}}_{a,b_2,\lambda }\). Once \(\psi _{a,b_1,\lambda }'(t)\le \psi _{a,b_2,\lambda }'(t)\) for all \(t>0\), it follows that \(\psi _{a,b_1,\lambda }'(t_{a,b_2,\lambda }^-(u))< \psi _{a,b_2,\lambda }'(t_{a,b_2,\lambda }^-(u))=0\) and hence, from Lemma A.1 we conclude that \(u\in {\mathcal {M}}_{a,b_1,\lambda }\).

(ii) Take \(u\in {\mathcal {M}}_{a,b,\lambda _1}\). Once \(\psi _{a,b,\lambda _2}'(t)\le \psi _{a,b,\lambda _1}'(t)\) for all \(t>0\), it follows that \(\psi _{a,b,\lambda _2}'(t_{a,b,\lambda _1}^-(u))< \psi _{a,b,\lambda _1}'(t_{a,b,\lambda _1}^-(u))=0\) and hence, from Proposition A.1 we conclude that \(u\in {\mathcal {M}}_{a,b,\lambda _1}\). \(\square \)

Proposition A.1

Fix \(a>0\) and let I be an interval. Then, the following holds true.

  1. (i)

    Fix \(b\in {\mathbb {R}}\). If \(c^-(a,b,\lambda )\) is defined for all \(\lambda \in I\), then it is non-increasing as a function of \(\lambda \).

  2. (ii)

    Fix \(\lambda \in {\mathbb {R}}\). If \(c^-(a,b,\lambda )\) is defined for all \(b\in I\), then it is non-decreasing as a function of b.

Proof

i) Indeed, fix \( \lambda _1<\lambda _2\) and \(u\in {\mathcal {M}}_{a,b,\lambda _1}\). Since from Lemma A.3 we have that \(u\in {\mathcal {M}}_{a,b,\lambda _2}\), it follows from Lemma A.2 that

$$\begin{aligned} c^-(a,b,\lambda _2)\le \psi _{a,b,\lambda _2}(t_{a,b,\lambda _2}^-(u))<\psi _{a,b,\lambda _1}(t_{a,b,\lambda _1}^-(u)), \forall u\in {\mathcal {M}}_{a,b,\lambda _1}. \end{aligned}$$
(23)

and hence \(c^-(a,b,\lambda _2)\le c^-(a,b,\lambda _1)\). The proof of ii) is similar. \(\square \)

Proposition A.2

Fix \(a>0\) and let I be an interval. Then, the following holds true.

  1. (i)

    Fix \(\lambda \in {\mathbb {R}}\). If \(c^-(a,b,\lambda )\) is defined for all \(b\in I\), then it is right continuous as a function of b.

  2. (ii)

    Fix \(b\in {\mathbb {R}}\). If \(c^-(a,b,\lambda )\) is defined for all \(\lambda \in I\), then it is right continuous as a function of \(\lambda \).

Proof

(i) Fix \(b_0\in I\). We claim that \(\lim _{b\downarrow b_0}c^-(a,b,\lambda )=c^-(a,b_0,\lambda )\). Indeed, once \(I\ni b\mapsto c^-(a,b,\lambda )\) is non-decreasing, we can assume that \(\lim _{b\downarrow b_0}c^-(a,b,\lambda )=c\ge c^-(a,b_0,\lambda )\). Suppose on the contrary that \(c>c^-(a,b_0,\lambda )\). Given \(\varepsilon >0\) choose \(u\in {\mathcal {M}}_{a,b_0,\lambda }\) such that \(\Phi _{a,b_0,\lambda }(t_{a,b_0,\lambda }^-(u)u)\in [c^-(a,b_0,\lambda ),c^-(a,b_0,\lambda )+\varepsilon )\) and \(c^-(a,b_0,\lambda )+\varepsilon <c\). From Lemma A.2 we conclude that for small \(\delta >0\)

$$\begin{aligned} c^-(a,b_0+\delta ,\lambda )\le \Phi _{a,b_0+\delta ,\lambda }(t_{a,b_0+\delta ,\lambda }^-(u)u)< c^-(a,b_0,\lambda )+\varepsilon <c, \end{aligned}$$

which is a contradiction and thus \(I\ni b\mapsto c^-(a,b,\lambda )\) is right continuous. The proof of (ii) is similar. \(\square \)

Appendix B: The case \(\lambda =0\)

We collect some results concerning the fiber maps \(\psi \) when \(\lambda =0\). The parameter now is \(b>0\), while \(a>0\) is fixed. For this reason, we write \(\psi _{b,u}\) and \(\Phi _b\) instead of \(\psi _{0,u}\) and \(\Phi _0\) and so on. As we already know, for each \(u\in H_0^1(\Omega ){\setminus }\{0\}\) the fiber map \(\psi _{b,u}\) has satisfies Proposition 4.2. One can see now that the systems \(\psi _{b,u}(t)=\psi _{b,u}'(t)=0\) and \(\psi '_{b,u}(t)=\psi _{b,u}''(t)=0\) admits a unique solution, with respect to tb, which are given respectively by (see [5] and [18])

$$\begin{aligned}&t_{0}(u)=\left( \frac{2^*a}{4-2^*}\frac{\Vert u\Vert ^2}{\Vert u\Vert _{2^*}^{2^*}}\right) ^{\frac{1}{2^*-2}}, \\&b_0(u)=a^{\frac{4-N}{2}}S_N^{\frac{N}{2}}C_1(N)\left( \frac{\Vert u\Vert _{2^*}}{\Vert u\Vert }\right) ^N, \end{aligned}$$

and

$$\begin{aligned}&t(u)=\left( \frac{2a}{4-2^*}\frac{\Vert u\Vert ^2}{\Vert u\Vert _{2^*}^{2^*}}\right) ^{\frac{1}{2^*-2}}, \\&b(u)=a^{\frac{4-N}{2}}S_N^{\frac{N}{2}}C_2(N)\left( \frac{\Vert u\Vert _{2^*}}{\Vert u\Vert }\right) ^N. \end{aligned}$$

As a conclusion of this analysis and similar to Propositions 2.4 and 3.7 we have

Proposition B.1

There holds

  1. (i)

    For each \(b\ge b_0(u)\) and each \(u\in H^1_0(\Omega ){\setminus } \{0\}\), \(\inf _{t>0}\psi _{b,u}(t)=0\); for each \(b<b_0(u)\) there exists \(u\in H^1_0(\Omega ){\setminus } \{0\}\) such that \(\Phi _b(u)<0\).

  2. (ii)

    For each \(b\ge b(u)\), the set \({\mathcal {N}}_b=\emptyset \); for each \(b<b(u)\), the sets \({\mathcal {N}}_b^+\), \({\mathcal {N}}_b^-\) and \({\mathcal {N}}_b^0\) are non empty.

Therefore:

Lemma B.1

The following holds true.

  1. (i)

    If \(a^\frac{N-4}{2}b<C_1(N)\), then there exists \(u\in H_0^1(\Omega ){\setminus }\{0\}\) such that \(\Phi _b(u)<0\).

  2. (ii)

    If \(a^\frac{N-4}{2}b\ge C_1(N)\), then \(\psi _{b ,u}(t)>0\) for all \(u\in H_0^1(\Omega ){\setminus }\{0\}\) and \(t>0\).

  3. (iii)

    If \(a^\frac{N-4}{2}b<C_2(N)\), then \({\mathcal {N}}^0_b, {\mathcal {N}}^-_b, {\mathcal {N}}^+_b\) are non-empty.

  4. (iv)

    If \(a^\frac{N-4}{2}b\ge C_2(N)\), then \({\mathcal {N}}_b=\emptyset \).

Remark B.1

Comparing Lemmas B.1 and 2.2 we see that

  1. (i)

    \(\Phi _b\) is weak lower semi-continuous if, and only, \(\Phi _b(u)\ge 0\) for all \(u\in H_0^1(\Omega )\).

  2. (ii)

    If \(\Phi _b'(u)u>0\) for all \(u\in H_0^1(\Omega ){\setminus }\{0\}\), then \(\Phi _b\) satisfies the Palais-Smale condition. Equivalently \({\mathcal {N}}_b=\emptyset \).

Corollary B.1

If \(a^\frac{N-4}{2}b<C_2(N)\), then for all \(\lambda >0\) we have \({\mathcal {N}}_{b}^-\ne \emptyset \).

Proof

Indeed, this is a consequence of Lemmas B.1 and A.3. We also refer the reader to [13, Lemma 2.6]. \(\square \)

The next lemma is an application of Lemma A.2 and Remark A.2:

Lemma B.2

Fix \(u\in H_0^1(\Omega ){\setminus }\{0\}\). The following holds true.

  1. (i)

    The function \((0,b(u))\ni \mapsto t_b^-(u)\) is continuous and increasing.

  2. (ii)

    The function \((0,b(u))\ni \mapsto t_b^+(u)\) is continuous and decreasing.

  3. (iii)
    $$\begin{aligned} \lim _{b\uparrow b(u)}t_b^-(u)=t(u)=\lim _{b\uparrow b(u)}t_b^+(u). \end{aligned}$$

The following proposition can be found in [18, 19] (with some adaptations). We give an outline of the proof (recall from Lemma B.1 that \({\mathcal {N}}^0_b, {\mathcal {N}}^-_b\) are not empty for all \(a,b>0\) satisfying \(a^\frac{N-4}{2}b<C_2(N)\)):

Proposition B.2

Suppose that \(a^\frac{N-4}{2}b<C_2(N)\), then

$$\begin{aligned} \Phi _b(u)=\frac{(2^*-2)^2a^2}{4\cdot 2^*(4-2^*)b}, \forall u\in {\mathcal {N}}^0_b. \end{aligned}$$

Moreover,

$$\begin{aligned} c^-(a,b,0)< \frac{(2^*-2)^2a^2}{4\cdot 2^*(4-2^*)b}=c^0(a,b,0). \end{aligned}$$

Proof

The first part is trivial. Now suppose on the contrary that there exists \(u\in {\mathcal {N}}^-_b\) such that

$$\begin{aligned} \Phi _b(u)\ge \frac{(2^*-2)^2a^2}{4\cdot 2^*(4-2^*)b}. \end{aligned}$$

From Lemma B.2 we have that \(t^-_b(u)=1<t^-_{b'}(u)<t^+_{b'}(u)<t^+_{b}(u)\) for each \(0<b<b'<b(u)\) and hence

$$\begin{aligned} \Phi _{b'}(t^-_{b'}(u)u)> & {} \Phi _{b'}(u)\\> & {} \Phi _{b}(u) \\\ge & {} \frac{(2^*-2)^2a^2}{4\cdot 2^*(4-2^*)b}, \end{aligned}$$

which implies that

$$\begin{aligned} \frac{(2^*-2)^2a^2}{4\cdot 2^*(4-2^*)b}<\lim _{b'\uparrow b(u)} \Phi _{b'}(t^-_{b'}(u)u)=\Phi _{b(u)}(t_b(u)u)=\frac{(2^*-2)^2a^2}{4\cdot 2^*(4-2^*)b(u)}, \end{aligned}$$

a contradiction since \(b<b(u)\). \(\square \)

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Faraci, F., Silva, K. On the Brezis–Nirenberg problem for a Kirchhoff type equation in high dimension. Calc. Var. 60, 22 (2021). https://doi.org/10.1007/s00526-020-01891-6

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Mathematics Subject Classification

  • 35J20
  • 35B33