Infinitely many large solutions to a variable order nonlocal singular equation

Abstract

The paper establishes the existence of infinitely many large energy solutions for a nonlocal elliptic problem involving a variable exponent fractional $p(\cdot )$-Laplacian and a singularity, provided a positive parameter incorporated in the problem is sufficiently small. A variational method can be implemented for an associated problem obtained by truncation related to the singularity. A comparison argument allows one to pass to the original singular problem.

5 Appendix

The aim of this section is to provide an explicit description in our nonlocal setting for the critical values $c_k$ of the $C^1$ functional $I_{\lambda }:W_0\rightarrow {\mathbb {R}}$ introduced in (3.3) that have been used in the proof of Theorem 1 by applying Theorem 3. This description can be obtained using the construction of minimax values relying on the classical notion of genus that we now recall in our underlying space $W_0$. The genus $\gamma (A)$ of a closed, symmetric (i.e., $A=-A$) subset A of $W_0$ with $0\notin A$ is defined to be the smallest positive integer k for which there exists an odd continuous map from A to ${\mathbb {R}}^{k}\setminus \{0\}$, and $\gamma (A)=\infty$ if no such k exists. For more details we refer to [17].

Let $\Gamma$ denote the family of all closed symmetric subsets of $W_0$ contained in $W_0\setminus \{0\}$. Let us note from Lemma 7 that taking $E=Y_k$ there exists $r_k>0$ such that for all $u\in Y_k$ with $\Vert u\Vert \ge r_k$ it holds $I_{\lambda }(u)\le 0$. We may suppose that the sequence $\{r_k\}$ is increasing. Denoting by $B_{r}$ the open ball in $W_0$ of radius $r>0$ centered at the origin, in line with [17, Chapter 9] we set

$$\begin{aligned} \Gamma _k=&\left\{ h(\overline{D_m\setminus S}): h\in G_m \text {and} \gamma (S)\le m-k \text {with an integer} m\ge k \right.&\nonumber \\&\left. \text {and some} S\in \Gamma \right\} , \end{aligned}$$

(5.1)

where $D_m={\overline{B}}_{r_m}\cap Y_m$ and $G_m=\{h\in C(D_m, W_0): h~\text {is odd and}~h=id~\text {on}~\partial {B_{r_m}}\cap {Y_m}\}$.

The next lemma patterns [17, Proposition 9.23].

Lemma 9

Assume that Z is a closed subspace of $W_0$ with ${\text {codim}}(Z)<k$. Then

$$\begin{aligned} A\cap \partial {B}_r\cap {Z}\ne \emptyset ~\text {for all}~A\in \Gamma _k~\text {and}~0<r<r_k. \end{aligned}$$

Proof

Let $A=h(\overline{D_m\setminus S})\in \Gamma _k$ according to (5.1) and let any $0<r<r_k$. Set

$$\begin{aligned} M=\{u\in D_m:\Vert u\Vert <r_k~\text {and}~h(u)\in B_r\}. \end{aligned}$$

We have that $0\in M$ (note that h is odd) and the component $M_0$ of 0 in M is a symmetric bounded neighbourhood of 0 in $Y_m$, so $\gamma (\partial M_0)=m$ because ${\text {dim}}(Y_m)=m$.

We have $\Vert h(u)\Vert =r$ for each $u\in \partial M_0$ since $M_0$ is a component of E, thus

$$\begin{aligned} \partial M_0\subset M_1:=\{u\in D_m:h(u)\in \partial B_r\}. \end{aligned}$$

It follows that $\gamma (M_1)\ge \gamma (\partial M_0)=m$, whence

$$\begin{aligned} \gamma (h(\overline{M_1\setminus S}))\ge \gamma (\overline{M_1\setminus S})\ge \gamma (M_1)-\gamma (S)\ge m-(m-k)=k. \end{aligned}$$

Using ${\text {codim}}(Z)<k$, we derive that $Z\cap h(\overline{M_1\setminus S})\ne \emptyset$. Hence there exists a $v\in \overline{M_1\setminus S}$ with $h(v)\in Z$. The definition of the sets A and $M_1$ renders that $h(v)\in A\cap \partial {B}_r\cap {Z}$, which completes the proof. $\square$

Now we can describe an unbounded sequence of critical values $c_k$ for the functional $I_{\lambda }$ used in the proof of Theorem 1.

Proposition 1

The minimax values

$$\begin{aligned} c_k=\inf \limits _{A\in \Gamma _k}\max \limits _{u\in {D}_k}I_{\lambda }(h(u)), \end{aligned}$$

(5.2)

with $A=h(\overline{D_m\setminus S})\in \Gamma _k$, are critical values of the $C^1$ functional $I_{\lambda }:W_0\rightarrow {\mathbb {R}}$ introduced in (3.3) and it holds $\lim \limits _{k\rightarrow \infty }c_k=+\infty$.

Proof

We first check that $\lim \limits _{k\rightarrow \infty }c_k=+\infty$. As in (3.13) we can find a sequence of positive numbers $\{\delta _k\}$ with $\lim \limits _{k\rightarrow \infty }\delta _k=0$ such that

$$\begin{aligned} I_{\lambda }(u)&\ge \frac{1}{p^+}\Vert u\Vert ^{p^-}-C_0(\lambda \Vert u\Vert ^{1-\gamma ^+} +\delta _k^{\alpha ^+}\Vert u\Vert ^{\alpha ^+}+\Vert u\Vert +1)~\text {for all}~u\in Z_{k-1}, \end{aligned}$$

(5.3)

with a constant $C_0>0$ and k large enough.

Through Lemma 9 we have $A\cap \partial {B}_r\cap {Z}_{k-1}\ne \emptyset$ for all $A\in \Gamma _k$ and $0<r<r_k$. Therefore

$$\begin{aligned} \max \limits _{u\in { A}}I_{\lambda }(h(u))\ge \inf \limits _{u\in \partial {B_r}\cap Z_{k-1}}I_{\lambda }(u)~\text {for all}~ A\in \Gamma _k~\text {and}~0<r<r_k, \end{aligned}$$

which implies

$$\begin{aligned} c_k\ge \inf \limits _{u\in \partial {B_r}\cap Z_{k-1}}I_{\lambda }(u)~\text {for all}~0<r<r_k. \end{aligned}$$

From (5.3) it turns out that

$$\begin{aligned} c_k\ge \frac{1}{p^+}r^{p^-}-C_0(\lambda r^{1-\gamma ^+}+\delta _k^{\alpha ^+}r^{\alpha ^+}+r+1) \end{aligned}$$

for all $0<r=\Vert u\Vert <r_k$ with $u\in {Z_{k-1}}$. In particular, for $r=r_k/2$ this becomes

$$\begin{aligned} c_k\ge \frac{1}{p^+2^{p^-}}r_k^{p^-}-C_0(\frac{\lambda }{2^{1-\gamma ^+}} r_k^{1-\gamma ^+}+\frac{\delta _k^{\alpha ^+}}{2^{\alpha ^+}}r_k^{\alpha ^+}+\frac{1}{2}r_k+1). \end{aligned}$$

(5.4)

With the choice $\delta _k=1/r_k$ (note that $\lim \limits _{k\rightarrow \infty }r_k=+\infty$), (5.4) gives

$$\begin{aligned} c_k\ge \frac{1}{p^+2^{p^-}}r_k^{p^-}-C_0(\frac{\lambda }{2^{1-\gamma ^+}} r_k^{1-\gamma ^+}+\frac{1}{2}r_k+1+\frac{1}{2^{\alpha ^+}}). \end{aligned}$$

Since $p^->1>1-\gamma ^+$ we infer that $\lim \limits _{k\rightarrow \infty }c_k=+\infty$.

Finally, let us show that $c_k$ is a critical value of $I_{\lambda }$. Suppose it fails to hold. Using Lemma 6, we can invoke the deformation lemma (see, e.g., [17, Theorem A.4]) to find constants $0<\epsilon <{\bar{\epsilon }}$ and a continuous map $\eta :[0,1]\times {W_0}\rightarrow {W_0}$ such that

• (a) $\eta (1,I_{\lambda }^{c_k+\epsilon })\subset I_{\lambda }^{c_k-\epsilon }$, where $I_{\lambda }^{r}:=\{u\in {W_0}:I_{\lambda }\le r\}$;

• (b) $\eta (t,u)=u,~\forall \,t\in [0,1]$, if $I_{\lambda }(u)\notin [c_k-{\bar{\epsilon }},c_k+{\bar{\epsilon }}]$;

• (c) $\eta (t,u)$ is odd in u.

Due to $\lim \limits _{k\rightarrow \infty }c_k=+\infty$, we may admit that

$$\begin{aligned} c_k>{\bar{\epsilon }},\quad \forall k. \end{aligned}$$

(5.5)

Fix $A=h(\overline{D_m\setminus S})\in \Gamma _k$, with h and S in (5.1), such that $\max \limits _{u\in A}I_{\lambda }(u)\le c_k+\epsilon$.

We claim that $\eta (1,h(\cdot ))\in G_k$. Indeed, $\eta (1,h(\cdot ))$ is odd thanks to property (c), whereas by (5.5) and the choice of $r_m$, we have $I_{\lambda }(u)\le 0< c_k-{\bar{\epsilon }}$ for all $u\in \partial {B_{r_m}}\cap {Y_m}$. Property (b) ensures the claim. In turn, it yields

$$\begin{aligned} \eta (1,A)\in \Gamma _k. \end{aligned}$$

(5.6)

On the other hand, property (a) entails

$$\begin{aligned} \max \limits _{v\in \eta (1,A)}{\bar{I}}_{\lambda }(v)\le c_k-\epsilon . \end{aligned}$$

In view of (5.6), this contradicts the definition of the minimax value $c_k$ in (5.2), which completes the proof. $\square$

We conclude this section by proving the following comparison principle.

Lemma 10

Assume (${\mathcal {A}}_p$), (${\mathcal {A}}_{\gamma }$) and (${\mathcal {A}}_f$) are true and $u, v\in W_0$. Suppose $u\le v$ on ${\mathbb {R}}^N\setminus \Omega$ and

$$\begin{aligned} (-\Delta )_{p(\cdot )}^{s}u-\frac{\lambda }{|u|^{\gamma (x)-1}u} \le (-\Delta )_{p(\cdot )}^{s}v-\frac{\lambda }{|v|^{\gamma (x)-1}v} \end{aligned}$$

weakly in $\Omega$ for a fixed $\lambda >0$ and $s\in (0,1)$, then $u\le v$ a.e. in ${\mathbb {R}}^N.$

Proof

On using the weak inequality as stated in the Lemma 10, we get

$$\begin{aligned} \langle (-\Delta )_{p(\cdot )}^sv,\phi \rangle -\int _{\Omega }\frac{\lambda \phi }{|v|^{\gamma (x)-1}v}dx&\ge \langle (-\Delta )_{p(\cdot )}^su,\phi \rangle -\int _{\Omega }\frac{\lambda \phi }{|u|^{\gamma (x)-1}u}dx, \end{aligned}$$

(5.7)

for all nonnegative $\phi \in W_0.$ Set

$$\begin{aligned} M(x,y)=\int _{0}^{1}|(v(x)-v(y))+t((u(x)-u(y))-(v(x)-v(y)))|^{p(x,y)-2}dt. \end{aligned}$$

From the definition we have $M(x, y)=M(y, x)\ge 0$ for $(x,y)\in {\mathbb {R}}^{N}\times {\mathbb {R}}^{N}$. By taking $a=v(x)-v(y)$, $b=u(x)-u(y)$ in the identity

$$\begin{aligned} |b|^{p(x,y)-2}b-|a|^{p(x,y)-2}a&=(p(x,y)-1)(b-a)\int _0^1|a+t(b-a)|^{p(x,y)-2}dt, \end{aligned}$$

we deduce

$$\begin{aligned}&|u(x)-u(y)|^{p(x,y)-2}(u(x)-u(y))-|v(x)-v(y)|^{p(x,y)-2}(v(x)-v(y))\\&=(p(x,y)-1)\{(u(x)-u(y))-(v(x)-v(y))\}M(x,y). \end{aligned}$$

Define $\psi :=u-v=(u-v)_{+}-(u-v)_{-}$, where $(u-v)_{\pm }=\max \{\pm (u-v),0\}$. Then, for $\phi =(u-v)_{+}$, we get

$$\begin{aligned}&[\psi (x)-\psi (y)][\phi (x)-\phi (y)]\nonumber \\&=(\phi (x)-\phi (y))^{2}+\phi (x)(u-v)_{-}(y)+\phi (y)(u-v)_{-}(x)\ge 0. \end{aligned}$$

(5.8)

On testing (5.7) with $\phi =(u-v)_{+}$, we obtain from (5.8) that

$$\begin{aligned}&0\ge \int _{\Omega }\lambda (u-v)_{+}\left( \frac{1}{|v|^{\gamma (x)-1}v}-\frac{1}{|u|^{\gamma (x)-1}u}\right) dx\\&\ge \langle (-\Delta )_{p(\cdot )}^su-(-\Delta )_{p(\cdot )}^sv,(u-v)_+\rangle \\&=\int _{{\mathbb {R}}^{N}\times {\mathbb {R}}^{N}}\frac{(p(x,y)-1)M(x,y)[\psi (x)-\psi (y)][\phi (x)-\phi (y)]}{|x-y|^{N+sp(x,y)}}dxdy\\&\ge 0. \end{aligned}$$

Hence $u\le v$ a.e. in ${\mathbb {R}}^N$, which completes the proof. $\square$