Existence and Hölder regularity of infinitely many solutions to a p-Kirchhoff-type problem involving a singular nonlinearity without the Ambrosetti–Rabinowitz (AR) condition

Abstract

We carry out an investigation of the existence of infinitely many solutions to a fractional p-Kirchhoff-type problem with a singularity and a superlinear nonlinearity with a homogeneous Dirichlet boundary condition. Further, the solution(s) will be proved to be bounded and a weak comparison principle has also been proved. A ‘$C^1$ versus $W_0^{s,p}$’ analysis has also been discussed.

Appendix

The appendix will address a few results that have been used in this article. Lemma 5.2 guarantees the existence of a positive solution to (3.7), and Lemma 5.5 will establish that the functional ${\bar{I}}$ verifies the mountain pass geometry, whereas Lemma 5.4 guarantees that a solution to (1.1) is greater than or equal to the solution to (3.7).

Remark 5.1

By saying ‘$u>0$ in $\Omega$,’ we will mean $\underset{V}{\text {ess}\inf } u>0$ for any compact set $V\subset \Omega$.

Lemma 5.2

Let $0<\gamma <1$, $\lambda ,\mu >0$. Then, the following problem

$$\begin{aligned} \left( a+b\int \limits _{{\mathbb {R}}^N}|u(x)-u(y)|^{p}K(x-y)\hbox {d}x\hbox {d}y\right) {\mathfrak {L}}_p^su= & {} \mu h(x)u^{-\gamma },~\text {in}~\Omega \nonumber \\ u> & {} 0,~\text {in}~\Omega \nonumber \\ u= & {} 0,~\text {in}~{\mathbb {R}}^N\setminus \Omega \end{aligned}$$

(5.1)

has a unique weak solution in $X_0$. This solution is denoted by ${\underline{u}}_{\mu }$, satisfying ${\underline{u}}_{\mu }\ge \epsilon _{\mu } v_0$ a.e. in $\Omega$, where $\epsilon _{\mu }>0$ is a constant.

Proof

We follow the proof in [54]. Firstly, we note that an energy functional on $X_0$ formally corresponding to (5.1) can be defined as follows.

$$\begin{aligned} E(u)&=\frac{a}{p}\Vert u\Vert ^p+\frac{b}{2p}\Vert u\Vert ^{2p}-\frac{\mu }{1-\gamma }\int \limits _{\Omega }h(x)(u^+)^{1-\gamma }\hbox {d}x \end{aligned}$$

(5.2)

for $u\in X_0$. By the Poincaré inequality, this functional is coercive and continuous on $X_0$. It follows that E possesses a global minimizer $u_0\in X_0$. Clearly, $u_0\ne 0$ since $E(0)=0>E(\epsilon v_0)$ for sufficiently small $\epsilon$ and some $v_0>0$ in $\Omega$.

Secondly, we have the decomposition $u=u^+-u^-$. Thus, if $u_0$ is a global minimizer for E, then so is $|u_0|$, by $E(|u_0|)\le E(u_0)$. Clearly enough, the equality holds iff $u_0^-=0$ a.e. in $\Omega$. In other words, we need to have $u_0\ge 0$, i.e., $u_0\in X_0$ where

$$\begin{aligned} X_0^+=\{u\in X_0:u\ge 0~\text {a.e. in}~\Omega \} \end{aligned}$$

is the positive cone in $X_0$.

Third, we will show that $u_0\ge \epsilon v_0>0$ holds a.e. in $\Omega$ for small enough $\epsilon$. Observe that

$$\begin{aligned} \begin{aligned} E'(tv_0)|_{t=\epsilon }=&a\epsilon ^{p-1}\Vert v_0\Vert ^p+b\epsilon ^{2p-1}\Vert v_0\Vert ^{2p}-\mu \epsilon ^{-\gamma }\int \limits _{\Omega }h(x)v_0^{1-\gamma }\hbox {d}x<0 \end{aligned} \end{aligned}$$

(5.3)

whenever $0<\epsilon \le \epsilon _{\mu }$ for some sufficiently small $\epsilon _{\mu }$. We now show that $u_0\ge \epsilon _{\mu }v_0$. On the contrary, suppose $w=(\epsilon _{\mu }v_0-u_0)^+$ does not vanish identically in $\Omega$. Denote

$$\begin{aligned} \Omega ^+=\{x\in \Omega :w(x)>0\}. \end{aligned}$$

We will analyze the function $\zeta (t)=E(u_0+tw)$ of $t\ge 0$. This function is convex owing to its definition over $X_0^+$ being convex. Further $\zeta '(t)=\langle E'(u_0+tw),w \rangle$ is nonnegative and nondecreasing for $t>0$. Consequently, for $0<t<1$ we have

$$\begin{aligned} \begin{aligned} 0\le \zeta '(1)-\zeta '(t)&=\langle E'(u_0+w)-E'(u_0+tw),w\rangle \\&=\int \limits _{\Omega ^+}E'(u_0+w)\hbox {d}x-\zeta '(t)\\&<0 \end{aligned} \end{aligned}$$

(5.4)

by inequality (5.3) and $\zeta '(t)\ge 0$ with $\zeta '(t)$ being nondecreasing for every $t>0$, which is a contradiction. Therefore, $w=0$ in $\Omega$, and hence, $u_0\ge \epsilon _{\mu }v_0$ a.e. in $\Omega$.

Finally, the functional E being strictly convex on $X_0^+$, we conclude that $u_0$ is the only critical point of E in $X_0^+$ with the property $\underset{V}{\text {ess}\inf }u_0>0$ for any compact subset $V\subset \Omega$. Therefore, we choose ${\underline{u}}_{\mu }=u_0$ in the cutoff functional. $\square$

Remark 5.3

We now perform an apriori analysis on a solution (if it exists). Suppose u is a solution to (1.1), then we observe the following

• 1. $I(u)=I(|u|)$. This implies that $u^-=0$ a.e. in $\Omega$.

• 2. In fact a solution to (1.1) can be considered to be positive, i.e., $u>0$ a.e. in $\Omega$ due to the presence of the singular term.

Thus, without loss of generality, we assume that the solution is positive.

Precisely, we now have the following result.

Lemma 5.4

(Apriori analysis). Fix a $\mu \in (0,\mu _0)$. Then, a solution of (1.1), say $u>0$, is such that $u\ge {\underline{u}}_{\lambda }$ a.e. in $\Omega$.

Proof

Fix $\mu \in (0,\mu _0)$ and let $u\in X_0$ be a positive solution to (1.1) and ${\underline{u}}_{\lambda }>0$ be a solution to (5.1). We will show that $u\ge {\underline{u}}_{\lambda }$ a.e. in $\Omega$. Thus, we let ${\underline{\Omega }}=\{x\in \Omega :u(x)<{\underline{u}}_{\lambda }(x)\}$ and from the equation satisfied by u, ${\underline{u}}_{\lambda }$, we have

$$\begin{aligned} 0&\le \langle (a+b\Vert {\underline{u}}_{\lambda }\Vert ^p){\mathfrak {L}}_p^s{\underline{u}}_{\lambda }-(a+b\Vert u\Vert ^p){\mathfrak {L}}_p^su,{\underline{u}}_{\lambda }-u\rangle _{{\underline{\Omega }}} +\lambda \int \limits _{{\underline{\Omega }}}g(x)u^{p-1}({\underline{u}}_{\lambda }-u)\hbox {d}x\nonumber \\&\le \mu \int \limits _{{\underline{\Omega }}}h(x)({\underline{u}}_{\lambda }^{-\gamma }-u^{-\gamma })({\underline{u}}_{\lambda }-u)\hbox {d}x\le 0. \end{aligned}$$

(5.5)

Further, we have

$$\begin{aligned} \langle (a+b\Vert {\underline{u}}_{\lambda }\Vert ^p){\mathfrak {L}}_p^s{\underline{u}}_{\lambda }-(a+b\Vert u\Vert ^p){\mathfrak {L}}_p^su,{\underline{u}}_{\lambda }-u\rangle _{{\underline{\Omega }}}&\ge 0. \end{aligned}$$

(5.6)

Hence, from (5.5) and (5.6), we obtain $u\ge {\underline{u}}_{\lambda }$ a.e. in $\Omega ^c$. $\square$

Lemma 5.5

The redefined functional ${\bar{I}}$ given in (3.8) verifies the mountain pass geometry for $\mu \in (0,\mu _0)$ with $\mu _0<\infty$.

Proof

By the Sobolev embedding, we obtain

$$\begin{aligned} {\bar{I}}(u)\ge \frac{a}{p}\Vert u\Vert ^p+\frac{b}{2p}\Vert u\Vert ^{2p}-\frac{\lambda \Vert g\Vert _{\infty }C_1}{p}\Vert u\Vert ^p-\frac{\mu \Vert h\Vert _{\infty }C_2}{1-\gamma }\Vert u\Vert ^{1-\gamma }-\int \limits _{\Omega }F(x,u)\hbox {d}x. \end{aligned}$$

where $C_1, C_2>0$ are uniform constants that are independent of the choice of u and $F(x,t)=\int \limits _{0}^{t}f(x,\omega )d\omega$. Now, for a pair $(\mu ,r)$, sufficiently small $\mu >0$ say $\mu _0$, we have that $\frac{a}{p}\Vert u\Vert ^p+\frac{b}{2p}\Vert u\Vert ^{2p}-\frac{\mu \Vert h\Vert _{\infty }C_2}{1-\gamma }\Vert u\Vert ^{1-\gamma }>0$ for each $\mu \in (0,\mu _0)$ and $\Vert u\Vert =r$ sufficiently small. Define $a(r)=\frac{a}{p}r^p+\frac{b}{2p}r^{2p}-\frac{\mu \Vert h\Vert _{\infty }C_2}{1-\gamma }r^{1-\gamma }$. Therefore, to sum it up we have

$$\begin{aligned} {\bar{I}}(u)\ge a(r)>0 \end{aligned}$$

for any $\mu \in (0,\mu _0)$ and for every u such that $\Vert u\Vert =r$. On the other hand, taking $u\in X_0$ and $t\ge 0$ we have ${\bar{I}}(tu)\rightarrow -\infty$ as $t\rightarrow \infty$. This verifies the second condition of the mountain pass theorem. $\square$