q-Laplace equation involving the gradient on general bounded and exterior domains

Abstract

The existence of positive singular solutions of

$$\begin{aligned} \left\{ \begin{array}{lcc} -\Delta _q u=(1+g(x))|\nabla u|^p &{}\quad \text {in}&{} B_1,\\ u=0&{}\quad \text {on}&{} \partial B_1, \end{array} \right. \end{aligned}$$

(1)

is proved, where $B_1$ is the unit ball in ${\mathbb {R}}^N$, $N \ge 3$, $2<q<N$, $\frac{N(q-1)}{N-1}<p<q$ and $g\ge 0$ is a Hölder continuous function with $g(0) = 0$. Also, the existence of positive singular solutions of

$$\begin{aligned} \left\{ \begin{array}{lcc} -\Delta _q u=|\nabla u|^p &{}\quad \text {in}&{} \Omega ,\\ u=0&{}\quad \text {on}&{} \partial \Omega . \end{array} \right. \end{aligned}$$

(2)

is proved, where $\Omega$ is a bounded smooth domain in ${\mathbb {R}}^N$, $N \ge 3$, $2< q<N$ and $\frac{N(q-1)}{N-1}<p<q$. Finally, the existence of a bounded positive classical solution of (2) with the additional property that $\nabla u(x) \cdot x > 0$ for large |x| is proved, in the case of $\Omega$ an exterior domain ${\mathbb {R}}^N$, $N\ge 3$ and $p >\frac{N(q-1)}{N-1}$.

Appendix

Here we have some of the computations.

Notice that $\alpha :=\frac{(N-1)(p-q+1)}{q-1},\ \beta :=\frac{p-q+1}{(q-1)(\alpha -1)},\ \varsigma =q-2$, thus

$$\begin{aligned} \begin{aligned} (q-1)(\alpha -1)=&(N-1)(p-q+1)-(q-1),\\ \\ N-2-\varsigma -\frac{p-q+2}{\beta }=&(N-1)-(q-1)-(p-q+2)\\ {}&\frac{(N-1)(p-q+1)-(q-1)}{p-q+1}\\ =&\left( 1-\left( p-q+2\right) \right) \left( N-1\right) -\left( q-1\right) \left( 1-\frac{p-q+2}{p-q+1}\right) \\ =&(q-1)(\frac{1}{p-q+1})-\left( p-q+1\right) \left( N-1\right) \end{aligned} \end{aligned}$$

(61)

Let $u_t$ is given by Example 1 then

$$\begin{aligned} \begin{aligned} {(u_t)}_r=\frac{-1}{(\beta r+tr^\alpha )^{\frac{1}{p-q+1}}},\ {(u_t)}_{rr}=\frac{1}{p-q+1}\frac{(\beta +t\alpha r^{\alpha -1})}{(\beta r+tr^\alpha )^{\frac{1}{p-q+1}+1}} \end{aligned} \end{aligned}$$

Also

$$\begin{aligned} \begin{aligned} (I)&\,\frac{(q-2)}{2} \nabla \phi \cdot \nabla | \nabla u_t|^2=(q-2){(u_t)}_{rr}{(u_t)}_r\,\frac{x\cdot \nabla \phi }{r},\\ {}&\\ (II)&\,\nabla u_t \cdot \nabla (\nabla u_t \cdot \nabla \phi )={(u_t)}_r^2\phi _{rr}+{(u_t)}_{rr}{(u_t)}_r\,\frac{x\cdot \nabla \phi }{r},\\ {}&\\ (III)&\,(\Delta u_t) (2 \nabla u_t \cdot \nabla \phi )=2{(u_t)}_{rr}{(u_t)}_r\,\frac{x\cdot \nabla \phi }{r}+2\frac{N-1}{r}{(u_t)}_r^2\,\frac{x\cdot \nabla \phi }{r}\\ {}&\\ (IV)&\,(p-q+4) | \nabla u_t|^{p-q+2} \nabla u_t \cdot \nabla \phi =(p-q+4){(u_t)}_r^{p-q+2}{(u_t)}_r\,\frac{x\cdot \nabla \phi }{r}. \end{aligned} \end{aligned}$$

(62)

The following lemma includes some fairly standard inequalities that are needed to prove the nonlinear mapping is a contraction. Note there are no smallness assumptions on the y and z terms. See, for instance, [3, 20, 21, 33] for a proof.

Lemma 8

Suppose $p>1$. There exists a constant C such that the following hold:

• (1) For all numbers $w>0$ and $\phi , {\tilde{\phi }} \in {\mathbb {R}}$,

  $$\begin{aligned} \left| |w+\phi |^p-pw^{p-1}\phi -w^p\right| \le C\left( w^{p-2}\phi ^2+|\phi |^p\right) , \end{aligned}$$

  and

  $$\begin{aligned}{} & {} \left| |w+{\tilde{\phi }}|^p-|w+\phi |^p-pw^{p-1}({\tilde{\phi }}-\phi )\right| \\{} & {} \le C\left( w^{p-2}\left( |\phi |+|{\tilde{\phi }}|\right) +|\phi |^{p-1}+|{\tilde{\phi }}|^{p-1}\right) \left| {\tilde{\phi }}-\phi \right| . \end{aligned}$$

• (2) For all $x,y,z\in {\mathbb {R}}^N$,

  $$\begin{aligned} \left| |x+y|^p-|x+z|^p\right| \le C\left( |x|^{p-1}+|y|^{p-1}+|z|^{p-1}\right) |y-z|. \end{aligned}$$

• (3) For $p>1$, there exists $C_p$ such that for and $x,y\in {\mathbb {R}}^N$ and ($x \ne 0$) and |y| small enough

  $$\begin{aligned} | |x+y|^p-|x|^p-p|x|^{p-2}x \cdot y|\le C_p\left( |y|^p+|x|^{p-2}|y|^2\right) . \end{aligned}$$

• (4) For $x,y, z\in {\mathbb {R}}^N$,

  $$\begin{aligned}{} & {} \left| |x+y|^p-|x+z|^p-p|x|^{p-2}x\cdot (y-z)\right| \\{} & {} \quad \le C\left( |x|^{p-2}\left( |y|+|z|\right) +|y|^{p-1}+|z|^{p-1}\right) \left| y-z\right| . \end{aligned}$$

Here we recall the following lemma from [4, Lemam 4].

Lemma 9

Suppose $1 < p\le 2$. Then there is some $C = C_p$ such that for all $x, y, z\in R^N$ one has

• (1) $0\le |x+y|^p-|x|^p-p|x|^{p-2}x\cdot y\le C|y|^p,$

• (2) $\left| |x+y|^p-p|x|^{p-2}x\cdot y-|x+z|^p+p|x|^{p-2}x\cdot z\right| \le C\left( |y|^{p-1}+|z|^{p-1}\right) |y-z|,$

• (3) $\left| |x+y|^p-|x+z|^p\right| \le C\left( |y|^{p-1}+|z|^{p-1}+|x|^{p-1}\right) |y-z|.$

We set $F_2=\Delta \phi (\nabla u_t\cdot \nabla \phi ), \, F_3=\Delta \phi (|\nabla \phi |^2),\, F_4(\phi )=\nabla u_t\cdot \nabla (|\nabla \phi |^2)$ and $F_5(\phi )=\nabla \phi \cdot \nabla (\nabla u_t \cdot \nabla \phi )$, then we have the following lemma (see [20, Lemma 16]).

Lemma 10

Let $\phi _1, \phi _2 \in X$. Then there exist $C_1$ and $C_2$ such that

• (1)
  $$\begin{aligned}{} & {} | F_2(\phi _2)-F_2(\phi _1)| \le 2 | \nabla u_t| | \Delta \phi _2| | \nabla \phi _2 - \nabla \phi _1|\nonumber \\{} & {} \quad + 2 | \nabla u_t| | \nabla \phi _2| | \Delta \phi _2 - \Delta \phi _1|, \end{aligned}$$

  (63)
• (2)
  $$\begin{aligned} \begin{array}{llll} | F_3(\phi _2)-F_3(\phi _1)|&{}\le | \Delta \phi _2|\left( | \nabla \phi _2 | + | \nabla \phi _1| \right) | \nabla \phi _2 - \nabla \phi _1| \\ &{} \; \; \; + | \nabla \phi _1|^2 | \Delta ( \phi _2 - \phi _1)| \end{array} \end{aligned}$$

  (64)
• (3)
  $$\begin{aligned}{} & {} | F_4(\phi _2)-F_4(\phi _1)|\le C_1\left( | \nabla u_t| | \nabla \phi _2| | D^2(\phi _2-\phi _1)|\right. \nonumber \\ {}{} & {} \left. + | \nabla u_t| | D^2 \phi _1| | \nabla \phi _2 - \nabla \phi _1|\right) , \end{aligned}$$

  (65)
• (4)
  $$\begin{aligned} \begin{array}{ll} &{}| F_5(\phi _2)-F_5(\phi _1)|\le C_2\left( | D^2 u_t| | \nabla \phi _2| | \nabla \phi _2 - \nabla \phi _1|+ | \nabla u_t| |D^2 \phi _2| | \nabla \phi _2 - \nabla \phi _1| \right. \\ &{} \qquad \left. + | \nabla \phi _1| |D^2 u_t| | \nabla \phi _2 - \nabla \phi _1| + | \nabla \phi _1| | \nabla u_t| |D^2(\phi _2 - \phi _1)|\right) . \end{array}\nonumber \\ \end{aligned}$$

  (66)
• (5)
  $$\begin{aligned} \begin{aligned}&| \nabla \phi _2 \cdot \nabla | \nabla \phi _2|^2-\nabla \phi _1 \cdot \nabla | \nabla \phi _1|^2|\\&=|(\nabla \phi _2 -\nabla \phi _1)\cdot \nabla | \nabla \phi _2|^2+\nabla \phi _1\cdot (\nabla | \nabla \phi _2|^2-\nabla | \nabla \phi _1|^2)\\&\le C |\nabla \phi _2 -\nabla \phi _1|\,|\nabla \phi _2|\, |D^2\phi _2|+|\nabla \phi _1|\, |\nabla \phi _2\\&\quad -\nabla \phi _1|\,|D^2\phi _2|+|\nabla \phi _1|^2|D^2(\phi _2-\phi _1)|\\ \end{aligned} \end{aligned}$$