Quasilinear Schrödinger Equations With Stein-Weiss Type Convolution and Critical Exponential Nonlinearity in ${\mathbb {R}}^N$

Abstract

In this article, we investigate the existence of positive solutions to the following class of quasilinear Schrödinger equations involving Stein-Weiss type convolution

$$\begin{aligned} \left\{ \begin{array}{l} {-}\Delta _N u {-}\Delta _N (u^{2})u +V(x)|u|^{N-2}u= \left( \displaystyle \int _{\mathbb {R}^N}\frac{F(y,u)}{|y|^\beta |x-y|^{\mu }}~dy\right) \displaystyle \frac{f(x,u)}{|x|^\beta } \; \;\;\;\text { in}\; {\mathbb {R}}^N, \end{array} \right. \end{aligned}$$

where $N\ge 2, 0<\mu <N,\, \beta \ge 0,$ and $2\beta +\mu < N.$ The potential $V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$ is a continuous function satisfying $0<V_0\le V(x)$ for all $x\in {\mathbb {R}}^N$ and some suitable assumptions. The nonlinearity $f:{\mathbb {R}}^N\times \mathbb {R}\rightarrow {\mathbb {R}}$ is a continuous function with critical exponential growth in the sense of the Trudinger-Moser inequality and $F(x,s)=\int _{0}^s f(x,t)dt$ is the primitive of f.

Appendix

Here, we prove the generalized Pohozaev identity for the following quasilinear equation:

where $1<p\le N,$ $0<\mu <N, \beta \ge 0,$ and $2\beta +\mu \le N.$ The nonlinearity $f:{\mathbb {R}}^N\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ is a continuous function and $F(x,s)=\int _{0}^s f(x,t)dt$ is the primitive of f. Similar results for the semilinear Choquard equations with Laplacian operator, readers are referred to [23, 39].

Proposition 6.1

Let $w\in W^{1,p}({\mathbb {R}}^N)\cap L_{loc}^\infty ({\mathbb {R}}^N)$ be a weak solution to problem ($Q_{*}$). Then

$$\begin{aligned}&\frac{(N-p)}{p}\int _{{\mathbb {R}}^N} |\nabla w|^p\;dx+\frac{N}{p} \int _{{\mathbb {R}}^N} |h(w)|^p dx\\&\quad \quad +\frac{(2N-\mu -2\beta )}{2}\int _{{{\mathbb {R}}^N}}\left( \int _{{\mathbb {R}^N}} \frac{F(y,h(w))}{|y|^\beta |x-y|^\mu }dy\right) \frac{F(x,h(w))}{|x|^\beta }dx=0. \end{aligned}$$

Proof

Using the idea of [49], let $w_{\epsilon }$ be the classical solution in $C^{3}_{loc}({\mathbb {R}}^N{\setminus } \{0\})$ of

$$\begin{aligned} \left\{ \begin{array}{l} -\text {div}(\epsilon +|\nabla w_\epsilon |^2)^{\frac{p-2}{2}} w_{\epsilon } +|h(w_\epsilon )|^{p-2}h(w_\epsilon )h'(w_\epsilon )\\ \quad = \left( \displaystyle \int _{\Omega }\frac{F(y, h(w_\epsilon ))}{|y|^\beta |x-y|^{\mu }}~dy\right) \frac{f(x, h(w_\epsilon ))}{|x|^\beta } h'(w_\epsilon )\; \;\;\;\text { in}\; {\mathbb {R}}^N. \end{array} \right. \end{aligned}$$

Then $w_{\epsilon }$ is bounded in $C^{1,\theta }({\mathbb {R}}^N{\setminus } \{0\})$ independently of $\epsilon \in (0,1]$ and converges to w in $C^{1,{\tilde{\theta }}}({\mathbb {R}}^N{\setminus } \{0\})$ for any $\theta < {{\tilde{\theta }}}$ as $\epsilon \rightarrow 0$. For $0<{\textrm{r}}<{\textrm{R}}$, define $\phi _{r,R}\in C_{c}^{\infty }({\mathbb {R}}^N)$ with $0\le \phi _{r,R}\le 1$, $\phi _{r,R}(x)=0$ in $|x|<\frac{r}{2}$ and $\phi _{r,R}(x)=1$ on $r<|x|< R$ with $|\nabla \phi _{r,R}(x)|<\frac{c}{R}$. By testing the equation against the function $\psi _{r, R}(x)= \phi _{r,R} (x \cdot \nabla w_\epsilon (x))$ and integrate over ${\mathbb {R}}^N$, we have

$$\begin{aligned}&\int _{{\mathbb {R}}^N} (\epsilon +|\nabla w_{\epsilon }|^2)^{\frac{p-2}{2}}\nabla w_{\epsilon }\cdot \nabla \psi _{r,R} dx + \int _{{\mathbb {R}}^N} |h(w_\epsilon )|^{p-2}h(w_\epsilon )h'(w_\epsilon ) \psi _{r,R} dx\\&\quad = \int _{\mathbb {R}^N}\left( \int _{\mathbb {R}^N}\frac{F(y,h(w_\epsilon ))}{|y|^\beta |x-y|^{\mu }}dy\right) \frac{f(x,h(w_\epsilon ))}{|x|^\beta } h'(w_\epsilon ) \psi _{r,R}(x) dx. \end{aligned}$$

Now we compute for every $R>r>0$,

$$\begin{aligned}&\int _{{\mathbb {R}}^N} |h(w_\epsilon )|^{p-2} h(w_\epsilon )h'(w_\epsilon ) \psi _{r,R}(x) dx\\&\quad = \int _{{\mathbb {R}}^N} |h(w_\epsilon )|^{p-2} h(w_\epsilon ) h'(w_{\epsilon }) \phi _{r,R}(x) x. \nabla w_{\epsilon }(x) dx \\&\quad = \int _{{\mathbb {R}}^N} \phi _{r,R}(x) x. \nabla \left( \frac{|h(w_\epsilon )|^p}{p}\right) (x) dx\\&\quad = - \int _{{\mathbb {R}}^N} (N \phi _{r,R}(x)+ \lambda x\cdot \nabla \phi _{r,R}) \frac{|h(w_\epsilon )|^p}{p} dx. \end{aligned}$$

Letting $\epsilon \rightarrow 0$, $r \rightarrow 0$ and $R\rightarrow \infty$, we obtain

$$\begin{aligned} \int _{{\mathbb {R}}^N} |h(w_\epsilon )|^{p-2} h(w_{\epsilon })h'(w_\epsilon )\psi _{r,R}(x) dx \rightarrow -\frac{N}{p} \int _{{\mathbb {R}}^N } |h(w)|^p dx, \end{aligned}$$

thanks to Lebesgue’s dominated convergence Theorem. Next, we have

$$\begin{aligned}&\int _{{\mathbb {R}}^N} |(\epsilon +|\nabla w_{\epsilon }(x)|^2)^{\frac{p-2}{2}} \nabla w_{\epsilon }(x)\cdot \nabla \psi _{r,R}(x) dx\\&\quad = \int _{{\mathbb {R}}^N} (\epsilon +|\nabla w_{\epsilon }(x)|^2)^{\frac{p-2}{2}} \nabla w_{\epsilon }(x) \nabla (\phi _{r,R}(x) x. \nabla w_{\epsilon }(x)) dx \\&\quad = \int _{{\mathbb {R}}^N} \phi _{r,R}(x) (\epsilon + |\nabla w_{\epsilon }|^2)^{\frac{p-2}{2}} |\nabla w_{\epsilon }(x)|^p - \frac{N}{p}\int _{{\mathbb {R}}^N} \phi _{r,R}(x) (\epsilon + |\nabla w_{\epsilon }|^2)^{\frac{p}{2}}\\&\quad \quad - \int _{{\mathbb {R}}^N} x. \nabla \phi _{r,R}(x) (\epsilon + |\nabla w_{\epsilon }|^2)^{\frac{p}{2}} dx. \end{aligned}$$

Similarly, taking $\epsilon \rightarrow 0$ in the left hand side of the last relation,

$$\begin{aligned}&\int _{{\mathbb {R}}^N} |(\epsilon +|\nabla w_{\epsilon }|^2)^{\frac{p-2}{2}} \nabla w_{\epsilon }(x)\cdot \nabla \psi _{r,R}(x) dx\\&\quad \rightarrow - \int _{{\mathbb {R}}^N} ((N-p) \phi _{r,R}(x)+ x\cdot \nabla \phi _{r,R}( x)) \frac{|\nabla w(x)|^p}{p} dx \end{aligned}$$

and using this,

$$\begin{aligned}{} & {} \lim _{r \rightarrow 0}\lim _{R\rightarrow \infty } \int _{{\mathbb {R}}^N}\\{} & {} |(\epsilon +|\nabla w_{\epsilon }(x)|^2)^{\frac{p-2}{2}} \nabla w_{\epsilon }(x)\cdot \nabla \psi _{r,R}(x) dx = -\frac{N-p}{p} \int _{{\mathbb {R}}^N } |\nabla w|^p dx. \end{aligned}$$

Finally, we get

$$\begin{aligned}&\int _{{{\mathbb {R}}^N}}\left( \int _{{{\mathbb {R}}^N}} \frac{F(y,h(w_\epsilon ))}{|y|^\beta |x-y|^\mu }dy\right) \frac{f(x,h(w_\epsilon )) }{|x|^\beta } \psi _{r,R}(x)h'(w_\epsilon )dx\\&\quad = \frac{1}{2} \int _{{\mathbb {R}}^N}\int _{{\mathbb {R}}^N} \frac{F(y,h(w_\epsilon ))f(x,h(w_\epsilon ))h'(w_\epsilon )\phi _{r,R}(x) x \cdot \nabla w_\epsilon (x) }{|y|^\beta |x-y|^{\mu } |x|^\beta }dxdy\\ {}&\qquad + \frac{1}{2} \int _{{\mathbb {R}}^N}\int _{\mathbb {R}^N}\frac{F(x,h(w_\epsilon ))f(y,h(w_\epsilon ))h'(w_\epsilon )\phi _{r,R}(y) y \cdot \nabla w_\epsilon (y) }{|y|^\beta |x-y|^{\mu } |x|^\beta }dxdy\\&\quad = -N \int _{{\mathbb {R}}^N}\int _{{\mathbb {R}}^N} \frac{F(x,h(w_\epsilon ))F(y,h(w_\epsilon ))\phi _{r,R}(x) }{|y|^\beta |x-y|^{\mu } |x|^\beta }dx dy\\&\quad \quad - \int _{{\mathbb {R}}^N}\int _{{\mathbb {R}}^N} \frac{F(x,h(w_\epsilon ))F(y,h(w_\epsilon )) x\cdot \nabla \phi _{r,R}(x) }{|y|^\beta |x-y|^{\mu } |x|^\beta }dx dy \\&\quad \quad + \frac{\mu }{2} \int _{{\mathbb {R}}^N}\int _{{\mathbb {R}}^N} \frac{F(x,h(w_\epsilon ))F(y,h(w_\epsilon )) }{|y|^\beta |x-y|^{\mu } |x|^\beta } \frac{(x-y)\cdot (x \phi _{r,R}(x)- y \phi _{r,R}(y))}{|x-y|^2}dx dy\\&\qquad + \beta \int _{{\mathbb {R}}^N}\int _{\mathbb {R}^N}\frac{F(x,h(w_\epsilon ))F(y,h(w_\epsilon ))\phi _{r,R}(x) }{|y|^\beta |x-y|^{\mu } |x|^\beta }dxdy. \end{aligned}$$

Taking $r \rightarrow 0$ and $R\rightarrow \infty$, by Lebesgue’s dominated convergence theorem,

$$\begin{aligned}&\int _{{\mathbb {R}}^N}\int _{{\mathbb {R}}^N} \frac{F(x,h(w_\epsilon ))F(y,h(w_\epsilon )) }{|y|^\beta |x-y|^{\mu } |x|^\beta } \frac{(x-y)\cdot (x \phi _{r,R}(x)- y \phi _{r,R}(y))}{|x-y|^2}dx dy\\&\quad \rightarrow \int _{\mathbb {R}^N}\int _{{\mathbb {R}}^N} \frac{F(x,h(w_\epsilon ))F(y,h(w_\epsilon ))}{|y|^\beta |x-y|^{\mu } |x|^\beta }dx dy;\\&\int _{{\mathbb {R}}^N}\int _{{\mathbb {R}}^N} \frac{F(x,h(w_\epsilon ))F(y,h(w_\epsilon )) x\cdot \nabla \phi _{r,R}(x) }{|y|^\beta |x-y|^{\mu } |x|^\beta }dx dy \rightarrow 0. \end{aligned}$$

Hence, combining the above estimates and taking $\epsilon \rightarrow 0$, we obtain

$$\begin{aligned}{} & {} \int _{{{\mathbb {R}}^N}}\left( \int _{{{\mathbb {R}}^N}} \frac{F(y,h(w_\epsilon ))}{|y|^\beta |x-y|^\mu }dy\right) \frac{f(x,h(w_\epsilon ))}{|x|^\beta }\psi _{r,R}(x) dx\\{} & {} \quad \rightarrow -\frac{2N-\mu -2\beta }{2} \int _{\mathbb {R}^N}\int _{{\mathbb {R}}^N} \frac{F(y,h(w))F(x,h(w))}{|y|^\beta |x-y|^{\mu } |x|^\beta }dxdy. \end{aligned}$$

This completes the proof of the proposition. $\square$