Existence of Positive Solutions for (N, p)-Laplace Equation with Exponential Nonlinearity Term and Convection Term in Dimension ${\mathbb {N}}$

Abstract

In this paper, we study the the following problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _N u-\mu \Delta _p u=\lambda (u^{r}+a|\nabla u|^{s})+f(x,u)&{}\quad \mathrm{in} \; \Omega ,\\ u>0 &{}\quad \mathrm{in} \; \Omega ,\\ u=0 &{}\quad \mathrm{on} \; \partial \Omega . \end{array}\right. } \end{aligned}$$

(0.1)

The term f could be exponential growth at $+\infty$. The convection term involved with $\nabla u$ makes the problem (0.1) nonvariational and the variational methods are not applicable. Under suitable conditions imposed on f, the approximation scheme is employed to obtain the existence of positive solutions for all $\lambda \in (0,\lambda ^*]$ with $\lambda ^*>0$.

Appendix

Lemma A.1

Assume that sequence $\{u_n\}\subset W_0^{1,N}(\Omega )$ satisfies $u_n\rightarrow u$ in $W_0^{1,N}(\Omega )$ as $n\rightarrow \infty$. Then there exists a subsequence $\{u_{n_j}\}$ and $\omega \in W_0^{1,N}(\Omega )$ such that

$$\begin{aligned} u_{n_j}(x)\rightarrow u(x) \quad \mathrm{a.e.} \; x\in \Omega , \end{aligned}$$

and

$$\begin{aligned} |u_{n_{j}}(x)|,|u(x)|\leqslant w(x) \; \mathrm{a.e.}\; x\in \Omega . \end{aligned}$$

Proof

Since $u_n\rightarrow u$ in $W_0^{1,N}(\Omega )$, going if necessary to a subsequence, we have

$$\begin{aligned} u_{n}\rightarrow u\quad \mathrm{in}\; L^{t}(\Omega ), \quad t\in [1,\infty ),\\ u_{n}(x)\rightarrow u(x) \quad \mathrm{a.e.} \; x\in \Omega . \end{aligned}$$

There exists a subsequence $\{u_{n_j}\}$ of $\{u_n\}$ such that

$$\begin{aligned} \Vert u_{n_{j+1}}-u_{n_j}\Vert \leqslant 2^{-j}. \end{aligned}$$

Set

$$\begin{aligned} w(x):=|u_{n_1}(x)|+\sum _{j=1}^{\infty }|u_{n_{j+1}(x)}-u_{n_{j}}(x)|, \quad x\in \Omega . \end{aligned}$$

It is clear that $w\in W_0^{1,N}(\Omega )$ and $|u_{n_{j}}(x)|,|u(x)|\leqslant w(x)$ a.e. $x\in \Omega$. $\square$

Lemma A.2

Suppose condition $\mathrm{(F)}$ holds. Let $u_n\rightarrow u$ in $W_0^{1,N}(\Omega )$ as $n\rightarrow \infty$. Then

$$\begin{aligned} f(\cdot ,u_n(\cdot ))\rightarrow f(\cdot ,u(\cdot )) \quad \mathrm{in}\; L^t(\Omega ), \quad t\in [1,\infty ). \end{aligned}$$

(A.1)

Proof

According to $u_n\rightarrow u$ in $W_0^{1,N}(\Omega )$ and Lemma A.1, going to a subsequence $\{u_{n_j}\}$ we have

$$\begin{aligned}&u_{n_j}\rightarrow u\quad \mathrm{in} \; L^{t}(\Omega ), \; t\in [1,\infty ),\nonumber \\&u_{n_j}(x)\rightarrow u(x) \quad \mathrm{a.e.} \; x\in \Omega , \end{aligned}$$

(A.2)

$$\begin{aligned}&|u_{n_{j}}(x)|,|u(x)|\leqslant w(x) \quad \mathrm{a.e.}\; x\in \Omega . \end{aligned}$$

(A.3)

On the one hand, by (A.2) and $f\in C(\Omega \times {\mathbb {R}})$, we get

$$\begin{aligned} |f(x,u_{n_j}(x))-f(x,u(x))|^t\rightarrow 0 \quad \mathrm{a.e.} \; x\in \Omega . \end{aligned}$$

On the other hand, by the condition $\mathrm{(F)}$, we have

$$\begin{aligned} |f(x,u_{n_j})-f(x,u)|^t&\leqslant 2^{t-1}\left( |f(x,u_{n_j})|^t+|f(x,u)|^t\right) \\&\leqslant 2^{t-1}C_0^t\bigg (|u_{n_j}|^{qt}\mathrm{exp}(t\alpha |u_{n_j}|^{N/(N-1)})\\&\quad +|u|^{qt}\mathrm{exp}(t\alpha |u|^{N/(N-1)})\bigg )\\&\leqslant 2^{t}C_0^t\left( w^{qt}\mathrm{exp}(t\alpha |w|^{N/(N-1)})\right) . \end{aligned}$$

By the Hölder inequality and Lemma 2.6, we have

$$\begin{aligned} \int _{\Omega }\left( w^{qt}\mathrm{exp}(t\alpha |w|^{N/(N-1)})\right) \leqslant |w|_{2 qt}^{qt}\left( \int _{\Omega }\mathrm{exp}(2t\alpha |w|^{{N/(N-1)}})\right) ^{1/2}<\infty . \end{aligned}$$

Therefore, by the Lebesgue dominated convergence theorem, we obtain

$$\begin{aligned} f(\cdot ,u_{n_j}(\cdot ))\rightarrow f(\cdot ,u(\cdot )) \quad \mathrm{in} \; L^t(\Omega ), \; t\in [1,\infty ). \end{aligned}$$

$\square$

Now, for $m\in {\mathbb {N}}$, we consider the mapping $A_m:W_m\rightarrow W_m^*$ defined by

$$\begin{aligned} \langle A_mu,\phi \rangle&=\int _{\Omega }|\nabla u|^{N-2}\nabla u\cdot \nabla \phi +\mu \int _{\Omega }|\nabla u|^{p-2}\nabla u\cdot \nabla \phi \\&\quad -\lambda \left( \int _{\Omega }(u^+)^{r}\phi +a\int _{\Omega }|\nabla u|^{s}\phi \right) -\int _{\Omega }f(x,u)\phi -\varepsilon \int _{\Omega }\phi , \end{aligned}$$

for $u,\phi \in W_m$.

Lemma A.3

Under the condition (F), $A_m$ is continuous.

Proof

Suppose $u_n\rightarrow u$ in $W_m$. For any $\phi \in W_m$ we get

$$\begin{aligned} \left\langle A_mu_n-A_mu,\phi \right\rangle&=\int _{\Omega }(|\nabla u_n|^{N-2}\nabla u_n-|\nabla u|^{N-2}\nabla u)\cdot \nabla \phi \\&\quad +\mu \int _{\Omega }(|\nabla u_n|^{p-2}\nabla u_n-|\nabla u|^{p-2}\nabla u)\cdot \nabla \phi \\&\quad -\lambda \left( \int _{\Omega }(u_n^+)^{r}\phi -\int _{\Omega }(u^+)^{r}\phi +a\int _{\Omega }|\nabla u_n|^{s}\phi \right. \\&\quad -\left. a\int _{\Omega }|\nabla u|^{s}\phi \right) -\int _{\Omega }(f(x,u_n)-f(x,u))\phi . \end{aligned}$$

By the Hölder inequality and the Sobolev embedding inequality, we deduce that

$$\begin{aligned} \left| \int _{\Omega }(|\nabla u_n|^{N-2}\nabla u_n-|\nabla u|^{N-2}\nabla u)\cdot \nabla \phi \right|&\leqslant \int _{\Omega }\left| |\nabla u_n|^{N-2}\nabla u_n-|\nabla u|^{N-2}\nabla u\right| |\nabla \phi |\\&\leqslant C\int _{\Omega }(|\nabla u_n|+|\nabla u|)^{N-2}|\nabla u_n-\nabla u||\nabla \phi |\\&\leqslant C(\Vert u_n\Vert ^{N-2}+\Vert u\Vert ^{N-2})\Vert u_n-u\Vert \Vert \phi \Vert ,\\ \left| \int _{\Omega }(|\nabla u_n|^{p-2}\nabla u_n-|\nabla u|^{p-2}\nabla u)\cdot \nabla \phi \right|&\leqslant \int _{\Omega }\left| |\nabla u_n|^{p-2}\nabla u_n-|\nabla u|^{p-2}\nabla u\right| |\nabla \phi |\\&\leqslant C\int _{\Omega }(|\nabla u_n|+|\nabla u|)^{p-2}|\nabla u_n-\nabla u||\nabla \phi |\\&\leqslant C(|\nabla u_n|_p^{p-2}+|\nabla u|_p^{p-2})|\nabla (u_n-u)|_p|\nabla \phi |_p\\&\leqslant C(\Vert u_n\Vert ^{p-2}+\Vert u\Vert ^{p-2})\Vert u_n-u\Vert \Vert \phi \Vert ,\\ \left| \int _{\Omega }(u_n^+)^{r}\phi -\int _{\Omega }(u^+)^{r}\phi \right|&\leqslant \left| (u_n^+)^{r}-(u^+)^{r}\right| _{2}|\phi |_2\\&\leqslant \gamma _2\left| (u_n^+)^{r}-(u^+)^{r}\right| _{2}\Vert \phi \Vert ,\\ \left| \int _{\Omega }|\nabla u_n|^{s}\phi -\int _{\Omega }|\nabla u|^{s}\phi \right|&\leqslant \left| |\nabla u_n|^{s}-|\nabla u|^{s}\right| _{N/s}|\phi |_{N/(N-s)}\\&\leqslant \gamma _{N/(N-s)}\left| |\nabla u_n|^{s}-|\nabla u|^{s}\right| _{N/s}\Vert \phi \Vert ,\\ \left| \int _{\Omega }f(x,u_n)\phi -\int _{\Omega }f(x,u)\phi \right|&\leqslant |f(x,u_n)-f(x,u)|_2|\phi |_2\\&\leqslant \gamma _2|f(x,u_n) -f(x,u)|_2\Vert \phi \Vert . \end{aligned}$$

Thus,

$$\begin{aligned} \Vert A_mu_n-A_mu\Vert _{W_m^*}&=\sup \limits _{\Vert \phi \Vert _{W_m}=1}|\langle A_mu_n-A_mu,\phi \rangle |\\&\leqslant C(\Vert u_n\Vert ^{N-2}+\Vert u\Vert ^{N-2})\Vert u_n-u\Vert \\&\quad +\mu C(\Vert u_n\Vert ^{p-2}+\Vert u\Vert ^{p-2})\Vert u_n-u\Vert \\&\quad +\lambda \gamma _2\left| (u_n^+)^{r}-(u^+)^{r}\right| _2+a\lambda \gamma _{N/(N-s)}\left| |\nabla u_n|^{s}\right. \\&\quad -\left. |\nabla u|^{s}\right| _{N/s}+\gamma _2|f(x,u_n)-f(x,u)|_2. \end{aligned}$$

Since $u_n\rightarrow u$ in $W_m$, going if necessary to subsequences, we have

$$\begin{aligned}&u_{n}\rightarrow u\quad \mathrm{in} \; L^{t}(\Omega ), \; t\in [1,\infty ), \\&u_{n}(x)\rightarrow u(x)\quad \mathrm{a.e.} \; x\in \Omega , \\&\nabla u_{n}(x)\rightarrow \nabla u(x)\quad \mathrm{a.e.} \; x\in \Omega . \end{aligned}$$

By the Lebesgue dominated convergence theorem and Lemma A.2, we have

$$\begin{aligned}&\lim _{n\rightarrow \infty }\int _{\Omega }\left| (u_n^+)^{r}-(u^+)^{r}\right| ^2=0, \\&\lim _{n\rightarrow \infty }\int _{\Omega }\left| |\nabla u_n|^{s}-|\nabla u|^{s}\right| ^{N/s}=0, \\&\lim _{n\rightarrow \infty }\int _{\Omega }\left| f(x,u_n)-f(x,u)\right| ^{2}=0. \end{aligned}$$

Therefore,

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert A_mu_n-A_mu\Vert _{W_m^*}=0. \end{aligned}$$

$\square$