A Quasilinear Elliptic Equation with Oscillating Nonlinearity

TL;DR

This paper establishes the existence of solutions for a quasilinear elliptic equation with oscillating nonlinearities near the origin, expanding understanding of such equations with unbounded nonlinear terms.

Contribution

It introduces a method to handle unbounded, oscillatory nonlinearities in quasilinear elliptic equations, providing new existence results.

Findings

Existence of solutions under oscillatory nonlinearities.

Handling unbounded nonlinearities near the origin.

Extension of solution theory for quasilinear elliptic equations.

Abstract

We find a solution of a quasilinear elliptic equation with Dirichlet's boundary condition on a smooth bounded domain and involving an unbounded continuous nonlinearity with oscillatory behavior near the origin.

Equations

$$\left\{\begin{array}[]{rcll}-div\big{(}\phi(|\nabla u|^{2})\nabla u\big{)}&=&f(x,u)\,\,\mbox{ in }\,\,\Omega,\\ u&=&0\,\,\mbox{ on }\,\,\partial\Omega,\end{array}\right.$$

$${\beta }_n\le {\gamma }_n,\forall n\in \mathbb{N};$$

$${\beta }_n\to 0^{+}\text{and}{\gamma }_n\to 0^{+};$$

$$f(x,{\gamma }_n)\le 0\text{mbox}in\Omega \forall n\in \mathbb{N};$$

$$\underset{n}{\mathrm{liminf}}\frac{f(x,{\beta }_n{\phi }_1)}{{\beta }_n}=+\infty \text{mbox}uniformlyin\Omega ,$$

$$\displaystyle\left\{\begin{array}[]{ccc}-\Delta\varphi_{1}=\lambda_{1}\varphi_{1}\,\,\mbox{in}\,\,\Omega,\\ \varphi_{1}=0\ \ \textrm{on}\ \ \partial\Omega.\end{array}\right.$$

$$\displaystyle f(x,t)=\left\{\begin{array}[]{lcl}\displaystyle-\lambda_{1}\sqrt{\frac{t}{\varphi_{1}(x)}}\sin\left(\frac{\sqrt{\varphi_{1}(x)}}{\sqrt{t}}\right)&,&t>0,\\ 0&,&t=0.\end{array}\right.$$

$${\beta }_n={\left(\frac{1}{2n\pi +\frac{3\pi }{2}}\right)}^2$$

$${\gamma }_n={\left(\frac{1}{2n\pi +\frac{\pi }{4}}\right)}^2.$$

$${\beta }_n\to 0^{+}\text{and}{\gamma }_n\to 0^{+};$$

$${\beta }_n\le {\gamma }_n,\forall n\in \mathbb{N};$$

$$f(x,{\gamma }_n)=-{\lambda }_1\frac{1}{\sqrt{{\phi }_1(x)}\left(2n\pi +\frac{\pi }{4}\right)}\sin \left(\sqrt{{\phi }_1(x)}\left(2n\pi +\frac{\pi }{4}\right)\right)\le 0\text{mbox}in\Omega \forall n\in \mathbb{N}$$

$$\frac{f(x,{\beta }_n{\phi }_1)}{{\beta }_n}={\lambda }_1\left(2n\pi +\frac{3\pi }{2}\right).$$

$$\gamma (\kappa +s{)}^{p-2}\le \varphi (s^2)\le \Gamma (\kappa +s{)}^{p-2}$$

$$(\gamma -\frac{1}{2})\varphi (s)\le {\varphi }^{\prime }(s)s\le \Gamma \varphi (s).$$

$${\int }_{\Omega }\varphi (|\nabla {\beta }_n{\phi }_1{|}^2)\nabla {\beta }_n{\phi }_1\nabla wdx\le {\int }_{\Omega }f(x,{\beta }_n{\phi }_1)wdx.$$

$$\begin{array}[]{rcl}\displaystyle\int_{\Omega}\phi(|\nabla\,\beta_{n}\varphi_{1}|^{2})\nabla\,\beta_{n}\varphi_{1}\nabla\,wdx&=&\displaystyle-\int_{\Omega}div\left(\phi(|\nabla\,\beta_{n}\varphi_{1}|^{2})\nabla\,\beta_{n}\varphi_{1}\right)wdx\\ &=&\displaystyle-2\beta_{n}^{3}\int_{\Omega}\left(\phi^{\prime}(|\nabla\,\beta_{n}\varphi_{1}|^{2})\sum_{i=1}^{N}\partial_{x_{i}}\varphi_{1}\left\langle\nabla\varphi_{1},\nabla(\partial_{x_{i}}\varphi_{1})\right\rangle\right)wdx\\ &+&\displaystyle\lambda_{1}\beta_{n}\int_{\Omega}\phi(|\nabla\,\beta_{n}\varphi_{1}|^{2})\varphi_{1}wdx.\end{array}$$

$$\begin{array}[]{c}\displaystyle-2\beta_{n}^{3}\int_{\Omega}\left(\phi^{\prime}(|\nabla\,\beta_{n}\varphi_{1}|^{2})\sum_{i=1}^{N}\partial_{x_{i}}\varphi_{1}\left\langle\nabla\varphi_{1},\nabla(\partial_{x_{i}}\varphi_{1})\right\rangle\right)wdx+\displaystyle\lambda_{1}\beta_{n}\int_{\Omega}\phi(|\nabla\,\beta_{n}\varphi_{1}|^{2})\varphi_{1}wdx\\ \displaystyle\leq\int_{\Omega}f(x,\beta_{n}\varphi_{1})wdx,\end{array}$$

$${\int }_{\Omega }\left[{\lambda }_1\left(2n\pi +\frac{3\pi }{2}\right)-{\lambda }_1\varphi (|\nabla {\beta }_n{\phi }_1{|}^2){\phi }_1+2{\beta }_n^2\left({\varphi }^{\prime }(|\nabla {\beta }_n{\phi }_1{|}^2)\sum _{i=1}^N{\partial }_{x_i}{\phi }_1⟨\nabla {\phi }_1,\nabla ({\partial }_{x_i}{\phi }_1)⟩\right)\right]wdx\ge 0$$

$$f(x,{\beta }_n{\phi }_1)={\lambda }_1{\beta }_n\left(2n\pi +\frac{3\pi }{2}\right).$$

$${\lambda }_1\left(2n\pi +\frac{3\pi }{2}\right)-{\lambda }_1\varphi (|\nabla {\beta }_n{\phi }_1{|}^2){\phi }_1+2{\beta }_n^2\left({\varphi }^{\prime }(|\nabla {\beta }_n{\phi }_1{|}^2)\sum _{i=1}^N{\partial }_{x_i}{\phi }_1⟨\nabla {\phi }_1,\nabla ({\partial }_{x_i}{\phi }_1)⟩\right)\ge 0$$

$$\displaystyle\left\{\begin{array}[]{ccc}-div\big{(}\phi(|\nabla u|^{2})\nabla u\big{)}&=&f(|x|,u)\,\,\mbox{ in }\,\,B_{R},\\ u&=&0\,\,\mbox{ on }\,\,\partial B_{R},\end{array}\right.$$

$$\underset{s\to +\infty }{\mathrm{liminf}}\frac{\varphi ((\sigma s{)}^2)}{\varphi (s^2)}>1\text{and}\underset{s\to +\infty }{\mathrm{limsup}}\frac{\varphi ((\sigma s{)}^2)}{\varphi (s^2)}<+\infty$$

$${\int }_{\Omega }\varphi (|\nabla u{|}^2)\nabla u\nabla wdx={\int }_{\Omega }f(x,u)wdx,$$

$$\left\{\begin{array}[]{rcll}-\Delta_{p}u&=&f(x,u)\,\,\mbox{ in }\,\,\Omega,\\ u&=&0\,\,\mbox{ on }\,\,\partial\Omega.\end{array}\right.$$

$$\beta (x)\ge 0\text{mbox}on\partial \Omega$$

$${\int }_{\Omega }\varphi (|\nabla \beta {|}^2)\nabla \beta \nabla wdx\ge {\int }_{\Omega }f(x,\beta )wdx$$

$$\alpha (x)\le 0\text{mbox}on\partial \Omega$$

$${\int }_{\Omega }\varphi (|\nabla \alpha {|}^2)\nabla \alpha \nabla wdx\le {\int }_{\Omega }f(x,\alpha )wdx$$

$$J(u)=\frac{1}{2}{\int }_{\Omega }\Phi (|\nabla u{|}^2)dx-{\int }_{\Omega }F(x,u)dx,$$

Full text

A Quasilinear Elliptic Equation with Oscillating Nonlinearity

**Anderson L. A. de Araujo

***Departamento de Matemática

Universidade Federal de Viçosa*,

*Av Peter Henry Rolfs, s/n, CEP 36570-900, Viçosa, MG, Brasil

e-mail: anderson.araujo@ufv.br

***Rafael dos Reis Abreu

***Departamento de Ciências Exatas e Educação do Centro de Blumenau

Universidade Federal de Santa Catarina*,

*Rua Pomerode, 710, CEP 89065-300, Blumenau, SC, Brasil

e-mail: rafael.abreu@ufsc.br*

(Received XX April XX

Communicated ……)

Abstract

We find a solution of the quasilinear elliptic equation -div\big{(}\phi(|\nabla u|^{2})\nabla u\big{)}=f(x,u) in $\Omega$ with Dirichlet’s boundary condition, where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$, $N\geq 1$ and $f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is an unbounded continuous function with oscillatory behavior near the origin.

2010 Mathematics Subject Classification. 35J25; 35J60; 35J62.

Key words. Dirichlet problem; oscillatory nonlinearity; a priori bounds; existence of solution.

1 Introduction

We study the quasilinear elliptic second order equation

[TABLE]

where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$, $N\geq 1$, -div\big{(}\phi(|\nabla u|^{2})\nabla u\big{)} is the $\phi$- Laplacian operator and $f:\Omega\times[0,+\infty)\rightarrow\mathbb{R}$ is an unbounded continuous function with oscillatory behavior near the origin in the sense that follows. There are two sequences of real numbers $\beta_{n}>0$ and $\gamma_{n}>0$ such that

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

where $\varphi_{1}$ is an eigenfunction of the Laplacian operator in $\Omega$ corresponding to the first eigenvalue $\lambda_{1}>0$, that is, $\varphi_{1}\in H_{0}^{1}(\Omega)\cap C^{\infty}(\overline{\Omega})$ satisfies $\varphi_{1}>0$ in $\Omega$ and

[TABLE]

By normalization, we can suppose $0<\varphi_{1}\leq 1$ in $\Omega$.

An example of a function which satisfies the above hypotheses is

[TABLE]

For each $n\in\mathbb{N}$, let us define

[TABLE]

and

[TABLE]

Hence

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Therefore, $f$, $\beta_{n}$ and $\gamma_{n}$ just defined satisfy (1.2), (1.3), (1.4) and (1.5).

We suppose that the function $\phi:[0,+\infty[\to[0,+\infty[$ is of class $C^{1}$ and satisfies the following ellipticity and growth conditions of Leray-Lions type (see [12]): there are constants $\gamma,\Gamma>0$, $\kappa\in[0,1]$ and $p\in[2,+\infty[$ such that, for every $s>0$,

[TABLE]

and

[TABLE]

Hence, we can define, for $s\geq 0$, the function $\Phi(s)=\int_{0}^{s}\phi(\xi)d\xi$.

We also suppose that there is $t_{0}>0$ such that, for every $w\in H_{0}^{1}(\Omega)$ satisfying $w\geq 0$ a.e. in $\Omega$ and for $0<\beta_{n}<t_{0}$, we have

[TABLE]

Note that, with the assumptions on $\phi$, the function $f$ and the sequence $(\beta_{n})$ defined in (1.9) and (1.10), respectively, satisfy (1.13). In fact, by integration by parts, we have

[TABLE]

Then, in this case, a sufficient condition to occur (1.13) is

[TABLE]

which is equivalent to

[TABLE]

since

[TABLE]

Provided that $\varphi_{1}\in C^{\infty}(\overline{\Omega})$, it follows from (1.5), (1.11) and (1.12) the existence of $n_{0}>0$ such that, for $n\geq n_{0}$, we have

[TABLE]

and, since $w\geq 0$, we verify the assumptions (1.13).

Quasilinear problems involving the $\phi$-Laplacian operator have been studied by some authors. In [9], the authors studied the following quasilinear elliptic problem in a ball involving an oscillatory nonlinearity:

[TABLE]

where $B_{R}$ denotes the open ball at the origin with radius $R$ in $\mathbb{R}^{N}$, $\phi:[0,+\infty[\rightarrow[0,+\infty[$ is continuous and satisfies

[TABLE]

for all $\sigma>1$ and $f:[0,R]\times\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function such that $s\mapsto\int^{s}_{0}f(|x|,\xi)d\xi$ has an oscillatory behaviour at $+\infty$. About quasilinear problems involving the $\phi$-Laplacian operator, we can also refer [2, 3, 11, 13].

It is also worth noting that problems with oscillatory nonlinearities have been of interest to many authors; see, for example, [1, 5, 7, 8, 10, 11].

In this work, we are concerned with the existence of a weak positive solution of (1.1). By a weak positive solution, we mean a function $u\in W^{1,p}_{0}(\Omega)\cap L^{\infty}(\Omega)$ satisfying $u>0$ a.e. in $\Omega$ and

[TABLE]

for every $w\in W_{0}^{1,p}(\Omega)$, where $p$ is the exponent which appears in (1.11). We emphasize that we complement the result obtained in [9] in the sense that we prove existence of a solution when $\Omega$ is not necessarily a ball and $f$ has an oscillatory behavior near the origin. Furthermore, our hypothesis on the $\phi$-Laplacian operator are different of those used there.

We state the main result.

Theorem 1.1

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$, $N\geq 1$. Assume that $\phi:[0,+\infty[\rightarrow[0,+\infty[$ satisfies (1.11), (1.12), $f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function satisfying (1.2), (1.3), (1.4) and (1.5) and (1.13) is satisfied. Then, there is $n_{0}\in\mathbb{N}$ such that, for every $n\geq n_{0}$, the problem (1.1) has at least one positive solution $u_{n}$. Furthermore, this solution satisfies $\|u_{n}\|_{C^{1}(\overline{\Omega})}\to 0$ as $n\to+\infty$.

The class of quasilinear differential operators which we can consider includes in particular the $p$-Laplacian operator $\Delta_{p}$, with $2\leq p<+\infty$; in this case, $\phi(s)=s^{\frac{p}{2}-1}$ and $\Phi(s^{2})=\frac{2}{p}|s|^{p}$. The Theorem 1.1 yields the following corollary.

Corollary 1.1

Assume the same assumptions in Theorem 1.1. Then, the same conclusions of Theorem 1.1 holds for the problem

[TABLE]

2 Proof of the main result

We say that a function $\beta\in C^{1}(\overline{\Omega})$ is a supersolution of (1.1) if

[TABLE]

and

[TABLE]

for every $w\in W_{0}^{1,p}(\Omega)$, with $w\geq 0$ a.e. in $\Omega$.

Similarly, $\alpha\in C^{1}(\overline{\Omega})$ is a subsolution of (1.1) if

[TABLE]

and

[TABLE]

for every $w\in W_{0}^{1,p}(\Omega)$, with $w\geq 0$ a.e. in $\Omega$.

We also introduce the functional $J:W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

where $\Phi(v)=\int_{0}^{v}\phi(s)ds$ and $F(x,u)=\int_{0}^{u}f(x,s)ds$.

The following existence and regularity result can be easily derived by combining [11, Lemma 2.1] and [6], see also [4, Lemma 3.1].

Lemma 2.1

Let $\Omega$ be as in Theorem 1.1, $\phi:[0,+\infty[\rightarrow[0,+\infty[$ satisfying (1.11), (1.12) and $f:\Omega\times[0,+\infty[\rightarrow~{}\mathbb{R}$ is a continuous function. Assume that there exist a subsolution $\alpha$ and a supersolution $\beta$ of (1.1) with $\alpha(x)\leq\beta(x)$ in $\Omega$. Then problem (1.1) has at least one solution $u$ such that $\alpha(x)\leq u(x)\leq\beta(x)$ in $\Omega$ and

[TABLE]

Moreover, there are constants $\sigma>0$ and $C>0$ continuously depending only on $\|\alpha\|_{\infty}$, $\|\beta\|_{\infty}$ and $\max\{|f(x,u)||x\in\overline{\Omega},\alpha(x)\leq u\leq\beta(x)\}$ such that, for every solution $u$ of (1.1) with $\alpha(x)\leq u(x)\leq\beta(x)$ in $\Omega$, we have $u\in C^{1,\sigma}(\overline{\Omega})$ and

[TABLE]

2.1 Proof of Theorem 1.1

Notice that for all $n\in\mathbb{N}$, $\gamma_{n}$ is a supersolution of (1.1). Indeed, $\gamma_{n}\in C^{1}(\overline{\Omega})$, $\gamma_{n}\geq 0$ in $\partial\Omega$ and, by (1.4),

[TABLE]

for every $w\in H_{0}^{1}(\Omega)$, with $w(x)\geq 0$ a.e. in $\Omega$. Therefore, $\gamma_{n}$ is a supersolution of (1.1).

On the other hand, notice that, for $n\in\mathbb{N}$ large enough, $0<\beta_{n}<t_{0}$ and it follows from (1.13) that $\beta_{n}\varphi_{1}$ is a subsolution of (1.1).

From (1.2) and the hypothesis of normalization of $\varphi_{1}$, it follows that

[TABLE]

for all $n\in\mathbb{N}$ large enough. Then, by Lemma 2.1, we have that there exists $n_{0}\in\mathbb{N}$ such that the problem (1.1) has, for each $n\geq n_{0}$, at least one solution $u_{n}$ such that

[TABLE]

[TABLE]

and

[TABLE]

where the constants $1\geq\tau>0$ and $C>0$ is independent of $n$. By Arzelà-Ascoli Theorem and (1.16), there is a subsequence $(u_{n_{k}})$ such that

[TABLE]

with $u=0$ em $\partial\Omega$. But letting $k\rightarrow+\infty$, we obtain $\beta_{n_{k}}\rightarrow 0$ and $\gamma_{n_{k}}\rightarrow 0$; then, by (1.15) , we conclude that $u_{n_{k}}\rightarrow 0$ uniformly in $\Omega$ and therefore $u=0$.

Hence, by (1.16), the sequence $(u_{n})$ has the property that every subsequence has a subsequence which converges to zero in $C^{1}(\overline{\Omega})$. In conclusion,

[TABLE]

From the above results, there exists $n_{0}\in\mathbb{N}$ such that, for $n\geq n_{0}$, we have

[TABLE]

and

[TABLE]

Consequently, for $n\geq n_{0}$, $u_{n}$ is a solution of (1.1) and

[TABLE]