ABSTRACT

This paper is devoted to the following high order elliptic problems under the Navier boundary condition: $\{\begin{array}{ll}-{\left(\Delta \right)}^{p}u=\lambda f\left(x,u\right),\hfill & x\in \Omega ,\hfill \\ u=\Delta u=\cdot \cdot \cdot ={\left(\Delta \right)}^{p-1}u=0,\hfill & x\in \partial \Omega .\hfill \end{array}$ Without assuming the standard subcritical polynomial growth condition ensuring the compactness of a bounded (P.S.) sequence, we show that the Navier boundary value problem has at least a weak nontrivial solution for all $\lambda >0$ by using mountain pass theorem.

Keywords:

Navier Boundary Value Problem, Polyharmonic, Variational Methods, Mountain Pass Theorem

1. Introduction and Main Results

Consider the Navier boundary value problem involving the polyharmonic:

$\{\begin{array}{ll}-{\left(\Delta \right)}^{p}u=\lambda f\left(x,u\right),\hfill & x\in \Omega ,\hfill \\ u=\Delta u=\cdot \cdot \cdot ={\left(\Delta \right)}^{p-1}u=0,\hfill & x\in \partial \Omega .\hfill \end{array}$ (1.1)

where $p\in {\mathbb{Z}}^{+}=\left\{1,2,3,\cdots \right\}$ , W is an open bounded domain in ${ℝ}^N$ ( $N\ge 2p+1$ ) with smooth boundary $\partial \Omega$ , $f\left(x\mathrm{,}t\right)$ is continuous on $\bar{\Omega }\times ℝ$ . We look for the weak solutions of (1.1) which are the same as the critical points of the functional $I_{\lambda }\mathrm{<mo>\; :\; </mo>}{H}_{\theta }^p\left(\Omega \right)\to ℝ$ defined by

$I_{\lambda }\left(u\right)=\frac{1}{2}{\Vert u\Vert }^2-\lambda {\displaystyle {\int }_{\Omega }}F\left(x\mathrm{,}u\right)\text{d}x\mathrm{,}$ (1.2)

where $F\left(x\mathrm{,}t\right)={\displaystyle {\int }_0^t}f\left(x\mathrm{,}s\right)\text{d}s$ , and

$H_{\theta }^p\left(\Omega \right):=\left\{u\in H^p\left(\Omega \right):{\Delta }^{i}u=0\text{if}x\in \partial \Omega ,\text{for}i<\frac{p}{2}\right\}$

is denoted by $E_p$ and its dual spaces by ${E^{\prime }}_p$ . The Hilbert space $E_p$ is equipped with the standard scalar product and norm, respectively, given by:

$\langle u,v\rangle =\{\begin{array}{ll}{\displaystyle {\int }_{\Omega }}\left({\Delta }^{l}u\right)\left({\Delta }^{l}v\right)\text{d}x,\hfill & \text{if}p=2l,\hfill \\ {\displaystyle {\int }_{\Omega }}\left(\nabla {\Delta }^{l}u\right)\left(\nabla {\Delta }^{l}v\right)\text{d}x,\hfill & \text{if}p=2l+1,\hfill \end{array}$

and

$\Vert u\Vert ={\langle u\mathrm{,}u\rangle }^{\frac{1}{2}}\mathrm{.}$

$I_{\lambda }$ is of $C^1$ with derivatives given by

$\langle {I^{\prime }}_{\lambda }\left(u\right)\mathrm{,}v\rangle =\langle u\mathrm{,}v\rangle -\lambda {\displaystyle {\int }_{\Omega }}f\left(x\mathrm{,}u\right)v\text{d}x\mathrm{,}u\mathrm{,}v\in E_p\mathrm{.}$

Motivated by the fact that such kinds of problems (1.1) are often used to describe an important class of physical problems, many authors have widely developed various methods and techniques, such as variational method, critical point theory, lower and upper solutions, and Brower degree for looking for multiple solutions of elliptic equations involving biharmonic, p-biharmonic and polyharmonic type operators (see, for example [1] - [8] and the references therein). In [1] , the author considered the existence of positive solutions in semilinear critical problems for polyharmonic operators and proved the existence result in some general domain under the appropriate assumptions by topological methods. In [2] , by variational methods, they obtain the existence of multiple weak solutions for a class of elliptic Navier boundary problems involving the p-biharmonic operator. By using Nehari manifold, Y.Y. Shang and L. Wang [3] considered (1.1) with critical growth and sublinear perturbation and obtained the existence of multiple nontrivial solutions. Moreover in [3] , a concrete example of application of such mathematical model to describe a physical phenomenon is also pointed out. By three critical points theorem obtained by B. Ricceri, Li and Tang [4] have obtained the existence of at least three weak solutions for a class of Navier boundary value problem involving the p-biharmonic. In [5] , combining the mountain pass theorem together with fountain theorem and with local linking theorem and symmetric mountain pass theorem, Li and Tang establish the existence of at least one solution and infinitely many solutions for a class of p(x)-biharmonic equations with Navier boundary condition, respectively. By using critical point theory, the authors [6] establish the existence of infinitely many weak solutions for a class of elliptic Navier boundary value problems depending on two parameters and involving the p-biharmonic operator. In [7] , a Navier boundary value problem is treated where the left-hand side of the equation involves an operator that is more general than the p-biharmonic. Further, by using the abstract and technical approach developed in [8] [9] [10] the authors are interested in looking for the existence of infinitely many weak solutions of perturbed p-biharmonic equations. For more results about (1.1) and its variants, we refer the interested readers to [11] - [16] and the references therein.

The aim of the manuscript is to consider the problem in a different case: based on the mountain pass theorem; we show that the Navier boundary value problem has at least a weak nontrivial solution for all $\lambda >0$ . We now assume the nonlinearity $f\left(x\mathrm{,}t\right)$ satisfies the following conditions.

(F₁) There exist constants $\alpha \ge \mathrm{0,}\beta \ge 0$ such that

$tf\left(x,t\right)-2F\left(x,t\right)\ge \alpha {\left|f\left(x,t\right)\right|}^{\frac{2N}{N+2p}}\text{for}\text{all}x\in \bar{\Omega },\forall \left|t\right|>\beta .$

(F₂)

$\underset{\left|t\right|\to \infty }{lim}\frac{f\left(x\mathrm{,}t\right)}{{\left|t\right|}^{p^{\mathrm{<mo>\; *\; </mo>}}-1}}=0\text{uniformly}\text{a}\text{.e}\text{.}x\in \Omega \mathrm{.}$

where $p^{*}=\frac{2N}{N-2p}$ is the Sobolev critical exponent.

(F₃)

$\underset{t\to 0}{\lim }\frac{f\left(x,t\right)}{t^{2p-1}}=0\text{uniformly}\text{a}\text{.e}\text{.}x\in \Omega .$

(F₄)

$\underset{\left|t\right|\to +\infty }{lim}\frac{F\left(x\mathrm{,}t\right)}{t^{2p}}=+\infty \text{uniformly}\text{a}\text{.e}\text{.}x\in \Omega \mathrm{.}$

The main theorem of this paper reads as follows.

Theorem 1. Suppose that (F₁)-(F₄) hold. Then Equation (1.1) has a weak nontrivial solution, for all $\lambda >0$ .

This paper is organized as follows. In Section 2, we give the proof of the Theorem 1. In the sequel, the letter C or $C_i\left(i=\mathrm{0,1,2,3,}\cdots \right)$ will be used to denote various positive constants whose exact value is irrelevant.

2. Proof of Theorem 1

Proof of Theorem 1. We will divide the proof of the theorem in three steps.

First step: the (P.S.) condition. We only prove Theorem 1 by taking $\lambda =1$ . The case of a general $\lambda >0$ will follow immediately. In fact, if $\lambda >0$ and $\lambda \ne 1$ , we only let $g\left(x,t\right)=\lambda f\left(x,t\right)$ . Then (1.1) becomes

$\{\begin{array}{ll}-{\left(\Delta \right)}^{p}u=g\left(x,u\right),\hfill & x\in \Omega ,\hfill \\ u=\Delta u=\cdot \cdot \cdot ={\left(\Delta \right)}^{p-1}u=0,\hfill & x\in \partial \Omega ,\hfill \end{array}$

The nonlinear term g also satisfies condition (F₁)-(F₄), then the same conclusion as in the case $\lambda =1$ holds. We denote I₁ as I for convenience.

Let $\left\{u_n\right\}$ be any sequence in $E_p$ such that $I\left(u_n\right)$ is bounded and $\Vert I^{\prime }\left(u_n\right)\Vert$ converges to zero, that is,

$I\left(u_n\right)\to C\mathrm{,}\Vert I^{\prime }\left(u_n\right)\Vert \to 0.$

Applying the Riesz-Fréchet representation theorem, there is $v_n\in E_p$ such that

$\langle I^{\prime }\left(u_n\right),\varphi \rangle =\langle v_n,\varphi \rangle ,\forall \varphi \in E_p,\text{and}\Vert I^{\prime }\left(u_n\right)\Vert =\Vert v_n\Vert .$

Thus, we obtain

$\langle u_n-v_n,\varphi \rangle ={\displaystyle {\int }_{\Omega }}f\left(x,u_n\right)\varphi \text{d}x,\forall \varphi \in E_p.$ (2.1)

In (2.1), taking $\varphi =u_n-v_n$ , we obtain

${\Vert u_n-v_n\Vert }^2={\displaystyle {\int }_{\Omega }}f\left(x\mathrm{,}{u}_n\right)\left(u_n-v_n\right)\text{d}x\mathrm{.}$

From Hölder’s inequality, we get:

${\Vert u_n-v_n\Vert }^{\frac{2N}{N+2p}}\le C_1{\displaystyle {\int }_{\Omega }}{\left|f\left(x,u_n\right)\right|}^{\frac{2N}{N+2p}}\text{d}x.$ (2.2)

On the other hand, since

${\displaystyle {\int }_{\Omega }}\left(u_{n}f\left(x\mathrm{,}{u}_n\right)-2F\left(x\mathrm{,}{u}_n\right)\right)\text{d}x=2I\left(u_n\right)-\langle I^{\prime }\left(u_n\right)\mathrm{,}{u}_n\rangle \mathrm{.}$ (2.3)

then (F₁) together with $2I\left(u_n\right)-\langle I^{\prime }\left(u_n\right)\mathrm{,}{u}_n\rangle \le C_2\left(\Vert u_n\Vert +1\right)$ , we have

$\begin{array}{l}{\displaystyle {\int }_{\Omega }}{\left|f\left(x\mathrm{,}{u}_n\right)\right|}^{\frac{2N}{N+2p}}\text{d}x\\ \le C_3{\displaystyle {\int }_{\Omega }}\left(u_{n}f\left(x\mathrm{,}{u}_n\right)-2F\left(x\mathrm{,}{u}_n\right)\right)\text{d}x\\ =2I\left(u_n\right)-\langle I^{\prime }\left(u_n\right)\mathrm{,}{u}_n\rangle \\ \le C_4\left(\Vert u_n\Vert +1\right)\\ \le C_5{\left(\Vert u_n\Vert +1\right)}^{\frac{2N}{N+2p}}\mathrm{.}\end{array}$ (2.4)

Since $\Vert I^{\prime }\left(u_n\right)\Vert =\Vert v_n\Vert$ is bounded and $\frac{2N}{N+2p}>1$ , by (2.2) and (2.4), we conclude that $\Vert u_n\Vert$ is bounded.

By (F₁), for any $\epsilon >0$ there exists $a\left(\epsilon \right)>0$ such that

${\left|f\left(x,t\right)\right|}^{\frac{2N}{N+2p}}\le \epsilon {\left|t\right|}^{\frac{2N}{N-2p}}+a\left(\epsilon \right)\text{for}t\in ℝ,x\in \Omega .$ (2.5)

Since $\left\{u_n\right\}$ is bounded in $E_p$ and $f\left(x\mathrm{,}t\right)$ is continuous on $\bar{\Omega }\times ℝ$ , then there exists a subsequence of $\left\{u_n\right\}$ , still denoted by $\left\{u_n\right\}$ and $u\in E_p$ such that

${\displaystyle {\int }_{\Omega }}f\left(x\mathrm{,}{u}_n\right)\text{d}x\to {\displaystyle {\int }_{\Omega }}f\left(x\mathrm{,}u\right)\text{d}x\mathrm{.}$ (2.6)

Moreover there is an $M>0$ such that

$\left|f\left(x\mathrm{,}t\right)\right|\le M\mathrm{,}\text{for}\text{all}\left(x\mathrm{,}t\right)\in \bar{\Omega }\times ℝ\mathrm{.}$ (2.7)

Combining (2.6) and (2.7) with the Egorov theorem, there exists ${\Omega }_{\epsilon }\subseteq \Omega$ such that $\left|\Omega \backslash {\Omega }_{\epsilon }\right|<\frac{\epsilon }{a\left(\epsilon \right)}$ and ${\left|f\left(x,u_n\right)-f\left(x,u\right)\right|}_{L^{\infty }\left({\Omega }_{\epsilon }\right)}$ converge to 0, and further

${\displaystyle {\int }_{{\Omega }_{\epsilon }}}{\left|f\left(x,u_n\right)-f\left(x,u\right)\right|}^{\frac{2N}{N+2p}}\text{d}x\to 0.$ (2.8)

Also, since $\left\{u_n\right\}$ is bounded in $E_p$ , we derive that

${\displaystyle {\int }_{\Omega }}\left({\left|u_n\right|}^{\frac{2N}{N-2p}}+{\left|u\right|}^{\frac{2N}{N-2p}}\right)\text{d}x\le C_4\mathrm{.}$ (2.9)

On one hand, we have

$\begin{array}{l}{\displaystyle {\int }_{\Omega \backslash {\Omega }_{\epsilon }}}{\left|f\left(x,u_n\right)-f\left(x,u\right)\right|}^{\frac{2N}{N+2p}}\text{d}x\\ \le 2^{\frac{2N}{N+2p}}\left({\displaystyle {\int }_{\Omega \backslash {\Omega }_{\epsilon }}}{\left|f\left(x,u_n\right)\right|}^{\frac{2N}{N+2p}}+{\left|f\left(x,u\right)\right|}^{\frac{2N}{N+2p}}\text{d}x\right).\end{array}$

Combining (2.5) and (2.9) with the fact $\left|\Omega \backslash {\Omega }_{\epsilon }\right|<\frac{\epsilon }{a\left(\epsilon \right)}$ , one has

${\displaystyle {\int }_{\Omega \backslash {\Omega }_{\epsilon }}}{\left|f\left(x,u_n\right)-f\left(x,u\right)\right|}^{\frac{2N}{N+2p}}\text{d}x\le C_5\epsilon .$ (2.10)

By (2.8), there exists an integer $N\left(\epsilon \right)$ such that for $n>N\left(\epsilon \right)$ we have

${\displaystyle {\int }_{{\Omega }_{\epsilon }}}{\left|f\left(x,u_n\right)-f\left(x,u\right)\right|}^{\frac{2N}{N+2p}}\text{d}x\le \epsilon .$ (2.11)

Taking account of (2.10), for $n>N\left(\epsilon \right)$ , we have

${\displaystyle {\int }_{\Omega }}{\left|f\left(x,u_n\right)-f\left(x,u\right)\right|}^{\frac{2N}{N+2p}}\text{d}x\le \left(C_5+1\right)\epsilon \mathrm{.}$ (2.12)

This proves that $u_n\to u$ in $E_p$ .

Second step: mountain-pass geometric structure.

I has a mountain pass geometry; i.e., there exists $u_1\in E_p$ and constants $r,\rho >0$ such that $I\left(u_1\right)<0,{\Vert u_1\Vert }_p>r$ and

$I\left(u\right)\ge \rho ,\text{when}{\Vert u\Vert }_p=r.$ (2.13)

Indeed, using the assumption (F₃), we have $t_0>0$ and ${\mu }_0\in \left(\mathrm{0,}{\mu }_1\right)$ such that

$\frac{f\left(x,t\right)}{t^{2p-1}}<{\mu }_0,\text{for}\left|t\right|<t_0,$ (2.14)

where ${\mu }_1$ denotes the lowest eigenvalue of ${\left(-\Delta \right)}^p$ with the Navier boundary condition. As shown in [16] , ${\mu }_1>0$ is the principal eigenvalue of

$\{\begin{array}{ll}{\left(-\Delta \right)}^{p}u=\mu u,\hfill & x\in \Omega ,\hfill \\ u=\Delta u=\cdot \cdot \cdot ={\left(\Delta \right)}^{p-1}u=0,\hfill & x\in \partial \Omega ,\hfill \end{array}$ (2.15)

and there is a corresponding eigenfunction ${\varphi }_1>0$ in W.

Hence, by (2.14), we have

$F\left(x\mathrm{,}t\right)\le \frac{{\mu }_0}{2}{t}^{2p}\mathrm{,}\text{for}\left|t\right|\le t_0\mathrm{.}$

This inequality together with the assumption (F₂) implies that

$F\left(x\mathrm{,}t\right)\le \frac{{\mu }_0}{2}{t}^{2p}+C{\left|t\right|}^{p^{\ast }}\mathrm{,}\text{for}t\in \text{R}$

with some $C>0$ . Since ${\mu }_1>0$ denotes the lowest eigenvalue of ${\left(-\Delta \right)}^p$ with the Navier boundary condition, it follows that ${\Vert u\Vert }_p^2\ge {\mu }_1{\Vert u\Vert }_p^2$ for $u\in E_p$ . Then the following estimates hold:

$\begin{array}{c}I\left(u\right)\ge \frac{1}{2}{\Vert u\Vert }_p^2-\frac{{\mu }_0}{2}{\Vert u\Vert }_p^2-C{\Vert u_n\Vert }_{p^{\ast }}^{p^{\ast }}\\ \ge \frac{{\mu }_1-{\mu }_0}{2{\mu }_1}{\Vert u\Vert }_p^2-C^{\prime }{\Vert u_n\Vert }_p^{p^{\ast }}.\end{array}$

This implies that there exist constants r and r such that

$I\left(u\right)\ge \rho \mathrm{,}\text{when}{\Vert u\Vert }_p=r\mathrm{.}$

From (F₄) follows that, for all $M>0$ there exists $C_M>0$ , such that

$F\left(x,t\right)\ge M{t}^{2p}-C_M,\forall x\in \Omega ,\forall t>0.$ (2.16)

By (3.4), we have

$\begin{array}{c}I\left(t{\varphi }_1\right)=\frac{1}{2}{t}^2{\Vert {\varphi }_1\Vert }_p^2-{\displaystyle {\int }_{\Omega }}F\left(x\mathrm{,}t{\varphi }_1\right)\text{d}x\\ \le \frac{1}{2}{t}^2{\Vert {\varphi }_1\Vert }_p^2-t^{2p}{\displaystyle {\int }_{\Omega }}M{\varphi }_1^{2p}\text{d}x+c\left|\Omega \right|\to -\infty \text{as}t\to \infty \mathrm{,}\end{array}$

where ${\varphi }_1$ is the principal eigenfunction of (2.15) corresponding to the principal eigenvalue ${\mu }_1$ . We fix $t>0$ so large that $I\left(t{\varphi }_1\right)<0$ and $t{\Vert {\varphi }_1\Vert }_p>r$ . Let $u_1:=t{\varphi }_1\in E_p$ and then constants $r,\rho >0$ such that $I\left(u_1\right)<0,{\Vert u_1\Vert }_p>r$ and satisfies (2.13), i.e. I has a mountain pass geometry.

Last step: critical value of I.

For $u_1$ in second step, we define

$\Gamma :=\left\{\gamma :C\left[0,1\right]\to E_p|\gamma \left(0\right)=0,\gamma \left(1\right)=u_1\right\},$

$c_0:=\underset{\gamma \in \Gamma }{\inf }\underset{0\le t\le 1}{\max }I\left(\gamma \left(t\right)\right).$

From Proposition 1, I satisfies the (P.S.) condition. By the Mountain Pass Theorem implies that $c_0$ is a critical value of I.

Acknowledgements

This work is supported by Natural Science Foundation of China (No. 11671331); Natural Science Foundation of Fujian Province (No. 2015J01585) and Scientific Research Foundation of Jimei University.

Cite this paper

Lan, Y.Y. (2018) Existence of Solutions to a Class of Navier Boundary Value Problem Involving the Polyharmonic. Advances in Pure Mathematics, 8, 373-379. https://doi.org/10.4236/apm.2018.84020

References