Variants of Bernstein’s theorem for variational integrals with linear and nearly linear growth

In [1] Bernstein proved that every $C^{2}$-solution $u=u(x)=u(x_{1},x_{2})$ of the non-parametric minimal surface equation

$$\operatorname{div}\Bigg{[}\frac{\nabla u}{\sqrt{1+|\nabla u|^{2}}}\Bigg{]}=0$$

(1.1)

over the entire plane must be an affine function, which means that with real numbers $a$, $b$, $c$ it holds

$$u(x_{1},x_{2})=ax_{1}+bx_{2}+c\,.$$

For a detailed discussion of this classical result the interested reader is referred for instance to [2], [3], [4], [5] and the references quoted therein.

Starting from Bernstein’s result the question arises to which classes of second order equations the Bernstein-property extends. More precisely, we replace (1.1) through the equation

$$L[u]=0$$

(1.2)

for a second order elliptic operator $L$ and assume that $u\in C^{2}(\mathbb{R}^{2})$ is an entire solution of (1.2) asking if $u$ is affine. To our knowledge a complete answer to this problem is open, however we have the explicit “Nitsche-criterion” established by J.C.C. Nitsche and J.A. Nitsche [6].

In our note we discuss equation (1.2) assuming that $L$ is the Euler-operator associated to the variational integral

$$J[u,\Omega]:=\int_{\Omega}f(\nabla u)\,{\rm d}x$$

with density $f$: $\mathbb{R}^{2}\to\mathbb{R}$ and for domains $\Omega\subset\mathbb{R}^{2}$, i.e. (1.2) is replaced by

$$\operatorname{div}\Big{[}Df(\nabla u)\Big{]}=0\,,$$

(1.3)

where in the minimal surface case (1.1) we have $f(p)=\sqrt{1+|p|^{2}}$, $p\in\mathbb{R}^{2}$, being an integrand of linear growth with repect to $\nabla u$. For this particular class of energy densities and under the additional assumption that $f$ is of type $f(p)=g(|p|)$, $p\in\mathbb{R}^{2}$, for a function $g\in C^{2}([0,\infty))$ such that

$$0<g^{\prime\prime}(t)\leq c(1+t)^{-\mu}\,,\quad t\geq 0\,,$$

with exponent $\mu\geq 3$ (including the minimal surface case) we proved the Bernstein-property in Theorem 1.2 of [7] benefiting from the work [8] of Farina, Sciunzi and Valdinoci.

Bernstein-type theorems under natural additional conditions to be imposed on the entire solutions of the Euler-equations for splitting-type variational integrals of linear growth have been established in the recent paper [9].

One of the main tools used in [9] is a Caccioppoli-inequality involving negative exponents which was already exploited in different variants in the papers [10], [11], [12], [13].

In fact we used this inequality to show that $\partial_{1}\nabla u\equiv 0$ which follows by considering the bilinear form $D^{2}f(\nabla u)$ with suitable weights. Since we are in two dimensions, the use of equation (1.3) then completes the proof of the splitting-type results in [9].

In the manuscript at hand we observe that even without splitting-structure it is possible to discuss the Caccioppoli-inequality with a directional weight obtaining $\partial_{1}\nabla u\equiv 0$ and to argue similar as before. We note that considering directional weights gives much more flexibility in choosing exponents than arguing with a full gradient (see Proposition 6.1 and Proposition 6.2 of [13]). We here already note that we also include a logarithmic variant of the Caccioppli-inequality as an approach to the limit case $\alpha=-1/2$.

Before going into these details let us have a brief look at power growth energy densities as for example

$$f(p)=(1+|p|^{2})^{\frac{s}{2}}\,,\quad p\in\mathbb{R}^{2}\,,$$

with exponent $s>1$. Then the Nitsche-criterion (compare [6], Satz) shows the existence of non-affine entire solutions to equation (1.3), and as we will shortly discuss in the Appendix the same reasoning applies to the nearly linear growth model

$$f(p)=|p|\ln(1+|p|)\,,\quad p\in\mathbb{R}^{2}\,,$$

(1.4)

which means that we do not have the Bernstein-property for equation (1.3) with density of the form (1.4).

However, as indicated above, we can establish a mild condition under which any entire solution is an affine function being valid for a large class of densities $f$ including the nearly linear and even the linear case.

Let us formulate our

Assumptions. The density $f$: $\mathbb{R}^{2}\to\mathbb{R}$ is of class $C^{2}$ such that

$$D^{2}f(p)(q,q)>0\,\quad\mbox{for all $p$, $q\in\mathbb{R}^{2}$, $q\not=0$.}$$

(1.5)

The results of Theorem 1.1 and Theorem 1.2 are not limited to the particular coordinate directions $e_{1}=(1,0)$ and $e_{2}=(0,1)$, more precisely it holds:

The proof of this result follows from the observation that the function $u$ is a local minimizer of the energy $\int f(\nabla u)\,{\rm d}x$ combined with a suitable linear transformation. If we let

	$\displaystyle T:$		$\displaystyle\mathbb{R}^{2}\to\mathbb{R}^{2}\,,\quad E_{\alpha}=T(e_{\alpha})\,,\quad\alpha=1,\,2\,,$
	$\displaystyle E_{\alpha\beta}$	$\displaystyle:=$	$\displaystyle E_{\alpha}\cdot E_{\beta}\,,\quad\alpha,\,\beta=1,\,2\,,$
	$\displaystyle\tilde{u}(x)$	$\displaystyle:=$	$\displaystyle u\big{(}T(x)\big{)}\,,\quad x\in\mathbb{R}^{2}\,,$
	$\displaystyle\tilde{f}(p)$	$\displaystyle:=$	$\displaystyle f\Bigg{(}T\Big{(}\big{(}E_{\alpha\beta}\big{)}^{-1}_{1\leq\alpha,\beta\leq 2}\,p\Big{)}\Bigg{)}\,,\quad p\in\mathbb{R}^{2}\,,$

then it holds

$$\partial_{\alpha}\tilde{u}(x)=\partial_{E_{\alpha}}u\big{(}T(x)\big{)}\,,\quad x\in\mathbb{R}^{2}\,,\quad\alpha=1,\,2\,,$$

and $\tilde{u}$ is an entire solution of equation (1.3) with $f$ being replaced by $\tilde{f}$, which follows from the local minimality of $\tilde{u}$ with respect to the energy $\int\tilde{f}(\nabla w)\,{\rm d}x$. Obviously the properties of $\tilde{f}$ required in Theorem 1.1 and Theorem 1.2, respectively, are consequences of the corresponding assumptions imposed on $f$, thus we can apply our previous results to $\tilde{u}$ (and $\tilde{f}$).

Our paper is organized as follows: in Section 2 we present the proof of Theorem 1.1 based on a Caccioppoli-inequality involving negative exponents, which has been established, for instance, in [13], Proposition 6.1.

Section 3 is devoted to the discussion of Theorem 1.2. We will make use of some kind of a limit version of Caccioppoli’s inequality, whose proof will be presented below. With the help of this result the claim of Theorem 1.2 follows along the lines of Section 2. We finish Section 3 by presenting a technical extension of Theorem 1.2, which just follows from an inspection of the arguments (compare Theorem 3.1).

For the reader’s convenience we discuss in an appendix equation (1.3) for the nearly linear growth case (1.4) and show that the Nitsche-criterion applies yielding non-affine solutions defined on the whole plane.

Let $f$ satisfy (1.5) and (1.6), let $u$ denote an entire solution of (1.3) and assume w.l.o.g. that (1.7) holds. We apply inequality (107) from Proposition 6.1 in [13] with the choices $l=1$, $i=1$ and

$$\Omega=B_{2R}=\big{\{}x\in\mathbb{R}^{2}:\,|x|<2R\big{\}}$$

to obtain for any $\alpha>-1/2$ and $\eta\in C^{\infty}_{0}(B_{2R})$, $0\leq\eta<1$,

	$\displaystyle\int_{B_{2R}}\eta^{2}D^{2}f(\nabla u)\big{(}\nabla\partial_{1}u,\nabla\partial_{1}u\big{)}\Gamma_{1}^{\alpha}\,{\rm d}x$				(2.1)
		$\displaystyle\leq$	$\displaystyle c\int_{B_{2R}}D^{2}f(\nabla u)(\nabla\eta,\nabla\eta)\Gamma_{1}^{\alpha+1}\,{\rm d}x\,,\quad\Gamma_{1}:=1+|\partial_{1}u|^{2}\,,$

with a finite constant independent of $R$.

Letting $\eta=1$ on $B_{R}$ and assuming $|\nabla\eta|\leq c/R$ we apply (1.6) to the r.h.s. of (2.1) and get

	$\displaystyle\int_{B_{R}}D^{2}f(\nabla u)\big{(}\nabla\partial_{1}u,\nabla\partial_{1}u)\Gamma_{1}^{\alpha}\,{\rm d}x$				(2.2)
		$\displaystyle\leq$	$\displaystyle cR^{-2}\int_{R<|x|<2R}\Gamma_{1}^{1+\alpha}\ln\big{(}2+|\nabla u|\big{)}\big{(}1+|\nabla u|\big{)}^{-1}\,{\rm d}x\,.$

On account of (1.7) we deduce for any $\varepsilon>0$

	$\displaystyle\Gamma_{1}^{1+\alpha}\ln\big{(}2+|\nabla u|\big{)}\big{(}1+|\nabla u|\big{)}^{-1}$	$\displaystyle\leq$	$\displaystyle c(\varepsilon)\big{(}1+|\partial_{2}u|^{2}\big{)}^{m(1+\alpha)}\big{(}1+|\nabla u|\big{)}^{\varepsilon-1}$
		$\displaystyle\leq$	$\displaystyle\tilde{c}(\varepsilon)\big{(}1+|\partial_{2}u|\big{)}^{2m(1+\alpha)}\big{(}1+|\partial_{2}u|\big{)}^{\varepsilon-1}$
		$\displaystyle=$	$\displaystyle\tilde{c}(\varepsilon)\big{(}1+|\partial_{2}u|\big{)}^{2m(1+\alpha)-1+\varepsilon}\,.$

Recall that $m<1$, hence $2m(1+\alpha)-1<0$ for $\alpha>-1/2$ sufficiently close to $-1/2$. We fix $\alpha$ with this property and finally select $\varepsilon>0$ such that $2m(1+\alpha)-1+\varepsilon\leq 0$ to obtain

$$\Gamma_{1}^{1+\alpha}\ln\big{(}2+|\nabla u|\big{)}\big{(}1+|\nabla u|\big{)}^{-1}\leq const<\infty\,.$$

(2.3)

Combining (2.2) with (2.3) it is shown that

$$\int_{\mathbb{R}^{2}}D^{2}f(\nabla u)\big{(}\nabla\partial_{1}u,\nabla\partial_{1}u\big{)}\Gamma_{1}^{\alpha}\,{\rm d}x<\infty\,.$$

(2.4)

We quote equation (108) from [13] again with the previous choices $l=1$, $i=1$, $\Omega=B_{2R}$, $\eta\in C^{\infty}_{0}(B_{2R})$, $0\leq\eta\leq 1$, and with $\alpha$ as fixed above. The same calculations as carried out after (108) then yield

	$\displaystyle\int_{B_{2R}}D^{2}f(\nabla u)\big{(}\nabla\partial_{1}u,\nabla\partial_{1}u\big{)}\eta^{2}\Gamma_{1}^{\alpha}\,{\rm d}x$				(2.5)
		$\displaystyle\leq$	$\displaystyle c\Bigg{|}\int_{B_{2R}-B_{R}}D^{2}f(\nabla u)\big{(}\nabla\partial_{1}u,\nabla\eta^{2}\big{)}\partial_{1}u\Gamma_{1}^{\alpha}\,{\rm d}x\Bigg{|}\,.$

On the r.h.s. of (2.5) we apply the Cauchy-Schwarz inequality to the bilinear form $D^{2}f(\nabla u)$, hence

	l.h.s. of (2.5)	$\displaystyle\leq$	$\displaystyle\Bigg{[}\int_{B_{2R}-B_{R}}D^{2}f(\nabla u)\big{(}\nabla\partial_{1}u,\nabla\partial_{1}u\big{)}\Gamma_{1}^{\alpha}\eta^{2}\,{\rm d}x\Bigg{]}^{\frac{1}{2}}$		(2.6)
			$\displaystyle\quad\cdot\Bigg{[}\int_{B_{2R}-B_{R}}D^{2}f(\nabla u)(\nabla\eta,\nabla\eta)\Gamma_{1}^{1+\alpha}\,{\rm d}x\Bigg{]}^{\frac{1}{2}}$
		$\displaystyle=:$	$\displaystyle T_{1}(R)^{\frac{1}{2}}\cdot T_{2}(R)^{\frac{1}{2}}\,.$

Here $\eta$ has been chosen in such a way that $\eta\equiv 1$ on $B_{R}$ and therefore $\operatorname{spt}(\nabla\eta)\subset B_{2R}-B_{R}$. By (2.4) we have

$$\lim_{R\to\infty}T_{1}(R)=0\,,$$

whereas the calculations carried out after (2.2) imply the boundedness of $T_{2}(R)$. Thus (2.5) and (2.6) imply

$$\int_{\mathbb{R}^{2}}D^{2}f(\nabla u)\big{(}\nabla\partial_{1}u,\nabla\partial_{1}u\big{)}\Gamma_{1}^{\alpha}\,{\rm d}x=0\,,$$

hence $\nabla\partial_{1}u=0$ on account of (1.5). This shows $\partial_{1}u=a$ for some number $a\in\mathbb{R}$ and since

$$u(x_{1},x_{2})-u(0,x_{2})=\int_{0}^{x_{1}}\frac{{\rm d}}{\,{\rm d}t}u(t,x_{2})\,{\rm d}t=ax_{1}$$

we can write

$$u(x_{1},x_{2})=\varphi(x_{2})+ax_{1}\,,\quad\varphi(x_{2}):=u(0,x_{2})\,.$$

Equation (1.3) gives

$$\frac{{\rm d}}{{\rm d}t}\frac{\partial f}{\partial p_{2}}\big{(}a,\varphi^{\prime}(t)\big{)}=0$$

so that

$$\frac{\partial f}{\partial p_{2}}\big{(}a,\varphi^{\prime}(t)\big{)}=c$$

for a constant $c$. Finally we observe that the function

$$y\mapsto\frac{\partial f}{\partial p_{2}}(a,y)$$

is strictly increasing (recall (1.5)), which shows the constancy of $\varphi^{\prime}$ and therefore

$$u(0,x_{2})=bx_{2}+c$$

for some numbers $b$, $c\in\mathbb{R}$. Altogether we have shown that $u$ is affine finishing the proof of Theorem 1.1. ∎

Let the assumptions of Theorem 1.2 hold and consider an entire solution $u\in C^{2}(\mathbb{R}^{2})$ of (1.3) without requiring (1.10) or (1.11) for the moment. If we use condition (1.9) in inequality (2.1) and if we assume that the choice $\alpha=-1/2$ is admissible in (2.1), then the calculations of Section 2 would immediately imply that $\nabla^{2}u=0$ yielding Bernstein’s theorem, i.e. the entire solution $u$ is an affine function without adding further hypotheses on $u$.

However, we do not have (2.1) in the case that $\alpha=-1/2$ and hence we provide a weaker version involving conditions like (1.10) or (1.11) in order to conclude that $u$ is affine.

To be precise, we assume the validity of (1.10), let $l$, $i$, $\Omega$ and $\eta$ as stated in front of (2.1) recalling

$$\int_{B_{2R}}D^{2}f(\nabla u)\big{(}\nabla\partial_{1}u,\nabla\psi\big{)}\,{\rm d}x=0$$

for the choice $\psi:=\eta^{2}\partial_{1}u\Phi(\Gamma_{1})$, where

$$\Phi(t):=\frac{\ln(e^{2}-1+t)}{\sqrt{t}}\,,\quad t\geq 1\,.$$

(3.1)

Note that the choice $\alpha=-1/2$ is compensated by the logarithm. Obviously $\Phi(1)=2$, $\Phi(\infty)=0$ together with

$$\Phi^{\prime}(t)=-\frac{1}{2}\frac{1}{t^{\frac{3}{2}}}\ln(e^{2}-1+t)+\frac{1}{\sqrt{t}(e^{2}-1+t)}<0\,,\quad t\geq 1\,,$$

(3.2)

where the negative sign for $\Phi^{\prime}(t)$ follows from $\ln(e^{2}-1+t)\geq 2$ for $t\geq 1$.

With $\psi$ from above and $\Phi$ defined according to (3.1) we obtain

	$\displaystyle\int_{B_{2R}}\eta^{2}D^{2}f(\nabla u)\big{(}\nabla\partial_{1}u,\nabla\partial_{1}u\big{)}\Phi(\Gamma_{1})\,{\rm d}x$				(3.3)
		$\displaystyle+$	$\displaystyle\int_{B_{2R}}\eta^{2}D^{2}f(\nabla u)\big{(}\nabla\partial_{1}u,\partial_{1}u\nabla\Phi(\Gamma_{1})\big{)}\,{\rm d}x$
			$\displaystyle=-2\int_{B_{2R}}\eta D^{2}f(\nabla u)\big{(}\nabla\partial_{1}u,\nabla\eta\big{)}\partial_{1}u\Phi(\Gamma_{1})\,{\rm d}x\,.$

Using the identity

	$\displaystyle\partial_{1}u\nabla\Phi(\Gamma_{1})$	$\displaystyle=$	$\displaystyle 2\nabla\partial_{1}u(\partial_{1}u)^{2}\Phi^{\prime}(\Gamma_{1})$
		$\displaystyle=$	$\displaystyle 2\nabla\partial_{1}u\Bigg{[}\Gamma_{1}\Phi^{\prime}(\Gamma_{1})-\Phi^{\prime}(\Gamma_{1})\Bigg{]}\,,$

the left-hand side of (3.3) equals

$$\int_{B_{2R}}\eta^{2}D^{2}f(\nabla u)\big{(}\nabla\partial_{1}u,\nabla\partial_{1}u\big{)}\Big{[}\Phi(\Gamma_{1})+2\Gamma_{1}\Phi^{\prime}(\Gamma_{1})-2\Phi^{\prime}(\Gamma_{1})\Big{]}\,{\rm d}x\,.$$

From (3.2) it follows (recalling $\Phi^{\prime}(t)\leq 0$ for $t\geq 1$)

	$\displaystyle\Phi(\Gamma_{1})+2\Gamma_{1}\Phi^{\prime}(\Gamma_{1})-2\Phi^{\prime}(\Gamma_{1})$
		$\displaystyle\geq$	$\displaystyle\Phi(\Gamma_{1})+2\Gamma_{1}\Phi^{\prime}(\Gamma_{1})$
		$\displaystyle=$	$\displaystyle\frac{\ln(e^{2}-1+\Gamma_{1})}{\sqrt{\Gamma_{1}}}+2\Gamma_{1}\Bigg{[}-\frac{1}{2}\frac{\ln(e^{2}-1+\Gamma_{1})}{\Gamma_{1}^{\frac{3}{2}}}+\frac{1}{\sqrt{\Gamma_{1}}(e^{2}-1+\Gamma_{1})}\Bigg{]}$
		$\displaystyle=$	$\displaystyle 2\frac{\sqrt{\Gamma_{1}}}{e^{2}-1+\Gamma_{1}}\geq c\frac{1}{\sqrt{\Gamma_{1}}}$

for some constant $c>0$. Altogether we deduce from (3.3) the inequality of Caccioppoli-type

	$\displaystyle\int_{B_{2R}}\eta^{2}D^{2}f(\nabla u)\big{(}\nabla\partial_{1}u,\nabla\partial_{1}u\big{)}\Gamma_{1}^{-\frac{1}{2}}\,{\rm d}x$				(3.4)
		$\displaystyle\leq$	$\displaystyle-2\int_{B_{2R}}\eta D^{2}f(\nabla u)\big{(}\nabla\partial_{1}u,\nabla\eta\big{)}\partial_{1}u\Gamma_{1}^{-\frac{1}{2}}\ln(e^{2}-1+\Gamma_{1})\,{\rm d}x$
		$\displaystyle=:$	$\displaystyle-2S\,.$

To the quantity $S$ we apply the Cauchy-Schwarz inequality valid for the bilinear form $D^{2}f(\nabla u)$ and get

	$\displaystyle|S|$	$\displaystyle\leq$	$\displaystyle\int_{B_{2R}}\Bigg{|}D^{2}f(\nabla u)\big{(}\eta\Gamma_{1}^{-\frac{1}{4}}\nabla\partial_{1}u,\partial_{1}u\Gamma_{1}^{-\frac{1}{4}}\ln(e^{2}-1+\Gamma_{1})\nabla\eta\big{)}\Bigg{|}\,{\rm d}x$		(3.5)
		$\displaystyle\leq$	$\displaystyle\Bigg{[}\int_{B_{2R}}D^{2}f(\nabla u)\big{(}\nabla\partial_{1}u,\nabla\partial_{1}u\big{)}\eta^{2}\Gamma_{1}^{-\frac{1}{2}}\,{\rm d}x\Bigg{]}^{\frac{1}{2}}$
			$\displaystyle\qquad\cdot\Bigg{[}\int_{B_{2R}}D^{2}f(\nabla u)(\nabla\eta,\nabla\eta)\Gamma_{1}^{\frac{1}{2}}\ln^{2}(e^{2}-1+\Gamma_{1})\,{\rm d}x\Bigg{]}^{\frac{1}{2}}\,.$

On the right-hand side of (3.5) we make use of Young’s inequality yielding for any $\varepsilon>0$

	$\displaystyle|S|$	$\displaystyle\leq$	$\displaystyle\varepsilon\int_{B_{2R}}\eta^{2}\Gamma_{1}^{-\frac{1}{2}}D^{2}f(\nabla u)\big{(}\nabla\partial_{1}u,\nabla\partial_{1}u\big{)}\,{\rm d}x$		(3.6)
			$\displaystyle+c(\varepsilon)\int_{B_{2R}}D^{2}f(\nabla u)(\nabla\eta,\nabla\eta)\Gamma_{1}^{\frac{1}{2}}\ln^{2}(e^{2}-1+\Gamma_{1})\,{\rm d}x\,.$

Finally we combine (3.6) and (3.4), thus for a fixed $\varepsilon$ being sufficiently small it holds

	$\displaystyle\int_{B_{2R}}D^{2}f(\nabla u)\big{(}\nabla\partial_{1}u,\nabla\partial_{1}u\big{)}\eta^{2}\Gamma_{1}^{-\frac{1}{2}}\,{\rm d}x$				(3.7)
		$\displaystyle\leq$	$\displaystyle c\int_{B_{2R}}D^{2}f(\nabla u)(\nabla\eta,\nabla\eta)\Gamma_{1}^{\frac{1}{2}}\ln^{2}(e^{2}-1+\Gamma_{1})\,{\rm d}x\,.$

The properties of $\eta$ as stated after (2.1) imply that the right-hand side of (3.7) is bounded by (recall (1.9))

	$\displaystyle cR^{-2}\int_{B_{2R}}|D^{2}f(\nabla u)|\Gamma_{1}^{\frac{1}{2}}\ln^{2}(e^{2}-1+\Gamma_{1})\,{\rm d}x$
		$\displaystyle\leq$	$\displaystyle cR^{-2}\int_{B_{2R}}\frac{1}{1+|\nabla u|}\Gamma_{1}^{\frac{1}{2}}\ln^{2}(e^{2}-1+\Gamma_{1})\,{\rm d}x$
		$\displaystyle\leq$	$\displaystyle cR^{-2}\int_{B_{2R}}\frac{1}{1+|\partial_{2}u|}(1+|\partial_{1}u|)\ln^{2}(1+|\partial_{1}u|)\,{\rm d}x\,.$

Quoting (1.10) and returning to (3.7) we find that

$$\int_{\mathbb{R}^{2}}D^{2}f(\nabla u)\big{(}\nabla\partial_{1}u,\nabla\partial_{1}u\big{)}\eta^{2}\Gamma_{1}^{-\frac{1}{2}}\,{\rm d}x<\infty\,.$$

(3.8)

With (3.8) and on account of estimate (3.5) it is immediate (recall (3.4)) that actually

$$\int_{\mathbb{R}^{2}}D^{2}f(\nabla u)\big{(}\nabla\partial_{1}u,\nabla\partial_{1}u\big{)}\Gamma_{1}^{-\frac{1}{2}}=0\,,$$

hence $\nabla\partial_{1}u=0$ and we can follow the lines of Section 2 to prove our claim. ∎

We leave the details to the reader just adding the obvious remark that clearly we can interchange the roles of the partial derivatives $\partial_{1}u$ and $\partial_{2}u$ or even work with arbitrary directional derivatives as done in Theorem 1.3.

We shortly discuss the Nitsche-criterion (see [6], Satz) for the model case

$$\int|\nabla u|\ln\big{(}1+|\nabla u|\big{)}\,{\rm d}x\,.$$

With a slight abuse of notation but in accordance with the terminology of [6] we let

$$g(t):=t\ln(1+t)\,,\quad t\geq 0\,,\qquad f(t):=g\big{(}\sqrt{t}\big{)}\,,$$

so that

$$J[u]:=\int|\nabla u|\ln\big{(}1+|\nabla u|\big{)}\,{\rm d}x=\int f\big{(}|\nabla u|^{2}\big{)}\,{\rm d}x\,.$$

Introducing the function $\lambda(t):=2f^{\prime\prime}(t)/f^{\prime}(t)$ again for $t\geq 0$ we claim

$$\int_{1}^{\infty}\frac{1+t\lambda(t)}{2+t\lambda(t)}\frac{1}{t}\,{\rm d}t=\infty\,.$$

(4.1)

From (4.1) it follows that the Euler-equation associated to the functional $J$ admits entire non-affine solutions.

For (4.1) we observe the formula

$$\frac{1}{t}\frac{1+t\lambda(t)}{2+t\lambda(t)}=\frac{1}{1+\frac{\sqrt{t}}{g^{\prime\prime}(\sqrt{t})}g^{\prime}(\sqrt{t})}=:\Theta(t)$$

and remark that for $t\gg 1$ it holds

$$c_{1}t\ln\big{(}1+\sqrt{t}\big{)}\leq\frac{\sqrt{t}g^{\prime}(\sqrt{t})}{g^{\prime\prime}(\sqrt{t})}\leq c_{2}t\ln\big{(}1+\sqrt{t}\big{)}$$

as well as $t\leq c_{3}t\ln\big{(}1+\sqrt{t}\big{)}$, hence

$$\Theta(t)\geq c_{4}\frac{1}{t\ln\big{(}1+\sqrt{t}\big{)}}\geq c_{5}\frac{1}{t\ln\big{(}1+t\big{)}}$$

again for $t\gg 1$. Since

$$\int_{1}^{\infty}\frac{{\rm d}t}{t\ln\big{(}1+t\big{)}}=\infty\,,$$

the claim (4.1) follows. ∎