Solvability conditions for some linear and nonlinear non-Fredholm elliptic problems

Abstract

We obtain solvability conditions in $H^{2}$ for some elliptic equations in cylindrical domains using the methods of spectral theory and scattering theory for Schrödinger type operators. We prove the existence of standing solitary waves in $H^{2}$ for some nonlinear equations. Both linear and nonlinear problems involve second order differential operators without Fredholm property.

Appendix

We investigate solvability conditions in $H^{2}(\mathbb{R }^{d}), \ d\in \mathbb{N }$ equipped with the norm

$$\begin{aligned} \Vert u\Vert _{H^{2}(\mathbb{R }^{d})}^{2}:=\Vert u\Vert _{L^{2}(\mathbb{R }^{d})}^{2}+ \Vert \Delta u\Vert _{L^{2}(\mathbb{R }^{d})}^{2} \end{aligned}$$

(5.1)

of the linear equation

$$\begin{aligned} -\Delta \phi -\omega \phi =-h(x),\quad \ \omega \ge 0 \end{aligned}$$

(5.2)

with a square integrable right side. Apparently, the uniqueness of solutions for this problem comes from the fact that the free Laplacian operator in the whole space does not have nontrivial square integrable eigenfunctions. Obviously,

$$\begin{aligned} \widehat{\phi }(p)=-\frac{\widehat{h}(p)}{p^{2}-\omega },\quad \ p\in \mathbb{R }^{d} \end{aligned}$$

(5.3)

with the hat symbol standing for the standard Fourier transform such that

$$\begin{aligned} \widehat{h}(p):=\frac{1}{{(2\pi )}^{\frac{d}{2}}}\int \limits _{\mathbb{R }^{d}}h(x) e^{-ipx}dx. \end{aligned}$$

We have the following statement in one dimension.

Lemma 5.

Let $h(x)\in L^{2}(\mathbb R )$.

• (a) When $\omega >0$ and $xh(x)\in L^{1}(\mathbb R )$ problem (5.2) admits a unique solution in $H^{2}(\mathbb R )$ if and only if

  $$\begin{aligned} \Bigg (h(x), \frac{e^{\pm i\sqrt{\omega }x}}{\sqrt{2\pi }}\Bigg ) _{L^{2}(\mathbb R )}=0. \end{aligned}$$

  (5.4)

• (b) When $\omega =0$ and $x^{2}h(x)\in L^{1}(\mathbb R )$ problem (5.2) admits a unique solution in $H^{2}(\mathbb R )$ if and only if

  $$\begin{aligned} (h(x), 1)_{L^{2}(\mathbb R )}=0,\quad \ (h(x), x)_{L^{2}(\mathbb R )}=0. \end{aligned}$$

  (5.5)

 

Proof.

Let us start with case (a) and introduce the auxiliary set in the Fourier space

$$\begin{aligned} A_{\delta }:=[-\sqrt{\omega }-\delta ,\ -\sqrt{\omega }+\delta ]\cup [\sqrt{\omega }-\delta ,\ \sqrt{\omega }+\delta ]:=A_{\delta }^{-}\cup A_{\delta }^{+}, \end{aligned}$$

with $0<\delta <\sqrt{\omega }$, such that

$$\begin{aligned} \widehat{\phi }(p)=-\frac{\widehat{h}(p)}{p^{2}-\omega }\chi _{A_{\delta }} -\frac{\widehat{h}(p)}{p^{2}-\omega }\chi _{A_{\delta }^{c}}. \end{aligned}$$

(5.6)

Here and below $\chi _{A}$ stands for the characteristic function of a set $A$ and $A^{c}$ for its complement. The second term in the right side of (5.6) is not singular and can be easily estimated above in the absolute value by ${\frac{|\widehat{h}(p)|}{{\delta }^{2}}\in L^{2}(\mathbb{R })}$. To study the behavior of the first term in the right side of (5.6) on $A_{\delta }^{+}$ we use the representation formula

$$\begin{aligned} \widehat{h}(p)=\widehat{h}(\sqrt{\omega })+\int \limits _{\sqrt{\omega }}^{p} \frac{d\widehat{h}(s)}{ds}ds \end{aligned}$$

and ${|\frac{d\widehat{h}(p)}{dp}|\le \frac{1}{\sqrt{2\pi }}\Vert xh\Vert _{L^{1}(\mathbb{R })}, \ p\in \mathbb{R }},$ which yields

$$\begin{aligned} \Bigg |\frac{\int \nolimits _{\sqrt{\omega }}^{p}\frac{d\widehat{h}(s)}{ds}ds}{p^{2}-\omega } \chi _{A_{\delta }^{+}}\Bigg |\le C \frac{\chi _{A_{\delta }^{+}}}{2\sqrt{\omega }- \delta }\in L^{2}(\mathbb R ). \end{aligned}$$

Similarly near the negative singularity

$$\begin{aligned} \widehat{h}(p)=\widehat{h}(-\sqrt{\omega })+\int \limits _{-\sqrt{\omega }}^{p} \frac{d\widehat{h}(s)}{ds}ds, \end{aligned}$$

such that

$$\begin{aligned} \Bigg |\frac{\int \nolimits _{-\sqrt{\omega }}^{p}\frac{d\widehat{h}(s)}{ds}ds}{p^{2}-\omega }\chi _{A_{\delta }^{-}}\Bigg |\le C \frac{\chi _{A_{\delta }^{-}}}{2\sqrt{\omega }-\delta }\in L^{2}(\mathbb R ). \end{aligned}$$

Therefore, it remains to investigate the square integrability of the sum of the two terms

$$\begin{aligned} \frac{\widehat{h}(\sqrt{\omega })}{p^{2}-\omega }\chi _{A_{\delta }^{+}}+ \frac{\widehat{h}(-\sqrt{\omega })}{p^{2}-\omega }\chi _{A_{\delta }^{-}}, \end{aligned}$$

for which the square of the $L^{2}(\mathbb{R })$ norm can be easily bounded below by

$$\begin{aligned} \frac{1}{({2\sqrt{\omega }+\delta )}^{2}}\Bigg [\int \limits _{-\sqrt{\omega }-\delta }^ {-\sqrt{\omega }+\delta }\frac{|\widehat{h}(-\sqrt{\omega })|^{2}}{(p+\sqrt{\omega })^{2}}dp+ \int \limits _{\sqrt{\omega }-\delta }^ {\sqrt{\omega }+\delta }\frac{|\widehat{h}(\sqrt{\omega })|^{2}}{(p-\sqrt{\omega })^{2}}dp \Bigg ]. \end{aligned}$$

The expression above is finite if and only if $\widehat{h}(\pm \sqrt{\omega })$ vanish which is equivalent to orthogonality relations (5.4). Then using formula (5.3) we easily obtain

$$\begin{aligned} p^{2}\widehat{\phi }(p)=-\widehat{h}(p)+\omega \widehat{\phi }(p)\in L^{2}(\mathbb{R }) \end{aligned}$$

under the conditions of the lemma such that $\phi (x)\in H^{2}(\mathbb{R })$. In the case when parameter $\omega$ vanishes we write

$$\begin{aligned} \widehat{\phi }(p)= -\frac{\widehat{h}(p)}{p^{2}}\chi _{\{p\in \mathbb{R }: \ |p|\le 1\}} -\frac{\widehat{h}(p)}{p^{2}}\chi _{\{p\in \mathbb{R }: \ |p|>1\}}. \end{aligned}$$

(5.7)

The second term in the right side of (5.7) can be bounded above in the absolute value by $|\widehat{h}(p)|\in L^{2}$, which will be true in higher dimensions studied in the following lemma as well. Let us expand the Fourier transform

$$\begin{aligned} \widehat{h}(p)=\widehat{h}(0)+\frac{d\widehat{h}}{dp}(0)p+ \int \limits _{0}^{p}\left(\int \limits _{0}^{s}\frac{d^{2}\widehat{h}(q)}{dq^{2}}dq\right)ds \end{aligned}$$

with the second derivative ${ |\frac{d^{2}\widehat{h}(q)}{dq^{2}}|\le \frac{1}{\sqrt{2\pi }} \Vert x^{2}h\Vert _{L^{1}(\mathbb{R })}<\infty , \ q\in \mathbb{R }}.$ Hence we estimate

$$\begin{aligned} \Bigg |\frac{\int \nolimits _{0}^{p}\Bigg (\int \nolimits _{0}^{s}\frac{d^{2}\widehat{h}(q)}{dq^{2}} dq\Bigg )ds}{p^{2}}\chi _{\{p\in \mathbb{R }: \ |p|\le 1\}}\Bigg |\le C \chi _{\{p\in \mathbb{R }: \ |p|\le 1\}}\in L^{2}(\mathbb{R }). \end{aligned}$$

The remaining sum of the two terms

$$\begin{aligned} \frac{\widehat{h}(0)}{p^{2}}\chi _{\{p\in \mathbb{R }: \ |p|\le 1\}}+ \frac{\frac{d\widehat{h}}{dp}(0)}{p}\chi _{\{p\in \mathbb{R }: \ |p|\le 1\}} \end{aligned}$$

is square integrable if and only if both $\widehat{h}(0)$ and ${\frac{d\widehat{h}}{dp}(0)}$ vanish which yields orthogonality relations (5.5). Clearly $p^{2}\widehat{\phi }(p)=-\widehat{h}(p)\in L^{2}(\mathbb{R })$ which completes the proof of the lemma in case (b). $\square$

 

Remark.

The proof of the fact that $\Delta \phi$ is square integrable, $\omega \ge 0$ given above is independent of the dimension and therefore, will be omitted in the proof of Lemma 6 below.

Then we turn our attention to the solvability conditions for Eq. (5.2) in higher dimensions. Note that the orthogonality relations derived below will be dependent upon the value of $d\ge 2$.

 

Lemma 6.

Let $h(x)\in L^{2}(\mathbb{R }^{d}), \ d\ge 2.$

• (a) When $\omega >0$ and $xh(x)\in L^{1}(\mathbb{R }^{d})$ problem (5.2) admits a unique solution in $H^{2}(\mathbb{R }^{d})$ if and only if

  $$\begin{aligned} \Bigg (h(x),\frac{e^{ipx}}{(2\pi )^{\frac{d}{2}}}\Bigg )_{L^{2}(\mathbb{R }^{d})} =0,\quad \ p\in S_{\sqrt{\omega }}^{d} \ a.e., \ d\ge 2. \end{aligned}$$

  (5.8)

• (b) When $\omega =0$ and $|x|^{2}h(x)\in L^{1}(\mathbb{R }^{2})$ problem (5.2) admits a unique solution in $H^{2}(\mathbb{R }^{2})$ if and only if

  $$\begin{aligned} (h(x), 1)_{L^{2}(\mathbb{R }^{2})}=0,\quad \ (h(x), x_{k})_ {L^{2}(\mathbb{R }^{2})}=0,\ 1\le k\le 2. \end{aligned}$$

  (5.9)

• (c) When $\omega =0$ and $|x|h(x)\in L^{1}(\mathbb{R }^{d}), \ d=3,4$ problem (5.2) admits a unique solution in $H^{2}(\mathbb{R }^{d})$ if and only if

  $$\begin{aligned} (h(x), 1)_{L^{2}(\mathbb{R }^{d})}=0,\quad \ d=3,4. \end{aligned}$$

  (5.10)

• (d) When $\omega =0$ and $|x|h(x)\in L^{1}(\mathbb{R }^{d}), \ d\ge 5$ problem (5.2) possesses a unique solution in $H^{2}(\mathbb{R }^{d})$.

 

Proof.

We start with the case of $\omega >0$ and introduce the spherical layer set in the space of $d$ dimensions

$$\begin{aligned} B_{\delta }:=\{ p\in \mathbb{R }^{d} \ | \ \sqrt{\omega }-\delta \le |p|\le \sqrt{\omega }+\delta \},\quad \ 0<\delta <\sqrt{\omega }. \end{aligned}$$

Thus

$$\begin{aligned} \widehat{\phi }(p)=-\frac{\widehat{h}(p)}{p^{2}-\omega }\chi _{B_{\delta }} -\frac{\widehat{h}(p)}{p^{2}-\omega }\chi _{B_{\delta }^{c}}. \end{aligned}$$

(5.11)

The second term in the right side of (5.11) can be easily estimated above in the absolute value by ${\frac{|\widehat{h}(p)|}{\delta \sqrt{\omega }}\in L^{2}(\mathbb{R }^{d})}$. To study the singular part of the expression above we will use the representation formula

$$\begin{aligned} \widehat{h}(p)=\widehat{h}(\sqrt{\omega },\sigma )+\int \limits _{\sqrt{\omega }}^{|p|} \frac{\partial \widehat{h}(|s|,\sigma )}{\partial |s|}d|s|, \end{aligned}$$

where $\sigma$ denotes the variables on the sphere. Clearly, $|\frac{\partial \widehat{h}}{\partial |p|}|\le \frac{1}{(2\pi )^{\frac{d}{2}}}\Vert xh(x)\Vert _{L^{1}(\mathbb{R }^{d})} <$ $\infty$ by the assumption of the lemma. This yields

$$\begin{aligned} \Bigg |\frac{\int \nolimits _{\sqrt{\omega }}^{|p|}\frac{\partial \widehat{h}}{\partial |s|} (|s|,\sigma )d|s|}{p^{2}-\omega }\chi _{B_{\delta }}\Bigg |\le \frac{C}{|p|+\sqrt{\omega }}\chi _{B_{\delta }}\in L^{2}(\mathbb{R }^{d}). \end{aligned}$$

Thus it remains to estimate the norm

$$\begin{aligned} \Bigg \Vert \frac{\widehat{h}(\sqrt{\omega },\sigma )}{p^{2}-\omega } \chi _{B_{\delta }}\Bigg \Vert _{L^{2}(\mathbb{R }^{d})}^{2}&= \int \limits _{S^{d}}d\sigma \int \limits _{\sqrt{\omega }-\delta }^{\sqrt{\omega }+\delta }\frac{|\widehat{h}(\sqrt{\omega },\sigma )|^{2}}{(p^{2}-\omega )^{2}}|p|^{d-1}d|p|\\&\ge \frac{(\sqrt{\omega }-\delta )^{d-1}}{(2\sqrt{\omega }+\delta )^{2}} \int \limits _{\sqrt{\omega }-\delta }^{\sqrt{\omega }+\delta }\frac{d|p|}{(|p|-\sqrt{\omega })^{2}}\int \limits _{S^{d}}d\sigma |\widehat{h}(\sqrt{\omega },\sigma )|^{2}, \end{aligned}$$

which is finite if and only if the Fourier image $\widehat{h}(p)$ vanishes a.e. on the sphere $S_{\sqrt{\omega }}^{d}$. This is equivalent to orthogonality relations (5.8).

When $\omega$ vanishes and the problem is in two dimensions we use the formula analogous to (5.7) in which our primary concern will be the first term in the right side. In the polar coordinates $x=(|x|, \theta _{x})$ and $p=(|p|, \theta _{p})$. We will make use of the expansion

$$\begin{aligned} \widehat{h}(p)=\widehat{h}(0)+|p|\frac{\partial \widehat{h}}{\partial |p|} (0, \theta _{p})+\int \limits _{0}^{|p|}\Bigg (\int \limits _{0}^{s}\frac{\partial ^{2}}{\partial |q|^{2}}\widehat{h}(|q|,\theta _{p})d|q|\Bigg )ds \end{aligned}$$

with

$$\begin{aligned} \widehat{h}(p)=\frac{1}{2\pi }\int \limits _{\mathbb{R }^{2}}h(x)e^{-i|p||x|cos \theta } dx, \end{aligned}$$

(5.12)

where $\theta$ here and below stands for the angle between vectors $x$ and $p$ in $\mathbb{R }^{2}$. Thus for the derivatives we have ${\frac{\partial \widehat{h}}{\partial |p|}(0,\theta _{p})\!=\! -\frac{i}{2\pi }\int _{\mathbb{R }^{2}}h(x)|x|cos \theta dx}$ and $|\frac{\partial ^{2}}{\partial |p|^{2}}\widehat{h}(p)| \le \frac{1}{2\pi }\Vert x^{2}h(x)\Vert _{L^{1}(\mathbb{R }^{2})}<\infty$ by the assumption of the lemma. Clearly

$$\begin{aligned} \Bigg |\frac{\int \nolimits _{0}^{|p|}\Bigg (\int \nolimits _{0}^{s}\frac{\partial ^{2}}{\partial |q|^{2}}\widehat{h}(|q|,\theta _{p})d|q|\Bigg )ds}{p^{2}} \chi _{\{p\in \mathbb{R }^{2}: |p|\le 1\}}\Bigg | \le C \chi _{\{p\in \mathbb{R }^{2}: |p|\le 1\}}\in L^{2}(\mathbb{R }^{2}) \end{aligned}$$

and it remains to estimate the terms

$$\begin{aligned} -\frac{\widehat{h}(0)}{p^{2}}\chi _{\{p\in \mathbb{R }^{2}: |p|\le 1\}}+ \frac{i \int \nolimits _{\mathbb{R }^{2}}|x|h(x)cos(\theta _{p}-\theta _{x})dx}{2\pi |p|} \chi _{\{p\in \mathbb{R }^{2}: |p|\le 1\}}, \end{aligned}$$

which can be written as

$$\begin{aligned} -\frac{\widehat{h}(0)}{p^{2}}\chi _{\{p\in \mathbb{R }^{2}: |p|\le 1\}}+ \frac{i}{2\pi }\frac{\sqrt{R_{1}^{2}+R_{2}^{2}}cos(\theta _{p}-\beta )}{|p|} \chi _{\{p\in \mathbb{R }^{2}: |p|\le 1\}}, \end{aligned}$$

where $R_{k}:=\int _{\mathbb{R }^{2}}x_{k}h(x)dx, \ k=1,2$ and ${tan\beta :=\frac{R_{2}}{R_{1}}}$. Note that the case of $R_{1}=0$ and $R_{2}\ne 0$ corresponds to the situation when the argument ${\beta =\frac{\pi }{2}}$ or ${-\frac{\pi }{2}}$. Evaluation of the square of the $L^{2}$ norm of the sum above yields

$$\begin{aligned} 2\pi |\widehat{h}(0)|^{2}\int \limits _{0}^{1}\frac{d|p|}{|p|^{3}}+ \frac{R_{1}^{2}+R_{2}^{2}}{4\pi ^{2}}\int \limits _{0}^{1}\frac{d|p|}{|p|} \int \limits _{0}^{2\pi }d\theta _{p}cos^{2}(\theta _{p}-\beta ), \end{aligned}$$

which is finite if and only if $\widehat{h}(0)$ along with $R_{1,2}$ vanish. This is equivalent to relations (5.9).

When $\omega =0$ and the equation is studied in $\mathbb{R }^{3}$ we will use the formula

$$\begin{aligned} \widehat{h}(p)=\widehat{h}(0)+\int \limits _{0}^{|p|}\frac{\partial \widehat{h}}{\partial |s|}(|s|, \sigma )d|s|. \end{aligned}$$

(5.13)

Let us investigate the square integrability of the sum

$$\begin{aligned} \frac{\widehat{h}(0)}{p^{2}}\chi _{\{p\in \mathbb{R }^{3}: \ |p|\le 1 \}}+ \frac{\int \nolimits _{0}^{|p|}\frac{\partial \widehat{h}}{\partial |s|}(|s|, \sigma ) d|s|}{p^{2}}\chi _{\{p\in \mathbb{R }^{3}: \ |p|\le 1 \}}. \end{aligned}$$

Using the three dimensional analog of (5.12) we obtain $|\frac{\partial \widehat{h}}{\partial |p|}(p)|\le \frac{1}{(2\pi )^{\frac{3}{2}}}$ $\Vert |x|h(x)\Vert _{L^{1}(\mathbb{R }^{3})}\!<\!\infty$ by the assumption of the lemma. Hence

$$\begin{aligned} \Bigg | \frac{\int \nolimits _{0}^{|p|}\frac{\partial \widehat{h}}{\partial |s|}(|s|, \sigma )d|s|}{p^{2}}\chi _{\{p\in \mathbb{R }^{3}: \ |p|\le 1 \}}\Bigg |\le \frac{C}{|p|}\chi _{\{p\in \mathbb{R }^{3}: \ |p|\le 1 \}}\in L^{2} (\mathbb{R }^{3}). \end{aligned}$$

(5.14)

The square of the $L^{2}$ norm of the remaining term will be given by $4\pi |\widehat{h}(0)|^{2}\int _{0}^{1}\frac{d|p|}{|p|^{2}} <$ $\infty$ if and only if $\widehat{h}(0)=0$, which is equivalent to relation (5.10) in three dimensions. For $\omega =0$ in $\mathbb{R }^{4}$ the argument will be similar to the three dimensional one.

When the parameter $\omega$ vanishes and $d\ge 5$ we will make use of the representation formula analogous to (5.13) and the upper bound similar to (5.14). Thus the square of the $L^{2}$ norm which remains to estimate will be equal to

$$\begin{aligned} \int \limits _{0}^{1}\frac{|\widehat{h}(0)|^{2}}{|p|^{4}}|S^{d}||p|^{d-1}d|p|= |S^{d}||\widehat{h}(0)|^{2}\int \limits _{0}^{1}|p|^{d-5}d|p|<\infty , \end{aligned}$$

which proves that when $\omega =0$ the orthogonality conditions in dimensions five and higher are not needed for solving Eq. (5.2). $\square$

The final proposition of the article is another, even simpler way to look at the solvability conditions for Eq. (5.2) in higher dimensions.

Lemma 7.

Let $\omega =0$ and $h(x)\in L^{1}(\mathbb{R }^{d})\cap L^{2}(\mathbb{R }^{d})$ with $d\ge 5$. Then problem (5.2) admits a unique solution in $H^{2}(\mathbb{R }^{d})$.

 

Proof.

Obviously,

$$\begin{aligned} |\widehat{h}(p)|\le \frac{1}{(2\pi )^{\frac{d}{2}}}\Vert h(x)\Vert _{L^{1}(\mathbb{R }^{d})}< \infty ,\quad \ p\in \mathbb{R }^{d}. \end{aligned}$$

It is sufficient to consider the first term in the right side of the higher dimensional analog of formula (5.7). For it we have the upper bound in the absolute value as

$$\begin{aligned} \frac{C}{p^{2}}\chi _{\{p\in \mathbb{R }^{d}: \ |p|\le 1 \}}\in L^{2}(\mathbb{R }^{d}),\quad \ d\ge 5, \end{aligned}$$

which completes the proof of the lemma. $\square$