1 Introduction

Fractional differential operators are modern tools in the modeling of many phenomena of mechanics, electric circuits and material sciences. However, this notion has several meanings starting from fractional derivatives of Riemann–Liouville, Caputo or other kinds, through pseudo-differential and singular integral operators up to definitions given by the Spectral Theory. Some definitions are equivalent though they seem to be completely different, see [16]. They have rather global nature in a sense that functions being their arguments have to be defined on the whole \({\mathbb {R}}^k.\) If such operators are applied in equations on an open bounded set \(\Omega \subset {\mathbb {R}}^k\) with the null Dirichlet’s boundary condition, then the authors set \(u=0\) outside \(\Omega .\) Operators of this kind are generalizations of the Laplace operator and they show some probabilistic motivation (Lévy processes) of models considered [2, 16, 17, 24].

In this paper, we use operator defined by the Dirichlet Laplacian \(-\Delta :L^2(\Omega )\supset \text {dom}(-\Delta )\rightarrow L^2(\Omega )\) by using the Spectral Theorem and functional calculus given in [8]. When g is a real function defined on the spectrum of \(-\Delta ,\) then \(g(-\Delta )\) is a certain self-adjoint operator on \(L^2(\Omega ).\) This approach includes positive powers \((-\Delta )^s,\) but operators defined in this way are different from the ones considered above. The arguments in [23] show that eigenfunctions of global \((-\Delta )^s\) differ from the eigenfunctions of the Laplacian which are the same for \(g(-\Delta )\) for any function g;  earlier the eigenvalues of these both operators were compared in [6]. Some existence and regularity results for equations involving spectral fractional Laplacians are given in [1, 4, 5, 14, 21, 25]. It would be interesting to compare results of this paper with theorems from the above mentioned articles even for fractional Laplacians. However, [1, 4, 5, 25] consider superlinear nonlinearities and our approach admits only ones with linear or sublinear growth (comp. (3.2)). In [21], properties of the fractional Laplacians with Navier and Dirichlet boundary conditions are studied as linear operators on different Sobolev spaces. The closest to our setting is [14], where the fractional Laplacian is replaced by

$$\begin{aligned} \sum _{i,j} \alpha _i\alpha _j (-\Delta )^{\beta _i+\beta _j}. \end{aligned}$$

The operator acts in \(L^2(\Omega )\) as in our approach and assumptions on parameters give the compactness of the inverse operator. However, the nonlinear term f is potential, since the method used in [14] is variational (here, we use the fixed point and topological degree methods). As in the case of usual Laplace operator, the results obtained by using the variational method need slightly weaker assumptions as obtained by the topological one. We also refer to some applications of fractional Laplacian [3, 18] and fractional p-Laplacian [7, 11, 26, 27].

There is no reason to restrict to power functions \(g(z)=z^s,\) since engineers using fractional operators rely on experimental arguments: numerical results obtained with fractional powers of differential operators are closer to experimental data. It is possible that some deviations from power functions will give even better results. The paper is organised as follows: in the next section we remind some facts concerning the Dirichlet Laplacian and the Spectral Theory. In Sect. 3, the existence of solutions for \(g(-\Delta )u=f(x,u)\) with invertible \(g(-\Delta )\) will be obtained. The proof of these theorems are simple. More interesting is the resonant case, in which this operator has a nontrivial kernel. It happens when g vanishes on some eigenvalue of the Laplacian. The existence theorem is then given with conditions similar to these due to Landesman and Lazer [15] and the proof is based on the coincidence degree technique of Mawhin. The last section is devoted to regularity of solutions to such equations. The results obtained there are not satisfactory in general case when any bounds of the sequence of \(L^2\) and supremum norms of derivatives of every order are not known. In fact, we can use these results for cuboids.

The theorems of this paper are completely analogous to those available for case of usual Dirichlet Laplacians \(g(s)=s\). The existence and uniqueness of a solution obtained by the Schauder Fixed Point Theorem or the Banach Contraction Principle (Theorems 1 and 2) are essentially weaker than the known ones in the case of the fractional Laplacians \(g(s)=s^{\beta }\), however the novelty of this paper is about the replacement of the power function by an arbitrary one. The main interesting resonant case (Theorem 4) cannot be applied to fractional Laplacians since \(s^{\beta }=0\) only for \(s=0\) which is not an eigenvalue of the Dirichlet Laplacian.

The authors are very grateful to the Reviewers for their helpful comments, corrections of misprints and improvements of the text. Special acknowledgements are due to the Reviewer 1 who noticed essential error in the proof of Theorem 5 and proposed a solution.

2 Dirichlet–Laplace operator and its generalizations

Let \(\Omega \) be an open bounded subset of \({\mathbb {R}}^k\) with Lipschitzian boundary and let \(-\Delta :L^2(\Omega )\supset \text {dom}(-\Delta ) \rightarrow L^2(\Omega )\) be the weak Dirichlet–Laplacian (compare [14]), i.e. it is Friedrich’s extension of the classical Dirichlet–Laplace operator to a self-adjoint operator. If \((-\Delta )_{s}: L^2(\Omega )\supset H^1_0(\Omega )\cap H^2(\Omega )\rightarrow L^2(\Omega )\) is the strong Dirichlet–Laplace, then \(H^1_0(\Omega )\cap H^2(\Omega )\subset \text {dom}(-\Delta )\) and \((-\Delta )u=(-\Delta )_s u\) for \(u\in H^1_0(\Omega )\cap H^2(\Omega )\). Moreover, for \(\Omega \) being of class \(C^2\) or \(\Omega \) being cuboid we have \(-\Delta =(-\Delta )_s\).

The operator \(-\Delta \) is self-adjoint and positive, its spectrum \(\sigma (-\Delta )\) consists of positive eigenvalues \(\lambda _n\), \(n\in {\mathbb {N}},\) with finite multiplicities being a nondecreasing sequence tending to infinity. If we put each eigenvalue as many times in the sequence as its multiplicity and by \(e_n\) denote a corresponding normalized in \(L^2\) eigenfunction than the operator can be written as

$$\begin{aligned} -\Delta u =\sum _{n=1}^{\infty } \lambda _n\langle u,e_n\rangle e_n. \end{aligned}$$
(2.1)

The spectral theory [8] provides us, so called, the operational calculus. If \(g:\sigma (-\Delta )\rightarrow {\mathbb {R}}\) is any function then a new operator \(g(-\Delta )\) can be defined by the formula

$$\begin{aligned} g(-\Delta ) u:= \sum _{n=1}^{\infty } g(\lambda _n)\langle u,e_n\rangle e_n \end{aligned}$$
(2.2)

with the domain

$$\begin{aligned} {\text {dom}} g(-\Delta ):=\left\{ u\in L^2:\; \sum _{n=1}^{\infty } g(\lambda _n)^2|\langle u,e_n\rangle |^2<\infty \right\} . \end{aligned}$$

In the special case \(g(s)=s^\beta \) this operator is the fractional Laplacian. Generally, the correspondence \(g\mapsto g(-\Delta )\) has the following algebraic properties:

$$\begin{aligned}&(g_1+g_2)(-\Delta )\supset g_1(-\Delta )+g_2(-\Delta ),\qquad (cg)(-\Delta )=cg(-\Delta ),\\&(g_1\cdot g_2)(-\Delta )\supset g_1(-\Delta ) g_2(-\Delta ), \end{aligned}$$

where \(g_1,g_2\) are real functions on the spectrum, c is a real constant and \(C\supset B\) means that operator C is an extension of operator B.

\(g(-\Delta )\) is a self-adjoint operator for any real function g,  it is positive, if g takes positive values, it is invertible, if \(g(\lambda _n)\ne 0\) for any n. If \(b:=\inf _n |g(\lambda _n)|>0,\) then \(g(-\Delta )^{-1}\) is defined on the whole \(L^2(\Omega ):\)

$$\begin{aligned} g(-\Delta )^{-1}u=\sum _{n=1}^{\infty } \frac{1}{g(\lambda _n)}\langle u,e_n\rangle e_n \end{aligned}$$
(2.3)

and it is bounded. Moreover, if

$$\begin{aligned} \lim _{n\rightarrow \infty } |g(\lambda _n)|=\infty , \end{aligned}$$
(2.4)

then this inverse operator is compact. However, if (2.4) holds but g vanishes for a finite set \(G:=\{ n\in {\mathbb {N}}: g(\lambda _n)=0\},\) then the resolvent

$$\begin{aligned} R(\lambda ,g(-\Delta )):=(g(-\Delta )-\lambda I)^{-1}=\sum _{n=1}^{\infty } \frac{1}{g(\lambda _n)-\lambda }\langle u,e_n\rangle e_n \end{aligned}$$

is meromorphic with the pole 0 an it has the form

$$\begin{aligned} R(\lambda ,g(-\Delta ))=\sum _{n\in G} c_n(\lambda )\langle u,e_n\rangle e_n+R_0(\lambda ), \end{aligned}$$

where \(\lim _{\lambda \rightarrow 0}|c_n(\lambda )|=\infty \) and \(R_0\) is holomorphic and compact in a neighborhood of 0.

We also need some properties of eigenfunctions \(e_n.\) They are of \(C^{\infty }\)-class in \(\Omega \) (even analytic, see [12], Cor. 8.11). It is well-known that the first eigenvalue is simple, hence \(\lambda _1<\lambda _2,\) and the corresponding eigenfunction \(e_1\) is positive in \(\Omega .\)

In the one dimensional case eigenvalues can be found explicitely. Assume for simplicity \(\Omega =(0,\pi )\), \(-\Delta u=-u''\), then \(\lambda _n=n^2\), \(e_n(x)=\sqrt{2/\pi }\sin (nx)\). In multidimensional case, one can find them for cuboids, some other polytopes or for balls. We write down the eigenvalues and the eigenfunctions in the case \(\Omega =(0,\pi )^k:\)

$$\begin{aligned} n:=(n_1,\ldots , n_k),\quad \lambda _n=\sum _j n_j^2, \quad e_n(x_1,\ldots , x_k)=\sqrt{\frac{2^k}{\pi ^k}}\prod _j \sin n_j x_j \end{aligned}$$
(2.5)

(one can find eigenvalues and eigenfunctions for general cuboids, since they can be obtained from \((0,\pi )^k\) by using an appropriate homothety, translation and orthogonal mappings) and in the case of 2-dimensional disk \(\Omega =\{ x\in {\mathbb {R}}^2: \; x_1^2+x_2^2<1\}\), in polar coordinates \((r,\theta ):\)

$$\begin{aligned} \begin{array}{ll} \lambda _{0,m}^2, &{} \qquad e_{0,m} (r,\theta )=\frac{1}{\sqrt{\pi }J'_0(\lambda _{0,m})}J_0(\lambda _{0,m}r),\\ \lambda _{n,m}^2, &{} \qquad e_{n,m,1}(r,\theta ) =\frac{1}{\sqrt{\pi }J'_n(\lambda _{n,m})}J_n(\lambda _{n,m}r)\cos n\theta ,\\ &{} \qquad e_{n,m,2}(r,\theta )= \frac{1}{\sqrt{\pi }J'_n(\lambda _{n,m})} J_n(\lambda _{n,m}r)\sin n\theta , \end{array} \end{aligned}$$
(2.6)

where \(\lambda _{n,m}\) stands for m-th positive zero of the Bessel function \(J_n,\) for \(n=1,2,\ldots \) they are double eigenvalues (comp. [13]).

In a general case \(\Omega \subset {\mathbb {R}}^k\) [13], we only know that if \(\Omega _1\subset \Omega _2,\) then

$$\begin{aligned} \lambda _n(\Omega _2)\le \lambda _n(\Omega _1). \end{aligned}$$

Moreover, there are two positive constants depending on \(\Omega \) such that

$$\begin{aligned} c(\Omega ) n^{\frac{2}{k}}\le \lambda _n(\Omega )\le d(\Omega ) n^{\frac{2}{k}}. \end{aligned}$$
(2.7)

3 Elliptic equations with generalized fractional Laplacian

In this section we are interested in the following equations:

$$\begin{aligned} g(-\Delta ) u= f(x,u), \end{aligned}$$
(3.1)

where \(f:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is measurable with respect to \(x\in \Omega ,\) continuous w.r.t. \(u\in {\mathbb {R}}\) and

$$\begin{aligned} |f(x,u)|\le a(x)+b|u|,\qquad a\in L^2(\Omega ),\; b\ge 0. \end{aligned}$$
(3.2)

The last condition is necessary and suffient for the Nemytskii operator defined by f,  i.e. \(\mathbf {f}(u)(x)=f(x,u(x))\) to act in \(L^2(\Omega )\) as a continuous mapping and map bounded sets into bounded ones.

Theorem 1

If \(g:\sigma (-\Delta )\rightarrow {\mathbb {R}}\) satisfies (2.4) and

$$\begin{aligned} \inf _n|g(\lambda _n)|>b, \end{aligned}$$

where b is a constant from (3.2), then equation (3.1) has a solution.

Proof

It is easily seen that (3.1) is equivalent to the existence of a fixed point of operator \(g(-\Delta )^{-1}\mathbf {f}: L^2(\Omega )\rightarrow L^2(\Omega ).\) Due to (3.2),

$$\begin{aligned} \Vert \mathbf {f}(u)\Vert \le \Vert a\Vert +b\Vert u\Vert , \end{aligned}$$
(3.3)

where \(\Vert \cdot \Vert \) stands for the \(L^2\)-norm. Since \(\Vert g(-\Delta )^{-1}\Vert =\sup _n |g(\lambda _n)|^{-1}< b^{-1}\) by our assumption, hence

$$\begin{aligned} \Vert g(-\Delta )^{-1}\mathbf {f} (u)\Vert \le \Vert g(-\Delta )^{-1}\Vert (\Vert a\Vert +b\Vert u\Vert ) \le \Vert u\Vert \end{aligned}$$

for \(\Vert u\Vert \le R\) with sufficiently large R. The operator \(g(-\Delta )^{-1}\mathbf {f}\) is compact on any bounded set as both mappings of the composition are continuous, \(\mathbf {f}\) maps bounded sets into bounded ones and \(g(-\Delta )^{-1}\) is compact. The existence of a solution to (3.1) is a consequence of the Schauder Fixed Point Theorem for the ball centered at 0 with radius R. \(\square \)

Uniqueness of solutions can be obtained under another assumptions.

Theorem 2

Let f be a Carathéodory function, \(f(\cdot ,0)\in L^2(\Omega )\) and there exists \(L>0\) such that

$$\begin{aligned} |f(x,u)-f(x,v)|\le L|u-v| \end{aligned}$$

for every \(u,v\in {\mathbb {R}}\) and a.e. \(x\in \Omega .\) If

$$\begin{aligned} L<\inf _n |g(\lambda _n)|, \end{aligned}$$

then equation (3.1) has a unique solution.

Proof

Notice that assumptions on f from this theorem imply (3.2) with \(a(\cdot )=f(\cdot ,0)\) and \(b=L.\) Moreover

$$\begin{aligned}&\Vert g(-\Delta )^{-1}\mathbf {f}(u)-g(-\Delta )^{-1}\mathbf {f}(v)\Vert \le \Vert g(-\Delta )^{-1}\Vert \Vert \mathbf {f}(u)-\mathbf {f}(v)\Vert \\&\le (\inf |g(\lambda _n)|)^{-1} L\Vert u-v\Vert \end{aligned}$$

and our assumption gives \(g(-\Delta )^{-1}\mathbf {f}\) is a contraction on \(L^2(\Omega )\). The assertion follows from the Banach Contraction Principle. \(\square \)

The above theorems consider equations with invertible linear part \(g(-\Delta ).\) We can also study resonant equations, when function g vanishes on some eigenvalues. The first result for the Laplace operator was obtained by Landesman and Lazer in 1970 and we generalize it for our operators. The most satisfactory result can be obtained in the case \(g(\lambda _1)=0\), g satisfies condition (2.4) and \(\inf _{n\ge 2}|g(\lambda _n)|>0.\)

Lemma 1

Assume \(f:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a Carathéodory function satisfying (3.2). If there exist limits

$$\begin{aligned} \lim _{u\rightarrow \pm \infty } f(x,u)=:f_{\pm }(x), \end{aligned}$$
(3.4)

uniform with respect to x and \(f_{\pm }\in L^{2}(\Omega ),\) then there is a function \(h\in L^{2}(\Omega )\) such that

$$\begin{aligned} |f(x,u)|\le h(x) \end{aligned}$$

for any (x, u).

Proof

Due to (3.4) there exists \(c>0\) such that

$$\begin{aligned}&|f(x,u)-f_+(x)|\le 1,\quad u>c, \quad {\text {a.e.}} x\in \Omega ,\\&|f(x,u)-f_-(x)|\le 1,\quad u<-c, \quad {\text {a.e.}} x\in \Omega . \end{aligned}$$

Then the assertion holds with

$$\begin{aligned} h(x)=\max \{ |f_+(x)|+1,|f_-(x)|+1,a(x)+bc\} . \end{aligned}$$

\(\square \)

In 1972 Jean Mawhin [19] introduced a topological degree for pairs of maps (L, N),  where L was a linear Fredholm map of index 0, \(L:X\supset {\text {dom}}L\rightarrow Z,\) and \(N:X\rightarrow Z\) which generalized the Leray-Schauder theory for pairs (I, N) with N completely continuous maps. We follow two later monographs of this theory [10, 20]. Let X and Z be Banach spaces, \(L:X\supset Y\rightarrow Z\) be a linear operator such that \(\dim \ker L=\text {codim} \,\text {im} L>0.\) (Such operators are called Fredholm with index 0. More generally, Fredholm operators have finite dimensional kernels, closed images with finite dimensional \(Z/\text {im}L\). The index of the operator is the difference of these dimensions.) Let \(N:X\rightarrow Z\) be continuous (and nonlinear). Let P be a linear projector on \(\ker L\) and Q a linear projector in Z along \({{\,\mathrm{im}\,}}{L}.\) Denote by \(K_P\) the inverse of

$$\begin{aligned} L|(\ker P\cap Y):\ker P\cap Y\rightarrow {{\,\mathrm{im}\,}}{L}. \end{aligned}$$

The operator N is called L-compact if QN and \(K_P(I-Q)N\) are completely continuous (usually \(K_P\) is completely continuous and N maps bounded sets into bounded ones).

Theorem 3

(Mawhin’s Continuation Theorem [10]) Let \(U\subset X\) be open and bounded, L be a Fredholm operator of index 0, N be L-compact operator and J be an arbitrary isomorphism of \({{\,\mathrm{im}\,}}{Q}\) onto \(\ker L.\) Suppose that equations

$$\begin{aligned} Lx=\mu N(x) \end{aligned}$$

have no solutions on \(\partial U \cap Y\) for any \(\mu \in (0,1]\) and the Brouwer degree

$$\begin{aligned} \deg (JQN|\ker L,U\cap \ker L,0)\ne 0. \end{aligned}$$

Then equation \(Lx=N(x)\) has a solution in \(\bar{U}.\) The Brouwer degree in the main assumption is often called coincidence degree of L and N.

Here, \(X=Z=L^2(\Omega )\), \(L=g(-\Delta )\), \(N=\mathbf {f}\), \(\ker L={\text {Lin}}\{ e_1\}\), \({{\,\mathrm{im}\,}}{L}=\ker L^{\bot }\), \(Z/{{\,\mathrm{im}\,}}{L}\) can be identified with \(\ker L\), P, Q are orthogonal projectors, \(J=I\), \(K_P=g(-\Delta )|_{\ker L^{\bot }}^{-1}:{{\,\mathrm{im}\,}}{L}\rightarrow \ker L^{\bot }.\)

Theorem 4

Suppose \(g(\lambda _1)=0\), g satisfies condition (2.4) and \(\inf _{n\ge 2}|g(\lambda _n)|>0\) and f is a Carathéodory function satisfying (3.2) and (3.4). If the following integrals have opposite signs:

$$\begin{aligned} \int _{\Omega } f_+\cdot e_1,\qquad \int _{\Omega } f_-\cdot e_1, \end{aligned}$$
(3.5)

then (3.1) has a solution.

Proof

It is obvious that \(g(-\Delta )\) has index 0 with kernel and cokernel being one-dimensional spaces spanned by \(e_1.\) Using the notation of Mawhin’s Continuation Theorem

$$\begin{aligned} K_P(I-Q)N (u)=\sum _{n\ge 2} \frac{1}{g(\lambda _n)}\langle \mathbf {f} (u),e_n\rangle e_n, \end{aligned}$$

where \(\Vert \mathbf {f} (u)\Vert \le \Vert h\Vert \) for any \(u\in L^2(\Omega )\) by the lemma. This operator is then compact due to

$$\begin{aligned} \frac{1}{g(\lambda _n)}\rightarrow 0,\qquad n\rightarrow \infty . \end{aligned}$$

We shall show that equations \(g(-\Delta )u=\mu \mathbf {f}(u)\) have no solution for \(\mu \in (0,1]\) with the norm sufficiently large. Suppose the converse: there are \(\mu _j\in (0,1]\) and \(u_j\in L^2,\) such that

$$\begin{aligned} g(-\Delta ) u_j=\mu _j \mathbf {f}(u),\qquad \lim _{j\rightarrow \infty } \Vert u_j\Vert = \infty . \end{aligned}$$
(3.6)

Denoting by \(\tilde{u}\) the orthogonal projection of u on \(\ker L^{\bot },\) we can write \(u_j=\tilde{u}_j+d_j e_1.\) Then we have

$$\begin{aligned}&\Vert \tilde{u}_j\Vert ^2=\sum _{n\ge 2} \frac{\mu _j^2}{g(\lambda _n)^2} \langle \mathbf {f}(u_j),e_n\rangle ^2 \le \Vert (g(-\Delta ) |_{e_1^{\bot }}) ^{-1}\mathbf {f}(u_j)\Vert ^2 \\&\le \frac{1}{\inf _{n\ge 2} g(\lambda _n)^2}\Vert h\Vert ^2=:M, \end{aligned}$$

hence the norms of \(\tilde{u}_j\) are uniformly bounded so the sequence \(|d_j|\) tends to the infinity. We shall show that both possibilities: there is a subsequence \(d_j\rightarrow +\infty \) (resp. \(d_j\rightarrow -\infty )\) lead to a contradiction in the same way. Take the first option into account and let \((d_j)\) stands for the corresponding subsequence.

Denote \(A_j:=\{ x\in \Omega : \; |\tilde{u}_j(x)|>\frac{1}{2}d_j e_1(x)\}.\) Since

$$\begin{aligned} \Vert \tilde{u}_j\Vert ^2\ge \int _{A_j} |\tilde{u}_j|^2\ge \frac{d_j^2}{4}\int _{A_j} e_1^2, \end{aligned}$$

the sequence of integrals of \(e_1^2\) over \(A_j\) tends to 0. Take any \(\varepsilon >0.\) Hence

$$\begin{aligned} \mu _k\{x\in A_j:\; e_1(x)>\varepsilon \} \varepsilon ^2\le \int _{A_j} e_1^2\rightarrow 0, \end{aligned}$$

where \(\mu _k\) denotes the k-dimensional Lebesgue measure, thus \(\mu _k(A_j)\rightarrow 0,\) since \(\varepsilon >0\) is arbitrary. Passing to a subsequence, we can assume

$$\begin{aligned} \sum _{j=1}^{\infty }\mu _k(A_j)<\infty . \end{aligned}$$

Due to the Borel-Cantelli Lemma

$$\begin{aligned} \mu _k\left( \bigcap _i\bigcup _{j\ge i} A_j\right) =0 \end{aligned}$$

and

$$\begin{aligned} \Omega \setminus \bigcap _i\bigcup _{j\ge i} A_j=\bigcup _i \bigcap _{j\ge i}(\Omega \setminus A_j)\subset \bigcup _i \bigcap _{j\ge i} \{x:\; \tilde{u}_j(x)+d_j e_1(x)>\frac{1}{2}d_j e_1(x)\}. \end{aligned}$$

On the other hand \(e_1\) is positive on \(\Omega ,\) thus \(\frac{1}{2}d_je_1(x)\rightarrow +\infty \) as \(j\rightarrow \infty \) and, therefore \(\tilde{u}_j(x)+d_je_1(x)\rightarrow +\infty \) a.e. and

$$\begin{aligned} \lim _{j\rightarrow \infty } f(x,\tilde{u}_j(x)+d_je_1(x))=f_+(x),\quad {\text {a.e.}}\; x. \end{aligned}$$

Let us multiply both sides of equation (3.6) by \(e_1\) and integrate it over \(\Omega \) and divide by \(\mu _j.\) The left-hand integral vanishes since the image of \(g(-\Delta )\) is orthogonal to \(e_1. \) The right-hand side tends to \(\int _{\Omega } f_+ e_1\) by Lebesgue’s Dominated Convergence Theorem, so it is nonzero for sufficiently large j. We have a contradiction. The second option gives the same with \(f_-\) instead of \(f_+.\)

Thus, we can take \(R>0\) so large that all solutions of \(g(-\Delta )u=\mu \mathbf {f}(u)\) sit in the open ball B(0, R) in \(L^2(\Omega ).\) We should calculate the Brouwer degree of the mapping

$$\begin{aligned} {\mathbb {R}}\ni d\mapsto \int _{\Omega } \mathbf{f}(de_1)\cdot e_1 \end{aligned}$$

on the interval \((-R, R).\) But these mappings take values with the same signs as (3.5) on both ends of this interval. Therefore this degree is \(+1\) or \(-1\) in the cases

$$\begin{aligned} \int _{\Omega } f_+\cdot e_1>0>\int _{\Omega } f_-\cdot e_1, \qquad \int _{\Omega } f_+\cdot e_1<0< \int _{\Omega } f_-\cdot e_1, \end{aligned}$$

respectively. By Mawhin’s Continuation Theorem we get the assertion. \(\square \)

Similar theorem can be obtained replacing \(\lambda _1\) by any simple eigenvalue \(\lambda _m.\) Since the corresponding eigenfunction \(e_m\) change signs, the main assumption (3.5) should be changed by: the integrals

$$\begin{aligned}&\int _{\{ x\in \Omega : e_m(x)>0\} } f_+ e_m+\int _{\{ x\in \Omega : e_m(x)<0\} } f_- e_m,\\&\int _{\{ x\in \Omega : e_m(x)<0\} } f_+ e_m+\int _{\{ x\in \Omega : e_m(x)>0\} } f_- e_m \end{aligned}$$

have opposite signs.

One can also replace the limits in definitions \(f_{\pm }\) by limes inferior and superior as in [9]. Namely, if

$$\begin{aligned} \int _{\Omega } \liminf _{u\rightarrow +\infty } f(\cdot ,u) e_1>0 >\int _{\Omega } \limsup _{u\rightarrow -\infty } f(\cdot ,u) e_1 \end{aligned}$$

or

$$\begin{aligned} \int _{\Omega } \liminf _{u\rightarrow -\infty } f(\cdot ,u) e_1>0 >\int _{\Omega } \limsup _{u\rightarrow +\infty } f(\cdot ,u) e_1 \end{aligned}$$

and the limit superior and limit inferior exist uniformly with respect to x,  then a solution exists.

4 Some regularity results

There could be obtained two kinds of regularity theorems. Both of them rely on some boundedness properties of the sequence of eigenfunctions \(e_n\) which can be checked in some cases. However, in general, they are difficult to verify. Assume that u is a solution of (3.1) where function g can vanish for a finite number of \(\lambda _n\) and f satisfies (3.2). We consider the question when a solution u belongs to the Sobolev space \(H^p(\Omega ).\) In this section, we assume that all eigenfunctions are sufficiently smooth up to the boundary.

Let \(\nabla ^p u\) denote the \(p+1\)-tensor \(D^{\alpha }u,\) where \(\alpha =(\alpha _1,\ldots .\alpha _k)\in {\mathbb {N}}^k,\) with entries

$$\begin{aligned} \frac{\partial ^{|\alpha |}u}{\partial x_1^{\alpha _1}\ldots \partial x_k^{\alpha _k}},\qquad |\alpha |=\sum _j \alpha _j=p \end{aligned}$$

and

$$\begin{aligned} \Vert \nabla ^p u\Vert ^2:=\int _{\Omega } \sum _{|\alpha |=p} |D^{\alpha } u|^2 \end{aligned}$$

is given by corresponding scalar products. We need an estimate:

$$\begin{aligned} \Vert \nabla ^p e_n \Vert ^2\le C\lambda _n^p, \qquad p\in {\mathbb {N}}, \end{aligned}$$
(4.1)

for a positive constant C depending on \(\Omega \) only. In particular, it means that \(e_n\in H^p(\Omega )\) for \(n\in {\mathbb {N}}.\) The second assumption is the orthogonality of vectors \(\nabla ^p e_n\) and \(\nabla ^p e_m\) for \(n\ne m\) and any p : 

$$\begin{aligned} \langle \nabla ^p e_n ,\nabla ^p e_m\rangle =0, \qquad n\ne m,\quad p\in {\mathbb {N}}. \end{aligned}$$
(4.2)

If \(p=1\), we have only

$$\begin{aligned} \Vert \nabla e_n\Vert ^2 =\lambda _n, \qquad \langle \nabla e_n,\nabla e_m\rangle =0,\;\;{\text {for}}\; n\ne m \end{aligned}$$

obtained with using integration by parts. For \(p>1,\) the question is more complicated.

Lemma 2

If \(\partial \Omega \) is piecewise smooth,

$$\begin{aligned}&\int _{\partial \Omega }\left\langle \frac{\partial }{\partial \nu } \nabla ^{p-1}e_n,\nabla ^{p-1}e_n\right\rangle _e \le 0, \\&\int _{\partial \Omega }\left\langle \frac{\partial }{\partial \nu } \nabla ^{p-1}e_n,\nabla ^{p-1}e_m\right\rangle _e = 0 \end{aligned}$$

for any \(p>1\), \(n\in {\mathbb {N}}\) and \(m\ne n,\) then (4.1) holds with \(C=1\) and (4.2) is satisfied too, where \(\langle \cdot ,\cdot \rangle _e\) stands for the scalar product in Euclidean spaces.

Proof

We can integrate by parts:

$$\begin{aligned}&\int _{\Omega }\langle \nabla ^p e_n,\nabla ^p e_m\rangle _e =\sum _{i_1,\ldots ,i_p}\int _{\Omega }\partial _{x_{i_1},\ldots ,x_{i_p}} e_n\cdot \partial _{x_{i_1},\ldots ,x_{i_p}} e_m \\&\quad =\sum _{i_1,\ldots ,i_{p-1}}\left( \int _{\Omega } \partial _{i_1,\ldots ,i_{p-1}}(-\Delta e_n) \cdot \partial _{i_1,\ldots ,i_{p-1}}(e_m)+\int _{\partial \Omega } \frac{\partial }{\partial \nu } \partial _{i_1,\ldots ,i_{p-1}}(e_n) \cdot \partial _{i_1,\ldots ,i_{p-1}}(e_m)\right) . \end{aligned}$$

The first summands equals 0 for \(m\ne n\) or equals \(\lambda _n \Vert \nabla ^{p-1}e_n\Vert \) for \(m=n\) by induction and the second ones are given by the assumptions of this lemma. \(\square \)

It is known that if domain is \(C^{p,\alpha }\), then \(e_n\in C^{p,\alpha }(\overline{\Omega })\) (compare Theorem 6.19 in [12]), hence \(\Vert \nabla ^{p}e_n\Vert \) is finite. In general, \(\Vert \nabla ^p e_n\Vert \) need not to be finite if \(p\ge 2\). We will not focus on the question of the finiteness of this norm, since (4.1), (4.2) require much more than this problem. We have written eigenvalues and eigenfunctions for normalized cuboids (2.5). The eigenfunctions are from \(C^{\infty }(\overline{\Omega }).\) One can easily check that (4.1) holds for cuboids with \(C=1\) (both sides are then equal). Simple calculations show that (4.2) is also true for these normalized eigenfunctions. General cuboids are images of the normalized one by an appropriate affine mapping being a composition of an orthogonal map, a homothety and a translation. These maps change eigenvalues multiplying them by \(c^k,\) where c is the ratio of the homothety. The eigenfunctions are the composition of the normalized one with the above affine map. Thus, \(e_n\in C^{\infty }(\overline{\Omega })\), \(n\in {\mathbb {N}},\) and relations (4.1), (4.2) hold for a general cuboid.

We know also the explicite form of eigenfunctions for a disk. However, in this case the problem is with the orthogonality of higher derivatives of different eigenfunctions and we are not able to check the needed conditions.

Finally, we want to emphasize that, to the best of our knowledge, conditions (4.1) and (4.2) do not follow from known properties of eigenfunctions for general domains. Only the explicit form of eigenfunctions can be used to obtain estimates (4.1) and (4.2).

Theorem 5

If u is a solution of (3.1), (4.1), (4.2) are satisfied, g vanishes on \(\lambda _n\) with \(n\in G\), \({{\,\mathrm{card}\,}}(G) <\infty ,\) and sequence

$$\begin{aligned} \frac{\lambda _n^p}{|g(\lambda _n)|^2},\qquad n\notin G, \end{aligned}$$
(4.3)

is bounded, then \(u\in H^p(\Omega ).\)

Proof

Let \(\tilde{u}\) denotes the orthogonal projection of u on \(\ker g(-\Delta )^{\bot }\) and \(u=\tilde{u}+\sum _{n\in G} d_n e_n.\) Since eigenfunctions \(e_n\), \(n\notin G,\) belong to \(H^p(\Omega ),\) we should only check if \(\tilde{u}\in H^p(\Omega ).\) But

$$\begin{aligned} \tilde{u}=\sum _{n\notin G} \frac{1}{g(\lambda _n)} \langle \mathbf {f}(u), e_n\rangle e_n \end{aligned}$$

where all entries of the series sit in \(H^p(\Omega ).\) By (4.2),

$$\begin{aligned} \Vert \nabla ^m\tilde{u}\Vert ^2=\sum _{n\notin G} \frac{1}{|g(\lambda _n)|^2}| \langle \mathbf {f}(u) ,e_n\rangle |^2 \Vert \nabla ^m e_n\Vert ^2 \end{aligned}$$

for \(m\le p\), we have from (4.1) and (4.3) that, for \(m\le p\),

$$\begin{aligned} \Vert \nabla ^m\tilde{u}\Vert ^2 = \sum _{n\notin G} \frac{\Vert \nabla ^m e_n\Vert ^2}{|g(\lambda _n)|^2}|\langle \mathbf {f}(u), e_n\rangle |^2\le C \Vert \mathbf {f}(u)\Vert ^2, \end{aligned}$$

thus we are done. \(\square \)

Remark

If function g is nondecreasing then one can use estimates

$$\begin{aligned} c_1 n^{2/k}\le \lambda _n \le c_2 n^{2/k} \end{aligned}$$

since \(\Omega \) contains a cube and it is a subset of another cube. Thus the assumptions guaranteeing regularity of both kinds do not require an explicit knowledge of eigenvalues but estimates on the growth of norms \(\nabla ^p e_n\) are still needed. (4.3) is then equivalent to the boundedness of the sequence

$$\begin{aligned} \frac{n^{p/k}}{|g(n^{2/k})|},\qquad n\notin G. \end{aligned}$$

Corollary 1

Assume (4.1) and (4.2). For the fractional Laplacian \((-\Delta )^{\beta },\) condition \(p\le 2\beta +k,\) guarantees \(u\in H^p(\Omega ).\)

The second kind of solution regularity is \(u\in C^p(\Omega ).\) It needs some information about the sequence of supremum norms of \(\nabla ^p e_n\), \(p=0,1,2,\ldots :\)

$$\begin{aligned} \Vert \nabla ^p e_n\Vert _{\infty }:=\max _{|\alpha |\le p}\sup _{x\in \Omega } |D^{\alpha }e_n (x)|. \end{aligned}$$

These numbers can be calculated for the case of cuboids and disks directly and the following result can be applied for such simple domains:

Theorem 6

If u is a solution of (3.1), where g vanishes only on \(\lambda _n\) with n from a finite set G and f is a Carathéodory function satisfying (3.2), then \(u\in C^p(\Omega )\) provided there exists \(C>0\) such that

$$\begin{aligned} \Vert \nabla ^p e_n\Vert _{\infty }\le C \lambda _n^{p/2} \end{aligned}$$
(4.4)

and the series

$$\begin{aligned} \sum _{n\notin G} \frac{\lambda _n^{p/2}}{|g(\lambda _n)|} <\infty . \end{aligned}$$

The proof is obvious since all partial derivatives of \(\tilde{u}\) up to the order p are sums of uniformly convergent series of continuous functions – multiples of \(D^{\alpha } e_n.\) The crucial assumption (4.4) holds for cubes in any dimension.

Corollary 2

If \(g(s)=s^\beta ,\) then \(u\in C^p\) for \(p<2\beta -k\).

5 Examples

Let \(\Omega :=(0,\pi )\subset {\mathbb {R}}\), \(-\Delta u=-u'',\) then \(\lambda _n=n^2\), \(n\in {\mathbb {N}}.\) Consider a fractional power: \(g(s):=s^{\beta },\) where \(\beta >0.\) Then \(\inf |g(\lambda _n)|=1\) and we can use Theorem 1 for the following nonlinearities:

  • \(f(x,u)=h_1(x)|u|^{\alpha }+h_2(x),\) where \(\alpha \in (0,1)\), \(h_1\in L^{\infty }\), \(h_2\in L^2\);

  • \(f(x,u)=h_1(x)u+h_2(x),\) where \(h_1\in L^{\infty }\), \(\Vert h_1\Vert _{\infty }<1\), \(h_2\in L^2\);

  • \(f(x,u)=h_1(x) u\arctan u +h_2(x),\) where \(h_1\in L^{\infty }\), \(\Vert h_1\Vert _{\infty }<2/\pi \), \(h_2\in L^2\).

The function f given by the last two formulas satisfy assumptions of Theorem 2, thus we have exactly one solution in these cases.

Now, we consider the same problem with \(g(s)=s\) and \(f(x,u)=u,\) then all assumptions of Theorem 1 will be satisfied except \(b<\inf _n g(\lambda _n)\) – here we have the equality instead the sharp inequality. The same statement is true for Theorem 2 and the assumption \(L<\inf _n g(\lambda _n).\) However the problem has an obvious solution \(u(x)=\sin x.\) If we change the nonlinear term and put \(f(x,u)=u-1,\) then the assumptions of both Theorem 1 and 2 are satisfied except the sharp inequalities but the problem has no solution.

Theorem 4 can be applied to functions \(g(s)=s^{\beta }-1,\) where \(\beta >0.\) Then for

$$\begin{aligned} f(x,u)=h_1(x) \arctan u+h_2(x), \end{aligned}$$

where \(h_1\in L^{\infty }(0,\pi )\), \(h_2\in L^2(0,\pi ),\) condition (3.5) has the following form:

$$\begin{aligned} \int _0^{\pi } \left( h_2(x)+\frac{\pi }{2} h_1(x)\right) \sin x\, dx>0>\int _0^{\pi } \left( h_2(x)-\frac{\pi }{2} h_1(x)\right) \sin x\, dx. \end{aligned}$$

Another example show that the existence of solutions for a resonant case can be obtained also for unbounded nonlinearities f with infinite limits \(f_{\pm }\) as \(u\rightarrow \pm \infty \) but satisfying (3.5). For usual Laplacian, such a generalization of Theorem 4 is known. Take \(\Omega =(0,\pi )\), \(g(s)=\sqrt{s} -1\), \(f(x,u)=u+\sqrt{2/\pi }\sin x.\) In this case \(f_{\pm }(x)=\pm \infty \) and (3.5) holds with the integrals \(\pm \infty .\) Then \(u=-e_1\) is a solution.