Asymmetric Robin problems with indefinite potential and concave terms

TL;DR

This paper investigates a complex Robin boundary value problem involving an indefinite potential and asymmetric nonlinearities, establishing multiple solutions using advanced variational and topological methods.

Contribution

It introduces new multiplicity results for Robin problems with indefinite potentials and asymmetric nonlinearities, employing variational, truncation, perturbation, and Morse theory techniques.

Findings

Proved existence of four solutions for small positive parameters.

Established five solutions under certain conditions.

Demonstrated the effectiveness of variational and Morse theory methods.

Abstract

We consider a parametric semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential. In the reaction, we have the competing effects of a concave term appearing with a negative sign and of an asymmetric asymptotically linear term which is resonant in the negative direction. Using variational methods together with truncation and perturbation techniques and Morse theory (critical groups) we prove two multiplicity theorems producing four and five respectively nontrivial smooth solutions when the parameter $\lambda>0$ is small.

Equations

$$P_{\lambda }$$

$$\displaystyle\begin{array}[]{c}``\mbox{Every sequence}\ \{u_{n}\}_{n\geqslant 1}\subseteq X\ \mbox{such that}\\ \{\varphi(u_{n})\}_{n\geqslant 1}\subseteq\mathbb{R}\ \mbox{is bounded and}\ (1+||u_{n}||)\varphi^{\prime}(u_{n})\rightarrow 0\ \mbox{in}\ X^{*}\ \mbox{as}\ n\rightarrow\infty,\\ \mbox{admits a strongly convergent subsequence}."\end{array}$$

$$\max \{\phi (u_0),\phi (u_1)\}<\inf [\phi (u):||u-u_0||=r]=m_r$$

$$‘‘u_n\stackrel{w}{\to }u\text{mbox}inX,||u_n||\to ||u||\Rightarrow u_n\to u\text{mbox}inX."$$

$$H^1(\Omega ),C^1(\overline{\Omega }),L^r(\partial \Omega )(1⩽r⩽\infty ).$$

$$(u,h)={\int }_{\Omega }(Du,Dh{)}_{{\mathbb{R}}^N}dz+{\int }_{\Omega }uhdz\text{mbox}forallu,h\in H^1(\Omega ).$$

$$||u||={\left[||u|{|}_2^2+||Du|{|}_2^2\right]}^{1/2}\text{mbox}forallu\in H^1(\Omega ).$$

$$C_{+}=\{u\in C^1(\overline{\Omega }):u(z)⩾0\text{mbox}forallz\in \overline{\Omega }\}.$$

$$D_{+}=\{u\in C_{+}:u(z)>0\text{mbox}forallz\in \overline{\Omega }\}\subseteq \mathrm{int}{C}_{+}.$$

$${\gamma }_0(u)=u{|}_{\partial \Omega }\text{mbox}forallu\in H^1(\Omega )\cap C^1(\overline{\Omega }).$$

$$\mathrm{im}{\gamma }_0=H^{\frac{1}{2},2}(\partial \Omega )\text{mbox}and\ker {\gamma }_0=H_0^1(\Omega ).$$

$$\left\{\begin{array}[]{l}-\Delta u(z)+\xi(z)u(z)=\hat{\lambda}u(z)\ \mbox{in}\ \Omega,\\ \frac{\partial u}{\partial n}+\beta(z)(u)=0\ \mbox{on}\ \partial\Omega.\end{array}\right\}$$

$$\gamma (u)=||Du|{|}_2^2+{\int }_{\Omega }\xi (z)u^{2}dz+{\int }_{\partial \Omega }\beta (z)u^{2}d\sigma \text{mbox}forallu\in H^1(\Omega ).$$

$$\widehat{{\lambda }_1}=\inf \left\{\frac{\gamma (u)}{||u|{|}_2^2}:u\in H^1(\Omega ),u\ne 0\right\}.$$

$$\gamma (u)+\mu ||u|{|}_2^2⩾c_0||u|{|}^2\text{mbox}forsome{c}_0>0,\text{mbox}forallu\in H^1(\Omega ).$$

$$E({\hat{\lambda }}_k)\subseteq C^1(\overline{\Omega })\text{mbox}forallk\in \mathbb{N}.$$

$$H^1(\Omega )={\overline{H}}_m\oplus {\hat{H}}_m.$$

$${\hat{\lambda }}_m=$$

$$=$$

$$c_1||u|{|}^2⩽\gamma (u)-{\int }_{\Omega }\vartheta (z)u^{2}dz\text{mbox}forallu\in {\hat{H}}_{m-1}.$$

$$\gamma (u)-{\int }_{\Omega }\vartheta (z)u^{2}dz⩽c_2||u|{|}^2\text{mbox}forallu\in {\overline{H}}_m.$$

$$\left\{\begin{array}[]{ll}-\Delta u(z)+\xi(z)u(z)=\tilde{\lambda}m(z)u(z)\ \mbox{in}\ \Omega,\\ \frac{\partial u}{\partial n}+\beta(z)u=0\ \mbox{on}\ \partial\Omega.\end{array}\right\}$$

$$|f_0(z,x)⩽a_0(z)[1+|x{|}^{r-1}]|\text{mbox}foralmostallx\in \mathbb{R},$$

$${\phi }_0(u)=\frac{1}{2}\gamma (u)-{\int }_{\Omega }{F}_0(z,u)dz\text{mbox}forallu\in H^1(\Omega ).$$

$${\phi }_0(u_0)⩽{\phi }_0(u_0+h)\text{mbox}forallh\in C^1(\overline{\Omega }),||h|{|}_{C^1(\overline{\Omega })}⩽{\rho }_1.$$

$${\phi }_0(u_0)⩽{\phi }_0(u_0+h)\text{mbox}forallh\in C^1(\overline{\Omega }),||h||⩽{\rho }_2.$$

$$\displaystyle\begin{array}[]{ll}\varphi^{c}=\{u\in X:\varphi(u)\leqslant c\},\\ K_{\varphi}=\{u\in X:\varphi^{\prime}(u)=0\},\\ K^{c}_{\varphi}=\{u\in K_{\varphi}:\varphi(u)=c\}.\end{array}$$

$$C_k(\phi ,u)=H_k({\phi }^c\cap U,{\phi }^c\cap U\backslash \{u\})\text{mbox}forallk\in {\mathbb{N}}_0,$$

$$C_k(\phi ,u)={\delta }_{k,0}\mathbb{Z}\text{mbox}forallk\in {\mathbb{N}}_0.$$

$$\displaystyle\delta_{k,m}=\left\{\begin{array}[]{ll}1&\mbox{if}\ k=m,\\ 0&\mbox{if}\ k\neq m.\end{array}\right.$$

$$u^{\pm }\in W^{1,p}(\Omega ),u=u^{+}-u^{-},|u|=u^{+}+u^{-}.$$

Full text

aa 2010 AMS Subject Classification: 35J20, 35J60

1. Introduction

Let $\displaystyle\Omega\subseteq\mathbb{R}^{N}$ be a bounded domain with a $\displaystyle C^{2}$-boundary $\displaystyle\partial\Omega$. In this paper we study the following parametric Robin problem:

[TABLE]

In this problem, the potential function $\displaystyle\xi\in L^{s}(\Omega)$ ($\displaystyle s>N$) is indefinite (that is, sign changing). In the reaction (right-hand side), the function $\displaystyle f(z,x)$ is Carathéodory (that is, for all $\displaystyle x\in\mathbb{R}$ the function $\displaystyle z\mapsto f(z,x)$ is measurable and for almost all $\displaystyle z\in\Omega$ the function $\displaystyle x\mapsto f(z,x)$ is continuous) and $\displaystyle f(z,\cdot)$ has linear growth near $\displaystyle\pm\infty$. However, the asymptotic behaviour of $\displaystyle f(z,\cdot)$ as $\displaystyle x\rightarrow\pm\infty$ is asymmetric. More precisely, we assume that the quotient $\displaystyle\frac{f(z,x)}{x}$ as $\displaystyle x\rightarrow+\infty$ stays the principal eigenvalue $\displaystyle\hat{\lambda}_{1}$ of the differential operator $\displaystyle u\mapsto-\Delta u+\xi(z)u$ with Robin boundary condition, while as $\displaystyle x\rightarrow-\infty$ the quotient $\displaystyle\frac{f(z,x)}{x}$ stays below $\displaystyle\hat{\lambda}_{1}$ with possible interaction (resonance) with respect to $\displaystyle\hat{\lambda}_{1}$ from the left. So, $\displaystyle f(z,\cdot)$ is a crossing (jumping) nonlinearity. In the term $\displaystyle-\lambda|u|^{q-2}u$, $\displaystyle\lambda>0$ is a parameter and $\displaystyle 1<q<2$. Hence this term is a concave nonlinearity. Therefore in the reaction we have the competing effects of resonant and concave terms. However, note that in our problem the concave nonlinearity enters with a negative sign. Such problems were considered by Perera [12], de Paiva & Massa [6], de Paiva & Presoto [7] for Dirichlet problems with zero potential (that is, $\displaystyle\xi\equiv 0$). Of the aforementioned works, only de Paiva & Presoto [7] have an asymmetric reaction of special form, which is superlinear in the positive direction and linear and nonresonant in the negative direction. Recently, problems with asymmetric reaction were studied by D’Agui, Marano & Papageorgiou [2] (Robin problems), Papageorgiou & Rădulescu [8, 11] (Neumann and Robin problems) and Recova & Rumbos [14] (Dirichlet problems).

We prove two multiplicity results in which we show that for all small $\displaystyle\lambda>0$ the problem has four and five nontrivial smooth solutions, respectively. Our approach uses variational tools based in the critical point theory, together with suitable truncation, perturbation and comparison techniques and Morse theory (critical groups).

2. Mathematical background and hypotheses

Let $\displaystyle X$ be a Banach space. We denote by $\displaystyle X^{*}$ the topological dual of $\displaystyle X$ and by $\displaystyle\langle\cdot,\cdot\rangle$ the duality brackets for the pair $\displaystyle(X^{*},X)$. Given $\displaystyle\varphi\in C^{1}(X,\mathbb{R})$, we say that $\displaystyle\varphi$ satisfies the “Cerami condition” (the “C-condition” for short) if the following property holds:

[TABLE]

This compactness-type condition on $\displaystyle\varphi(\cdot)$, is crucial in deriving the minimax theory of the critical values of $\displaystyle\varphi$. One of the main results in that theory is the so-called “mountain pass theorem”, which we recall below.

Theorem 1**.**

Assume that $\displaystyle\varphi\in C^{1}(X,\mathbb{R})$ satisfies the $\displaystyle C$-condition, $\displaystyle u_{0},u_{1}\in X$, $\displaystyle||u_{1}-u_{0}||>r$,

[TABLE]

and $\displaystyle c=\inf_{\gamma\in\Gamma}\max_{0\leqslant t\leqslant 1}\varphi(\gamma(t))$ with $\displaystyle\Gamma=\{\gamma\in C([0,1],X):\gamma(0)=u_{0},\gamma(1)=u_{1}\}$. Then $\displaystyle c\geqslant m_{r}$ and $\displaystyle c$ is a critical value of $\displaystyle\varphi$ (that is, there exists $\displaystyle u\in X$ such that $\displaystyle\varphi(u)=c,\varphi^{\prime}(u)=0$).

Recall that a Banach space $\displaystyle X$ has the “Kadec-Klee property”, if the following holds:

[TABLE]

It is an easy consequence of the parallelogram law, that every Hilbert space has the Kadec-Klee property (see Gasinski & Papageorgiou [3]).

In the study of problem ($\displaystyle P_{\lambda}$), we will use the following three spaces:

[TABLE]

The Sobolev space $\displaystyle H^{1}(\Omega)$ is a Hilbert space with inner product given by

[TABLE]

We denote by $\displaystyle||\cdot||$ the corresponding norm on $\displaystyle H^{1}(\Omega)$. So, we have

[TABLE]

The space $\displaystyle C^{1}(\overline{\Omega})$ is an ordered Banach space with positive (order) cone

[TABLE]

This cone has a nonempty interior. Note that

[TABLE]

In fact, $\displaystyle D_{+}$ is the interior of $\displaystyle C_{+}$ when the latter is furnished with the relative $\displaystyle C(\overline{\Omega})$-norm topology.

On $\displaystyle\partial\Omega$ we consider the $\displaystyle(N-1)$-dimensional Hausdorff (surface) measure $\displaystyle\sigma(\cdot)$. Using this measure on $\displaystyle\partial\Omega$, we can define in the usual way the “boundary” Lebesgue spaces $\displaystyle L^{r}(\partial\Omega)$ (for $\displaystyle 1\leqslant r\leqslant\infty$). From the theory of Sobolev spaces, we know that there exists a unique continuous linear map $\displaystyle\gamma_{0}:H^{1}(\Omega)\rightarrow L^{2}(\partial\Omega)$ known as the “trace map” such that

[TABLE]

So, the trace map assigns “boundary values” to every Sobolev function. The trace map is compact into $\displaystyle L^{p}(\partial\Omega)$ for all $\displaystyle 1\leqslant p<\frac{2(N-1)}{N-2}$ if $\displaystyle N\geqslant 3$ and into $\displaystyle L^{p}(\partial\Omega)$ for all $\displaystyle 1\leqslant p\leqslant\infty$ if $\displaystyle N=1,2$. Also we have

[TABLE]

In what follows, for the sake of notational simplicity, we drop the use of the trace map $\displaystyle\gamma_{0}$. All restrictions of Sobolev functions on $\displaystyle\partial\Omega$ are understood in the sense of traces.

Next, we consider the following linear eigenvalue problem:

[TABLE]

This problem was studied by D’Agui, Marano & Papageorgiou [2]. We impose the following conditions on the potential function $\displaystyle\xi(\cdot)$ and on the boundary coefficient $\displaystyle\beta(\cdot)$.

$\displaystyle H(\xi):$ $\displaystyle\xi\in L^{s}(\Omega)$ with $\displaystyle s>N$.

Remark 1**.**

The potential function is both unbounded and sign-changing.

$\displaystyle H(\beta):$ $\displaystyle\beta\in W^{1,\infty}(\partial\Omega)$ and $\displaystyle\beta(z)\geqslant 0$ for all $\displaystyle z\in\partial\Omega$.

Remark 2**.**

If $\displaystyle\beta\equiv 0$, then we recover the Neumann problem.

Let $\displaystyle\gamma:H^{1}(\Omega)\rightarrow\mathbb{R}$ be the $\displaystyle C^{2}$ functional defined by

[TABLE]

Problem (1) admits a smallest eigenvalue $\displaystyle\hat{\lambda}_{1}\in\mathbb{R}$ given by

[TABLE]

Moreover, there exists $\displaystyle\mu>0$ such that

[TABLE]

Using (3) and the special theorem for compact self-adjoint operators on Hilbert spaces, we produce the full spectrum of (2). This consists of a sequence $\displaystyle\{\hat{\lambda}_{k}\}_{k\in\mathbb{N}}$ of distinct eigenvalues such that $\displaystyle\hat{\lambda}_{k}\rightarrow+\infty$. Let $\displaystyle E(\hat{\lambda}_{k})$ denote the eigenspace corresponding to the eigenvalue $\displaystyle\hat{\lambda}_{k}$. From the regularity theory of Wang [15], we have

[TABLE]

Each eigenspace has the “Unique Continuation Property” (UCP for short). This means that if $\displaystyle u\in E(\hat{\lambda}_{k})$ vanishes on a set of positive Lebesgue measure, then $\displaystyle u\equiv 0$.

Let $\displaystyle\overline{H}_{m}=\bigoplus^{m}_{k=1}E(\hat{\lambda}_{k})$ and $\displaystyle\hat{H}_{m}=\overline{H}^{\perp}_{m}=\overline{\bigoplus_{k\geqslant m+1}E(\hat{\lambda}_{k})}$. We have

[TABLE]

Moreover, for every $\displaystyle m\geqslant 2$, we have variational characterizations for the eigenvalues for the eigenvalues $\displaystyle\hat{\lambda}_{m}$ analogus to that for $\displaystyle\hat{\lambda}_{1}$ (see (2)):

[TABLE]

In (2) the infimum is realized on $\displaystyle E(\hat{\lambda}_{1})$, while in (4) both the infimum and the supremum are realized on $\displaystyle E(\hat{\lambda}_{m})$. We know that $\displaystyle\dim E(\hat{\lambda}_{1})=1$ (that is, the first eigenvalue $\displaystyle\hat{\lambda}_{1}$ is simple). Hence the elements of $\displaystyle E(\hat{\lambda}_{1})$ have constant sign. We denote by $\displaystyle\hat{u}_{1}\in C_{+}\backslash\{0\}$ the positive $\displaystyle L^{2}$-normalized eigenfunction (that is, $\displaystyle||\hat{u}_{1}||_{2}=1$) corresponding to $\displaystyle\hat{\lambda}_{1}$. From the strong maximum principle we have $\displaystyle\hat{u}_{1}(z)>0$ for all $\displaystyle z\in\Omega$ and if $\displaystyle\xi^{+}\in L^{\infty}(\Omega)$ (that is, the potential function is bounded above), then by the Hopf boundary point theorem we have $\displaystyle\hat{u}_{1}\in D_{+}$ (see Pucci & Serrin [13, p. 120]).

Using (2), (4) and the above properties, we have the following useful inequalities.

Proposition 2**.**

• (a)

If $\displaystyle\vartheta\in L^{\infty}(\Omega),\ \vartheta(z)\leqslant\hat{\lambda}_{m}$ for almost all $\displaystyle z\in\Omega$, $\displaystyle\vartheta\not\equiv\hat{\lambda}_{m}$ $\displaystyle(m\in\mathbb{N})$, then there exists $\displaystyle c_{1}>0$ such that

[TABLE]

• (b)

If $\displaystyle\vartheta\in L^{\infty}(\Omega),\ \vartheta(z)\geqslant\hat{\lambda}_{m}$ for almost all $\displaystyle z\in\Omega$, $\displaystyle\vartheta\not\equiv\hat{\lambda}_{m}$ $\displaystyle(m\in\mathbb{N})$, then there exists $\displaystyle c_{2}>0$ such that

[TABLE]

Note that if $\displaystyle\xi\equiv 0,\ \beta\equiv 0$, then $\displaystyle\hat{\lambda}_{1}=0$, while if $\displaystyle\xi\geqslant 0$ and either $\displaystyle\xi\equiv 0$ or $\displaystyle\beta\equiv 0$, then $\displaystyle\hat{\lambda}_{1}>0$. Also, the elements of $\displaystyle E(\hat{\lambda}_{k})$ for $\displaystyle k\geqslant 2$ are nodal (that is, sign-changing).

In addition to the eigenvalue problem (1), we can consider a weighted version of it. So, let $\displaystyle m\in L^{\infty}(\Omega),\ m(z)\geqslant 0$ for almost all $\displaystyle z\in\Omega,\ m\not\equiv 0$ and consider the following linear eigenvalue problem

[TABLE]

This eigenvalue problem exhibits the same properties as (1). So, the spectrum consists of a sequence $\displaystyle\{\tilde{\lambda}_{k}(m)\}_{k\in\mathbb{N}}$ of distinct eigenvalues such that $\displaystyle\tilde{\lambda}_{k}(m)\rightarrow+\infty$ as $\displaystyle k\rightarrow+\infty$. As for (1), the first eigenvalue $\displaystyle\tilde{\lambda}_{1}(m)$ is simple and the elements of $\displaystyle E(\tilde{\lambda}_{1}(m))\subseteq C^{1}(\overline{\Omega})$ have fixed sign, while the elements of $\displaystyle E(\tilde{\lambda}_{k}(m))\subseteq C^{1}(\overline{\Omega})$ (for all $\displaystyle k\geqslant 2$) are nodal. We have variational characterisations for all the eigenvalues as in (2) and (4) only now the Rayleigh quotient is $\displaystyle\frac{\gamma(u)}{\int_{\Omega}m(z)u^{2}dz}$. Moreover, the eigenspaces have the UCP property. These properties lead to the following monotonicity property for the map $\displaystyle m\mapsto\tilde{\lambda}_{k}(m)$, $\displaystyle k\in\mathbb{N}$.

Proposition 3**.**

If $\displaystyle m_{1},\,m_{2}\in L^{\infty}(\Omega)$, $\displaystyle 0\leqslant m_{1}(z)\leqslant m_{2}(z)$ for almost all $\displaystyle z\in\Omega$, $\displaystyle m_{1}\not\equiv 0$, $\displaystyle m_{2}\not\equiv m_{1}$, then $\displaystyle\tilde{\lambda}_{k}(m_{2})<\tilde{\lambda}_{k}(m_{1})$ for all $\displaystyle k\in\mathbb{N}$.

Let $\displaystyle f_{0}:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ be a Carathéodory function such that

[TABLE]

with $\displaystyle a_{0}\in L^{\infty}(\Omega)$ and \displaystyle 1<r\leqslant 2^{*}=\left\{\begin{array}[]{ll}\frac{2N}{N-2}&\mbox{if}\ N\geqslant 3\\ +\infty&\mbox{if}\ N=1,2\end{array}\right. (the critical Sobolev exponent). Let $\displaystyle F_{0}(z,x)=\int^{x}_{0}f_{0}(z,s)ds$ and consider the $\displaystyle C^{1}$-functional $\displaystyle\varphi_{0}:H^{1}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

As in Papageorgiou & Rădulescu [10, Proposition 8], using the regularity theory of Wang [15], we have the following result.

Proposition 4**.**

Assume that $\displaystyle u_{0}\in H^{1}(\Omega)$ is a local $\displaystyle C^{1}(\overline{\Omega})$-minimizer of $\displaystyle\varphi_{0}(\cdot)$, that is, there exists $\displaystyle\rho_{1}>0$ such that

[TABLE]

Then $\displaystyle u_{0}\in C^{1,\alpha}(\overline{\Omega})$ with $\displaystyle 0<\alpha<1$ and $\displaystyle u_{0}$ is also a local $\displaystyle H^{1}(\Omega)$-minimizer of $\displaystyle\varphi_{0}$, that is, there exists $\displaystyle\rho_{2}>0$ such that

[TABLE]

Now we will recall some definitions and facts from Morse theory (critical groups). So, let $\displaystyle X$ be a Banach space, $\displaystyle\varphi\in C^{1}(X,\mathbb{R})$ and $\displaystyle c\in\mathbb{R}$. We introduce the following sets:

[TABLE]

Given a topological pair $\displaystyle(Y_{1},Y_{2})$ such that $\displaystyle Y_{2}\subseteq Y_{1}\subseteq X$, for every $\displaystyle k\in\mathbb{N}_{0}$, we denote by $\displaystyle H_{k}(Y_{1},Y_{2})$ the $\displaystyle k$th-relative singular homology group for the pair $\displaystyle(Y_{1},Y_{2})$ with integer coefficients. Suppose that $\displaystyle u\in K^{c}_{\varphi}$ is isolated. The critical groups of $\displaystyle\varphi$ at $\displaystyle u$ are defined by

[TABLE]

with $\displaystyle U$ being a neighbourhood of $\displaystyle u$ such that $\displaystyle K_{\varphi}\cap\varphi^{c}\cap U=\{u\}$. The excision property of singular homology, implies that the above definition of critical groups is independent of the isolating neighbourhood $\displaystyle U$. If $\displaystyle u$ is a local minimizer of $\displaystyle\varphi$, then

[TABLE]

Here, $\displaystyle\delta_{k,m}$ denotes the Kronecker symbol defined by

[TABLE]

Next, let us fix our notation. If $\displaystyle x\in\mathbb{R}$, we set $\displaystyle x^{\pm}=\max\{\pm x,0\}$. Then for $\displaystyle u\in W^{1,p}(\Omega)$, we define $\displaystyle u^{\pm}(\cdot)=u(\cdot)^{\pm}$. We know that

[TABLE]

Given a measurable function $\displaystyle g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ (for example, a Carathéodory function), we denote by $\displaystyle N_{g}(\cdot)$ the Nemitsky (superposition) map defined by

[TABLE]

Also, $\displaystyle A\in\mathscr{L}(H^{1}(\Omega),H^{1}(\Omega)^{*})$ is defined by

[TABLE]

The hypotheses on the nonlinearity $\displaystyle f(z,x)$, are the following:

$\displaystyle H(f):$ $\displaystyle f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function such that $\displaystyle f(z,0)=0$ for almost all $\displaystyle z\in\Omega$ and

• (i)

for every $\displaystyle\rho>0$, there exists $\displaystyle a_{\rho}\in L^{\infty}(\Omega)$ such that

[TABLE]

• (ii)

there exist functions $\displaystyle\eta,\hat{\eta}\in L^{\infty}(\Omega)$ and $\displaystyle m\in\mathbb{N},m\geqslant 2$ such that

[TABLE]

and there exists $\displaystyle\tilde{\eta}>0$ such that

[TABLE]

• (iii)

if $\displaystyle F(z,x)=\int^{x}_{0}f(z,s)ds$, then

[TABLE]

• (iv)

there exist functions $\displaystyle\vartheta,\hat{\vartheta}\in L^{\infty}(\Omega)$ and $\displaystyle l\in\mathbb{N},l\geqslant m$ such that

[TABLE]

Remark 3**.**

Hypothesis $\displaystyle H(f)(ii)$ implies that $\displaystyle f(z,\cdot)$ has asymmetric behaviour as $\displaystyle x\rightarrow\pm\infty$ (jumping nonlinearity). Moreover, as $\displaystyle x\rightarrow-\infty$ we can have resonance with respect to the principal eigenvalue $\displaystyle\hat{\lambda}_{1}$. Hypothesis $\displaystyle H(f)(iii)$ implies that this resonance is from the left of $\displaystyle\hat{\lambda}_{1}$ in the sense that

[TABLE]

Note that hypotheses $\displaystyle H(f)(i),(ii),(iv)$ imply that

[TABLE]

For every $\displaystyle\lambda>0$, let $\displaystyle\varphi_{\lambda}:H^{1}(\Omega)\rightarrow\mathbb{R}$ be the energy functional for problem ($\displaystyle P_{\lambda}$) defined by

[TABLE]

Evidently, $\displaystyle\varphi_{\lambda}\in C^{1}(H^{1}(\Omega),\mathbb{R})$.

Let $\displaystyle\mu>0$ be as in (3). We introduce the following truncations-perturbations of the reaction in problem ($\displaystyle P_{\lambda}$):

[TABLE]

Both are Carathéodory functions. We set $\displaystyle K^{\pm}_{\lambda}(z,x)=\int^{x}_{0}k^{\pm}_{\lambda}(z,s)ds$ and consider the $\displaystyle C^{1}$-functionals $\displaystyle\hat{\varphi}^{\pm}_{\lambda}:H^{1}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

3. Compactness conditions for the functionals

We consider the functionals $\displaystyle\hat{\varphi}^{\pm}_{\lambda},\varphi_{\lambda}$ and we show that they satisfy the compactness-type condition

Proposition 5**.**

If hypotheses $\displaystyle H(\xi),\ H(\beta),\ H(f)$ hold, then for every $\displaystyle\lambda>0$ the functional $\displaystyle\hat{\lambda}^{+}_{\lambda}$ satisfies the C-condition.

Proof.

We consider a sequence $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq H^{1}(\Omega)$ such that

[TABLE]

From (9) we have

[TABLE]

In (10) we choose $\displaystyle h=-u_{n}^{-}\in H^{1}(\Omega)$. Then

[TABLE]

From(10) and (11), we have

[TABLE]

We show that $\displaystyle\{u^{+}_{n}\}_{n\geqslant 1}\subseteq H^{1}(\Omega)$ is bounded. Arguing by contradiction, suppose that

[TABLE]

Let $\displaystyle y_{n}=\frac{u^{+}_{n}}{||u^{+}_{n}||},\ n\in\mathbb{N}$. Then $\displaystyle||y_{n}||=1,\ y_{n}\geqslant 0$ for all $\displaystyle n\in\mathbb{N}$. So, we may assume that

[TABLE]

Using (12) we obtain

[TABLE]

From (6) we see that

[TABLE]

So, by passing to a subsequence if necessary and using hypothesis $\displaystyle H(f)(ii)$, we have

[TABLE]

(see Aizicovici, Papageorgiou & Staicu [1], proof of Proposition 16).

If in (3) we choose $\displaystyle h=y_{n}-y\in H^{1}(\Omega)$, pass to the limit as $\displaystyle n\rightarrow\infty$ and use (13), (14), (16) and the fact that $\displaystyle q<2$, then

[TABLE]

In (3) we pass to the limit as $\displaystyle n\rightarrow\infty$ and use (17). We obtain

[TABLE]

From (17) and Proposition 3, we have

[TABLE]

Then (19), (20) and the fact that $\displaystyle||y||=1$ (see (18)) imply that

[TABLE]

But this contradicts (14). Therefore

[TABLE]

We may assume that

[TABLE]

In (10) we choose $\displaystyle h=u_{n}-u\in H^{1}(\Omega)$, pass to the limit as $\displaystyle n\rightarrow\infty$ and use (21) and (6). Then

[TABLE]

The proof is now complete. ∎

Proposition 6**.**

If hypotheses $\displaystyle H(\xi),\ H(\beta),\ H(f)$ hold, then for every $\displaystyle\lambda>0$ the functional $\displaystyle\hat{\varphi}^{-}_{\lambda}$ is coercive.

Proof.

According to hypothesis $\displaystyle H(f)(iii)$ given any $\displaystyle\rho>0$, we can find $\displaystyle M_{2}=M_{2}(\rho)>0$ such that

[TABLE]

We have

[TABLE]

From hypothesis $\displaystyle H(f)(ii)$ we have

[TABLE]

If in (23) we let $\displaystyle v\rightarrow-\infty$ and use (24), then

[TABLE]

We proceed by contradiction and assume that $\displaystyle\hat{\lambda}^{-}_{\lambda}$ is not coercive. This means that we can find $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq H^{1}(\Omega)$ such that

[TABLE]

Let $\displaystyle v_{n}=\frac{u_{n}}{||u_{n}||},\ n\in\mathbb{N}$. Then $\displaystyle||v_{n}||=1$ for all $\displaystyle n\in\mathbb{N}$ and so we may assume that

[TABLE]

From (26) we have

[TABLE]

From (6) we obtain

[TABLE]

Hence by the Dunford-Pettis theorem and hypothesis $\displaystyle H(f)(ii)$ we have

[TABLE]

We return to (28), pass to the limit as $\displaystyle n\rightarrow\infty$ in (26), (27), (29). Since $\displaystyle\gamma(\cdot)$ is sequentially weakly lower semicontinuous on $\displaystyle H^{1}(\Omega)$, we obtain

[TABLE]

First we assume that $\displaystyle\tilde{e}\not\equiv\hat{\lambda}_{1}$ (see (29)). Then from (3) and Proposition 2 we have

[TABLE]

Then on account of (27) and (31), we have

[TABLE]

In (28) we pass to the limit as $\displaystyle n\rightarrow\infty$ and use (32), (3) and the sequential weak lower semicontinuity of $\displaystyle\gamma(\cdot)$. We obtain

[TABLE]

From (28) we obtain

[TABLE]

which contradicts the fact that $\displaystyle||v_{n}||=1$ for all $\displaystyle n\in\mathbb{N}$.

Next we assume that $\displaystyle\tilde{e}(z)=\hat{\lambda}_{1}$, for almost all $\displaystyle z\in\Omega$. From (3) and (2), we have

[TABLE]

If $\displaystyle\tau=0$, then $\displaystyle v\geqslant 0$ and arguing as above (see the part of the proof after (31)), we obtain $\displaystyle v=0$, contradicting the fact that $\displaystyle||v_{n}||=1$ for all $\displaystyle n\in\mathbb{N}$.

If $\displaystyle\tau>0$, then from (33) we have

[TABLE]

This means that

[TABLE]

But this contradicts (26).

We conclude that $\displaystyle\hat{\varphi}^{-}_{\lambda}$ is coercive. ∎

This proposition leads to the following corollary (see Marano & Papageorgiou [4, Proposition 2.2]).

Corollary 7**.**

If hypotheses $\displaystyle H(\xi),H(\beta),H(f)$ hold, then for every $\displaystyle\lambda>0$ the functional $\displaystyle\hat{\varphi}^{-}_{\lambda}$ satisfies the C-condition.

Now we turn our attention to the energy functional $\displaystyle\varphi_{\lambda},\ \lambda>0$.

Proposition 8**.**

If hypotheses $\displaystyle H(\xi),H(\beta),H(f)$ then for every $\displaystyle\lambda>0$ the functional $\displaystyle\varphi_{\lambda}$ satisfies the C-condition.

Proof.

We consider a sequence $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq H^{1}(\Omega)$ such that

[TABLE]

From (35) we have

[TABLE]

In (36) we choose $\displaystyle h=u_{n}\in H^{1}(\Omega)$. Then

[TABLE]

On the other hand, from (34) we have

[TABLE]

We add (37) and (38). Recalling that $\displaystyle q<2$, we obtain

[TABLE]

Using hypothesis $\displaystyle H(f)(iii)$, we see that

[TABLE]

We use (39) to show that $\displaystyle\{u^{-}_{n}\}_{n\geqslant 1}\subseteq H^{1}(\Omega)$ is bounded. Arguing by contradiction, we may assume that

[TABLE]

Let $\displaystyle y_{n}=\frac{u^{-}_{n}}{||u^{-}_{n}||},\ n\in\mathbb{N}$. Then $\displaystyle||y_{n}||=1,\ y_{n}\geqslant 0$ for all $\displaystyle n\in\mathbb{N}$. We may assume that

[TABLE]

In (36) we choose $\displaystyle h=-u^{-}_{n}\in H^{1}(\Omega)$. Then

[TABLE]

From (6) we see that

[TABLE]

So, by passing to a subsequence if necessary and using hypothesis $\displaystyle H(f)(ii)$ we have

[TABLE]

Returning to (42), passing to the limit as $\displaystyle n\rightarrow\infty$ and using (40) (recall that $\displaystyle q<2$), (41), (43) and the sequential weak lower semicontinuity of $\displaystyle\gamma(\cdot)$, we obtain

[TABLE]

First we assume that $\displaystyle\tilde{e}\not\equiv\hat{\lambda}_{1}$ (see (43)). Then from (44) and Proposition 2, we have

[TABLE]

From this and (42), we infer that

[TABLE]

which contradicts the fact that $\displaystyle||y_{n}||=1$ for all $\displaystyle n\in\mathbb{N}$.

We now assume that $\displaystyle\tilde{e}(z)=\hat{\lambda}_{1}$ for almost all $\displaystyle z\in\Omega$. Then from (44) and (2) we have

[TABLE]

If $\displaystyle\tau=0$, then $\displaystyle y=0$ and as above we have

[TABLE]

a contradiction since $\displaystyle||y_{n}||=1$ for all $\displaystyle n\in\mathbb{N}$.

If $\displaystyle\tau>0$, then $\displaystyle y(z)>0$ for all $\displaystyle z\in\Omega$ and so

[TABLE]

This contradicts (39). Therefore

[TABLE]

Next, we show that $\displaystyle\{u^{+}_{n}\}_{n\geqslant 1}\subseteq H^{1}(\Omega)$ is bounded. From (36) and (45), we have

[TABLE]

Using this bound and a contradiction argument as in the proof of Proposition 5, we show that

[TABLE]

From this, as before (see the proof of Proposition 5), via the Kadec-Klee property, we conclude that $\displaystyle\varphi_{\lambda}$ satisfies the C-condition. ∎

4. Multiplicity theorems

In this section using variational methods, truncation and perturbation techniques and Morse theory, we prove two multiplicity theorems for problem ($\displaystyle P_{\lambda}$) when $\displaystyle\lambda>0$ is small. In the first result, we produce four nontrivial smooth solutions, while in the second theorem, under stronger conditions on $\displaystyle f(z,\cdot)$, we establish the existence of five nontrivial smooth solutions.

We start with a result which allows us to satisfy the mountain-pass geometry (see Theorem 1) and also distinguish the solutions we produce from the trivial one.

Proposition 9**.**

If hypotheses $\displaystyle H(\xi),\ H(\beta),\ H(f)$ hold, then for every $\displaystyle\lambda>0$, $\displaystyle u=0$ is a local minimizer of $\displaystyle\varphi_{\lambda}$ and of $\displaystyle\hat{\varphi}^{\pm}_{\lambda}$.

Proof.

We do the proof for the functional $\displaystyle\varphi_{\lambda}$. The proofs for $\displaystyle\hat{\varphi}^{\pm}_{\lambda}$ are similar.

Recall that

[TABLE]

Then for $\displaystyle u\in C^{1}(\overline{\Omega})\backslash\{0\}$ we have

[TABLE]

So, if $\displaystyle||u||_{\infty}\leqslant||u||_{C^{1}(\overline{\Omega})}<\left(\frac{\lambda}{qc_{4}}\right)^{\frac{1}{2-q}}$, then $\displaystyle\varphi_{\lambda}(u)>0=\varphi_{\lambda}(0)$. Hence

[TABLE]

Similarly for the functionals $\displaystyle\hat{\varphi}^{\pm}_{\lambda}$. ∎

With the next proposition we guarantee that for small $\displaystyle\lambda>0$ the functional $\displaystyle\hat{\varphi}^{+}_{\lambda}(\cdot)$ satisfies the mountain pass geometry (see Theorem 1).

Proposition 10**.**

If hypotheses $\displaystyle H(\xi),\ H(\beta),\ H(f)$ hold, then we can find $\displaystyle\lambda^{*}>0$ such that for all $\displaystyle\lambda\in(0,\lambda^{*})$, there is $\displaystyle t_{0}=t_{0}(\lambda)>0$ for which we have $\displaystyle\hat{\varphi}^{+}_{\lambda}(t_{0}\hat{u}_{1})<0$.

Proof.

Let $\displaystyle r>2$. From hypothesis $\displaystyle H(f)(iv)$ and (46), we see that given $\displaystyle\epsilon>0$ we can find $\displaystyle c_{5}=c_{5}(\epsilon,r)>0$ such that

[TABLE]

Then for all $\displaystyle t>0$, we have

[TABLE]

Note that

[TABLE]

Choosing $\displaystyle\epsilon\in(0,k_{*})$, we see from (48) that

[TABLE]

Consider the function

[TABLE]

Evidently, $\displaystyle\mathcal{J}_{\lambda}\in C^{1}(0,+\infty)$ and since $\displaystyle 1<q<2<r$, we see that

[TABLE]

So, we can find $\displaystyle t_{0}\in(0,+\infty)$ such that

[TABLE]

Then

[TABLE]

Since $\displaystyle\frac{2-q}{r-q}<1$, we see that

[TABLE]

So, we can find $\displaystyle\lambda^{*}>0$ such that

[TABLE]

Then from (49) it follows that

[TABLE]

This completes the proof of Proposition 10. ∎

Remark 4**.**

In fact, a careful reading of the above proof reveals that

[TABLE]

Proposition 11**.**

If hypotheses $\displaystyle H(\xi),\ H(\beta),\ H(f)$ hold and $\displaystyle\lambda\in(0,\lambda^{*})$, then there exists $\displaystyle u_{0}\in C^{1}(\overline{\Omega})$ with $\displaystyle u_{0}(z)<0$ for all $\displaystyle z\in\Omega$ and

[TABLE]

Proof.

From Proposition 6 we know that $\displaystyle\hat{\varphi}^{-}_{\lambda}$ is coercive. Also, the Sobolev embedding theorem and the compactness of the trace map, imply that $\displaystyle\hat{\varphi}^{-}_{\lambda}$ is sequentially weakly lower semicontinuous. Hence, by the Weierstrass-Tonelli theorem, we can find $\displaystyle u_{0}\in H^{1}(\Omega)$ such that

[TABLE]

From (50) we see that

[TABLE]

From (51) we have

[TABLE]

In (4) we choose $\displaystyle h=u^{+}_{0}\in H^{1}(\Omega)$. Then

[TABLE]

From (4) and (7) it follows that

[TABLE]

Let $\displaystyle\tau_{\lambda}(z,x)=f(z,x)-\lambda|x|^{q-2}x$ and $\displaystyle\hat{k}_{\lambda}(z)=\frac{\tau_{\lambda}(z,u_{0}(z))}{1+|u_{0}(z)|}$, for $\displaystyle\lambda>0$. Hypotheses $\displaystyle H(f)(i),(ii)$ imply that

[TABLE]

From (53) we have

[TABLE]

(recall that $\displaystyle u_{0}\leqslant 0$). Since $\displaystyle(\xi-\hat{k}_{\lambda})(\cdot)\in L^{s}(\Omega)$ (for $\displaystyle s>N$), we deduce by Lemma 5.1 of Wang [15] that

[TABLE]

Then the Calderon-Zygmund estimates (see Wang [15, Lemma 5.2]) imply that

[TABLE]

Moreover, the Harnack inequality (see Theorem 7.2.1 in Pucci & Serrin [13, p. 163]), implies that

[TABLE]

This completes the proof. ∎

Remark 5**.**

The negative sign of the concave term does not allow us to conclude that $\displaystyle u_{0}\in-D_{+}$ when $\displaystyle\xi^{+}\in L^{\infty}(\Omega)$ (by Hopf’s boundary point theorem, see Pucci & Serrin [13, p. 120]).

Now we can state and prove our first multiplicity theorem.

Theorem 12**.**

Assume that hypotheses $\displaystyle H(\xi),\ H(\beta),\ H(f)$ hold. Then there exists $\displaystyle\hat{\lambda}>0$ such that for all $\displaystyle\lambda\in(0,\hat{\lambda})$ problem ($\displaystyle P_{\lambda}$) has at least four nontrivial solutions

[TABLE]

Proof.

From Proposition 11 and its proof (see (53)), we already have one solution

[TABLE]

This solution is a global minimizer of the functional $\displaystyle\hat{\varphi}^{-}_{\lambda}$.

Claim 1**.**

$\displaystyle u_{0}$* is a local minimizer of the energy functional $\displaystyle\varphi_{\lambda}$.*

We first show that $\displaystyle u_{0}$ is a local $\displaystyle C^{1}(\overline{\Omega})$-minimizer of $\displaystyle\varphi_{\lambda}$. Arguing by contradiction, suppose that we could find a sequence $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq C^{1}(\overline{\Omega})$ such that

[TABLE]

Then for all $\displaystyle n\in\mathbb{N}$, we have

[TABLE]

From (54) we have

[TABLE]

Therefore we can find $\displaystyle n_{0}\in\mathbb{N}$ such that

[TABLE]

a contradiction.

Hence we have that

[TABLE]

This proves the claim.

Using (7) and the regularity theory of Wang [15], we can see that

[TABLE]

On account of (56) we see that we may assume that both critical sets $\displaystyle K_{\hat{\varphi}^{-}_{\lambda}}$ and $\displaystyle K_{\hat{\varphi}^{+}_{\lambda}}$ are finite or, otherwise, we already have an infinity of nontrivial smooth solutions of constant sign and so we are done.

From Proposition 9 we know that for all $\displaystyle\lambda>0,u=0$ is a local minimizer of $\displaystyle\hat{\varphi}^{-}_{\lambda}$. Since $\displaystyle K_{\hat{\varphi}^{-}_{\lambda}}$ is finite, we can find $\displaystyle\rho\in(0,||u_{0}||)$ small such that

[TABLE]

(see Aizicovici, Papageorgiou & Staicu [1], proof of Proposition 29).

From Corollary 7 we know that

[TABLE]

Then (57) and (58) permit the use of Theorem 1 (the mountain pass theorem). So, we can find $\displaystyle\hat{u}\in H^{1}(\Omega)$ such that

[TABLE]

It follows that

[TABLE]

As before, Harnack’s inequality implies that

[TABLE]

Now we use once more Proposition 9 to find $\displaystyle\rho_{0}\in(0,t_{0})$ small enough such that

[TABLE]

Proposition 10 implies that we can find $\displaystyle\lambda^{*}>0$ such that

[TABLE]

Moreover, Proposition 5 implies that

[TABLE]

Then on account of (59), (60), (61), we can apply Theorem 1 (the mountain pass theorem) and produce $\displaystyle v_{0}\in H^{1}(\Omega)$ such that

[TABLE]

Once again, Harnack’s inequality guarantees that

[TABLE]

Let $\displaystyle l\in\mathbb{N}$ be as in hypothesis $\displaystyle H(f)(iv)$ and set

[TABLE]

We have

[TABLE]

Consider $\displaystyle u\in\overline{H}_{l}$. We have

[TABLE]

(see Proposition 2). Choosing $\displaystyle\epsilon\in(0,c_{2})$ we have

[TABLE]

Reasoning as in the proof of Proposition 11, we can find $\displaystyle\hat{\lambda}\in(0,\lambda^{*}]$ such that for all $\displaystyle\lambda\in(0,\hat{\lambda}]$ there exists $\displaystyle\rho_{\lambda}>0$ for which we have

[TABLE]

For $\displaystyle u\in\hat{H}_{l}$ we have

[TABLE]

Finally, consider the half-space

[TABLE]

Exploiting the orthogonality of $\displaystyle\hat{H}_{l}$ and $\displaystyle\overline{H}_{l}$, for every $\displaystyle u\in H_{+}$, we have

[TABLE]

Then (62), (63), (64) permit the use of Theorem 3.1 of Perera [12]. So, we can find $\displaystyle y_{0}\in H^{1}(\Omega)$ such that

[TABLE]

From (4) it is clear that $\displaystyle y_{0}\neq 0$. Recall that

[TABLE]

Therefore from (4) it follows that

[TABLE]

Also, from the claim we have that $\displaystyle u_{0}$ is a local minimizer of $\displaystyle\varphi_{\lambda}$.

Hence

[TABLE]

Note that $\displaystyle d_{l}\geqslant 2$ (since $\displaystyle l\geqslant m\geqslant 2$). Therefore

[TABLE]

and so from (4) and (67), we infer that

[TABLE]

So, we conclude that $\displaystyle y_{0}\in C^{1}(\overline{\Omega})\backslash\{0\}$ is a fourth nontrivial solution of ($\displaystyle P_{\lambda}$) (for all $\displaystyle\lambda\in(0,\hat{\lambda})$) distinct from $\displaystyle u_{0},\hat{u},v_{0}$. ∎

If we strengthen the hypotheses on $\displaystyle f(z,\cdot)$ we can improve the above multiplicity theorem and produce a fifth nontrivial smooth solution.

The new conditions on the nonlinearity $\displaystyle f(z,x)$ are the following:

$\displaystyle H(f)^{\prime}$: $\displaystyle f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a measurable function such that for almost all $\displaystyle z\in\Omega$, $\displaystyle f(z,0)=0$, $\displaystyle f(z,\cdot)\in C^{1}(\mathbb{R})$, hypotheses $\displaystyle H(f)^{\prime}(i),(ii),(iii)$ are the same as the corresponding hypotheses $\displaystyle H(f)(i),(ii),(iii)$ and

(iv) there exist $\displaystyle l\in\mathbb{N},\ l\geqslant m$ such that

[TABLE]

Theorem 13**.**

If hypotheses $\displaystyle H(\xi),\ H(\beta),\ H(f)^{\prime}$ hold, then there exists $\displaystyle\hat{\lambda}>0$ such that for all $\displaystyle\lambda\in(0,\hat{\lambda})$ problem ($\displaystyle P_{\lambda}$) has at least five nontrivial solutions

[TABLE]

Proof.

Now we have $\displaystyle\varphi_{\lambda}\in C^{2}(H^{1}(\Omega)\backslash\{0\},\mathbb{R})$. Similarly, $\displaystyle\hat{\varphi}^{\pm}_{\lambda}\in C^{2}(H^{1}(\Omega)\backslash\{0\},\mathbb{R})$.

The solutions $\displaystyle u_{0},\hat{u},v_{0},y_{0}$ are a consequence of Theorem 12. From Proposition 9 and (67), we have

[TABLE]

Also, from the proof of Theorem 12, we know that

[TABLE]

Invoking Corollary 6.102 of Motreanu, Motreanu & Papageorgiou [5], we have

[TABLE]

The continuity in the $\displaystyle C^{1}$-norm of the critical groups (see Theorem 5.126 in Gasinski & Papageorgiou [3, p. 836]), implies that

[TABLE]

From (69), (70), (71) it follows that

[TABLE]

The fourth nontrivial solution $\displaystyle y_{0}\in C^{1}(\overline{\Omega})$ was produced by using Theorem 3.1 of Perera [12]. According to that theorem, we can also find another function $\displaystyle\hat{y}\in H^{1}(\Omega)$, $\displaystyle\hat{y}\neq y_{0}$ such that

[TABLE]

From (68), (72), (73) we conclude that

[TABLE]

is the fifth nontrivial solution of problem ($\displaystyle P_{\lambda}$), for all $\displaystyle\lambda\in(0,\hat{\lambda})$. ∎

Acknowledgements. This research was supported by the Slovenian Research Agency grants P1-0292, J1-8131, J1-7025, N1-0064, and N1-0083. V.D. Rădulescu acknowledges the support through a grant of the Romanian Ministry of Research and Innovation, CNCS–UEFISCDI, project number PN-III-P4-ID-PCE-2016-0130, within PNCDI III.