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1534-0392

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1553-5258

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## Communications on Pure & Applied Analysis

2014 , Volume 13 , Issue 6

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2014, 13(6): 2141-2154
doi: 10.3934/cpaa.2014.13.2141

*+*[Abstract](289)*+*[PDF](214.2KB)**Abstract:**

We deal with the problem \begin{eqnarray} \Delta^2 u +V(|x|)u = f(u), u\in D^{2,2}(R^N) \end{eqnarray} where $\Delta^2$ is biharmonic operator and the potential $V > 0 $ is measurable, singular at the origin and may also have a continuous set of singularities. The nonlinearity is continuous and has a super-linear power-like behaviour; both sub-critical and super-critical cases are considered. We prove the existence of nontrivial radial solutions. If $f$ is odd, we show that the problem has infinitely many radial solutions.

2014, 13(6): 2155-2175
doi: 10.3934/cpaa.2014.13.2155

*+*[Abstract](342)*+*[PDF](477.7KB)**Abstract:**

We give the first natural examples of Calderón-Zygmund operators in the theory of analysis on post-critically finite self-similar fractals. This is achieved by showing that the purely imaginary Riesz and Bessel potentials on nested fractals with $3$ or more boundary points are of this type. It follows that these operators are bounded on $L^{p}$, $1 < p < \infty$ and satisfy weak $1$-$1$ bounds. The analysis may be extended to infinite blow-ups of these fractals, and to product spaces based on the fractal or its blow-up.

2014, 13(6): 2177-2210
doi: 10.3934/cpaa.2014.13.2177

*+*[Abstract](352)*+*[PDF](574.2KB)**Abstract:**

In this paper we prove local-in-time Strichartz estimates with loss of derivatives for Schrödinger equations with variable coefficients and potentials, under the conditions that the geodesic flow is nontrapping and potentials grow polynomially at infinity. This is a generalization to the case with variable coefficients and improvement of the result by Yajima-Zhang [40]. The proof is based on microlocal techniques including the semiclassical parametrix for a time scale depending on a spatial localization and the Littlewood-Paley type decomposition with respect to both of space and frequency.

2014, 13(6): 2211-2228
doi: 10.3934/cpaa.2014.13.2211

*+*[Abstract](322)*+*[PDF](402.8KB)**Abstract:**

This paper is concerned with large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping. For the nonlinear damping case, i.e. $\beta \neq 0,$ results for the linear damping case are extended to the case of nonlinear damping. Compared with the results obtained by Marcati and Pan, better decay estimates are obtained in this paper.

2014, 13(6): 2229-2252
doi: 10.3934/cpaa.2014.13.2229

*+*[Abstract](290)*+*[PDF](512.2KB)**Abstract:**

In this article we study the relations between the long-time dynamics of the 3D Allen-Cahn-LANS-$\alpha$ model and the exact 3D Allen-Cahn-Navier-Stokes system. Following the idea of [26], we prove that bounded set of solutions of the Allen-Cahn-LANS-$\alpha$ model converge to the trajectory attractor $\mathcal{U}_0 $ of the 3D Allen-Cahn-Navier-Stokes system as time goes to $+ \infty $ and $\alpha$ approaches $0^+. $ In particular we show that the trajectory attractors $\mathcal{U}_{\alpha} $ of the 3D Allen-Cahn-LANS-$\alpha$ model converges to the trajectory attractor $\mathcal{U}_0 $ of the 3D Allen-Cahn-Navier-Stokes as $ \alpha $ approaches $0^+. $ Let us mention that the strong nonlinearity that results from the coupling of the convective Allen-Cahn system and the LANS-$\alpha$ equations makes the analysis of the problem considered in this article more involved.

2014, 13(6): 2253-2272
doi: 10.3934/cpaa.2014.13.2253

*+*[Abstract](401)*+*[PDF](453.3KB)**Abstract:**

In this work, we present a Hodge-type decomposition for variable exponent spaces of Clifford-valued functions, where one of the components is the kernel of the parabolic-type Dirac operator.

2014, 13(6): 2273-2287
doi: 10.3934/cpaa.2014.13.2273

*+*[Abstract](370)*+*[PDF](427.0KB)**Abstract:**

This paper is concerned with the existence of positive solutions for a class of quasilinear Schrödinger equations in $R^N$ with critical growth and potential vanishing at infinity. By using a change of variables, the quasilinear equations are reduced to semilinear one. Since the potential vanish at infinity, the associated functionals are still not well defined in the usual Sobolev space. So we have to work in the weighted Sobolev spaces, by Hardy-type inequality, then the functionals are well defined in the weighted Sobolev space and satisfy the geometric conditions of the Mountain Pass Theorem. Using this fact, we obtain a Cerami sequence converging weakly to a solution $v$. In the proof that $v$ is nontrivial, the main tool is classical arguments used by H. Brezis and L. Nirenberg in [13].

2014, 13(6): 2289-2303
doi: 10.3934/cpaa.2014.13.2289

*+*[Abstract](531)*+*[PDF](420.6KB)**Abstract:**

In this paper, we study the existence of positive solution to $p-$Kirchhoff type problem \begin{eqnarray} &(a+\mu(\int_{\mathbb{R}^N}(|\nabla u|^p+V(x)|u|^p)dx)^{\tau})(-\Delta_pu+V(x)|u|^{p-2}u)=|u|^{m-2}u\\ &+\lambda |u|^{q-2}u, \; {\rm in}\; \mathbb{R}^N \\ &u(x)>0, \;\;{\rm in}\;\; \mathbb{R}^N,\;\; u\in W^{1,p}(\mathbb{R}^N), \end{eqnarray} where $a, \mu>0, \tau\ge 0, \lambda\in \mathbb{R} $ and $1 < p < N, p < q < m < p^*=\frac{pN}{N-p}$. The potential $V(x)\in C(\mathbb{R}^N)$ and $0 < \inf_{x\in\mathbb{R}^N}V(x) < \sup_{x\in\mathbb{R}^N}V(x) < \infty$. The existence of solution will be obtained by the Nehari manifold and variational method.

2014, 13(6): 2305-2316
doi: 10.3934/cpaa.2014.13.2305

*+*[Abstract](321)*+*[PDF](358.0KB)**Abstract:**

In the paper, we apply the moving plane method to prove that if the right hand sides of equation and Neumann boundary condition are both independent of one variable, the domain and the solution to the Hessian over-determined problem are mirror symmetric. Our result generalizes the previous results on radial symmetry. In the end, we get the mirror symmetry of over-determined problems for more general equations, which include Weingarten curvature equation.

2014, 13(6): 2317-2330
doi: 10.3934/cpaa.2014.13.2317

*+*[Abstract](356)*+*[PDF](434.3KB)**Abstract:**

In this paper, we present a new proof on the existence of infinitely many sign-changing solutions for the following Brézis-Nirenberg problem \begin{eqnarray} -\Delta u=\lambda u+|u|^{2^{*}-2}u \quad \textrm{in}\, \Omega, \qquad u=0 \quad \textrm{on}\,\partial\Omega, \end{eqnarray} for each fixed $\lambda>0$, under the assumptions that $N\geq 7$, where $\Omega$ is a bounded smooth domain of $\mathbb{R}^{N}$, $2^{*}=\frac{2N}{N-2}$ is the critical Sobolev exponent. In order to construct sign-changing solutions, we will use a combination of invariant sets method and Ljusternik-Schnirelman type minimax method, which is much simpler than the proof of [20] depending on the estimates of Morse indices of nodal solutions to obtain the same result.

2014, 13(6): 2331-2350
doi: 10.3934/cpaa.2014.13.2331

*+*[Abstract](314)*+*[PDF](428.3KB)**Abstract:**

We prove the solution of the full compressible fluid models of Korteweg type with centered rarefaction wave data of large strength exists globally in time. As the viscosity, heat-conductivity and capillary coefficients tend to zero, the global solution converges to the centered rarefaction wave solution of the corresponding Euler equations uniformly when the initial perturbation is small. Our analysis is based on the energy method.

2014, 13(6): 2351-2358
doi: 10.3934/cpaa.2014.13.2351

*+*[Abstract](259)*+*[PDF](345.6KB)**Abstract:**

In this article we are interested in the propagation speed for solutions of hyperbolic boundary value problems in the $WR$ class. Using the Holmgren principle, we show that this speed is finite and we are able to give an explicit expression for the maximal speed. Due to a propagation phenomenon along the boundary that is specific to the $WR$ class, the maximal speed can be larger than the propagation speed for the Cauchy problem. This is consistent with previous examples of the litterature.

2014, 13(6): 2359-2376
doi: 10.3934/cpaa.2014.13.2359

*+*[Abstract](411)*+*[PDF](470.2KB)**Abstract:**

We study the concentration phenomenon for solutions of the fractional nonlinear Schrödinger equation, which is nonlocal. We mainly use the Lyapunov-Schmidt reduction method. Precisely, consider the nonlinear equation \begin{eqnarray} (-\varepsilon^2\Delta)^sv+Vv-|v|^{\alpha}v=0\quad\mbox{in}\quad\mathbf R^n, \end{eqnarray} where $n =1, 2, 3$, $\max\{\frac{1}{2}, \frac{n}{4}\}< s < 1$, $1 \leq \alpha < \alpha_*(s,n)$, $V\in C^3_{b}(\mathbf{R}^n)$. Here the exponent $\alpha_*(s,n)=\frac{4s}{n-2s}$ for $0 < s < \frac{n}{2}$ and $\alpha_*(s,n)=\infty$ for $s \geq\frac{n}{2}$. Then for each non-degenerate critical point $z_0$ of $V$, there is a nontrivial solution of equation (1) concentrating to $z_0$ as $\varepsilon\to 0$.

2014, 13(6): 2377-2394
doi: 10.3934/cpaa.2014.13.2377

*+*[Abstract](371)*+*[PDF](452.2KB)**Abstract:**

We consider the Cauchy problem for the $L^2$-critical nonlinear Schrödinger equation with a nonlinear damping. According to the power of the damping term, we prove the global existence or the existence of finite time blowup dynamics with the log-log blow-up speed for $\|\nabla u(t)\|_{L^2}$.

2014, 13(6): 2395-2406
doi: 10.3934/cpaa.2014.13.2395

*+*[Abstract](489)*+*[PDF](426.5KB)**Abstract:**

The aim of this paper is to study radial symmetry properties for ground state solutions of elliptic equations involving a regional fractional Laplacian, namely \begin{eqnarray} (-\Delta)_{\rho}^{\alpha}u + u = f(u) \quad \mbox{in} \ \mathbb{R}^{n}, \ \ \mbox{for} \ \ \alpha\in (0,1). \end{eqnarray} In [9], the authors proved that problem (1) has a ground state solution. In this work we prove that the ground state level is achieved by a radially symmetry solution. The proof is carried out by using variational methods jointly with rearrangement arguments.

2014, 13(6): 2407-2443
doi: 10.3934/cpaa.2014.13.2407

*+*[Abstract](285)*+*[PDF](677.2KB)**Abstract:**

We consider the free boundary problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region, the flow is governed by the usual compressible MHD equations, while in the vacuum region we consider the Maxwell system for the electric and the magnetic fields, in order to investigate the well-posedness of the problem, in particular in relation with the electric field in vacuum. At the free interface, driven by the plasma velocity, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary.

Under suitable stability conditions satisfied at each point of the plasma-vacuum interface, we derive a basic a priori estimate for solutions to the linearized problem in the Sobolev space $H^1_{\tan}$ with conormal regularity. The proof follows by a suitable secondary symmetrization of the Maxwell equations in vacuum and the energy method.

An interesting novelty is represented by the fact that the interface is characteristic with variable multiplicity, so that the problem requires a different number of boundary conditions, depending on the direction of the front velocity (plasma expansion into vacuum or viceversa). To overcome this difficulty, we recast the vacuum equations in terms of a new variable which makes the interface characteristic of constant multiplicity. In particular, we don't assume that plasma expands into vacuum.

2014, 13(6): 2445-2464
doi: 10.3934/cpaa.2014.13.2445

*+*[Abstract](328)*+*[PDF](438.0KB)**Abstract:**

We study the differential system which describes the steady flow of an electrically conducting fluid in a saturated porous medium, when the fluid is subjected to the action of a magnetic field. The system consists of the stationary Brinkman-Forchheimer equations and the stationary magnetic induction equation. We prove existence of weak solutions to the system posed in a bounded domain of $\mathbb{R}^3$ and equipped with boundary conditions. We also prove uniqueness in the class of small solutions, and regularity of weak solutions. Then we establish a convergence result, as the Brinkman coefficient (viscosity) tends to 0, of the weak solutions to a solution of the system formed by the Darcy-Forchheimer equations and the magnetic induction equation.

2014, 13(6): 2465-2474
doi: 10.3934/cpaa.2014.13.2465

*+*[Abstract](342)*+*[PDF](313.5KB)**Abstract:**

In this paper we study an elliptic eigenvalue problem with non-local boundary condition. We prove the existence of the principal eigenvalue and its main properties. As consequence, we show the existence and uniqueness of positive solution of a nonlinear problem arising from population dynamics.

2014, 13(6): 2475-2492
doi: 10.3934/cpaa.2014.13.2475

*+*[Abstract](357)*+*[PDF](432.4KB)**Abstract:**

In this paper, we first derive an equation for a single species population with two age stages and a fixed maturation period living in the half plane such as ocean and big lakes. By adopting the compact open topology, we establish some a priori estimate for nontrivial solutions after describing asymptotic properties of the nonlocal delayed effect, which enables us to show the permanence of the equation. Then we can employ standard dynamical system theoretical arguments to establish the global dynamics of the equation under appropriate conditions. Applying the main results to the model with Ricker's birth function and Mackey-Glass's hematopoiesis function, we obtain threshold results for the global dynamics of these two models.

2014, 13(6): 2493-2508
doi: 10.3934/cpaa.2014.13.2493

*+*[Abstract](359)*+*[PDF](390.7KB)**Abstract:**

In this paper, we study the structure of solutions of a fourth order elliptic equation with a singular nonlinearity. For different boundary values $\kappa$, we establish the global bifurcation branches of solutions to the equation. More precisely, we show that $\kappa=1$ is a critical boundary value to change the structure of solutions to this problem.

2014, 13(6): 2509-2542
doi: 10.3934/cpaa.2014.13.2509

*+*[Abstract](355)*+*[PDF](548.8KB)**Abstract:**

We consider a heat transmission problem for a composite material which fills the $n$-dimensional Euclidean space. The composite has a periodic structure and consists of two materials. In each periodicity cell one material occupies a cavity of size $\epsilon$, and the second material fills the remaining part of the cell. We assume that the thermal conductivities of the materials depend nonlinearly upon the temperature. We show that for $\epsilon$ small enough the problem has a solution, \textit{i.e.}, a pair of functions which determine the temperature distribution in the two materials. Then we analyze the behavior of such a solution as $\epsilon$ approaches $0$ by an approach which is alternative to those of asymptotic analysis. In particular we prove that if $n\geq 3$, the temperature can be expanded into a convergent series expansion of powers of $\epsilon$ and that if $n=2$ the temperature can be expanded into a convergent double series expansion of powers of $\epsilon$ and $\epsilon \log \epsilon$.

2014, 13(6): 2543-2557
doi: 10.3934/cpaa.2014.13.2543

*+*[Abstract](384)*+*[PDF](453.8KB)**Abstract:**

Dissipative problems in electrorheological fluids, porous media and image processing often involve spatially dependent exponents. They also include time-dependent terms as in equation \begin{eqnarray} \frac{\partial u_\lambda}{\partial t}(t)-\textrm{div}(D_\lambda(t)|\nabla u_\lambda(t)|^{p(x)-2}\nabla u_\lambda(t))+|u_\lambda(t)|^{p(x)-2}u_\lambda(t) = B(t,u_\lambda(t)) \end{eqnarray} on a bounded smooth domain $\Omega$ in $\mathbb{R}^n$, $n\geq 1$, with a homogeneous Neumann boundary condition, where the exponent $p(\cdot)\in C(\bar{\Omega}, \mathbb{R}^+)$ satisfying $p^-$ $:=$ $\min p(x)$ $>$ $2$, and $\lambda$ $\in$ $[0,\infty)$ is a parameter.

The existence and upper semicontinuity of pullback attractors are established for this equation under the assumptions, amongst others, that $B$ is globally Lipschitz in its second variable and $D_\lambda$ $ \in $ $L^\infty([\tau,T] \times \Omega, \mathbb{R}^+)$ is bounded from above and below, is monotonically nonincreasing in time and continuous in the parameter $\lambda$. The global existence and uniqueness of strong solutions is obtained through results of Yotsutani.

2014, 13(6): 2559-2587
doi: 10.3934/cpaa.2014.13.2559

*+*[Abstract](395)*+*[PDF](287.0KB)**Abstract:**

In this paper we consider the global gradient estimates for weak solutions of $p(x)$-Laplacian type equation with small BMO coefficients in a $\delta$-Reifenberg flat domain. The modified Vitali covering lemma, good $\lambda$-inequalities, the maximal function technique and the appropriate localization method are the main analytical tools. The global Caldéron--Zygmund theory for such equations is obtained. Moreover, we generalize the regularity estimates in the Lebesgue spaces to the Orlicz spaces.

2014, 13(6): 2589-2608
doi: 10.3934/cpaa.2014.13.2589

*+*[Abstract](311)*+*[PDF](1399.6KB)**Abstract:**

We study the bifurcation curve and exact multiplicity of positive solutions of the combustion problem with general Arrhenius reaction-rate laws \begin{eqnarray} u^{\prime \prime }(x)+\lambda (1+\epsilon u)^{m}e^{\frac{u}{1+\epsilon u}}=0, -1 < x < 1, \\ u(-1)=u(1)=0, \end{eqnarray} where the bifurcation parameters $\lambda, \epsilon >0$ and $-\infty < m <1$. We prove that, for $(-4.103\approx)$ $\tilde{m}\leq m < 1$ for some constant $\tilde{m}$, the bifurcation curve is S-shaped on the $(\lambda, \|u\|_{\infty })$-plane if $0<\epsilon \leq \frac{6}{7}\epsilon _{\text{tr}}^{\text{Sem}}(m)$, where \begin{eqnarray} \epsilon _{\text{tr}}^{\text{Sem}}(m)=\left\{ \begin{array}{l} (\frac{1-\sqrt{1-m}}{m})^{2}\ \text{ for }-\infty < m < 1, m \neq 0, \\ \frac{1}{4}\ \text{for}\ m=0, \end{array}\right. \end{eqnarray} is the Semenov transitional value for general Arrhenius kinetics. In addition, for $-\infty < m < 1$, the bifurcation curve is S-like shaped if $0<\epsilon \leq \frac{8}{9} \epsilon _{\text{tr}}^{\text{Sem}}(m).$ Our results improve and extend those in Wang (Proc. Roy. Soc. London Sect. A, 454 (1998), 1031--1048.)

2014, 13(6): 2609-2639
doi: 10.3934/cpaa.2014.13.2609

*+*[Abstract](410)*+*[PDF](1590.3KB)**Abstract:**

The main objective of this paper is to describe the dynamic transition of the incompressible MHD equations in a three dimensional (3D) rectangular domain from a perspective of pattern formation. We aim to classify the formations of roll, rectangle and hexagonal patterns at the first critical Rayleigh number.

When the first eigenvalue of the linearized operator is real and simple, the critical eigenvector has either a roll structure or a rectangle structure. In both cases we find that the transition is continuous or jump depending on a non-dimensional number computed explicitly in terms of system parameters.

When the critical eigenspace has dimension two corresponding to two real eigenvalues, we study the transitions of hexagonal pattern. In this case, we show that all three types of transitions--continuous, jump and mixed--can occur in eight different transition scenarios.

Finally, we study the case where the first eigenvalue is complex, simple and corresponding eigenvector has a roll structure. In this case, we find that both continuous and jump transitions are possible.

We give several bounds on the parameters which separate the parameter space into regions of different transition scenarios.

2014, 13(6): 2641-2673
doi: 10.3934/cpaa.2014.13.2641

*+*[Abstract](263)*+*[PDF](982.0KB)**Abstract:**

In this paper we continue the study of ``large" small-amplitude limit cycles in slow-fast codimension 3 elliptic bifurcations which is initiated in [8]. Our treatment is based on blow-up and good normal forms.

2014, 13(6): 2675-2692
doi: 10.3934/cpaa.2014.13.2675

*+*[Abstract](307)*+*[PDF](286.7KB)**Abstract:**

The Jacobian elliptic functions are generalized to functions including the generalized trigonometric functions. The paper deals with the basis property of the sequence of generalized Jacobian elliptic functions in any Lebesgue space. In particular, it is shown that the sequence of the classical Jacobian elliptic functions is a basis in any Lebesgue space if the modulus $k$ satisfies $0 \le k \le 0.99$.

2014, 13(6): 2693-2712
doi: 10.3934/cpaa.2014.13.2693

*+*[Abstract](413)*+*[PDF](465.0KB)**Abstract:**

The paper is concerned with the asymptotic behavior of two species population whose densities are described by Kolmogorov systems of predator-prey type in random environment. We study the omega-limit set and find conditions ensuring the existence and attractivity of a stationary density. Some applications to the predator-prey model with Beddington-DeAngelis functional response are considered to illustrate our results.

2014, 13(6): 2713-2731
doi: 10.3934/cpaa.2014.13.2713

*+*[Abstract](553)*+*[PDF](632.3KB)**Abstract:**

We study positive radial solutions to the boundary value problem \begin{eqnarray} -\Delta u = \lambda K(|x|)f(u), \quad x \in \Omega, \\ \frac{\partial u}{\partial \eta}+\tilde{c}(u)u = 0, \quad |x|=r_0, \\ u(x) \rightarrow 0, \quad |x|\rightarrow \infty, \end{eqnarray} where $\Delta u=div \big(\nabla u\big)$ is the Laplacian of $u$, $\lambda$ is a positive parameter, $\Omega=\{x \in \mathbb{R}^N| N>2, |x|> r_0 \mbox{ with }r_0>0\}$, $K:[r_0, \infty)\rightarrow(0,\infty)$ is a continuous function such that $\lim_{r \rightarrow \infty}K(r)=0$, $\frac{\partial}{\partial \eta}$ is the outward normal derivative, and $\tilde{c}:[0,\infty) \rightarrow (0,\infty)$ is a continuous function. We consider various $C^1$ classes of the reaction term $f:[0,\infty) \rightarrow \mathbb{R}$ that are sublinear at $\infty$ $(i.e. \lim_{s \rightarrow \infty}\frac{f(s)}{s}=0)$. In particular, we discuss existence and multiplicity results for classes of $f$ with $(a)$ $f(0)>0$, $(b)$ $f(0)<0$, and $(c)$ $f(0)=0$. We establish our existence and multiplicity results via the method of sub-super solutions. We also discuss some uniqueness results.

2014, 13(6): 2733-2748
doi: 10.3934/cpaa.2014.13.2733

*+*[Abstract](316)*+*[PDF](479.5KB)**Abstract:**

The collapse of a wedge-shaped dam containing fluid initially with a uniform velocity can be described as an expansion problem of gas into vacuum. It is an important class of binary interaction of rarefaction waves in the two dimensional Riemann problems for the compressible Euler equations. In this paper, we present various characteristic decompositions of the two dimensional pseudo-steady Euler equations for the generalized Chaplygin gas and obtain some priori estimates. By these estimates, we prove the global existence of solution to the expansion problem of a wedge of gas into vacuum with the half angle $\theta\in(0,\pi/2)$ for the generalized Chaplygin gas.

2014, 13(6): 2749-2766
doi: 10.3934/cpaa.2014.13.2749

*+*[Abstract](340)*+*[PDF](420.9KB)**Abstract:**

We consider a nonlinear Dirichlet problem driven by the sum of a $p$-Laplacian ($p>2$) and a Laplacian, with a reaction that is ($p-1$)-sublinear and exhibits an asymmetric behavior near $\infty$ and $-\infty$, crossing $\hat{\lambda}_1>0$, the principal eigenvalue of $(-\Delta_p, W^{1,p}_0(\Omega))$ (crossing nonlinearity). Resonance with respect to $\hat{\lambda}_1(p)>0$ can also occur. We prove two multiplicity results. The first for a Caratheodory reaction producing two nontrivial solutions and the second for a reaction $C^1$ in the $x$-variable producing three nontrivial solutions. Our approach is variational and uses also the Morse theory.

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