Constant sign and nodal solutions for logistic-type equations with equidiffusive reaction

We consider a logistic-type equation driven by the p-Laplace differential operator with an equidiffusive reaction term. Combining variational methods based on critical point theory together with truncation techniques and Morse theory, we show that when λ > λ₁, the problem has extremal solutions of constant sign and when λ > λ₂ it has also a nodal (sign-changing) solution. Here λ₁ < λ₂ are the first two eigenvalues of the negative Dirichlet p-Laplacian. In the semilinear case (i.e. p = 2) we produce two nodal solutions.     

Critical exponents for the evolution p-Laplacian equation with a localized reaction

This paper deals with the large time behavior of nonnegative solutions to the equation $$u_t = div\left( {\left| {\nabla u} \right|^{p - 2} \nabla u} \right) + a\left( x \right)u^q ,\left( {x,t} \right) \in R^N \times (0,T),$$ where p > 2, q > 0, and the function a(x) ?? 0 has a compact support. We obtain the critical exponent for global existence q ₀ and the Fujita exponent q _c . In one-dimensional case N = 1, we have $q_0 = \frac{{2(p - 1)}} {p}$ and q _c = 2(p ? 1). Particularly, all solutions are global in time if 0 < q ?? q _o, but blow up if q ₀ < q ?? q _c ; while if q > q _c both blowing up solutions and global solutions exist. However, for the case N ?? p > 2, these two critical exponents are exactly the same. Namely, q ₀ = p ? 1 = q _c .     

On the existence of a continuous branch of nodal solutions of elliptic equations with convex-concave nonlinearities

We study the existence of nodal solutions of a parametrized family of Dirichlet boundary value problems for elliptic equations with convex-concave nonlinearities. In the main result, we prove the existence of nodal solutions u _λ for λ ∈ (?∞, λ*₀). The critical value λ*₀ >0 is found by a spectral analysis procedure according to Pokhozhaev’s fibering method. We show that the obtained solutions form a continuous branch (in the sense of level lines of the energy functional) with respect to the parameter λ. Moreover, we prove the existence of an interval \(( - \infty ,\tilde \lambda )\) , where \(\tilde \lambda > 0\) , on which this branch consists of solutions with exactly two nodal domains.     

On the limit behavior of the magnetic Zakharov system

In this paper,we prove that the solutions of magnetic Zakharov system converge to those of generalized Zakharov system in Sobolev space H s,s > 3/2,when parameter β→∞.Further,when parameter (α,β) →∞ together,we prove that the solutions of magnetic Zakharov system converge to those of Schro¨dinger equation with magnetic effect in Sobolev space H s,s > 3/2.Moreover,the convergence rate is also obtained.     

An application of the maximum principle to describe the layer behavior of large solutions and related problems

This work is devoted to the analysis of the asymptotic behavior of positive solutions to some problems of variable exponent reaction-diffusion equations, when the boundary condition goes to infinity (large solutions). Specifically, we deal with the equations ??u = u ^p(x), ??u = ?m(x)u?+?a(x)u ^p(x) where a(x)??? a ₀ >?0, p(x)??? 1 in ??, and ??u = e ^p(x) where p(x)??? 0 in ??. In the first two cases p is allowed to take the value 1 in a whole subdomain ${\Omega_c\subset \Omega}$ , while in the last case p can vanish in a whole subdomain ${\Omega_c\subset \Omega}$ . Special emphasis is put in the layer behavior of solutions on the interphase ?? _i :?= ??? _c ???. A similar study of the development of singularities in the solutions of several logistic equations is also performed. For example, we consider ???u = ?? m(x)u?a(x) u ^p(x) in ??, u = 0 on ???, being a(x) and p(x) as in the first problem. Positive solutions are shown to exist only when the parameter ?? lies in certain intervals: bifurcation from zero and from infinity arises when ?? approaches the boundary of those intervals. Such bifurcations together with the associated limit profiles are analyzed in detail. For the study of the layer behavior of solutions the introduction of a suitable variant of the well-known maximum principle is crucial.     

Existence of Infinitely Many Non-Trivial Bifurcation Points

We consider the non-linear two point boundary value problem where λ > 0,f ∈ C², f′ ≥ 0, f(0) < 0 and lim_{u → ∞} f(u) > 0. By considering the non-negative as well as all sign changing solutions, we establish the existence of infinitely many non-trivial bifurcation points. Further, when f is superlinear, we prove that there exists a constant λ* > 0, such that for each λ ∈ (0, λ*) there are exactly two solutions with m interior zeros for every m = 1,2, …We apply our results to the case when f(u) = u ³ - k; k > 0, and also discuss the evolution of the bifurcation diagram as k → 0.     

Bifurcation phenomena for nonlinear superdiffusive Neumann equations of logistic type

We consider a nonlinear Neumann logistic equation driven by the p-Laplacian with a general Carathéodory superdiffusive reaction. We are looking for positive solutions of such problems. Using minimax methods from critical point theory together with suitable truncation techniques, we show that the equation exhibits a bifurcation phenomenon with respect to the parameter λ > 0. Namely, we show that there is a λ_* > 0 such that for λ < λ_*, the problem has no positive solution; for λ = λ_*, it has at least one positive solution; and for λ > λ_*, it has at least two positive solutions.     

Uniqueness of large radial solutions and existence of nonradial solutions for a superlinear Dirichlet problem in annulii

Here we establish the existence of infinitely many nonradial solutions for a superlinear Dirichlet problem in annulii. Our proof relies on estimating the number of radial solutions having a prescribed number of nodal regions. We prove that, for k>0 large, there exist exactly two radial solutions with k nodal regions (connected components of ). The problem need not be homogeneous.     

Multiple solutions for semilinear elliptic equations with Neumann boundary condition and jumping nonlinearities

We obtain nonconstant solutions of semilinear elliptic Neumann boundary value problems with jumping nonlinearities when the asymptotic limits of the nonlinearity fall in the type (Il), l>2 and (IIl), l?1 regions formed by the curves of the Fucik spectrum. Furthermore, we have at least two nonconstant solutions in every order interval under resonance case. In this paper, we apply the sub-sup solution method, Fucik spectrum, mountain pass theorem in order intervals, degree theory and Morse theory to get the conclusions.     

On exact solutions of Stokes second problem for a Burgers�� fluid, II. The cases �� =?��2/4 and �� > ��2/4

In this study, the exact solutions of the Stokes second problem for a Burgers?? fluid are presented when the relaxation time satisfies the conditions ?? =???²/4 and ?? >???²/4. The velocity field and the associated tangential stress, when only one initial condition is necessary for velocity, are determined by means of the Laplace transform. The physical interpretation for the emerging parameters is discussed with the help of graphical illustrations. The similar solutions for the Stokes?? first problem are obtained as the limiting cases of our solutions.     

Diophantine equations with products of consecutive terms in Lucas sequences

In this paper, we show that if (u_n)_n?1 is a Lucas sequence, then the Diophantine equation in integers n?1, k?1, m?2 and y with |y|>1 has only finitely many solutions. We also determine all such solutions when (u_n)_n?1 is the sequence of Fibonacci numbers and when u_n=(xⁿ-1)/(x-1) for all n?1 with some integer x>1.     

Critical Partitions and Nodal Deficiency of Billiard Eigenfunctions

The paper addresses the nodal count (i.e., the number of nodal domains) for eigenfunctions of Schr?dinger operators with Dirichlet boundary conditions in bounded domains. The classical Sturm theorem states that in dimension one, the nodal and eigenfunction counts coincide: the nth eigenfunction partitions the interval into n nodal domains. The Courant Nodal Theorem claims that in any dimension, the number of nodal domains ?? _n of the nth eigenfunction cannot exceed n. However, it follows from an asymptotically stronger upper bound by Pleijel that in dimensions higher than 1 the equality can hold for only finitely many eigenfunctions. Thus, in most cases a ??nodal deficiency?? d _n ?=?n??? _n arises. One can say that the nature of the nodal deficiency has not been understood. It was suggested in recent years that, rather than starting with eigenfunctions, one can look at partitions of the domain into ?? sub-domains, asking which partitions can correspond to eigenfunctions, and what would be the corresponding deficiency. To this end one defines an ??energy?? of a partition, for example, the maximum of the ground state energies of the sub-domains. One notices that if a partition does correspond to an eigenfunction, then the ground state energies of all the nodal domains are the same, i.e., it is an equipartition. It was shown in a recent paper by Helffer, Hoffmann-Ostenhof and Terracini that (under some natural conditions) partitions minimizing the energy functional correspond to the ??Courant sharp?? eigenfunctions, i.e. to those with zero nodal deficiency. In this paper it is shown that it is beneficial to restrict the domain of the functional to the equipartition, where it becomes smooth. Then, under some genericity conditions, the nodal partitions correspond exactly to the critical points of the functional. Moreover, the nodal deficiency turns out to be equal to the Morse index at the corresponding critical point. This explains, in particular, why the minimal partitions must be Courant sharp.     

A local mountain pass type result for a system of nonlinear Schr?dinger equations

We consider a singular perturbation problem for a system of nonlinear Schr?dinger equations: $$ \begin{array}{l} -\varepsilon^2\Delta v_1 +V_1(x)v_1 = \mu_1 v_1^3 + \beta v_1v_2^2 \quad {\rm in}\,\,{\bf R}^N, \\ -\varepsilon^2\Delta v_2 +V_2(x)v_2 = \mu_2 v_2^3 + \beta v_1^2v_2 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x) >0 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x)\in H^1({\bf R}^N), \end{array} \quad\quad\quad\quad\quad (*) $$ where N?=?2, 3, ?? ₁, ?? ₂, ?? > 0 and V ₁(x), V ₂(x): R ^N ?? (0, ??) are positive continuous functions. We consider the case where the interaction ?? > 0 is relatively small and we define for ${P\in{\bf R}^N}$ the least energy level m(P) for non-trivial vector solutions of the rescaled ??limit?? problem: $$ \begin{array}{l} -\Delta v_1 +V_1(P)v_1 = \mu_1 v_1^3 + \beta v_1v_2^2 \quad {\rm in}\,\,{\bf R}^N, \\ -\Delta v_2 +V_2(P)v_2 = \mu_2 v_2^3 + \beta v_1^2v_2 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x) >0 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x)\in H^1({\bf R}^N). \end{array} \quad\quad\quad\quad\quad\quad (**) $$ We assume that there exists an open bounded set ${\Lambda\subset{\bf R}^N}$ satisfying $$ {\mathop {\rm inf} _{P\in\Lambda} m(P)} < {\mathop {\rm inf}_{P\in\partial\Lambda} m(P)}. $$ We show that (*) possesses a family of non-trivial vector positive solutions ${\{(v_{1\varepsilon}(x), v_{2\varepsilon} (x))\}_{\varepsilon\in (0,\varepsilon_0]}}$ which concentrates??after extracting a subsequence ?? _n ?? 0??to a point ${P_0\in\Lambda}$ with ${m(P_0)={\rm inf}_{P\in\Lambda}m(P)}$ . Moreover (v _1?? (x), v _2?? (x)) converges to a least energy non-trivial vector solution of (**) after a suitable rescaling.     

Nonlinear resonant periodic problems with concave terms

We consider a nonlinear periodic problem, driven by the scalar p-Laplacian with a concave term and a Caratheodory perturbation. We assume that this perturbation f(t,x) is (p−1)-linear at ±∞, and resonance can occur with respect to an eigenvalue λm+1, m?2, of the negative periodic scalar p-Laplacian. Using a combination of variational techniques, based on the critical point theory, with Morse theory, we establish the existence of at least three nontrivial solutions. Useful in our considerations is an alternative minimax characterization of λ₁>0 (the first nonzero eigenvalue) that we prove in this work.     

Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations

We consider the Dirichlet problem for positive solutions of the equation −Δ_m(u)=f(u) in a bounded smooth domain Ω, with f locally Lipschitz continuous, and prove some regularity results for weak solutions. In particular when f(s)>0 for s>0 we prove summability properties of , and Sobolev's and Poincaré type inequalities in weighted Sobolev spaces with weight |Du|^m−2. The point of view of considering |Du|^m−2 as a weight is particularly useful when studying qualitative properties of a fixed solution. In particular, exploiting these new regularity results we can prove a weak comparison principle for the solutions and, using the well known Alexandrov-Serrin moving plane method, we then prove a general monotonicity (and symmetry) theorem for positive solutions u of the Dirichlet problem in bounded (and symmetric in one direction) domains when f(s)>0 for s>0 and m>2. Previously, results of this type in general bounded (and symmetric) domains had been proved only in the case 1<m<2.     

Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations

In this paper we consider a class of planar autonomous systems having an isolated limit cycle x₀ of smallest period T>0 such that the associated linearized system around it has only one characteristic multiplier with absolute value 1. We consider two functions, defined by means of the eigenfunctions of the adjoint of the linearized system, and we formulate conditions in terms of them in order to have the existence of two geometrically distinct families of T-periodic solutions of the autonomous system when it is perturbed by nonsmooth T-periodic nonlinear terms of small amplitude. We also show the convergence of these periodic solutions to x₀ as the perturbation disappears and we provide an estimation of the rate of convergence. The employed methods are mainly based on the theory of topological degree and its properties that allow less regularity on the data than that required by the approach, commonly employed in the existing literature on this subject, based on various versions of the implicit function theorem.     

Nonlinear Nonhomogeneous Dirichlet Equations Involving a Superlinear Nonlinearity

We consider a nonlinear elliptic Dirichlet equation driven by a nonlinear nonhomogeneous differential operator involving a Carathéodory function which is (p?1)-superlinear but does not satisfy the Ambrosetti–Rabinowitz condition. First we prove a three-solutions-theorem extending an earlier classical result of Wang (Ann Inst H Poincaré Anal Non Linéaire 8(1):43–57, 1991). Subsequently, by imposing additional conditions on the nonlinearity \({f(x,\cdot)}\), we produce two more nontrivial constant sign solutions and a nodal solution for a total of five nontrivial solutions. In the special case of (p, 2)-equations we prove the existence of a second nodal solution for a total of six nontrivial solutions given with complete sign information. Finally, we study a nonlinear eigenvalue problem and we show that the problem has at least two nontrivial positive solutions for all parameters \({\lambda > 0}\) sufficiently small where one solution vanishes in the Sobolev norm as \({\lambda \to 0^+}\) and the other one blows up (again in the Sobolev norm) as \({\lambda \to 0^+}\).     

Qualitative Phenomena for Some Classes of Quasilinear Elliptic Equations with Multiple Resonance

We consider nonlinear nonhomogeneous Dirichlet problems driven by the sum of a p-Laplacian and a Laplacian. The hypotheses on the reaction term incorporate problems resonant at both ±∞ and zero. We consider both cases p>2 and 1<p<2 (singular case) and we prove four multiplicity theorems producing three or four nontrivial solutions. For the case p>2 we provide precise sign information for all the solutions. Our approach uses critical point theory, truncation and comparison techniques, Morse theory and the Lyapunoff-Schmidt reduction method.     

Estimation of the remainder of a cubature formula on a Chebyshev grid for two-variable functions

For a cubature formula of the form $$\int\limits_0^{2\pi } {\int\limits_0^{2\pi } {f(x,y)dxdy = \frac{{4\pi ^2 }} {{mn}}\sum\limits_{i = 0}^{n - 1} {\sum\limits_{j = 0}^{m - 1} {f\left( {\frac{{2\pi i}} {n},\frac{{2\pi j}} {m}} \right) + R_{n,m} (f)} } } }$$ on a Chebyshev grid, the remainder R _n,m (f) is proved to satisfy the sharp estimate $$\mathop {\sup }\limits_{f \in H\left( {r_1 ,r_2 } \right)} \left| {R_{n,m} (f)} \right| = O\left( {n^{ - r_1 + 1} + m^{ - r_1 + 1} } \right)$$ in some class of functions H(r ₁, r ₂) defined by a generalized shift operator. Here, r ₁, r ₂ > 1; ??^?1 ?? n/m ?? ?? with ?? > 0; and the constant in the O-term depends only on ??.     

Liouville type results for a <Emphasis Type="Italic">p</Emphasis>-Laplace equation with negative exponent

Positive entire solutions of the equation \(\Delta _p u = u^{ - q} in \mathbb{R}^N (N \geqslant 2)\) where 1 < p ≤ N, q > 0, are classified via their Morse indices. It is seen that there is a critical power q = q _c such that this equation has no positive radial entire solution that has finite Morse index when q > q _c but it admits a family of stable positive radial entire solutions when 0 < q ≤ q _c. Proof of the stability of positive radial entire solutions of the equation when 1 < p < 2 and 0 < q ≤ q _c relies on Caffarelli–Kohn–Nirenberg’s inequality. Similar Liouville type result still holds for general positive entire solutions when 2 < p ≤ N and q > q _c. The case of 1 < p < 2 is still open. Our main results imply that the structure of positive entire solutions of the equation is similar to that of the equation with p = 2 obtained previously. Some new ideas are introduced to overcome the technical difficulties arising from the p-Laplace operator.