1,720,976 research outputs found
Elliptic problems involving the fractional Laplacian in R^N
No full textWe study the existence and multiplicity of solutions for elliptic equations in R^N, driven by a non-local integro-differential operator, which main prototype is the fractional Laplacian. The model under consideration, denoted by (P), depends on a real parameter and involves two superlinear nonlinearities, one of which could be critical or even supercritical. The main theorem of the paper establishes the existence of three critical values of \lambda which divide the real line in different intervals, where (P) admits no solutions, at least one nontrivial non-negative entire solution and two nontrivial non-negative entire solutions
Kirchhoff Systems with nonlinear source and boundary damping terms
No full textAbstract. In this paper we treat the question of the non–existence of global solutions, or their long time behavior, of nonlinear hyperbolic Kirchhoff systems. The main p–Kirchhoff operator may be affected by a perturbation and the systems also involve an external force f
and a nonlinear boundary damping Q. When p = 2, we consider some problems involving a higher order dissipation term, under dynamic boundary conditions. Special subcases of f and Q, interesting in applications, are presented in Sections 4, 5 and 6
Existence of entire solutions for a class of quasilinear elliptic equations
No full textThe paper deals with the existence of entire solutions for a quasilinear equation (Eλ) in RN, depending on a real parameter λ, which involves a general elliptic operator in divergence form A and two main nonlinearities. The competing nonlinear terms combine each other, being the first subcritical and the latter supercritical. We prove the existence of a critical value λ*>0 with the property that (Eλ) admits nontrivial non-negative entire solutions if and only if λ≥λ*. Furthermore, when λ>λ**≥λ*, the existence of a second independent nontrivial non-negative entire solution of (Eλ) is proved under a further natural assumption on A
SADDLE TYPE SOLUTIONS FOR A CLASS OF REVERSIBLE ELLIPTIC EQUATIONS
No full textThis paper is concerned with the existence of saddle type solutions for a class of semilinear elliptic equations of the type
−∆u(x)+Fu(x,u) = 0, x ∈ Rn, n ≥ 2, (PDE)
where F is a periodic and symmetric nonlinearity. Under a non degen- eracy condition on the set of minimal periodic solutions, saddle type solutions of (PDE) are found by a renormalized variational procedure
Kirchhoff systems with dynamic boundary conditions
No full textWe are interested in the study of the global non-existence of solutions of hyperbolic nonlinear problems, governed by the p-Kirchhoff operator, under dynamic boundary conditions, when p > p_n with p_n < 2. The systems involve nonlinear external forces and may be affected by a perturbation. Several models already treated in the literature are covered in special subcases, and concrete examples are provided for the
source term f and the external nonlinear boundary damping Q
Global Nonexistence for Nonlinear Kirchhoff Systems
No full textIn this paper we consider the problem of non-continuation of solutions of dissipative nonlinear Kirchhoff systems, involving the p(x)-Laplacian operator and governed by nonlinear driving forces f = f (t, x, u), as well as nonlinear external damping terms Q = Q(t, x, u, u_t ), both of which could significantly dependent on the time t . The theorems are obtained through the study of the natural energy Eu associated to the solutions u of the systems. Thanks to a new approach of the classical potential well and concavity methods, we show the nonexistence of global solutions, when the initial energy is controlled above by a critical value; that is, when the initial data belong to a specific region in the phase plane. Several consequences, interesting in applications, are given in particular subcases. The results are original also for the scalar standard wave equation when p(x)=2 and even for problems linearly damped
Existence theorems for quasilinear elliptic eigenvalue problems in unbounded domains
No full textThe paper focuses on the existence of nontrivial solutions of a nonlinear eigenvalue problem depending on a real parameter λ under Robin boundary conditions in unbounded domains, with (possibly noncompact) smooth boundary. The problem involves a weighted p-Laplacian operator and subcritical nonlinearities and even in the case p=2 the main existence results are new. Denoted by λ1 the first eigenvalue of the underlying Robin eigenvalue problem, we prove the existence of (weak) solutions, with different methods, according to the case λ≥λ1 or λ<λ1. In the first part of the paper we show the existence of a nontrivial solution for all λ in R for the problem under Ambrosetti-Rabinowitz type conditions on the nonlinearities involved in the model. In details, we apply the Mountain Pass theorem of Ambrosetti and Rabinowitz if λ<λ1, while we use mini-max methods and linking structures over cones, as in Degiovanni [On topological and metric critical point theory, J. Fixed Point Theory Appl. 7 (2010), 85-102] and in Degiovanni and Lancelotti [Linking over cones and nontrivial solutions for p-Laplacian equations with p-superlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), 907-919], if λ≥λ1. In the latter part of the paper we do not require any longer the Ambrosetti-Rabinowitz condition at infinity, but the so called Szulkin-Weth conditions and we obtain the same result for all λ in R. More precisely, using the Nehari manifold method for C1 functionals developed by Szulkin and Weth [The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, D.Y. Gao and D. Motreanu eds., International Press, Boston, 2010, 597-632], we prove existence of ground states, multiple solutions and least energy sign-changing solutions, whenever λ<λ1. On the other hand, in the case λ≥λ1, we establish the existence of solutions again by linking methods
Existence theorems for quasilinear elliptic eigenvalue problems in unbounded domains
No full textThe paper focuses on the existence of nontrivial solutions of a nonlinear eigenvalue problem depending on a real parameter λ under Robin boundary conditions in unbounded domains, with (possibly noncompact) smooth boundary. The problem involves a weighted p-Laplacian operator and subcritical nonlinearities and even in the case p=2 the main existence results are new. Denoted by λ1 the first eigenvalue of the underlying Robin eigenvalue problem, we prove the existence of (weak) solutions, with different methods, according to the case λ≥λ1 or λ<λ1. In the first part of the paper we show the existence of a nontrivial solution for all λ in R for the problem under Ambrosetti-Rabinowitz type conditions on the nonlinearities involved in the model. In details, we apply the Mountain Pass theorem of Ambrosetti and Rabinowitz if λ<λ1, while we use mini-max methods and linking structures over cones, as in Degiovanni [On topological and metric critical point theory, J. Fixed Point Theory Appl. 7 (2010), 85-102] and in Degiovanni and Lancelotti [Linking over cones and nontrivial solutions for p-Laplacian equations with p-superlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), 907-919], if λ≥λ1. In the latter part of the paper we do not require any longer the Ambrosetti-Rabinowitz condition at infinity, but the so called Szulkin-Weth conditions and we obtain the same result for all λ in R. More precisely, using the Nehari manifold method for C1 functionals developed by Szulkin and Weth [The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, D.Y. Gao and D. Motreanu eds., International Press, Boston, 2010, 597-632], we prove existence of ground states, multiple solutions and least energy sign-changing solutions, whenever λ<λ1. On the other hand, in the case λ≥λ1, we establish the existence of solutions again by linking methods
Asymptotic stability for Kirchhoff systems in variable exponent Sobolev spaces
No full textWe deal with the question of global and local asymptotic stability, as time tends to infinity, of solutions of dissipative anisotropic Kirchhoff systems, governed by the p(x)-Laplacian operator, in the framework of the variable exponent Sobolev spaces. Concrete applications are presented in special subcases of the external force f and the distributed damping Q involved in the systems
Existence results for singular nonlinear BVPs in the critical regime
Full text linkWe study the existence of solutions for a class of boundary value problems on the half line, associated to a third order ordinary differential equation governed by the
Φ-Laplacian operator. The equation contains a Carathéodory function satisfying a weak growth condition of Winter-Nagumo type which is assumed to be
continuous and it may vanish in a subset of zero Lebesgue measure, so that the problem
can be singular. The approach we follow is based on fixed point techniques
combined with the upper and lower solutions method
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