Mountain pass theorem
View article: Infinitely many solutions for the stationary Kirchhoff problems involving the fractional<i>p</i>-Laplacian
Infinitely many solutions for the stationary Kirchhoff problems involving the fractional<i>p</i>-Laplacian Open
The aim of this paper is to establish the multiplicity of weak solutions for a Kirchhoff-type problem driven by a fractional p-Laplacian operator with homogeneous Dirichlet boundary conditions.
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Combined effects for fractional Schrödinger–Kirchhoff systems with critical nonlinearities Open
In this paper, we investigate the existence of solutions for critical Schrödinger–Kirchhoff type systems driven by nonlocal integro–differential operators. As a particular case, we consider the following system: [see formula in PDF] where …
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<i>p</i> -fractional Hardy–Schrödinger–Kirchhoff systems with critical nonlinearities Open
This paper deals with the existence of nontrivial solutions for critical Hardy–Schrödinger–Kirchhoff systems driven by the fractional p -Laplacian operator. Existence is derived as an application of the mountain pass theorem and the Ekelan…
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Fractional elliptic problems with critical growth in the whole of $\R^n$ Open
We study the following nonlinear and nonlocal elliptic equation in~$\R^n$ $$ (-Δ)^s u = ε\,h\,u^q + u^p \ {\mbox{ in }}\R^n, $$ where~$s\in(0,1)$, $n>2s$, $ε>0$ is a small parameter, $p=\frac{n+2s}{n-2s}$, $q\in(0,1)$, and~$h\in L^1(\R^n)\…
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Multiplicity and concentration of positive solutions for the fractional Schrödinger–Poisson systems with critical growth Open
In this paper, we study the multiplicity and concentration of solutions for the following critical fractional Schrödinger–Poisson system: where ϵ> 0 is a small parameter, (− △ )α denotes the fractional Laplacian of order α = s,t ∈ (0,1), w…
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Nonlocal Schrödinger-Kirchhoff equations with external magnetic field Open
The paper deals with the existence and multiplicity of solutions of the fractional Schrödinger-Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case $\begin{equation*}…
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Nonlocal Kirchhoff Problems with Singular Exponential Nonlinearity Open
In this paper, we first develop the fractional Trudinger–Moser inequality in singular case and then we use it to study the existence and multiplicity of solutions for a class of perturbed fractional Kirchhoff type problems with singular ex…
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Fractional elliptic equations with critical exponential nonlinearity Open
We study the existence of positive solutions for fractional elliptic equations of the type (-Δ) 1/2 u = h ( u ), u > 0 in (-1,1), u = 0 in ℝ∖(-1,1) where h is a real valued function that behaves like e u 2 as u → ∞ . Here (-Δ) 1/2 is the f…
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Existence of solutions for a class of p(x)-laplacian equations involving a concave-convex nonlinearity with critical growth in R^{N} Open
We prove the existence of solutions for a class of quasilinear problems involving variable exponents and\nwith nonlinearity having critical growth. The main tool used is the variational method, more precisely, Ekeland's Variational Princip…
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Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity Open
It is shown that the planar Schrödinger-Poisson system with a general nonlinear interaction function has a nontrivial solution of mountain-pass type and a ground state solution of Nehari-Pohozaev type. The conditions on the nonlinear funct…
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Saddle point solutions for non-local elliptic operators Open
The paper deals with equations driven by a non-local integrodifferential operator Lk with homogeneous Dirichlet boundary conditions. These equations have a variational structure and we find a solution for them using the Saddle Point Theore…
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Solutions of nonlinear problems involving <i>p</i> ( <i>x</i> )-Laplacian operator Open
In the present paper, by using variational principle, we obtain the existence and multiplicity of solutions of a nonlocal problem involving p ( x )-Laplacian. The problem is settled in the variable exponent Sobolev space W 0 1, p ( x ) (Ω)…
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Existence and symmetry result for fractional <i>p</i>-Laplacian in $\mathbb{R}^{n}$ Open
In this article we are interested in the following fractional $p$-Laplacian equation in $\mathbb{R}^n$$(-\Delta)_{p}^{s}u + V(x)|u|^{p-2}u = f(x,u) \mbox{ in } \mathbb{R}^{n},$where $p\geq 2$, $0 < s < 1$, $n\geq 2$ and $f$ is $p$-superlin…
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Boundary value problem with fractional <i>p</i> -Laplacian operator Open
The aim of this paper is to obtain the existence of solution for the fractional p -Laplacian Dirichlet problem with mixed derivatives t D T α (| 0 D t α u ( t )| p -2 0 D t α u ( t )) = f ( t , u ( t )), t ∈ [0, T ], u (0) = u ( T ) = 0, w…
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Fractional elliptic problems with critical growth in the whole of (R^n) Open
We study a nonlinear and nonlocal elliptic equation. The problem has a variational structure, and this allows us to find a positive solution by looking at critical points of a suitable energy functional. In particular, in this paper, we fi…
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Noncoercive resonant (<i>p</i>,2)-equations with concave terms Open
We consider a nonlinear Dirichlet problem driven by the sum of a p -Laplace and a Laplacian (a (p,2) {(p,2)} -equation). The reaction exhibits the competing effects of a parametric concave term plus a Caratheodory perturbation which is res…
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Existence and multiplicity of nontrivial solutions for a nonlocal problem Open
In this paper, we study the existence and multiplicity of nontrivial solutions for a new nonlocal problem. Variational method, mountain pass lemma. Some existence and multiplicity results of nontrivial solutions are obtained.
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( <i>p</i> ,2)-equations asymmetric at both zero and infinity Open
We consider a ( p , 2 ) {(p,2)} -equation, that is, a nonlinear nonhomogeneous elliptic equation driven by the sum of a p -Laplacian and a Laplacian with p > 2 {p>2} . The reaction term is ( p - 1 ) {(p-1)} -linear, but exhib…
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Existence of solutions for critical (p,q)-Laplacian equations in ℝN Open
In this paper, we are mainly interested in existence properties for a class of nonlinear PDEs driven by the ([Formula: see text])-Laplace operator where the reaction combines a power-type nonlinearity at critical level with a subcritical t…
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Critical Schrödinger-Hardy systems in the Heisenberg group Open
The paper is focused on existence of nontrivial solutions of a Schrödinger-Hardy system in the Heisenberg group, involving critical nonlinearities. Existence is obtained by an application of the mountain pass theorem and the Ekeland variat…
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Existence of positive radial solutions for a problem involving the weighted Heisenberg $ p(\cdot) $-Laplacian operator Open
A variational principle is applied to examine a Muckenhoupt weighted $ p(\cdot) $-Laplacian equation on the Heisenberg groups. The existence of at least one positive radial solution to the problem under the Dirichlet boundary condition bel…
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Combined Effects of Concave-Convex Nonlinearities in a Fourth-Order Problem with Variable Exponent Open
We study two classes of nonhomogeneous elliptic problems with Dirichlet boundary condition and involving a fourth-order differential operator with variable exponent and power-type nonlinearities. The first result of this paper establishes …
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Three nontrivial solutions for nonlinear fractional Laplacian equations Open
We study a Dirichlet-type boundary value problem for a pseudodifferential equation driven by the fractional Laplacian, proving the existence of three nonzero solutions. When the reaction term is sublinear at infinity, we apply the second d…
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Constant sign and nodal solutions for superlinear ( <i>p</i> , <i>q</i> )–equations with indefinite potential and a concave boundary term Open
We consider a nonlinear elliptic equation driven by the ( p , q )–Laplacian plus an indefinite potential. The reaction is ( p − 1)–superlinear and the boundary term is parametric and concave. Using variational tools from the critical point…
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Existence of positive solutions for a class of semipositone quasilinear problems through Orlicz-Sobolev space Open
In this paper we show the existence of weak solutions for a class of semipositone problems of the type \begin{equation}\tag {P} \left \{ \begin {array}{rclcl} -\Delta _{\Phi } u & = & f(u)-a & \mbox {in} & \Omega , \\ u(x)& > & 0 & \mbox {…
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A Priori Bounds and Existence of Solutions for Slightly Superlinear Elliptic Problems Open
We consider the semilinear elliptic problem where a is a continuous function which may change sign and f is superlinear but does not satisfy the standard Ambrosetti-Rabinowitz condition. We show that if f is regularly varying of index one …
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MULTIPLICITY OF SOLUTIONS FOR A CLASS OF NON-LOCAL ELLIPTIC OPERATORS SYSTEMS Open
In this paper, we investigate the existence and multiplicity of solutions for systems driven by two non-local integrodifferential operators with homogeneous Dirichlet boundary conditions. The main tools are the Saddle point theorem, Ekelan…
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Nonlocal Schrödinger-Kirchhoff equations with external magnetic field Open
The paper deals with existence and multiplicity of solutions of the fractional Schrödinger--Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case \begin{equation*} (a+…
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An existence theorem for a non-autonomous second order nonlocal multivalued problem Open
FIGURES 28–36. Jilinga asymmetrica sp. nov.: 28, Male pygofer side, lateral view; 29, Anal tube, ventral view; 30, Valve, ventral view; 31, Subgenital plate, ventral view; 32, Aedeagus and dorsal connective, lateral view; 33, Aedeagus and …
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On multiple solutions to a nonlocal fractional $p(\cdot )$-Laplacian problem with concave–convex nonlinearities Open
The aim of this paper is to examine the existence of at least two distinct nontrivial solutions to a Schrödinger-type problem involving the nonlocal fractional $p(\cdot )$ -Laplacian with concave–convex nonlinearities when, in general…