The existence of a positive solution to asymptotically linear scalar field equations

2000 ◽  
Vol 130 (1) ◽  
pp. 81-105 ◽  
Author(s):  
G. Li ◽  
H.-S. Zhou
Keyword(s):  
Positive Solution ◽  
Nonlinear Term ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Field Equations ◽  
Usual Condition ◽  
Image Position ◽  
Scalar Field Equations ◽  
Constraint Method ◽  
Linear Scalar

We consider the following elliptic equation: where m > 0, f(x, u)/u tends to a positive constant as u → + ∞. Here, the nonlinear term f(x, u) does not satisfy the usual condition, that is, for some θ > 0, which is important in using the mountain pass theorem. The aim of this paper is to discuss how to use the mountain pass theorem to show the existence of a positive solution to the present problem when we lose the above condition. Furthermore, if f(x, u) ≡ f(u), we also prove that the above problem has a ground state by using the artificial constraint method.

Quasilinear Scalar Field Equations Involving Critical Sobolev Exponents and Potentials Vanishing at Infinity

2018 ◽  
Vol 61 (3) ◽  
pp. 705-733 ◽  
Author(s):  
Athanasios N. Lyberopoulos
Keyword(s):  
Scalar Field ◽  
Weak Solutions ◽  
Bound States ◽  
Critical Sobolev Exponent ◽  
Field Equations ◽  
Critical Sobolev Exponents ◽  
Sobolev Exponent ◽  
Image Position ◽  
Scalar Field Equations

AbstractWe are concerned with the existence of positive weak solutions, as well as the existence of bound states (i.e. solutions inW1,p(ℝN)), for quasilinear scalar field equations of the form$$ - \Delta _pu + V(x) \vert u \vert ^{p - 2}u = K(x) \vert u \vert ^{q - 2}u + \vert u \vert ^{p^ * - 2}u,\qquad x \in {\open R}^N,$$where Δpu: =div(|∇u|p−2∇u), 1 <p<N,p*: =Np/(N−p) is the critical Sobolev exponent,q∈ (p, p*), whileV(·) andK(·) are non-negative continuous potentials that may decay to zero as |x| → ∞ but are free from any integrability or symmetry assumptions.

scholarly journals Multiplicity results for Kirchhoff type elliptic problems with Hardy potential

2019 ◽  
Vol 38 (4) ◽  
pp. 31-50
Author(s):  
M. Bagheri ◽  
Ghasem A. Afrouzi
Keyword(s):  
Local Minimum ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Mountain Pass Theorem ◽  
Elliptic Problems ◽  
Mountain Pass ◽  
Hardy Potential ◽  
Multiplicity Results ◽  
Kirchhoff Type ◽  
Minimum Theorem

In this paper, we are concerned with the existence of solutions for fourth-order Kirchhoff type elliptic problems with Hardy potential. In fact, employing a consequence of the local minimum theorem due to Bonanno and mountain pass theorem we look into the existence results for the problem under algebraic conditions with the classical Ambrosetti-Rabinowitz (AR) condition on the nonlinear term. Furthermore, by combining two algebraic conditions on the nonlinear term using two consequences of the local minimum theorem due to Bonanno we ensure the existence of two solutions, applying the mountain pass theorem given by Pucci and Serrin we establish the existence of third solution for our problem.

Decaying Solutions Of 2mth Order Elliptic Problems

1991 ◽  
Vol 43 (3) ◽  
pp. 449-460 ◽  
Author(s):  
W. Allegretto ◽  
L. S. Yu
Keyword(s):  
Positive Solution ◽  
Elliptic Problem ◽  
Mountain Pass Theorem ◽  
Elliptic Problems ◽  
Mountain Pass ◽  
Weighted Spaces ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Regularity Estimates ◽  
Open Question

AbstractWe consider a semilinear elliptic problem , (n > 2m). Under suitable conditions on f, we show the existence of a decaying positive solution. We do not employ radial arguments. Our main tools are weighted spaces, various applications of the Mountain Pass Theorem and LP regularity estimates of Agmon. We answer an open question of Kusano, Naito and Swanson [Canad. J. Math. 40(1988), 1281-1300] in the superlinear case: , and improve the results of Dalmasso [C. R. Acad. Sci. Paris 308(1989), 411-414] for the case .

Multiple bound states of higher topological type for semi-classical Choquard equations

2020 ◽  
pp. 1-27
Author(s):  
Xiaonan Liu ◽  
Shiwang Ma ◽  
Jiankang Xia
Keyword(s):  
Bound States ◽  
Local Minimum ◽  
Mountain Pass Theorem ◽  
Riesz Potential ◽  
Topological Type ◽  
Mountain Pass ◽  
Choquard Equation ◽  
Symmetric Mountain Pass Theorem ◽  
Image Position ◽  
Minimum Points

Abstract We are concerned with the semi-classical states for the Choquard equation $$-{\epsilon }^2\Delta v + Vv = {\epsilon }^{-\alpha }(I_\alpha *|v|^p)|v|^{p-2}v,\quad v\in H^1({\mathbb R}^N),$$ where N ⩾ 2, I α is the Riesz potential with order α ∈ (0, N − 1) and 2 ⩽ p < (N + α)/(N − 2). When the potential V is assumed to be bounded and bounded away from zero, we construct a family of localized bound states of higher topological type that concentrate around the local minimum points of the potential V as ε → 0. These solutions are obtained by combining the Byeon–Wang's penalization approach and the classical symmetric mountain pass theorem.

A BERESTYCKI–LIONS THEOREM REVISITED

2012 ◽  
Vol 14 (05) ◽  
pp. 1250033 ◽  
Author(s):  
JIANJUN ZHANG ◽  
WENMING ZOU
Keyword(s):  
Scalar Field ◽  
Ground State ◽  
Mountain Pass ◽  
Field Equations ◽  
Nonlinear Scalar Field ◽  
Growth Restrictions ◽  
Scalar Field Equations ◽  
Least Energy Solutions ◽  
Least Energy

In 1983, Berestycki and Lions [Nonlinear scalar field equations I. Existence of a ground state, Arch. Ration. Mech. Anal.82 (1983) 313–346] studied the following elliptic problem: [Formula: see text] where N ≥ 3, g is subcritical at infinity. They proved the existence of a ground state under some appropriate growth restrictions on g. In the present paper, we improve this result by showing that under the critical growth assumption on g the problem admits a ground state. In addition we study a mountain pass characterization of the least energy solutions of the problem. Without the assumption of the monotonicity of the function [Formula: see text], we show that the mountain pass value gives the least energy level.

Multiplicity of solutions to the weighted critical quasilinear problems

2012 ◽  
Vol 55 (1) ◽  
pp. 181-195 ◽  
Author(s):  
Sihua Liang ◽  
Jihui Zhang
Keyword(s):  
Bounded Domain ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Concentration Compactness ◽  
Positive Parameter ◽  
Symmetric Mountain Pass Theorem ◽  
Concentration Compactness Principle ◽  
Compactness Principle ◽  
Quasilinear Problems ◽  
Image Position

AbstractWe consider a class of critical quasilinear problemswhere 0 ∈ Ω ⊂ ℝN, N ≥ 3, is a bounded domain and 1 < p < N, a < N/p, a ≤ b < a + 1, λ is a positive parameter, 0 ≤ μ < $\bar{\mu}$ ≡ ((N − p)/p − a)p, q = q*(a, b) ≡ Np/[N − pd] and d ≡ a+1 − b. Infinitely many small solutions are obtained by using a version of the symmetric Mountain Pass Theorem and a variant of the concentration-compactness principle. We deal with a problem that extends some results involving singularities not only in the nonlinearities but also in the operator.

Schrödinger–Poisson system with potential of critical growth

2016 ◽  
Vol 09 (04) ◽  
pp. 1650086
Author(s):  
Abdessamad Hassani ◽  
Khalid Iskafi
Keyword(s):  
Variational Principle ◽  
Positive Solution ◽  
Mountain Pass Theorem ◽  
Critical Growth ◽  
Ekeland Variational Principle ◽  
Mountain Pass ◽  
Positive Energy ◽  
Poisson System ◽  
Nonlinear Potential

In this paper, we consider a Schrödinger–Poisson system with nonlinear potential of critical growth, and we prove existence of positive solution with positive energy by using the Ekeland variational principle and the Mountain-Pass theorem.

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