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Radial entire solutions of a class of quasilinear elliptic equations
... The equations of this kind were considered by many authors in Euclidean spaces (see, for instance, [5]- [8]). The functions ( ) most frequently appearing in studies are of the following types, ( ) = −2 , > 1; ...
... Proof. Suppose the opposite, i.e., there exists no solution ( ) to equation (8) described in the statement of the lemma, but at the same time there exists a positive entire solution ( , ) of inequality (1). The positivity of solution to inequality in particular implies that ( ) > 0. Let ∈ (0, ( )), and [0; ) be the maximal segment of the existence for the solution to equation (8) subject to the conditions (0) = and ′ (0) = 0. ...
... Suppose the opposite, i.e., there exists no solution ( ) to equation (8) described in the statement of the lemma, but at the same time there exists a positive entire solution ( , ) of inequality (1). The positivity of solution to inequality in particular implies that ( ) > 0. Let ∈ (0, ( )), and [0; ) be the maximal segment of the existence for the solution to equation (8) subject to the conditions (0) = and ′ (0) = 0. By the assumption < ∞. ...
... For bounded domains, under hypotheses on f at the origin and infinite, but also considering the positone case, the existence of multiple positive solutions is proved by Fukagai and Narukawa, in [11], using the variational arguments in the Orlicz-Sobolev spaces. Related results can be found in [2,4,6,14,16]. ...
... Here, the constant C * can be taken positive due to (14). Thus, the hypothesis (f 4 ) is verified. ...
We prove an existence result concerning positive radial solutions for a nonlinear ϕ-Laplacian problem by using a shooting technique. The nonlinearity satisfies adequate conditions and has semipositone behaviour. Our result generalizes a previous result in the literature.
... For closely related results the reader is referred to the papers [4] [6] [7] [8] in which equations of the form (A) with different nonlinear functions φ are considered and oscillation theory for such equations is so designed as to apply to the partial differential equation ...
... The importance of such equations has been widely recognized in connection with the study of capillarity and surfaces with prescribed mean curvature; see e.g., [6] [7] [8]. ...
... known as the prescribed mean curvature equation or the capillary surface equation. Such a kind of problems has been deeply studied in the recent years: existence and non-existence results of solutions decaying to zero at infinity have been proved by [2,8,10,11,12,14,19,20,22], among others, under different assumptions on the nonlinearity f and on the function φ. Moreover, for bounded domains, we recall [7,13,17,18]. ...
... This, together with (14), implies that ...
In this paper we prove the existence of a nontrivial non-negative radial solution for the quasilinear elliptic problem (Formula presented.) where N ≥ 2, φ(t) behaves like tq/2 for small t and tp/2 for large t, 1 < p < q < N, 1 < α ≤ p*q′/p′ and max{q,α} < s < p*, being p* = pN/N-p and p′ and q′ the conjugate exponents, respectively, of p and q. Our aim is to approach the problem variationally by using the tools of critical points theory in an Orlicz-Sobolev space. A multiplicity result is also given.
... and for some generalizations can be found, among others, in [3,10,12,18,19,21,25,26]. More recently, a lot of attention has been paid on the prescribed mean curvature equation in the Lorentz-Minkowski case ...
This paper deals with the prescribed mean curvature equations both in the Euclidean case and in the Lorentz-Minkowski case in presence of a nonlinearity g such that . We show the existence of oscillating solutions, namely with an unbounded sequence of zeros. We show the existence of oscillating solutions, namely with an unbounded sequence of zeros. Moreover these solutions are periodic, if N=1, while they are radial symmetric and decay to zero at infinity with their derivatives, if .
... Generalizations of problems (1.2) and (1.3), including reactions involving a competition between power type nonlinearities, have been deeply studied in recent years in the context of nonlinear equations on bounded domains with different types of boundary conditions (see [3-5, 9, 10, 13, 17, 18, 21, 23, 25-27, 32] and the references therein), as well as either in the entire R N or in unbounded domains for various classes of nonlinearities (see [1,10,12,15,19,20,24,25,30]). ...
In this paper we study existence and nonexistence of positive radial solutions of a Dirichlet problem for the prescribed mean curvature operator with weights in a ball with a suitable radius. Because of the presence of different weights, possibly singular or degenerate, the problem under consideration appears rather delicate, it requires an accurate qualitative analysis of the solutions, as well as the use of Liouville type results based on an appropriate Pohozaev type identity. In addition, sufficient conditions for global solutions to be oscillatory are given.
... In this paper, we discuss the solvability of the generalized Hessian inequality where σ k (λ) = 1≤i 1 <···<i k ≤n λ i 1 · · · λ i k , λ = (λ 1 , λ 2 , · · · , λ n ) ∈ R n , k = 1, 2, · · · , n is the k-th elementary symmetric function, λ (D i (A (|Du|) D j u)) denotes the eigenvalues of the symmetric matrix of (D i (A (|Du|) D j u)), and A, f are two given positive continuous functions on (0, +∞). The generalized Hessian operator σ k (λ (D i (A (|Du|) D j u))), introduced by many authors [1,2,3], is an important class of fully nonlinear operator. It is a generalization of some typical operators we shall be interested in as follows: the m-k-Hessian operator for the case A(p) = p m−2 , m > 1 is treated by Trudinger and Wang [4]; the k-mean curvature operator for the case A(p) = (1 + p 2 ) − 1 2 is treaded by Concus and Finn [16] and Peletier and Serrin [17]; the generalized k-mean curvature operator for the case A(p) = (1 + p 2 ) −α , α < 1 2 and A(p) = p 2m−2 (1 + p 2m ) − 1 2 , m > 1 is treated by Tolksdorf [5], Usami [6] and Suzuki [7], respectively. ...
In this paper, we discuss the more general Hessian inequality including the Laplacian, p-Laplacian, mean curvature, Hessian, k-mean curvature operators, and provide a necessary and sufficient condition on the global solvability, which can be regarded as generalized Keller-Osserman conditions.
... For any q > 1, there exist radial solutions to (1.1), see [13,16]. We know that the pointwise differential inequality has exerted a great influence on the theory of elliptic partial differential equations. ...
In this paper, we are inspired by Ng\^{o}, Nguyen and Phan's [15] study of the pointwise inequality for positive -solutions of biharmonic equations with negative exponent by using the growth condition of solutions. They propose an open question of whether the growth condition is necessary to obtain the pointwise inequality. We give a positive answer to this open question. We establish the following local pointwise inequality for positive -solutions of the biharmonic equations with negative exponent where denotes the ball centered at with radius R, , , and some constants , , .
... − div{φ(|∇u| 2 )∇u} = g (u) have been studied, for example, in [2,7,9,13,14,22]. Especially the assumption (φ2) is related with so-called Δ 2 -condition in the literature. ...
In this paper, we study a class of quasilinear elliptic equations which appears in nonlinear optics. By using the mountain pass theorem together with a technique of adding one dimension of space (Hirata et al. in Topol Methods Nonlinear Anal 35:253–276, 2010; Jeanjean in Nonlinear Anal Theory Methods Appl 28:1633–1659, 1997), we prove the existence of a non-trivial weak solution for general nonlinear terms of Berestycki–Lions’ type. The existence of a radial ground state solution and a ground state solution is also established under stronger assumptions on the quasilinear term.
... In this case, it is well known that standing waves reproduce solitary waves traveling 93 in the direction ξ . 94 In [123] a ground state solution for problem (6) has been proved to hold under 95 suitable assumptions on the parameter h and the potential V . The problem reduces 96 to a semi-linear elliptic equation ...
The present chapter is concerned with a whole review of the well known
Schrödinger equation in a mixed case of nonlinearities. We precisely consider
a general nonlinear model characterized by a superposition of linear, sub-linear,
super-linear sometimes concave–convex power laws on the form f (u) = |u|p−1u±
|u|p−1u. In a first part, we develop theoretical results on existence, uniqueness,
classification as well as the behavior of the solutions of the ground state radial
problem according to the power laws and the initial value. Next, in a second part,
some examples are developed with graphical illustrations to confirm the theoretical
results exposed previously. The graphs show coherent states between the theoretical
findings and the numerical illustrations.
The chapter in its whole aim is a review of existing results about the studied
problems reminiscent of some few cases that are not previously developed. We
aim thus it will constitute a good reference especially for beginners in the field
of nonlinear analysis of PDEs.
... − div{φ(|∇u| 2 )∇u} = g(u) have been studied, for example, in [2,7,9,13,14,22]. Especially the assumption (φ2) is related with so-called ∆ 2 -condition in the literature. ...
In this paper, we study a class of quasilinear elliptic equations which appears in nonlinear optics. By using the mountain pass theorem together with a technique of adding one dimension of space, we prove the existence of a non-trivial weak solution for general nonlinear terms of Berestycki-Lions' type. The existence of a radial ground state solution and a ground state solution is also established under stronger assumptions on the quasilinear term.
... It has been proved in [6] that if 0 < p ≤ 1, the equation (1.1) admits no entire smooth solution. It is showed in [4,7] that for any p > 1, there exist radial solutions to (1.1). ...
In this note, we are interested in entire solutions to the semilinear biharmonic equation
\[
{\mathrm{\Delta }}^{2}u=-{u}^{-p},\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}u>0\phantom{\rule{1em}{0ex}}\text{in}\phantom{\rule{0.25em}{0ex}}{\mathbb{R}}^{N},
\] where and . In particular, the stability outside a compact set of the entire radial solutions will be completely studied, which resolves the remaining case in [5].
... and for some generalizations can be found, among others, in [3,10,11,17,18,20,24,25]. More recently, a lot of attention has been paid on the prescribed mean curvature equation in the Lorentz-Minkowski case ...
This paper deals with the prescribed mean curvature equations − div 1±∇u|∇u|2 = g(u) in RN , both in the Euclidean case, with the sign “+”, and in the Lorentz-Minkowski case, with the sign “−”, for N 1 under the assumption g(0) > 0. We show the existence of oscillating solutions, namely with an unbounded sequence of zeros. Moreover these solutions are periodic, if N = 1, while they are radial symmetric and decay to zero at infinity with their derivatives, if N 2. © 2018 American Institute of Mathematical Sciences. All Rights Reserved.
... Recently the problem (1.15) was discussed by Naito-Naito [17] for the sublinear case, and by Yanagida-Yotsutani [23] for the superlinear case. In both papers, it is shown that, under certain conditions on α and q(i), the problem (1.15) has an infinite sequence of solutions u k (r), fc = 1, 2, , such that u k (r) has exactly k — 1 zeros in (0, oo). ...
... When H is a radial function and behaves like c\x\ , /eR and c > 0, as |x| —@BULLET oo, Theorems A and B characterize the decaying order of H for (2) to admit positive entire solutions. Related results are found in [11] [12] [16]. ...
... The problem of existence and asymptotic behavior of positive radial entire solutions of (1.1) and more general equations has been extensively investigated by many authors (see, for example, [1][2][3][4][5][6] and the references therein). In ...
We consider the existence of entire solutions of a quasilinear elliptic equation div(|Du|^{p−2}Du)=k(x)f(u), x∈R^N, where p>1, N∈N. Conditions of the existence of entire solutions have been obtained by different authors. We prove an optimality of some of these results and give new sufficient conditions for the nonexistence of entire solutions.
... Existence, non-existence and multiplicity of positive solutions of problem (1) have been discussed by several authors in the last decades. The case where Ω is a ball and u is a classical radially symmetric solution has been studied, among others, by Ni and Serrin [32, 33, 34], Serrin [43], Peletier and Serrin [41], Atkinson, Peletier and Serrin [2], Ishimura [24], Kusano and Swanson [25], Clement, Manásevich and Mitidieri [11], Franchi, Lanconelli and Serrin [17], Bidaut-Veron [3], Conti and Gazzola [13], Chang and Zhang [9], del Pino and Guerra [16], also in relation with the existence of ground states. The one-dimensional problem has been rather thoroughly discussed in a series of recent papers by Bonheure, Habets, Obersnel and Omari [23, 5, 36, 6], Bereanu and Mawhin [4] and Pan [40]. ...
We discuss existence and multiplicity of positive solutions of the prescribed mean curvature problem in a general bounded domain Ω⊂RN, depending on the behavior at zero or at infinity of f(x,s), or of its potential . Our main effort here is to describe, in a way as exhaustive as possible, all configurations of the limits of F(x,s)/s2 at zero and of F(x,s)/s at infinity, which yield the existence of one, two, three or infinitely many positive solutions. Either strong, or weak, or bounded variation solutions are considered. Our approach is variational and combines critical point theory, the lower and upper solutions method and elliptic regularization.
... Most modern authors have focused on the existence, nonexistence and multiplicity of positive solutions. The case where Ω ⊆ R n , for n ≥ 2, has been studied by numerous authors: Concus and Finn [18, 19, 20, 21, 22, 23, 24]; Giusti [31, 32, 33, 34]; Gilbarg and Trudinger [30]; Ni and Serrin [48, 49, 50]; Finn [27, 28] (and the references therein); Peletier and Serrin [60]; Atkinson, Peletier and Serrin [3, 4]; Serrin [61]; Ishimura [37, 38]; Kusano and Swanson [40]; Nakao [47]; Noussair, Swanson and Jianfu [51]; Bidaut-Veron [9]; Clément, Manásevich and Mitidieri [16]; Coffman and Ziemer [17]; Conti and Gazzola [25]; Amster and Mariani [1]; Habets and Omari [35]; Le [42, 43]; Chang and Zhang [15]; del Pino and Guerra [26]; Moulton and Pelesko [45, 46]; Bereanu, Jebelean and Mawhin [7, 8]; Obersnel and Omari [53, 54]; Brubaker and Pelesko [13]; Brubaker and Lindsay [12]. Also, the case where n = 1 has been studied by numerous authors in a recent series of papers: Kusahara and Usami [39]; Benevieri, dò O and * Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA (Email addresses: de Medeiros [5, 6]; Bonheure, Habets, Obersnel and Omari [10, 11]; Habets and Omari [36]; Obersnel [52]; Pan [56]; Li and Liu [44]; Burns and Grinfeld [14]; Pan and Xing [58, 57]. ...
In this paper we analyze the classical solution set ({\lambda},u), for
{\lambda}>0, of a one-dimensional prescribed mean curvature equation on the
interval [-L,L]. It is shown that the solution set depends on the two
parameters, {\lambda} and L, and undergoes two bifurcations. The first is a
standard saddle node bifurcation, which happens for all L at {\lambda} =
{\lambda}*(L). The second is a splitting bifurcation; specifically, there
exists a value L* such that as L transitions from greater than or equal L* to
less than L* the upper branch of the bifurcation diagram splits into two parts.
In contrast, the solution set of the semilinear version of the prescribed mean
curvature equation is independent of L and exhibits only a saddle node
bifurcation. Therefore, as this analysis suggests, the splitting bifurcation is
a byproduct of the mean curvature operator coupled with the singular
nonlinearity.
In this paper, we study the existence of infinitely many
nodal solutions for the following quasilinear elliptic equation
\begin{equation*}
\begin{cases}
-\nabla\cdot\left[\phi'(|\nabla u|^2)\nabla u\right]+|u|^{\alpha-2}u=f(u) , \quad x\in \mathbb{R}^N,\\
u(x)\rightarrow 0, \quad \mbox{as} ~|x|\rightarrow \infty,
\end{cases}
\end{equation*}
where , behaves like for small t and for large t, $1
In this paper, we study a quasilinear equation in by a variational approach. The particular nature of the differential operator requires a preliminary study on the functional
space.
Our main result is the proof of the existence of a ground state.
In this article, we show that, under suitable assumptions, the generalized p-Laplacian boundary value problem has at least one solution.
We investigate the problem of implementation of the Liouville type theorems on the existence of positive solutions of some quasilinear elliptic inequalities on model (spherically symmetric) Riemannian manifolds. In particular, we find exact conditions for the existence and nonexistence of entire positive solutions of the studied inequalities on the Riemannian manifolds. The method is based on a study of radially symmetric solutions of an ordinary differential equation generated by the basic inequality and establishing the relationship of the existence of entire positive solutions of quasilinear elliptic inequalities and solvability of the Cauchy problem for this equation. In addition, in the paper we apply classical methods of the theory of elliptic equations and second order inequalities (the maximum principle, the principle of comparison, etc.). The obtained results generalize similar results obtained previously by Y. Naito and H.Usami for Euclidean space Rn, as well as some earlier results by A. G. Losev and E. A. Mazepa.
In this article, we prove a compact embedding theorem for the weighted Orlicz-Sobolev space of radially symmetric functions. Using the embedding theorem and critical points theory, we prove the existence of multiple radial solutions and radial ground states for the following modified capillary surface equation -div (vertical bar del u vertical bar(2p-2)del u/root 1+vertical bar del u vertical bar(2p)) + T(vertical bar x vertical bar)vertical bar u vertical bar(alpha-2)u = K(vertical bar x vertical bar)vertical bar u vertical bar(s-2)u, u > 0, x is an element of R-N, u(vertical bar x vertical bar) -> 0, as vertical bar x vertical bar -> infinity, where N >= 3, 1 < alpha < p < 2p < N, s satisfies some suitable conditions, K(vertical bar x vertical bar) and T(vertical bar x vertical bar) are continuous, nonnegative functions.
This paper treats the quasilinear elliptic inequality
where N ≥ 2, m > 1, σ > m − 1, and p :ℝ N → (0, ∞ ) is continuous. Sufficient conditions are given for this inequality to have no positive entire solutions. When p has radial symmetry, the existence of positive entire solutions can be characterized by our results and some known results.
We study positive radial entire solutions of second-order quasilinear elliptic systems of the form Delta(p)u = f(|x|, u, upsilon), Delta(q)upsilon = g(|x|, u, upsilon), x is an element of R-N. Sufficient and/or necessary conditions of f and g are obtained for (*) to have positive radial entire solutions with prescribed asymptotic behavior at infinity. We first study the upper and lower estimates for the nonlinear integral operator, which is "inverse" of one-dimensional polar form of the N-dimensional m-Laplace operator.
A version of the Sturm-Liouville theory is given for the one-dimensional p-Laplacian including the radical case. The treatment is modern but follows the strategy of Elbert’s early work. Topics include a Prüfer-type transformation, eigenvalue existence, asymptotics and variational principles, and eigenfunction oscillation.
We investigate the blow-up of solutions of nonuniformly parabolic equations. It will be shown that the gradient of the solution tends to infinity as time goes to infinity, even though theL∞-norm of the solution may remain uniformly bounded. This phenomenon is in contrast with a result for semilinear heat equations, where it follows from regularity theory that the solution converges to an equilibrium if it is uniformly bounded in theL∞-norm.
We shall consider here the question of existence and nonexistence of ground states for the prescribed mean curvature equation in ℝn
(n > 2), that is, we consider solutions of the problem \left. {\begin{array}{*{20}{c}} {div\left( {\frac{{Du}}{{{{\left( {1 + + {{\left| {Du} \right|}^2}} \right)}^{{1/2}}}}}} \right) + f(u) = 0\;in {\mathbb{R}^n}} \\ {u > 0\quad in\,{\mathbb{R}^n}} \\ {u(x) \to 0\;as\,x \to \infty } \\ \end{array} } \right\} (1.1) where Du denotes the gradient of u. The function f(u), defined for u > 0, will be assumed throughout to satisfy the following hypotheses: (H1) (H2) f(0) = 0, and there exists a number a ≥ 0 such that if a > 0 we require .
In this note we study the existence and the uniqueness of radially symmetric ground states for the quasilinear elliptic equation
Semilinear elliptic partial differential equations of the type
1
will be considered throughout real Euclidean N -space, where m ≧ 2 is a positive integer, Δ denotes the N -dimensional Laplacian, and f is a real-valued continuous function in [0, ∞) × (0, ∞). Detailed hypotheses on the structure of f are listed in Section 3.
Our objective is to prove the existence of radially symmetric positive entire solutions u(x) of (1) which are asymptotic to positive constant multiples of | x | 2 m −2 i as | x | → ∞ for every i = 1,…, m , N ≧ 2 i + 1.
On etudie les etats fondamentaux singuliers de l'equation de Poisson semilineaire Δu+f(u)=0 et de son analogue quasi-lineaire div(A(|Du|)Du)+f(u)=0 ou Δ est le laplacien a n dimensions et A(p) et f(u) sont des fonctions donnees
We study the singular solutions of the capillarity equation Hiv Dυ » *>, inR", l/l + |/>y|2 with a A' < 0. We prove the global existence of a rotationally symmetric solution. We prove the uniqueness of a symmetric solution negative and concave near the origin. Introduction. In this paper we study the existence and uniqueness of a singular solution of the capillarity equation in R^: (1) div( £>*;/(/l + \Dv\2) = Kυ9 with a K < 0. The situation is quite different from the case K > 0, where every isolated singularity is removable (4). We restrict our attention to the symmetric case where υ depends only on the distance r from the origin. Let
13) has a decaying positive radial entire solution in RN (N> 3) for all suhiciently small A > 0. (III) If 171 < 1, 0 f F(t) 60 in R
- Eq Shows
shows that Eq. (7.13) has a decaying positive radial entire solution in RN
(N> 3) for all suhiciently small A > 0.
(III) If 171 < 1, 0 f F(t) 60 in R,, and (7.15) holds, Theorems 7.2
14) holds, Theorem 5.1-5.3 imply that Eq. (7.13) has a radial entire solution in RN (N B 3) which is bounded above and below by positive constants in the following cases: (i) y = 1 and lI
- I If
I) If (7.14) holds, Theorem 5.1-5.3 imply that Eq. (7.13) has a
radial entire solution in RN (N B 3) which is bounded above and below by
positive constants in the following cases:
(i) y = 1 and lI( sufficiently small.