We prove the existence of a spherically symmetric solution for a Schrödinger equation with a nonlocal nonlinearity of Choquard type. This term is assumed to be subcritical and satisfy almost optimal assumptions. The mass of of the solution, described by its norm in the Lebesgue space, is prescribed in advance. The approach to this constrained problem relies on a Lagrange formulation and new deformation arguments. In addition, we prove that the obtained solution is also a ground state, which means that it realizes minimal energy among all the possible solutions to the problem.

We study existence of solutions for the fractional problemwhere N 2, s ∈ (0, 1), m > 0, μ is an unknown Lagrange multiplier and satisfies Berestycki-Lions type conditions. Using a Lagrangian formulation of the problem (P m ), we prove the existence of a weak solution with prescribed mass when g has L 2 subcritical growth. The approach relies on the construction of a minimax structure, by means of a Pohozaev's mountain in a product space and some deformation arguments under a new version of the Palais-Smale condition introduced in Hirata and Tanaka (2019 Adv. Nonlinear Stud. 19 263-90); Ikoma and Tanaka (2019 Adv. Differ. Equ. 24 609-46). A multiplicity result of infinitely many normalized solutions is also obtained if g is odd.

In this paper, we investigate semi-classical bound states for logarithmic Schrödinger type equations with a potential function which has a finite number of singularities of at most logarithmic strength. We construct localized solutions concentrating at a logarithmic type singular point of the potential, and we also characterize the asymptotic limiting profile of the localized solutions. To accomplish these, we develop new penalization techniques for treating the difficulties associated with the non-smoothness of the variational formulation and the singularity of the potentials.

<p lang="fr">&lt;abstract&gt;&lt;p&gt;Goal of this paper is to study the following doubly nonlocal equation&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document} $(- \Delta)^s u + \mu u = (I_\alpha*F(u))F'(u) \quad {\rm{in } }\;{\mathbb{R}^N}\qquad\qquad\qquad\qquad ({\rm{P } }) $ \end{document} &lt;/tex-math&gt; &lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;in the case of general nonlinearities $ F \in C^1(\mathbb{R}) $ of Berestycki-Lions type, when $ N \geq 2 $ and $ \mu &amp;gt; 0 $ is fixed. Here $ (-\Delta)^s $, $ s \in (0, 1) $, denotes the fractional Laplacian, while the Hartree-type term is given by convolution with the Riesz potential $ I_{\alpha} $, $ \alpha \in (0, N) $. We prove existence of ground states of (P). Furthermore we obtain regularity and asymptotic decay of general solutions, extending some results contained in &lt;sup&gt;[&lt;xref ref-type="bibr" rid="b23"&gt;23&lt;/xref&gt;,&lt;xref ref-type="bibr" rid="b61"&gt;61&lt;/xref&gt;]&lt;/sup&gt;.&lt;/p&gt;&lt;/abstract&gt;</p>

We study existence of radially symmetric solutions for the nonlocal problem: where (formula presetned) a unknown Lagrange multiplier. Using a Lagrange formulation of the problem (1 ), we develop new deformation arguments under a version of the Palais-Smale condition introduced in the recent papers (Hirata and Tanaka, Adv Nonlinear Stud 19:263–290, 2019; Ikoma and Tanaka, Adv Differ Equ 24:609–646, 2019) and we prove the existence of a ground state solution for the nonlinear Choquard equation with prescribed mass, when F satisfies Berestycki-Lions type conditions.

Abstract. We study the existence of L2 normalized solutions for nonlinear Schrödinger equations and systems. Under new Palais-Smale type conditions, we develop new deformation arguments for the constrained functional on (Formula Presented). As applications, we give other proofs to the results of [5, 8, 22]. As to the results of [5, 22], our deformation result enables us to apply the genus theory directly to the corresponding functional to obtain infinitely many solutions. As to the result [8], via our deformation result, we can show the existence of vector solution without using constraint related to the Pohozaev identity.

We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in ℝ N (N ≥ 2): (Equation Presented) where g(ξ) ∈ C(ℝ, ℝ), m > 0 is a given constant and μ ∈ ℝ is a Lagrange multiplier. We introduce a new approach using a Lagrange formulation of problem (∗)m. We develop a new deformation argument under a new version of the Palais-Smale condition. For a general class of nonlinearities related to [H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I: Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 313.345], [H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 347.375], [J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in ℝ N : Mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal. 35 (2010), no. 2, 253.276], it enables us to apply minimax argument for L 2 constraint problems and we show the existence of infinitely many solutions as well as mountain pass characterization of a minimizing solution of the problem (Equation Presented).

We study existence and multiplicity of semi-classical states for the nonlinear Choquard equation −ε2Δv + V (x) v = ε1α (Iα ∗ F(v))f(v) in RN , where N ≥ 3, α ∈ (0, N), Iα(x) = Aα/|x|N-α is the Riesz potential, F ∈ C1(R, R), F´(s) = f(s) and ε > 0 is a small parameter. We develop a new variational approach and we show the existence of a family of solutions concentrating, as ε → 0, to a local minima of V (x) under general conditions on F(s). Our result is new also for f(s) = |s|p-2s and applicable for p ∈ (NN+α, NN+-α2 ). Especially, we can give the existence result for locally sublinear case p ∈ (NN+α , 2), which gives a positive answer to an open problem arisen in recent works of Moroz and Van Schaftingen. We also study the multiplicity of positive single-peak solutions and we show the existence of at least cupl(K)+1 solutions concentrating around K as ε → 0, where K ⊂ Ω is the set of minima of V (x) in a bounded potential well Ω, that is, m0 ≡ infxεΩ V (x) < infxε∂Ω V (x) and K = {x ∈ Ω; V (x) = m0}.

We study spatially periodic logarithmic Schrodinger equations:
{ - Delta u + V(x)u = Q(x)u log u(2), u &gt; 0 in R-N, (LS)
u is an element of H-1 (R-N),
where N &gt;= 1 and V(x), Q(x) are spatially 1-periodic functions of class C-1. We take an approach using spatially 2L-periodic problems (L &gt;&gt; 1) and we show the existence of infinitely many multi-bump solutions of (LS) which are distinct under Z(N)-action.

We study, in the semiclassical limit, the singularly perturbed nonlinear Schrödinger equations LA,Vħu=f(|u|2)uinRNwhere N≥ 3 , LA,Vħ is the Schrödinger operator with a magnetic field having source in a C1 vector potential A and a scalar continuous (electric) potential V defined by LA,Vħ=-ħ2Δ-2ħiA·∇+|A|2-ħidivA+V(x).Here, f is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain Ω ⊂ RN such that (Formula presented.). For ħ> 0 small we prove the existence of at least cupl (K) + 1 geometrically distinct, complex-valued solutions to (0.1) whose moduli concentrate around K as ħ→ 0.

The Clark theorem is important in critical point theory. For a class of even functionals it ensures the existence of infinitely many negative critical values converging to 0 and it has important applications to sublinear elliptic problems. We study the convergence of the corresponding critical points and we give a characterization of accumulation points of critical points together with examples, in which critical points with negative critical values converges to nonzero critical point. Our results improve the abstract results in Kajikiya [4] and Liu-Wang [7].

We consider singularly perturbed nonlinear Schrödinger equations (Formula Presented) where (Formula Presented) and f is a nonlinear term which satisfies the so-called Berestycki–Lions conditions. We assume that there exists a bounded domain (Formula Presented) such that (Formula Presented) and we set (Formula Presented). For (Formula Presented) small we prove the existence of at least (Formula Presented) solutions to (0.1) concentrating, as (Formula Presented) around K. We remark that, under our assumptions of f, the search of solutions to (0.1) cannot be reduced to the study of the critical points of a functional restricted to a Nehari manifold.

We consider the boundary value problem(Pλ)u∈H01(Ω)∩L∞(Ω):-δu=λc(x)u+μ(x)|∇u|2+h(x), where Ω⊂RN, N≥3 is a bounded domain with smooth boundary. It is assumed that c{greater-than but not equal to}0, c, h belong to Lp(Ω) for some p>N/2 and that μ∈L∞(Ω). We explicitly describe a condition which guarantees the existence of a unique solution of (Pλ) when λ<0 and we show that these solutions belong to a continuum. The behaviour of the continuum depends in an essential way on the existence of a solution of (P0). It crosses the axis λ=0 if (P0) has a solution, otherwise it bifurcates from infinity at the left of the axis λ=0. Assuming that (P0) has a solution and strengthening our assumptions to μ(x)≥μ1>0 and h{greater-than but not equal to}0, we show that the continuum bifurcates from infinity on the right of the axis λ=0 and this implies, in particular, the existence of two solutions for any λ>0 sufficiently small.

In this note we present some uniqueness and comparison results for a class of problem of the form
-Lu = H(x, u, del u) + h(x), u is an element of H-0(1)(Omega) boolean AND L-infinity(Omega), (0.1)
where Omega subset of R-N, N &gt;= 2 is a bounded domain, L is a general elliptic second order linear operator with bounded coefficients and H is allowed to have a critical growth in the gradient. In some cases our assumptions prove to be sharp. (C) 2014 Elsevier Inc. All rights reserved.

In this paper we study the existence of multi-bump positive solutions of the following nonlinear elliptic problem:
-Delta u = u(p) in Omega(t) u=0 on partial derivative Omega(t).
Here 1 &lt; p &lt; N+2/N-2 when N &gt;= 3, 1 &lt; p &lt; infinity when N = 2 and Omega(t) and is a tubular domain which expands as t -&gt; infinity. See (1.6) below for a precise definition of expanding tubular domain. When the section D of Omega(t) is a ball, the existence of multi-bump positive solutions is shown by Dancer and Yan (Commun Partial Differ Equ, 27(1-2), 23-55, 2002) and by Ackermann et al. (Milan J Math, 79(1), 221-232, 2011) under the assumption of a non-degeneracy of a solution of a limit problem. In this paper we introduce a new local variational method which enables us to show the existence of multi-bump positive solutions without the non-degeneracy condition for the limit problem. In particular, we can show the existence for all N &gt;= 2 without the non-degeneracy condition. Moreover we can deal with more general domains, for example, a domain whose section is an annulus, for which least energy solutions of the limit problem are really degenerate.

We study a system of nonlinear Schrodinger equations with quadratic interaction in space dimension n &lt;= 6. The Cauchy problem is studied in L-2, in H-1, and in the weighted L-2 space &lt; x &gt; L--1(2) = F(H-1) under mass resonance condition, where &lt; x &gt; = (1 + vertical bar x vertical bar(2))(1/2) and F is the Fourier transform. The existence of ground states is studied by variational methods. Blow-up solutions are presented in an explicit form in terms of ground states under mass resonance condition, which ensures the invariance of the system under pseudo-conformal transformations. (c) 2012 Elsevier Masson SAS. All rights reserved.

In this article we study the existence and asymptotic profiles of high-energy rotation type solutions of the singularly perturbed forced pendulum equation ε2u'ε+sin uε= ε2ε in (-L, L). We prove that the asymptotic profile of these solutions is described in terms of an energy function which satisfy an integro-differential equation. Also we show that given a suitable energy function E satisfying the integro-differential equation, a family of solutions of the pendulum equation above exists having E as the asymptotic profile, when ε → 0. © 2013 IOP Publishing Ltd & London Mathematical Society.

We consider a singularly perturbed elliptic equation ε 2Δu - V (x)u + f(u) = 0, u(x) > 0 on ℝN, lim |x|→∞ u(x) = 0, where V (x) > 0 for any x ε ℝN: The singularly perturbed problem has corresponding limiting problems ΔU - cU + f(U) = 0, U(x) > 0 on ℝN, lim |x|→∞U(x) = 0, c > 0: Berestycki-Lions [3] found almost necessary and sufficient conditions on the nonlinearity f for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of the potential V under possibly general conditions on f ε C1 In this paper, we prove that under the optimal conditions of Berestycki-Lions on f 2 C1, there exists a solution concentrating around topologically stable positive critical points of V , whose critical values are characterized by minimax methods. © European Mathematical Society 2013.

We study the existence of a positive solution of a nonlinear elliptic equation Mathematical equation represented where k ≥ 2 and D is a bounded domain in Rℓ, ℓ ≥ 1. We give almost necessary and sufficient condition on f (ℰ) for the existence of a positive solution, which is inspired by the works of Berestycki-Lions [1] and Berestycki- Gallouët-Kavian [3].

We consider a singular perturbation problem for a system of nonlinear Schrödinger equations:where N = 2, 3, μ1, μ2, β > 0 and V1(x), V2(x): RN → (0, ∞) are positive continuous functions. We consider the case where the interaction β > 0 is relatively small and we define for P ε RN the least energy level m(P) for non-trivial vector solutions of the rescaled "limit" problem: We assume that there exists an open bounded set Λ ⊂ RN satisfying We show that (*) possesses a family of non-trivial vector positive solutions which concentrates-after extracting a subsequence e{open}n → 0-to a point P0 ε Λ with m(P0) = infPεΛm(P). Moreover (v1e{open}(x), v2e{open}(x)) converges to a least energy non-trivial vector solution of (**) after a suitable rescaling. © 2010 Springer-Verlag.

We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in ℝN : -Δu = g(u) in ℝN', u ∈ H1 (ℝN). We give an extension of the existence results due to H. Berestycki, T. GaIloüe't and O. Kavian [2]. We take a mountain pass approach in H1(ℝ N) and introduce a new method generating a Palais-Smale sequence with an additional property related to Pohozaev identity. ©2010 Juliusz Schauder Center for Nonlinear Studies.

We study the nonlinear Schrodinger equations:
(P(lambda)) -Delta u+(lambda(2)a(x)+1)u = vertical bar u vertical bar(p-1)u, u is an element of H(1)(R(N)),
where p &gt; 1 is a subcritical exponent, a(x) is a continuous function satisfying a(x) &gt;= 0,0 &lt; lim inf(vertical bar x vertical bar -&gt;infinity) a(x) &lt;= lim sup(vertical bar x vertical bar -&gt;infinity) a(x) &lt; infinity and a(-1)(0) consists of 2 connected bounded smooth components Omega(1) and Omega(2).
We study the existence of solutions (u(lambda)) of (P(lambda)) which converge to 0 in R(N) \ (Omega(1) boolean OR Omega(2)) and to a prescribed pair (v(1)(x), v(2)(x)) of solutions of the limit problem:
-Delta v(i) + v(i) = vertical bar v(i)vertical bar(p-1)v(i) in Omega(i)
(i = 1, 2) as lambda -&gt; infinity.

In this article we prove that the semi-linear elliptic partial differential equation- Δ u + u = up in Ω,u > 0 in Ω, u = 0 on ∂ Ω possesses a unique positive radially symmetric solution. Here p > 1 and Ω is the annulus {x ∈ RN | a < | x | < b}, with N ≥ 2, 0 < a < b ≤ ∞. We also show the positive solution is non-degenerate. © 2008 Elsevier Inc. All rights reserved.

In this article we construct a new type of solutions for the Gierer and Meinhardt system -ε2uxx + u &=&u 2/v in (0,L),- vxx+ v &=& u2 in (0, L) with boundary conditions u x (0) = u x (L) = 0 and v x (0) = v x (L) = 0. As ε approaches zero, we construct a family of positive solution (u ε , v ε ) such that the activator u ε oscillates c 0/ε times, with c 0 in an appropriate range, while the inhibitor remains close to a limiting profile, which is a strictly decreasing function. © 2007 Springer-Verlag.

We consider a noncompact hypersurface H in R2 N which is the energy level of a singular Hamiltonian of "strong force" type. Under global geometric assumptions on H, we prove that it carries a closed characteristic, as a consequence of a result by Hofer and Viterbo on the Weinstein conjecture in cotangent bundles of compact manifolds. Our theorem contains, as particular cases, earlier results on the fixed energy problem for singular Lagrangian systems of strong force type. © 2006 Elsevier Inc. All rights reserved.

This article is devoted to the nonlinear Schrödinger equation [InlineMediaObject not available: see fulltext.] when the parameter ε approaches zero. All possible asymptotic behaviors of bounded solutions can be described by means of envelopes, or alternatively by adiabatic profiles. We prove that for every envelope, there exists a family of solutions reaching that asymptotic behavior, in the case of bounded intervals. We use a combination of the Nehari finite dimensional reduction together with degree theory. Our main contribution is to compute the degree of each cluster, which is a key piece of information in order to glue them.

We study singularly perturbed 1D nonlinear Schrodinger equations (1.1). When V ( x) has multiple critical points, ( 1.1) has a wide variety of positive solutions for small epsilon and the number of positive solutions increases to infinity as epsilon --&gt; 0. We give an estimate of the number of positive solutions whose growth order depends on the number of local maxima of V (x). Envelope functions or equivalently adiabatic profiles of high frequency solutions play an important role in the proof.

We study the existence of multiple positive stable solutions for
-epsilon(2)Delta u(x) = u(x)(2) (b(x) - u(x)) in Omega, partial derivative u/partial derivative n (x) = 0 on partial derivative Omega.
Here epsilon &gt; 0 is a small parameter and b(x) is a piecewise continuous function which changes sign. These type of equations appear in a population growth model of species with a saturation effect in biology.

We study the existence of multiple positive stable solutions for
-epsilon(2)Delta u(x) = u(x)(2) (b(x) - u(x)) in Omega, partial derivative u/partial derivative n (x) = 0 on partial derivative Omega.
Here epsilon &gt; 0 is a small parameter and b(x) is a piecewise continuous function which changes sign. These type of equations appear in a population growth model of species with a saturation effect in biology.

In this note we study the existence of non-collision periodic solutions for singular Hamiltonian systems with weak force. In particular for potential V (t, q) similar to - 1/dist(q, D)(alpha), where D is a compact C-3-surface in R-N, we prove the existence of a non- collision periodic solution.

In this note we study the existence of non-collision periodic solutions for singular Hamiltonian systems with weak force. In particular for potential V (t, q) similar to - 1/dist(q, D)(alpha), where D is a compact C-3-surface in R-N, we prove the existence of a non- collision periodic solution.

We prove the existence of a positive solution for a class of equations of the form
-Delta u + V(x)u = f (u), u is an element of H-1 (R-N).
On the nonlinearity f, only conditions around 0 and at infinity are required.

We study the balanced Allen-Cahn problem in a singular perturbation setting. We are interested in the behaviour of clusters of layers, i.e. a family of solutions mu(epsilon) (x) with an increasing number of layers as epsilon -&gt; 0. In particular, we give a characterization of cluster of layers with asymptotically positive length by means of a limit energy function and, conversely, for a given admissible pattern, i.e. for a given a limit energy function, we construct a family of solutions with the corresponding behaviour.

We study the balanced Allen-Cahn problem in a singular perturbation setting. We are interested in the behaviour of clusters of layers, i.e. a family of solutions mu(epsilon) (x) with an increasing number of layers as epsilon -&gt; 0. In particular, we give a characterization of cluster of layers with asymptotically positive length by means of a limit energy function and, conversely, for a given admissible pattern, i.e. for a given a limit energy function, we construct a family of solutions with the corresponding behaviour.

We consider a class A equations of the form
-epsilon(2) Deltau + V(x)u = f(u), u is an element of H-1(R-N).
By variational methods, we show the existence of families of positive solutions concentrating around local minima of the potential V(x), as epsilon --&gt; 0. We do not require uniqueness of the ground state solutions of the associated autonomous problems nor the monotonicity of the function xi --&gt; f(xi)/xi. We deal with asymptotically linear as well as superlinear nonlinearities.

We consider a class A equations of the form
-epsilon(2) Deltau + V(x)u = f(u), u is an element of H-1(R-N).
By variational methods, we show the existence of families of positive solutions concentrating around local minima of the potential V(x), as epsilon --&gt; 0. We do not require uniqueness of the ground state solutions of the associated autonomous problems nor the monotonicity of the function xi --&gt; f(xi)/xi. We deal with asymptotically linear as well as superlinear nonlinearities.

We consider the equation
-u" = g(u), u(x) is an element of H-1 (R) (0.1)
Under general assumptions on the nonlinearity g we prove that the, unique up to translation, solution of (0.1) is at the mountain pass level of the associated functional. This result extends a corresponding result for least energy solutions when (0.1) is set on R-N.

We consider the equation
-u" = g(u), u(x) is an element of H-1 (R) (0.1)
Under general assumptions on the nonlinearity g we prove that the, unique up to translation, solution of (0.1) is at the mountain pass level of the associated functional. This result extends a corresponding result for least energy solutions when (0.1) is set on R-N.

We consider the multiple existence of positive solutions of the following nonlinear Schrodinger equation:
-Deltau + (lambdaa(x) + b(x))u = u(p), u &gt; 0 in R-N, (P-lambda)
where p is an element of (1, N+2/N-2) if N greater than or equal to 3 and p is an element of (1, infinity) if N = 1, 2, and a (x), b(x) are continuous functions. We assume that a(x) is nonnegative and has a potential well Omega := int a(-1) (0) consisting of k components Omega(1),..., Omega(k) and the first eigenvalues of -Delta + b(x) on Omega(j) under Dirichlet boundary condition are positive for all j = 1, 2,..., k. Under these conditions we show that (P-lambda) has at least 2(k) - 1 positive solutions for large lambda. More precisely we show that for any given non-empty subset J subset of {1, 2,... k}, (P-lambda) has a positive solutions u(lambda)(x) for large lambda. In addition for any sequence lambda(n) --&gt; infinity we can extract a subsequence lambda(ni) along which ulambda(ni) converges strongly in H-1 (R-N). Moreover the limit function u(x) = lim(i--&gt;infinity) ulambda(ni) satisfies
(i) For j is an element of J the restriction u\Omega(j) of u(x) to Omega(j) is a least energy solution of -Deltav + b(x)v = v(p) in Omega(j) an v = 0 on partial derivativeOmega(j).
(ii) u(x) = 0 for x is an element of R-N \ (boolean ORjis an element ofJ Omega(j)).

We consider the multiple existence of positive solutions of the following nonlinear Schrodinger equation:
-Deltau + (lambdaa(x) + b(x))u = u(p), u &gt; 0 in R-N, (P-lambda)
where p is an element of (1, N+2/N-2) if N greater than or equal to 3 and p is an element of (1, infinity) if N = 1, 2, and a (x), b(x) are continuous functions. We assume that a(x) is nonnegative and has a potential well Omega := int a(-1) (0) consisting of k components Omega(1),..., Omega(k) and the first eigenvalues of -Delta + b(x) on Omega(j) under Dirichlet boundary condition are positive for all j = 1, 2,..., k. Under these conditions we show that (P-lambda) has at least 2(k) - 1 positive solutions for large lambda. More precisely we show that for any given non-empty subset J subset of {1, 2,... k}, (P-lambda) has a positive solutions u(lambda)(x) for large lambda. In addition for any sequence lambda(n) --&gt; infinity we can extract a subsequence lambda(ni) along which ulambda(ni) converges strongly in H-1 (R-N). Moreover the limit function u(x) = lim(i--&gt;infinity) ulambda(ni) satisfies
(i) For j is an element of J the restriction u\Omega(j) of u(x) to Omega(j) is a least energy solution of -Deltav + b(x)v = v(p) in Omega(j) an v = 0 on partial derivativeOmega(j).
(ii) u(x) = 0 for x is an element of R-N \ (boolean ORjis an element ofJ Omega(j)).

We study a mountain pass characterization of least energy solutions of the following nonlinear scalar field equation in R-N:
-Deltau = g(u), u is an element of H-1(R-N),
where N greater than or equal to 2. Without the assumption of the monotonicity of t bar right arrow g(t)/t, we show that the mountain pass value gives the least energy level.

The existence of solutions with multiple transition layers for the spatially inhomogeneous phase transition problem is discussed. The location and multiplicity of the transition layers, especially the clustering layers and the boundary layers were studied. The existence of complicated dynamics, described in terms of symbolic sequence of integers was proved using Conley index theory.

We study a mountain pass characterization of least energy solutions of the following nonlinear scalar field equation in R-N:
-Deltau = g(u), u is an element of H-1(R-N),
where N greater than or equal to 2. Without the assumption of the monotonicity of t bar right arrow g(t)/t, we show that the mountain pass value gives the least energy level.

We consider the problem of finding standing waves to a nonlinear Schrödinger equation. This leads to searching for solutions of the equation -ε2u″ + V(x)u = |u|p-1u in R, p > 1, when s is a small parameter. Given any finite set of points x1 < X2 < ⋯ < xm constituted by isolated local maxima or minima of V, and corresponding arbitrary integers n i, i = 1,..., m, we prove that there is a finite energy solution exhibiting a cluster of n/ spikes concentrating around each xi as ε → 0. The clusters consist of spikes with alternating sign if the point is a minimum, and of constant sign if it is a maximum. This construction extends to infinitely many clusters of spikes under appropriate conditions. The proof follows an elementary variational matching approach, which resembles the so-called broken-geodesic method.

In this paper we establish the existence of a positive solution for an asymptotically linear elliptic problem on R-N. The main difficulties to overcome are the lack of a priori bounds for Palais-Smale sequences and a lack of compactness as the domain is unbounded. For the first one we make use of techniques introduced by Lions in his work on concentration compactness. For the second we show how the fact that the "Problem at infinity" is autonomous, in contrast to just periodic, can be used in order to regain compactness.

The existence of positive solutions for a class of nonhomogeneous elliptic equations in RN was analyzed. The mountain pass minimax value was strictly less than the first level of the break down of (PS)c-condition for I(u). Through the Mountain Pass Theorem, a critical point was obtained which was a positive solution of the nonhomogeneous elliptic equation.

In this paper we establish the existence of a positive solution for an asymptotically linear elliptic problem on R-N. The main difficulties to overcome are the lack of a priori bounds for Palais-Smale sequences and a lack of compactness as the domain is unbounded. For the first one we make use of techniques introduced by Lions in his work on concentration compactness. For the second we show how the fact that the "Problem at infinity" is autonomous, in contrast to just periodic, can be used in order to regain compactness.

We study the existence of multiple positive solutions for nonhomogeneous superlinear elliptic problems in R-N:
{-Deltau + u = g(x,u) + f(x) in R-N,
{u &gt; 0 in R-N,
{u is an element of H-1 (R-N),
where g(x,u) is a superlinear function such that g (x,0) equivalent to 0 and f is an element of H-1(R-N), f greater than or equal to 0, f not equivalent to 0. A typical example of g(x,u) is g(x,u) = a(x)u(P), where 1 &lt; p &lt; (N+2)/(N-2) if N greater than or equal to 3, 1 &lt; p &lt; infinity if N = 1, 2. We show the existence of multiple positive solutions under suitable conditions on g(x, u) and f(x).

We study the existence of multiple positive solutions for nonhomogeneous superlinear elliptic problems in R-N:
{-Deltau + u = g(x,u) + f(x) in R-N,
{u &gt; 0 in R-N,
{u is an element of H-1 (R-N),
where g(x,u) is a superlinear function such that g (x,0) equivalent to 0 and f is an element of H-1(R-N), f greater than or equal to 0, f not equivalent to 0. A typical example of g(x,u) is g(x,u) = a(x)u(P), where 1 &lt; p &lt; (N+2)/(N-2) if N greater than or equal to 3, 1 &lt; p &lt; infinity if N = 1, 2. We show the existence of multiple positive solutions under suitable conditions on g(x, u) and f(x).

We consider the existence of positive solutions of the following semilinear elliptic problem in R-N:
[GRAPHICS]
where 1 &lt; p &lt; N+2/N-2 (N greater than or equal to 3), 1 &lt; p &lt; infinity (N = 1, 2), a(x) is an element of C(R-N), f(x) is an element of H-1(R-N) and f(x) greater than or equal to 0. under the conditions;
1 degrees a(x) is an element of (0,1] for all x is an element of R-N,
2 degrees a(x) --&gt; 1 as \x\ --&gt; infinity,
3 degrees there exist delta &gt; 0 and C &gt; 0 such that
a(x) - 1 greater than or equal to -Ce-(2+delta)\x\ for all x is an element of R-N,
4 degrees a(x) not equal 1,
we show that (*) has at least four positive solutions for sufficiently small \\f\\(H-1(RN)) but f not equal 0.

We study Trudinger type inequalities in R-N and their best exponents alpha(N). We show for alpha is an element of (0, alpha(N)), alpha(N) = N omega(N-1)((N-1)) (omega(N-1) is the surface area of the unit sphere in R-N), there exists a constant C-alpha&gt;0 such that
[GRAPHICS]
for all u is an element of W-1,W-N (R-N) \ {0}. Here Phi(N) (xi) is defined by
[GRAPHICS]
is also shown that (*) with alpha greater than or equal to alpha(N) is false, which is different from the usual Trudinger's inequalities in bounded domains.

We consider the existence of positive solutions of the following semilinear elliptic problem in ℝN: (Formula Presented) where 1 &lt
p &lt
N + 2/N - 2 (N ≥ 3), 1 &lt
p &lt
∞ (N = 1, 2), a(x) ∈ C(ℝN), f(x) ∈ H-1 (ℝN) and f(x) ≥ 0. Under the conditions: 1° a(x) ∈ (0, 1) for all x ∈ ℝN, 2° a(x) → 1 as |x| → ∞, 3° there exist δ &gt
0 and C &gt
0 such that a(x) - 1 ≥ -Ce-(2+δ)|x| for all x ∈ ℝN, 4° a(x) ≢ 1, we show that (*) has at least four positive solutions for sufficiently small ||f||H-1(ℝN) but f ≢ 0.

We study the existence of unbounded solutions of singular Hamiltonian systems:
q + del V(q) = 0,
were V(q) similar to - 1/\q\alpha is a potential with a singularity. For a class of singular potentials with a strong force alpha &gt; 2, we show the existence of at least one hyperbolic-like solutions. More precisely, for given H &gt; 0 and theta(+), theta(-) is an element of SN-1, we find a solution q(t) of (*) satisfying
[GRAPHICS]

We study the existence of unbounded solutions of singular Hamiltonian systems:
q + del V(q) = 0,
were V(q) similar to - 1/\q\alpha is a potential with a singularity. For a class of singular potentials with a strong force alpha &gt; 2, we show the existence of at least one hyperbolic-like solutions. More precisely, for given H &gt; 0 and theta(+), theta(-) is an element of SN-1, we find a solution q(t) of (*) satisfying
[GRAPHICS]

We study the existence of periodic solutions of singular Hamiltonian systems as well as closed geodesics on non-compact Riemannian manifolds via variational methods.
For Hamiltonian systems, we show the existence of a periodic solution of prescribed-energy problem:
(q) double over dot + del V(q) = 0,
1/2\(q) over dot\(2) + V(q) = 0
under the conditions: (i) V(q) &lt; 0 for all q is an element of R-N \ {0}; (ii) V(q) similar to -l/\q\(2) as \q\ similar to 0 and \q\ similar to infinity.
For closed geodesics, we show the existence of a non-constant closed geodesic on (R x SN-1, g) under the condition:
g((s,x)) similar to ds(2) + h(0) as s similar to +/- infinity,
where h(0) is the standard metric on SN-1. (C) 2000 Editions scientifiques et medicales Elsevier SAS.

We consider the following semilinear elliptic equation involving critical Sobolev exponents:
-Delta u = K(\x\)u((n+2)/(n-2)) in R(n),
u(x) --&gt; 0 as \x\ --&gt; infinity,
where n greater than or equal to 3, K(r) is an element of C([0, infinity), R). We grove the existence of a positive radial solution with asymptotic behavior C/\x\(n-2) at \x\ = infinity under the conditions (i) K(r) &gt; 0 for all r &gt; 0, (ii) K(0) = K(infinity), and (iii) there exist C, delta &gt; 0 such that K(r) greater than or equal to K(0) - Cr-delta for small r &gt; 0 and K(r) greater than or equal to K(0) - Cr--delta for large r &gt; 0.

We consider the following semilinear elliptic equation involving critical Sobolev exponents:
-Delta u = K(\x\)u((n+2)/(n-2)) in R(n),
u(x) --&gt; 0 as \x\ --&gt; infinity,
where n greater than or equal to 3, K(r) is an element of C([0, infinity), R). We grove the existence of a positive radial solution with asymptotic behavior C/\x\(n-2) at \x\ = infinity under the conditions (i) K(r) &gt; 0 for all r &gt; 0, (ii) K(0) = K(infinity), and (iii) there exist C, delta &gt; 0 such that K(r) greater than or equal to K(0) - Cr-delta for small r &gt; 0 and K(r) greater than or equal to K(0) - Cr--delta for large r &gt; 0.

We consider T-periodic solutions of singular Hamiltonian systems with weak force q + delV(q, t) = 0, where V(q, t) approximately - 1/\q\alpha near q = 0 with alpha is-an-element-of (0, 2) . In particular, we study some properties of generalized T-periodic solutions, which were introduced by Bahri and Rabinowitz.

We consider T-periodic solutions of singular Hamiltonian systems with weak force In particular, we study some properties of generalized T-periodic solutions, which were introduced by Bahri and Rabinowitz. © 1994 American Mathematical Society.

We prove the existence of periodic solutions with prescribed energy for a class of N-body type problems with Keplerian like interaction.

We prove the existence of periodic solutions with prescribed energy for a class of N-body type problems with Keplerian like interaction.

Under a weak force type condition, we consider the existence of time periodic solutions of singular Hamiltonian systems:
[GRAPHICS] We assume V(q, t) &lt; 0 for all q, t and V(q, t), V(q)(q, t) --&gt; 0 as Absolute value of q --&gt; infinity. Moreover we assume V(q, t) is of a form: V(q, t) = - 1/\q\alpha + U(q,t) where 0 &lt; alpha &lt; 2 and U(q, t) is-an-element-of C2 ((R(N)\{0}) x R, R) is a T-periodic function in t such that \q\alpha U(q, t), \q\alpha+1 U(q)(q, t), \q\alpha+2 U(qq)(q, t), \q\alpha U(t)(q, t) --&gt; O as Absolute value of q --&gt; 0.
For alpha is-an-element-of (1, 2], we prove the existence of a non-collision solution of (HS). For alpha is-an-element-of (0, 1], we prove that the generalized solution of (HS), which is introduced in [BR], enters the singularity 0 at most one time in its period. Our argument depends on a minimax argument due to [BR] and an estimate of Morse index of corresponding functional, which will be obtained via re-scaling argument.

We study the existence of forced vibrations of nonlinear wave equation: {Mathematical expression} where g(ξ)∈C(R, R)is a function with superlinear growth and f(x, t) is a function which is 2π-periodic in t. Under the suitable growth condition on g(ξ), we prove the existence of infinitely many solution of (*) for any given f(x, t). © 1992 Fondazione Annali di Matimatica Pura ed Applicata.

We study the existence of forced vibrations of nonlinear wave equation: [GRAPHICS} where g(xi) is-an-element-of C(R, R) is a function with superlinear growth and f(x, t) is a function which is 2pi-periodic in t. Under the suitable growth condition on g(xi), we prove the existence of infinitely many solution of (*) for any given f(x, t).

We consider the existence of homoclinic orbits for a first order Hamiltonian system z ̇ = JHz(t, z). We assume H(t, z) is of form H(t, z) = 1 2(Az, z) + W(t, z), where A is a symmetric matrix with δ(JA)∩iR = ∅ and W(t, z) is 2π-periodic in t and has superquadratic growth in z. We prove the existence of a nontrivial homoclinic solution z∞(t) and subharmonic solutions (zT(t))Tε{lunate}N (i.e., 2πT-periodic solutions) of (HS) such that ZT(t) → Z∞(t) in Cloc1(R,R2N) as T → ∞. © 1991.

Many mathematicians are concerned with free vibrations—that is, they assume f(x, t) = 0 and find non trivial solutions. This chapter discusses the nonlinear vibrating string equation. © 1987, Publishing Committee of Lecture Notes in Numerical and Applied Analysis