Existence and shape of solutions for a class of elliptic systems on the critical hyperbola

Asymptotic behavior of least energy solutions to the Lane-Emden system near the critical hyperbola

In this paper, the existence of a nontrivial least energy solution is considered for the nonlinear fractional Schrödinger-Poisson systems (−Δ)su + V(x)u + ϕu = |u|p−1u and (−Δ)tϕ = u2 in R3, where (−Δ)α is the fractional Laplacian for α = s, t ∈ (0, 1) with s < t and 2s + 2t > 3. Under some appropriate assumptions on the non-constant potential V(x), we prove the existence of a nontrivial least energy solution when 2<p<2s*−1=(3+2s)/(3−2s) by using some new analytical skills and the Nehari-Pohožaev type manifold.

Part 1 Constrained minimization: preliminaries constrained minimization dual method minimizers with the least energy application of dual method multiple solutions of nonhomogeneous equation sets of constraints constrained minimization for Ff subcritical problem application to the p-Laplacian critical problem. Part 2 Applications of Lusternik-Schnirelman theory: Palais-Smale condition, case p not equal to q duality mapping Palais-Smale condition, case p=q the Lusternik-Schnirelman theory case p>q case p q set of constraints V application to a critical case p=n technical lemmas existence result for problem (3.34). Part 4 Potentials with covariance condition: preliminaries and constrained minimization dual method minimization subject to constraint V Sobol inequality mountain pass theorem and constrained minimization minimization problem for a system of equations. Part 5 Eigenvalues and level sets: level sets continuity and monotonicity of delta the differentiability properties of delta Schechters's version of the mountain pass theorem general condition for solvability of (5.11) properties of the function K(t) Hilbert space case application to elliptic equations. Part 6 Generalizations of the mountain pass theorem: version of a deformation lemma mountain pass alternative consequences of mountain pass alternative Hampwile alternative applicability of the mountain pass theorem mountain pass and Hampwile alternative. Part 7 Nondifferentiable functionals: concept of a generalized gradient generalized gradients in function spaces mountain pass theorem for locally Lipschitz functionals consequences of theorem 7.3.1 application to boundary value problems with discontinuous nonlinearity lower semicontinuous perturbation deformation lemma for functionals satisfying condition (L) application to variational inequalities. Part 8 Concentration-compactness principle - subcritical case: concentration-compactness principle at infinity - subcritical case constrained minimization - subcritical case constrained minimization b not equal const, subcritical case behaviour of the Palais-Smale sequences the exterior Dirichlet problem the Palais-Smale condition concentration-compactness principle 1. Part 9 Concentration-compactness principle - critical case: critical Sobolev exponent concentration-compactness principle 2 - loss of mass at infinity constrained minimization - critical case - Palais-Smale sequences in critica case symmetric solutions remarks on compact embeddings into L2*(Q) and L2*(R) appendix.

We study the existence of least energy solutions to the nonlinear scalar field equation: (1) − Δ u + λu + V ( x ) u = Q ( x ) | u | p u , u ∈ H 1 ( R N ) , Where V ( x ) , Q ( x ) ∈ L ∞ ( R N ) are real functions satisfying suitable assumptions. By considering the Nehari type constraint N λ := { u ∈ H 1 ( R N ) ∖ { 0 } : ∫ R N ( | ∇u | 2 + λ | u | 2 + V ( x ) | u | 2 ) d x = ∫ R N Q ( x ) | u | p + 2 d x } , it is shown that (1) exists at least a nontrivial least energy solutions if \\lambda _{*}:=-\\inf \\sigma (-\\Delta +V) $ ]]> λ > λ ∗ := − inf σ ( − Δ + V ) . In this argument, we point out λ ∗ is a critical value for the existence of least energy solutions restricted to N λ . That is, there exists global least energy solutions in N λ if \\lambda _{*} $ ]]> λ > λ ∗ , but not if λ < λ ∗ . Moreover, the asymptotic behavior of least energy solutions as λ → + ∞ is analyzed.

We consider the Hamiltonian elliptic system $$\displaystyle{\left \{\begin{array}{@{}l@{\quad }l@{}} \begin{array}{l} -\varDelta u = \vert v\vert ^{p-1}v \mbox{ in} -\varDelta v =\mu \vert u\vert ^{s-1}u + \vert u\vert ^{q-1}u \mbox{ in} u,v > \mbox{ in} \varOmega, u,v = \mbox{ on} \partial \varOmega,\end{array} \quad \end{array} \right.}$$ where \(\varOmega \subset \mathbb{R}^{N}\) is a bounded smooth domain, N ≥ 3 and μ > 0. We assume that the point (p, q) lies on the critical hyperbola $$\displaystyle{ \frac{1} {p + 1} + \frac{1} {q + 1} = \frac{N - 2} {N} \mbox{ and that $s$ satisfies} \frac{p + 1} {p} \leq s + 1 < q + 1.}$$ The main contributions in this paper are twofold: to indicate that the location, critical or noncritical, of the point (p, q) on the critical hyperbola can interfere on the existence of solutions of the above system; to prove that if Ω has a rich topology, described by its Lusternik-Schnirelmann category, then the system has multiple solutions, at least as many as cat Ω (Ω), in case the parameter μ > 0 is sufficiently small and if s satisfies some suitable and natural conditions which depends on the critical or noncritical location of (p, q).

We study the following Lane–Emden system: $$\begin{align*} &amp; -\Delta u=|v|^{q-1}v \quad \ \textrm{in}\ \Omega, \qquad -\Delta v=|u|^{p-1}u \quad \ \textrm{in}\ \Omega, \qquad u_{\nu}=v_{\nu}=0 \quad \ \textrm{on}\ \partial \Omega, \end{align*}$$with $\Omega $ a bounded regular domain of ${\mathbb{R}}^{N}$, $N \ge 4$, and exponents $p, q$ belonging to the so-called critical hyperbola $1/(p+1)+1/(q+1)=(N-2)/N$. We show that, under suitable conditions on $p, q$, least-energy (sign-changing) solutions exist, and they are classical. In the proof we exploit a dual variational formulation, which allows to deal with the strong indefinite character of the problem. We establish a compactness condition which is based on a new Cherrier-type inequality. We then prove such condition by using as test functions the solutions to the system in the whole space and performing delicate asymptotic estimates. If $N \ge 5$, $p=1$, the system above reduces to a biharmonic equation, for which we also prove existence of least-energy solutions. Finally, we prove some partial symmetry and symmetry-breaking results in the case $\Omega $ is a ball or an annulus.

We use variational methods to study the existence of non-trivial and radially symmetric solutions to the Hénon–Lane–Emden system with weights, when the exponents involved lie on the "critical hyperbola". We also discuss qualitative properties of solutions and non-existence results.

Least energy solutions for a weakly coupled fractional Schrödinger system

In this paper, we investigate the Hardy-Sobolev type integral systems (1) with Dirichlet boundary conditions in a half space $\mathbb{R}_+^n$. We use the method of moving planes in integral forms introduced by Chen, Li and Ou [12] to prove that each pair of integrable positive solutions $(u, v)$ of the above integral system is rotationally symmetric about $x_n$-axis in both subcritical and critical cases $\frac{n-t}{p+1}+\frac{n-s}{q+1}\geq n-2m$ (Theorem 1.1). We also derive the non-existence of nontrivial nonnegative solutions with finite energy in the subcritical case (Theorem 1.2). By slightly modifying our arguments for studying the integral system, we can prove by a similar but simpler way that the same conclusions also hold for a single integral equation of Hardy-Sobolev type in both critical and subcritical cases (Theorem 1.3).

In this work, we study the existence of non-trivial multiple solutions for a class of quasilinear elliptic systems equipped with concave-convex nonlinearities and critical growth terms in bounded domains. By using the variational method, especially Nehari manifold and Palais-Smale condition, we prove the existence and multiplicity results of positive solutions.

Existence of sign-changing solutions to a Hamiltonian elliptic system in [formula omitted

This paper is devoted to a modification of the classical Cahn–Hilliard equation proposed by some physicists. This modification is obtained by adding the second time derivative of the order parameter multiplied by an inertial coefficient ε > 0, which is usually small in comparison with the other physical constants. The main feature of this equation is the fact that even a globally bounded nonlinearity is ‘supercritical’ in the case of two and three space dimensions. Thus, the standard methods used for studying semilinear hyperbolic equations are not very effective in the present case. Nevertheless, we have recently proven the global existence and dissipativity of strong solutions in the 2D case (with a cubic controlled growth nonlinearity) and for the 3D case with small ε and arbitrary growth rate of the nonlinearity (see (Grasselli et al 2009 J. Evol. Eqns 9 371–404, Grasselli et al 2009 Commun. Partial Diff. Eqns 34 137–70)). The present contribution studies the long-time behaviour of rather weak (energy) solutions of that equation and it is a natural complement of the results of our previous papers (Grasselli et al 2009 J. Evol. Eqns 9 371–404, Grasselli et al 2009 Commun. Partial Diff. Eqns 34 137–70). In particular, we prove here that the attractors for energy and strong solutions coincide for both the cases mentioned above. Thus, the energy solutions are asymptotically smooth. In addition, we show that the non-smooth part of any energy solution decays exponentially in time and deduce that the (smooth) exponential attractor for the strong solutions constructed previously is simultaneously the exponential attractor for the energy solutions as well. It is worth noting that the uniqueness of energy solutions in the 3D case is not known yet, so we have to use the so-called trajectory approach which does not require uniqueness. Finally, we apply the obtained exponential regularization of the energy solutions for verifying the dissipativity of solutions of the 2D modified Cahn–Hilliard equation in the intermediate phase space of weak solutions (in between energy and strong solutions) without any restriction on ε.

The L_{p} -Minkowski problem deals with the existence of closed convex hypersurfaces in \mathbb{R}^{n+1} with prescribed p -area measures. It extends the classical Minkowski problem and embraces several important geometric and physical applications. Existence of solutions has been obtained in the sub-critical case {p&gt;-n-1} , but the problem remains open in the super-critical case {p&lt;-n-1} . In this paper, we introduce new ideas to solve the problem for all the super-critical exponents. A crucial ingredient in our proof is a topological method based on the calculation of the homology of a topological space of ellipsoids. Our results show that the L_{p} -Minkowski problem admits a solution in both the sub-critical and super-critical cases but does not have a solution in general in the critical case.

Semi-classical ground states concentrating on the nonlinear potential for a Dirac equation

Existence of breather solutions of the DNLS equations with unbounded potentials