Prescribed energy solutions to some scaled problems via a scaled Nehari manifold

In this work, we study the Nehari manifold and its application to the following sub-elliptic system involving strongly coupled critical terms and concave nonlinearities: \[ \left\{ \begin{aligned} -\Delta_{\mathbb{G}} u &= \frac{\eta_{1}\alpha_{1}}{2^*}\,|u|^{\alpha_{1}-2}|v|^{\beta_{1}}u +\frac{\eta_{2}\alpha_{2}}{2^*}\,|u|^{\alpha_{2}-2}|v|^{\beta_{2}}u \\ &+\lambda\, g(z)\,|u|^{q-2}u, && z\in\Omega, \\ -\Delta_{\mathbb{G}} v &= \frac{\eta_{1}\beta_{1}}{2^*}\,|u|^{\alpha_{1}}|v|^{\beta_{1}-2}v +\frac{\eta_{2}\beta_{2}}{2^*}\,|u|^{\alpha_{2}}|v|^{\beta_{2}-2}v \\ &+\mu\, h(z)\,|v|^{q-2}v, && z\in\Omega, \\ u &= v = 0, && z\in\partial\Omega, \end{aligned} \right. \] where \(\Omega\) is an open bounded subset of \(\mathbb{G}\) with smooth boundary, \(-\Delta_{\mathbb{G}}\) is the sub-Laplacian on a Carnot group \(\mathbb{G}\); \(\eta_1, \eta_2, \lambda, \mu\) are positive, \(\alpha_1+\beta_1=2^*\), \(\alpha_2+\beta_2=2^*\), \(1<q<2\), \(2^*=\frac{2Q}{Q-2}\) is the critical Sobolev exponent, and \(Q\) is the homogeneous dimension of \(\mathbb{G}\). By exploiting the Nehari manifold and variational methods, we prove that the system has at least two positive solutions.

The aim of this paper is to obtain an existence and multiplicity result for a strongly coupled concave-convex system for an {\it optimal} choice of involved real parameters via the Nehari manifold method. In the paper, we have obtained the parametric region which is optimal in the sense that the constraint minimization idea based on the Nehari manifold is no longer applicable if the parameters lie in the exterior of the optimal region. By applying a finer analysis of fibering maps, we have shown the existence of atleast two positive solutions for the parameters lying below and even above the parametric optimal curve, characterized variationally via nonlinear generalized Rayleigh quotient. The main result of the paper is complemented by the study of the problem for negative parameter values.

In this paper, we concern the existence of sign-changing solutions for the following Kirchhoff-type problems with critical growth: [Formula: see text] Here, [Formula: see text] is a smoothly bounded domain, [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. Using the nodal Nehari manifold approach combined with the decomposition method, we prove the existence of least-energy nodal solutions for the above problem. Notably, the decomposition technique plays a crucial role in analyzing the lower bound of the energy functional in the sublinear case, i.e. [Formula: see text] and [Formula: see text], where the attainability of the infimum is demonstrated. This provides a key step in establishing the existence of nodal solutions. Moreover, by employing topological degree theory and the quantitative deformation lemma, we demonstrate that the minimizer obtained from the constrained set is indeed a nodal solution to the above problem.

Purpose In this paper, we study a fourth-order singular problems of type Leray–Lions with singular weight and with no-flux boundary conditions. Design/methodology/approach We use the Nehari manifold to establish our result for our problem. Findings We prove that there exist at least two nontrivial positive weak solutions. Originality/value In particular, it investigates the problem in the presence of singular weights and variable exponents, a case that has not been extensively studied in the existing literature.

In this paper, we consider the following double phase Kirchhoff-type problems − ψ ( ∫ R N ( | ∇ u | p p + μ ( x ) | ∇ u | q q ) d x ) div ( | ∇u | p − 2 ∇u + μ ( x ) | ∇u | q − 2 ∇u ) = f ( x , u ) in R N , where N ≥ 2 , 1<p<N, p < q < p ∗ = Np N − p , 0 ≤ μ ( ⋅ ) ∈ C 0 , α ( R N ) with α ∈ ( 0 , 1 ] , ψ ( s ) = a 0 + b 0 s v − 1 for s ∈ R , with 0 $ ]]> a 0 ≥ 0 , b 0 > 0 and v ≥ 1 and f : R N × R → R is a Carathéodory superlinear and subcritical growth function. Using a variational tool based on critical point theory with the Nehari manifold method and genus theory, we obtain two constant sign solutions (one is positive and the other is one negative) and infinitely many solutions.

By using the Ekeland variational principle and Nehari manifold, we study the following fractional p-Laplacian Kirchhoff equations: M[u]s,pp+∫RNV(x)|u|pdx[(−Δ)psu+V(x)|u|p−2u]=λ|u|q−2uln|u|,x∈RN,(P). In these equations, λ∈R∖{0},p∈(1,+∞), s∈(0,1),sp&lt;N,ps*=NpN−sp, M(τ)=a+bτθ−1, a,b∈R+,1&lt;θ&lt;ps*p, V(x)∈C(RN,R) is a potential function and (−Δ)ps is the fractional p-Laplacian operator. The existence of solutions is deeply influenced by the positive and negative signs of λ. More precisely, (i) Equation (P) has one ground state solution for λ&gt;0 and pθ&lt;q&lt;ps*, with a positive corresponding energy value; and (ii) Equation (P) has at least two nontrivial solutions for λ&lt;0 and p&lt;q&lt;ps*, with positive and negative corresponding energy values, respectively.

We investigate a class of Schrödinger equations with saturable nonlinearity, including a nonzero intensity function g(x)∈C(RN,R). Under the suitable assumptions on g(x), we prove the existence of a bound state solution by using the Nehari manifold and the linking theorem.

We study the degenerate logistic problem with a non-linear term that is asymptotically linear at infinity. Existence and uniqueness of a steady state positive solution is proved, and in addition a solution that changes sign is obtained. In this work, we use the Nehari manifold to obtain a positive solution depending on the parameter λ. In order to find a sign-changing solution, the Mountain Pass Theorem on this natural constraint and weighted spectral theory are applied.

This article investigates the existence, non-existence, and multiplicity of weak solutions for a parameter-dependent nonlocal Schrödinger–Kirchhoff type problem on R N involving singular nonlinearity. By performing fine analysis based on Nehari submanifolds and fibre maps, our goal is to show the problem has at least two positive solutions even if λ lies beyond the extremal parameter λ ∗ .

Abstract In this paper we study logarithmic double phase problems with superlinear right-hand sides and nonlinear Neumann boundary condition. In particular, we show that the problem under consideration has a least energy sign-changing solution. The proof is based on the minimization of the energy functional over the related nodal Nehari manifold along with the Poincaré–Miranda existence theorem. As a result of independent interest, we prove the existence of a new and very general equivalent norm in the logarithmic Musielak–Orlicz Sobolev space. In addition, we present a priori bounds for a large class of logarithmic double phase problems involving convection terms for critical and subcritical situations.

For discrete nonlinear Schrödinger equations (DNLS) with double power nonlinearities we prove the existence of normalized ground state solutions by means of variational methods. Considering the corresponding constrained (with prescribed mass) minimization problem of least energy, respectively action, for DNLS where both nonlinear terms are of focusing type, the existence proof utilizes the Concentration Compactness Principle. For DNLS for which the leading nonlinearity is focusing while the lower order one is defocusing, the method of the Nehari manifold is used. Orbital stability of the ground state solutions is shown. Furthermore, we discuss the existence of excitation thresholds for the creation of ground state solutions focusing on the impact of the lower order nonlinearity on the threshold values of the mass for the excitation of ground state solutions.

ABSTRACT In this article, we study the ‐biharmonic equation involving Choqurd type nonlinearity with sign‐changing weight functions in a bounded domain with Dirichlet boundary condition. Using the Nehari manifold and fibering map analysis, we show the multiplicity results in subcritical case and existence results in critical case with respect to parameter . Moreover, we explore the concentration‐compactness principle for ‐biharmonic Choquard equation in critical case. The results obtained here are also new in case of ‐Laplacian.

In this work, we consider an initial-boundary value problem of a plate equation in a bounded domain of R n , with memory and the time-weighted function α ( t ) . We apply the Faedo-Galerkin method and the contraction mapping principle to establish the local existence of weak solutions. Subsequently, we explore the dynamics of these weak solutions, focusing on global existence and finite-time blow-up, using the Nehari manifold and modified concavity arguments. Furthermore, we derive the upper and lower bounds for the blow-up time of solutions with high-energy levels.

ABSTRACTIn this paper we confirm that with is exactly the critical exponent for the embedding from into () (see [3, 4]) and name it as the upper Hénon‐Sobolev critical exponent. Based on this fact, we study the ground‐state solutions of critical Hénon equations in via the Nehari manifold methods and the great idea of Brézis‐Nirenberg in [9]. We establish the existence of the positive radial ground‐state solutions for the problem with one single upper Hénon‐Sobolev critical exponent. We also deal with the existence of the nonnegative radial ground‐state solutions for the problems with multiple critical exponents, including Hardy‐Sobolev critical exponents or Sobolev critical exponents or the upper Hénon‐Sobolev critical exponents.

Nehari Manifold Optimization and Its Application for Finding Unstable Solutions of Semilinear Elliptic PDEs

This paper mainly studies the existence of sign-changing solutions for the following px-biharmonic Kirchhoff-type equations: −a+b∫RN1p(x)|Δu|p(x)dxΔp(x)2u+V(x)|u|p(x)−2u = Kxf(u),x∈RN where Δp(x)2u=Δ|Δu|p(x)−2Δu is the p(x) biharmonic operator, a,b&gt;0 are constants, N≥2, V(x),K(x) are positive continuous functions which vanish at infinity, and the nonlinearity f has subcritical growth. Using the Nehari manifold method, deformation lemma, and other techniques of analysis, it is demonstrated that there are precisely two nodal domains in the problem’s least energy sign-changing solution ub. In addition, the convergence property of ub as b→0 is also established.

Correction: On a minimization problem involving fractional Sobolev spaces on Nehari manifold

In this work, we investigate the existence of at least three solutions to a variable exponent double-phase problem with a reaction term that is only locally Lipschitz continuous. Additionally, we analyse the sign properties of these solutions. Specifically, we establish the existence of two constant-sign solutions, one positive and one negative, using the Mountain Pass Theorem. The third solution, which is nodal (i.e., it changes sign), is obtained via the Nehari manifold approach. Finally, we demonstrate that the nodal solution has exactly two nodal domains.

ABSTRACTVia the Nehari manifold method and the analysis of the fibering maps, we study in this paper the existence of two nontrivial weak solutions for such regular Kirchhoff problem driven by a nonlocal integro‐differential operator of regular elliptic type.