In this paper, we study the following Choquard–type problem [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] is a parameter, [Formula: see text] and [Formula: see text] is the Hardy–Littlewood–Sobolev critical exponent. The potential [Formula: see text] is assumed to be periodic, while [Formula: see text] is bounded, and [Formula: see text] is a [Formula: see text]–reaction term. Under suitable growth and monotonicity assumption on [Formula: see text], we establish the existence of ground state solution without assuming the Ambrosetti–Rabinowitz condition, provided that [Formula: see text] is sufficiently large. Our approach is variational, and relies on several key tools, including the Mountain Pass Theorem, the Nehari manifold method, the concentration–compactness principle, and the analysis of a suitable limiting problem.

The mountain pass theorem in terms of tangencies

This work investigates the existence and uniqueness of a solution to a discrete Robin boundary value problem involving the anisotropic \(\vec{p}\)-mean curvature operator. The existence result is established through variational methods, specifically by applying the Mountain Pass Theorem of Ambrosetti and Rabinowitz in combination with Ekeland’s Variational Principle. Uniqueness is obtained under the assumption of Lipschitz continuity on the nonlinear term.

We study the admissible values of the wave speed $c$ for which the beam equation with jumping nonlinearity possesses a travelling wave solution. In contrast to previously studied problems modelling suspension bridges, the presence of the term with negative part of the solution in the equation results in restrictions of $c$. In this paper, we provide the maximal wave speed range for which the existence of the travelling wave solution can be proved using the Mountain Pass Theorem. We also introduce its close connection with related Dirichlet problems and their Fučík spectra. Moreover, we present several analytical approximations of the main existence result with assumptions that are easy to verify. Finally, we formulate a conjecture that the infimum of the admissible wave speed range can be described by the Fučík spectrum of a simple periodic problem.

Given $s$, $q\in(0,1)$, and a bounded and integrable function $h$ which is strictly positive in an open set, we show that there exist at least two nonnegative solutions $u$ of the critical problem $$(-\Delta)^s u=\varepsilon h(x)u^q+u^{2^*_s-1},$$% as long as $\varepsilon> 0$ is sufficiently small. Also, if $h$ is nonnegative, these solutions are strictly positive. The case $s=1$ was established in \cite{MR1801341}, which highlighted, in the classical case, the importance of combining perturbative techniques with variational methods: indeed, one of the two solutions branches off perturbatively in $\varepsilon$ from $u=0$, while the second solution is found by means of the Mountain Pass Theorem. The case $s\in(0,1/2]$ was already established, with different methods, in \cite{MR3617721} (actually, in \cite{MR3617721} it was erroneously believed that the method would have carried through all the fractional cases $s\in(0,1)$, so, in a sense, the results presented here correct and complete the ones in \cite{MR3617721}).

Abstract We prove the existence of infinitely many nontrivial solutions for time-harmonic nonlinear Maxwell’s equations on bounded domains and on $$\mathbb {R}^3$$ R 3 using dual variational methods. In the dual setting we apply a new version of the Symmetric Mountain Pass Theorem that does not require the Palais-Smale condition.

This paper deals with the existence and multiplicity of nontrivial solutions for [Formula: see text]-Laplace equations with the Stein–Weiss reaction under critical exponential nonlinearity in the Heisenberg group [Formula: see text]. In addition, a weight function and two positive parameters have also been included in the nonlinearity. The developed analysis is significantly influenced by these two parameters. Further, the mountain pass theorem, the Ekeland variational principle, the Trudinger–Moser inequality, the doubly weighted Hardy–Littlewood–Sobolev inequality and a completely new Brézis–Lieb-type lemma for Choquard nonlinearity play key roles in our proofs.

ABSTRACT This paper is concerned with the existence of positive solutions for a class of Schrödinger‐Bopp‐Podolsky systems in three‐dimensional Euclidean space. The system couples a nonlinear Schrödinger equation with a fourth‐order elliptic equation governing the electrostatic potential. We establish existence under assumptions on an external potential, which is allowed to vanish at infinity, and on a nonlinearity with subcritical growth. A key feature of our work is demonstrating that existence hinges on a delicate interplay between the decay rate of the potential and the local behavior of the nonlinearity near the origin. The proof relies on variational methods. We employ a penalization technique to overcome the lack of compactness, establish the existence of a solution for the resulting auxiliary problem via the Mountain Pass Theorem, and use crucial ‐estimates to show this solution solves the original system for a suitable range of parameters.

ABSTRACT The paper deals with the existence and multiplicity of positive solutions for ‐Laplacian Kirchhoff‐type impulsive fractional differential equations. First, in the degenerate case, we prove that this equation has at least a positive solution based on the mountain pass theorem for any sufficiently large, and the solution converges to zero as . Then, by applying the Ekeland variational principle, at least a positive solution is obtained when is sufficiently small, and the solution converges to zero as . Moreover, in the nondegenerate case, we establish the existence of infinitely many solutions by using truncation arguments and Krasnoselskii's genus theory when is sufficiently small.

We study the elliptic equation { − Δ u = u p − 1 + λ u q − 1 ln ⁡ u , x ∈ Ω , u ≥ 0 , x ∈ Ω , u = 0 , x ∈ ∂Ω , where Ω ⊂ R N is a bounded smooth domain, N ≥ 3 , λ ≥ 0 is a parameter, 1 < q < 2 ∗ , 1 < p ≤ 2 ∗ . Under different conditions on the parameters λ , p and q, we investigate the existence and multiplicity of solutions to this problem utilizing the mountain pass Theorem and the Nehari technique. Compared to u q − 1 , some interesting phenomena occur since the sign of the logarithmic term u q − 1 ln ⁡ u is indefinite.

Abstract This article investigates the existence and multiplicity of nontrivial solutions for Schrödinger-Bopp-Podolsky systems with critical nonlinearity in R 3 {{\mathbb{R}}}^{3} . Under appropriate assumptions about the potential and nonlinear terms, the existence and multiplicity of solutions are obtained by using the concentration-compactness principle and the symmetric mountain pass theorem. To some extent, we generalize the previous results.

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.

Abstract We investigate the existence and multiplicity of solutions for a class of the generalized coupled system involving poly-Laplacian and the parameter λ \lambda on finite graphs. By using the Mountain pass lemma together with the cut-off technique, we obtain that system has at least a nontrivial weak solution ( u λ , v λ ) \left({u}_{\lambda },{v}_{\lambda }) for every large parameter λ \lambda when the nonlinear term F ( x , u , v ) F\left(x,u,v) satisfies superlinear growth conditions only in a neighborhood of origin point (0, 0). We also obtain a concrete form for the lower bound of λ \lambda and the trend of ( u λ , v λ ) \left({u}_{\lambda },{v}_{\lambda }) with the change of λ \lambda . Moreover, by using a revised Clark’s theorem together with cut-off technique, we obtain that system has a sequence of solutions tending to 0 for every λ &gt; 0 \lambda \gt 0 when the nonlinear term F ( x , u , v ) F\left(x,u,v) satisfies sublinear growth conditions only in a neighborhood of origin point (0, 0).

This work presents a solving method for problems of Ambrosetti-Prodi type involving p-Laplacian and p-pseudo-Laplacian operators by using adequate variational methods. A variant of the mountain pass theorem, together with a kind of Palais-Smale condition, is involved in order to obtain and characterize solutions for some mathematical physics issues. Applications of these results for solving some physical chemical problems evolved from the need to model real phenomena are displayed. Solutions for Dirichlet problems containing these two operators applied for modeling critical micellar concentration, as well as the volume fraction of liquid mixtures, have been drawn.

Abstract This paper investigates the existence of at least two distinct weak solutions using Bonanno’s theorem [G. Bonanno, Relations between the mountain pass theorem and local minima, Adv. Nonlinear Anal. 1 (2012), no. 3, 205–220, Theorem 3.2], and establishes the existence of no fewer than two nontrivial weak solutions through the Bonanno–D’Aguì theorem [G. Bonanno and G. D’Aguì, Two nonzero solutions for elliptic Dirichlet problems, Z. Anal. Anwend. 35 (2016), no. 4, 449–464, Theorem 2.1] applied to a general class of fourth-order Leray–Lions problems that include an s ⁢ ( x ) {s(x)} -Hardy potential and a nonlocal singular term.

The present paper discusses an elliptic equation with Steklov boundary conditions and $(p(x),q(x))$-Laplacian. Using mountain pass theorem together with Ekeland's variational principle, we prove, under appropriate conditions on the functions involved, that the problem admits at least two solutions.

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.

ABSTRACTBy means of the Ekeland variational principle coupled with the mountain pass lemma, we study a class of nonlinear second‐order parametric partial difference equations involving ‐Laplacian. Taking into account both the cases of large and small , we establish criteria to ensure the existence of two nontrivial homoclinic solutions for sufficiently large and one nontrivial homoclinic solution for all . Finally, three special examples are presented to demonstrate the applications of our results. Our assumptions relax some known ones, and results generalize some existing literature.

This paper presents several sufficient conditions for the existence of at least one solution and at least one nonzero solution to the boundary value problem Δn(p(k)Δnx(k-n))=λf(k,x(k)),k∈N[n,T]x(0)=x(1)=…=x(n-1)=0,x(T+1)=x(T+2)=…=x(T+n)=0,with a parameter λ. The technical approach is based on variational methods and the Mountain pass lemma.

In this paper, we are concerned with semiclassical states to the following fractional nonlinear elliptic equation $$ \varepsilon^{2s}(-\Delta)^s u + V(x) u=\mathcal{N}(|u|)u \quad \mbox{in } \mathbb R^N, $$ where $0&lt; s &lt; 1$, $\varepsilon&gt; 0$ is a small parameter, $N&gt; 2s$, $V \in C^1\big(\mathbb R^N, \mathbb R^+\big)$ and $\mathcal{N}\in C\big(\mathbb R, \mathbb R^+\big)$. The nonlinearity has Sobolev subcritical, critical or supercritical growth, i.e.\ $\mathcal{N}(t)=t^{p-2}$, $\mathcal{N}(t)=\mu t^{p-2} +t^{2^*_s-2}$ or $\mathcal{N}(t)=t^{p-2} + \lambda t^{r-2}$ for $t \geq 0$, where $2&lt; p&lt; 2^*_s&lt; r$ and $\mu, \lambda&gt; 0$. The fractional Laplacian $(-\Delta)^s$ is characterized as $\mathcal{F}((-\Delta)^{s}u)(\xi)=|\xi|^{2s} \mathcal{F}(u)(\xi)$ for $\xi \in \mathbb R^N$, where $\mathcal{F}$ denotes the Fourier transform. We construct positive semiclassical states and an infinite sequence of sign-changing semiclassical states with higher energies clustering near the local minimum points of the potential $V$. The solutions are of higher topological type, which are obtained from a minimax characterization of higher dimensional symmetric linking structure via the symmetric mountain pass theorem. They correspond to critical points of the underlying energy functional at energy levels where compactness condition breaks down. The proofs are variational and mainly rely on penalization methods, $s$-harmonic extension theories and blow-up arguments along with local type Pohozaev identities.