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In this paper, we focus on Kirchhoff type problems driven by a logarithmic double phase operator with variable exponents. Under very general assumptions on the nonlinearity and using variational tools, like the mountain pass theorem, we establish the existence of at least one nontrivial weak solution for the problem under consideration. Then, under different hypotheses on the reaction term, we are also able to derive a multiplicity result of solutions for our problem. We stress that in order to produce such a multiplicity result a key role is played by a variant of the symmetric mountain pass theorem. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of a double phase problem with variable exponents and Dirichlet boundary condition. Based on an abstract critical point theorem, we establish existence results under very general assumptions on the nonlinear term, such as a subcritical growth and a superlinear condition. In particular, we prove the existence of two bounded weak solutions with opposite energy sign and we state some special cases in which they turn out to be nonnegative.

We are concerned with the existence of nodal solutions for a third order boundary value problem u ′ ′ ′ (x) = g (u (x)) + p (x , u (x) , u ′ (x) , u ′ ′ (x)) , x ∈ (0 , 1) , u (0) = u (1) = u ′ (1) = 0 , where g : R → R is continuous and satisfies lim | ξ | → ∞ g (ξ) / ξ = ∞ (g is superlinear as | ξ | → ∞ ), p : [ 0 , 1 ] × R 3 → R is continuous and satisfies | p (x , ξ 0 , ξ 1 , ξ 2) | ≤ C + | ξ 0 | / 3 , x ∈ [ 0 , 1 ] , (ξ 0 , ξ 1 , ξ 2) ∈ R 3 , for some C > 0 . We obtain infinitely many solutions having specified nodal properties. The proof of our main result is based upon bifurcation techniques. [ABSTRACT FROM AUTHOR]

Abstract We study the following nonlinear mass supercritical Kirchhoff equation: − ( a + b ∫ R N | ∇ u | 2 ) △ u + μ u = f ( u ) in R N , $$ - \biggl(a+b \int _{\mathbb{R}^{N}} \vert \nabla u \vert ^{2} \biggr) \triangle u+ \mu u=f(u) \quad \text{in } {\mathbb{R}^{N}}, $$ where a , b , m > 0 $a ,b,m>0$ are prescribed, and the normalized constrain ∫ R N | u | 2 d x = m $\int _{\mathbb{R}^{N}}|u|^{2}\,dx =m$ is satisfied in the case 1 ≤ N ≤ 3 $1\leq N\leq 3$ . The nonlinearity f is more general and satisfies weak mass supercritical conditions. Under some mild assumptions, we establish the existence of ground state when 1 ≤ N ≤ 3 $1\leq N\leq 3$ and obtain infinitely many radial solutions when 2 ≤ N ≤ 3 $2\leq N\leq 3$ by constructing a particular bounded Palais–Smale sequence.

We study the following nonlinear mass supercritical Kirchhoff equation: − (a + b ∫ R N | ∇ u | 2) △ u + μ u = f (u) in R N , where a , b , m > 0 are prescribed, and the normalized constrain ∫ R N | u | 2 d x = m is satisfied in the case 1 ≤ N ≤ 3 . The nonlinearity f is more general and satisfies weak mass supercritical conditions. Under some mild assumptions, we establish the existence of ground state when 1 ≤ N ≤ 3 and obtain infinitely many radial solutions when 2 ≤ N ≤ 3 by constructing a particular bounded Palais–Smale sequence. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of wave equations with linear couplings. Such model arises when two waves propagate simultaneously. Based on variational methods and Galerkin approximation argument, we obtain some multiplicity results of periodic solutions for the problem when the linear coupling coefficient $ \varepsilon $ is small enough and the nonlinearities satisfy superlinear growth. [ABSTRACT FROM AUTHOR]

In this paper we consider a class of anisotropic $ (\vec{p}, \vec{q}) $-Laplacian problems with nonlinear right-hand sides that are superlinear at $ \pm\infty $. We prove the existence of two nontrivial weak solutions to this kind of problem by applying an abstract critical point theorem under very general assumptions on the data without supposing the Ambrosetti-Rabinowitz condition.

In this paper we consider a class of anisotropic (→p, →q)-Laplacian problems with nonlinear right-hand sides that are superlinear at 1. We prove the existence of two nontrivial weak solutions to this kind of problem by applying an abstract critical point theorem under very general assumptions on the data without supposing the Ambrosetti-Rabinowitz condition. [ABSTRACT FROM AUTHOR]

Multiplicity results are proved for solutions both with positive and negative energy, as well as nonexistence results, of a generalized quasilinear Schrödinger potential free equation in the entire R N involving a nonlinearity which combines a power-type term at a critical level with a subcritical term, both with weights. The equation has been derived from models of several physical phenomena such as superfluid film in plasma physics as well as the self-channelling of a high-power ultra-short laser in matter. Proof techniques, also in the symmetric setting, are based on variational tools, including concentration compactness principles, to overcome lack of compactness, and the use of a change of variable in order to deal with a well defined functional. [ABSTRACT FROM AUTHOR]

We study the existence, nonexistence, and multiplicity of a weak solution of the singular Neumann problem where , is a bounded smooth domain, is the -Laplace operator, , and is a constant. Here is a function having superlinear growth at infinity and . Using the sub-supersolution method and the variational method, under appropriate assumptions on and we show that there exists a domain bounded by the graph of a map such that admits at least two solutions for all at least one solution for , and no solution for all . [ABSTRACT FROM AUTHOR]

We construct multiple solutions to the nonlocal Liouville equation (-Δ)1/2 u = K(x)eu in R. More precisely, for K of the form K(x) = 1 + ε k (x) with ε Є (0, 1) small and k Є C1,α (R) L∞ (R) for some α >0, we prove the existence of multiple solutions to the above equation bifurcating from the bubbles. These solutions provide examples of flat metrics in the half-plane with prescribed geodesic curvature K(x) on its boundary. Furthermore, they imply the existence of multiple ground state soliton solutions for the Calogero--Moser derivative nonlinear Schrödinger equation. [ABSTRACT FROM AUTHOR]

We study positive solutions to the two point boundary value problem: $ \begin{equation*} \begin{matrix}Lu = -u'' = \lambda \bigg\{\dfrac{A}{u^\gamma}+M\big[u^\alpha+u^\delta\big]\bigg\} \; ;\; (0, 1) \\ u(0) = 0 = u(1)\; \; \; \; \; \; \; \; \; \; \end{matrix} \end{equation*} $ where $ A < 0 $, $ \alpha \in (0, 1), \delta > 1, \gamma \in (0, 1) $ are constants and $ \lambda > 0, M > 0 $ are parameters. We prove that the bifurcation diagram $ (\lambda \text{ vs } \|u\|_\infty) $ for positive solutions is at least a reversed S-shaped curve when $ M\gg1 $. Recent results in the literature imply that for $ M\gg1 $ there exists a range of $ \lambda $ where there exist at least two positive solutions. Here, when $ M\gg1 $, we prove the existence of a range of $ \lambda $ for which there exist at least three positive solutions and that the bifurcation diagram is at least a reversed S-shaped curve. Further, via a quadrature method and Python computations, for $ M\gg1 $, we show that the bifurcation diagram is exactly a reversed S-shaped curve. Also, when the operator $ L $ is replaced by a $ p $-Laplacian operator with $ p > 1 $, as well as $ p $-$ q $ Laplacian operator with $ p = 4 $ and $ q = 2 $, we show that the bifurcation diagram is again an exactly reversed S-shaped curve when $ M\gg1 $.

We study the following nonlinear mass supercritical Schrödinger–Poisson–Slater equation - ▵ u + μ u - γ (| x | - 1 ∗ | u | 2) u - a f (u) = 0 in R 3 satisfying the normalized constrain ∫ R 3 | u | 2 d x = m , and m > 0 prescribed. The nonlinearity f is super-critical. Under some mild assumptions, we establish the existence of ground state and infinitely many radial solutions by constructing a particular bounded Palais-Smale sequence when γ < 0 , a > 0 . Meanwhile, we obtain the non-existence result in the case γ < 0 , a < 0 and the existence result when γ > 0 , a < 0 via variational methods. [ABSTRACT FROM AUTHOR]

We study positive solutions to the two point boundary value problem: L u = − u ″ = λ { A u γ + M [ u α + u δ ] } ; (0 , 1) u (0) = 0 = u (1) where A < 0 , α ∈ (0 , 1) , δ > 1 , γ ∈ (0 , 1) are constants and λ > 0 , M > 0 are parameters. We prove that the bifurcation diagram (λ vs ‖ u ‖ ∞) for positive solutions is at least a reversed S-shaped curve when M ≫ 1. Recent results in the literature imply that for M ≫ 1 there exists a range of λ where there exist at least two positive solutions. Here, when M ≫ 1 , we prove the existence of a range of λ for which there exist at least three positive solutions and that the bifurcation diagram is at least a reversed S-shaped curve. Further, via a quadrature method and Python computations, for M ≫ 1 , we show that the bifurcation diagram is exactly a reversed S-shaped curve. Also, when the operator L is replaced by a p -Laplacian operator with p > 1 , as well as p - q Laplacian operator with p = 4 and q = 2 , we show that the bifurcation diagram is again an exactly reversed S-shaped curve when M ≫ 1. [ABSTRACT FROM AUTHOR]

Dirichlet problem in an n-dimensional billiard space is investigated. In particular, the system of ODEs x ¨ (t) = f (t , x (t)) together with Dirichlet boundary conditions x (0) = A , x (T) = B in an n-dimensional interval K with elastic impact on the boundary of K is considered. The existence of multiple solutions having prescribed number of impacts with the boundary is proved. As a consequence the existence of infinitely many solutions is proved, too. The problem is solved by reformulating it into non-impulsive problem with a discontinuous right-hand side. This auxiliary problem is regularized and the Schauder Fixed Point Theorem is used. [ABSTRACT FROM AUTHOR]

The objective of this work is to investigate a nonlocal problem involving singular and critical nonlinearities: ([ u ] s , p p) σ − 1 (− Δ) p s u = λ u γ + u p s ∗ − 1 in Ω , u > 0 , in Ω , u = 0 , in R N ∖ Ω , where Ω is a bounded domain in R N with the smooth boundary ∂ Ω , 0 < s < 1 < p < ∞ , N > s p , 1 < σ < p s ∗ / p , with p s ∗ = N p N − p s , (− Δ) p s is the nonlocal p-Laplace operator and [ u ] s , p is the Gagliardo p-seminorm. We combine some variational techniques with a truncation argument in order to show the existence and the multiplicity of positive solutions to the above problem. [ABSTRACT FROM AUTHOR]

We prove multiplicity results for solutions, both with positive and negative energy, for a class of singular quasilinear Schrödinger equations in the entire R N R N involving a critical term, nontrivial weights and positive parameters λ λ , β β , covering several physical models, coming from plasma physics as well as high-power ultra short laser in matter. Also the symmetric setting is investigated. Our proofs relay on variational tools, including concentration compactness principles because of the delicate situation of the double lack of compactness. In addition, a necessary reformulation of the original problem in a suitable variational setting, produces a singular function, delicate to be managed. [ABSTRACT FROM AUTHOR]

In this paper we study double phase problems with nonlinear boundary condition and gradient dependence. Under quite general assumptions we prove existence results for such problems where the perturbations satisfy a suitable behavior in the origin and at infinity. Our proofs make use of variational tools, truncation techniques and comparison methods. The obtained solutions depend on the first eigenvalues of the Robin and Steklov eigenvalue problems for the p-Laplacian.

Abstract In this paper, we consider the existence of nontrivial solutions for a fractional p-Laplacian equation in a bounded domain. Under different assumptions of nonlinearities, we give existence and multiplicity results respectively. Our approach is based on variational methods and some analytical techniques.

In this paper, we use the Leray–Schauder degree theory to study the following singular periodic problems: x ″ + f (x ′) + g (t , x) = s , x (0) − x (T) = 0 = x ′ (0) − x ′ (T) , where f : ℝ → ℝ is a continuous function with f (0) = 0 , function g : ℝ / T ℤ × ℝ + → ℝ is continuous with an attractive singularity at the origin, and s is a constant. We consider the case where the friction term f satisfies a local superlinear growth condition but not necessarily of the Nagumo type, and function g does not need to satisfy coercivity conditions. An Ambrosetti–Prodi type result is obtained. [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of multiple positive solutions to the following elliptic problem involving a nonhomogeneous operator with nonstandard growth of p p - q q type and singular nonlinearities ⎧ ⎩ ⎨ − L p , q u u = λ f (u) u γ , u > 0 = 0 in Ω , on ∂ Ω , { − L p , q u = λ f (u) u γ , u > 0 in Ω , u = 0 on ∂ Ω , where Ω Ω is a bounded domain in R N R N with C 2 C 2 boundary, N ≥ 1 N ≥ 1 , λ > 0 λ > 0 is a real parameter, L p , q u : = d i v (| ∇ u | p − 2 ∇ u + | ∇ u | q − 2 ∇ u) , L p , q u := d i v (| ∇ u | p − 2 ∇ u + | ∇ u | q − 2 ∇ u) , 1 < p < q < ∞ 1 < p < q < ∞ , γ ∈ (0 , 1) γ ∈ (0 , 1) , and f f is a continuous nondecreasing map satisfying suitable conditions. By constructing two distinctive pairs of strict sub and super solution, and using fixed point theorems by Amann [ 1 ], we prove existence of three positive solutions in the positive cone of C δ (Ω ¯ ¯ ¯ ¯) C δ (Ω ¯) and in a certain range of λ λ. [ABSTRACT FROM AUTHOR]

Stimulated by a multiple solvability of periodic boundary value problems to differential equations with time–dependent impulses, we recall a related definition of parametric topological entropy for a sequence of continuous self–maps on a compact metric space. The main simple idea consists in replacing the iterates of a single map by the compositions of several various maps. For an equicontinuous countable family of self–maps on a compact connected polyhedron, we develop a lower estimate of this entropy in terms of the asymptotic Nielsen numbers of their compositions. This Ivanov–type equality is then applied, via the associated Poincaré translation operators, to differential equations with time–dependent impulses on tori. If the supporting space differs from a homotopy type of tori, then the situation becomes more delicate. Nevertheless, on compact connected punctured surfaces, we are still able to apply in a similar way the Artin braid group theory to planar differential equations with a finite number of homeomorphic impulses. Some further possibilities are commented in remarks and several illustrative examples are supplied. [ABSTRACT FROM AUTHOR]

We prove existence and multiplicity results in R N for an elliptic problem of (p, q)-Laplacian type with a nonlinearity involving both a critical term and a subcritical term with a positive real parameter λ . In particular, nonnegative nontrivial weights satisfying some symmetry conditions with respect to a certain group T are included in the nonlinearity. We prove first the existence of at least one solution with positive energy for λ sufficiently small using Mountain Pass Theorem, then we obtain the existence of infinitely many weak solutions with positive (finite) energy for every λ positive applying Fountain Theorem. Our proofs use variational methods and concentration compactness principles. [ABSTRACT FROM AUTHOR]

The existence and multiplicity of T-periodic solutions to a class of differential equations with attractive singularities at the origin are investigated in the paper. The approach is based on a new method of construction of strict upper and lower functions. The multiplicity results of Ambrosetti–Prodi type are established using a priori estimates and certain properties of topological degree. [ABSTRACT FROM AUTHOR]

We study existence and multiplicity of positive ground states for the scalar curvature equation Δ u + K (| x |) u n + 2 n - 2 = 0 , x ∈ R n , n > 2 , when the function K : R + → R + is bounded above and below by two positive constants, i.e. 0 < K ̲ ≤ K (r) ≤ K ¯ for every r > 0 , it is decreasing in (0 , R) and increasing in (R , + ∞) for a certain R > 0 . We recall that in this case ground states have to be radial, so the problem is reduced to an ODE and, then, to a dynamical system via Fowler transformation. We provide a smallness non perturbative (i.e. computable) condition on the ratio K ¯ / K ̲ which guarantees the existence of a large number of ground states with fast decay, i.e. such that u (| x |) ∼ | x | 2 - n as | x | → + ∞ , which are of bubble-tower type. We emphasize that if K(r) has a unique critical point and it is a maximum the radial ground state with fast decay, if it exists, is unique. [ABSTRACT FROM AUTHOR]

We deal with the periodic boundary value problem associated with the parameter-dependent second-order nonlinear differential equation

In the present paper we initiate the variational analysis of a super sinh-Gordon system on compact surfaces, yielding the first example of non-trivial solution of min-max type. The proof is based on a linking argument jointly with a suitably defined Nehari manifold and a careful analysis of Palais-Smale sequences. We complement this study with a multiplicity result exploiting the symmetry of the problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of nontrivial solutions for a fractional p-Laplacian equation in a bounded domain. Under different assumptions of nonlinearities, we give existence and multiplicity results respectively. Our approach is based on variational methods and some analytical techniques. [ABSTRACT FROM AUTHOR]

We give a lower estimate of the number of anti-periodic solutions of implicit differential inclusions on tori. Our approach is based on the application of the topological essential fixed point theory, jointly with the Nielsen theory for multivalued admissible maps. Since one of the conditions is rather technical (zero topological dimension of a fixed point set), some simple illustrative examples to the main theorem are supplied. In the single-valued case of implicit differential equations, relevant arguments are still discussed in concluding remarks. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the following autonomous Kirchhoff-type equation:

In this paper, we deal with the Schrödinger's problems, in the first part we study the theoretical side, we show the existence of at least three weak solutions, our main tools are based on variational inequalities, more precisely, using the three critical points theorem due to Ricceri, existence and multiplicity results are established. In the second part, we are interested in the application side, more exactly, we examine some computational problems on the discretization of finite elements of the p(x)-Laplacian, we propose a quasi-Newton minimization approach for the solution, our numerical tests show that these algorithms are able to resolve the problems with p(x)-Laplacian, for different values of p(x). [ABSTRACT FROM AUTHOR]

We study existence and multiplicity of radial ground states for the scalar curvature equation Δ u + K (| x |) u n + 2 n - 2 = 0 , x ∈ R n , n > 2 , when the function K : R + → R + is bounded above and below by two positive constants, i.e. 0 < K ̲ ≤ K (r) ≤ K ¯ for every r > 0 , it is decreasing in (0, 1) and increasing in (1 , + ∞) . Chen and Lin (Commun Partial Differ Equ 24:785–799, 1999) had shown the existence of a large number of bubble tower solutions if K is a sufficiently small perturbation of a positive constant. Our main purpose is to improve such a result by considering a non-perturbative situation: we are able to prove multiplicity assuming that the ratio K ¯ / K ̲ is smaller than some computable values. [ABSTRACT FROM AUTHOR]

This article concerns the existence of non-trivial weak solutions for a class of non-homogeneous Neumann problems. The approach is through variational methods and critical point theory in Orlicz-Sobolev spaces. We investigate the existence of two solutions for the problem under some algebraic conditions with the classical Ambrosetti-Rabinowitz condition on the nonlinear term and using a consequence of the local minimum theorem due to Bonanno and mountain pass theorem. Furthermore, by combining two algebraic conditions on the nonlinear term and employing two consequences of the local minimum theorem due Bonanno we ensure the existence of two solutions, by applying the mountain pass theorem of Pucci and Serrin, we set up the existence of the third solution for the problem.

In this paper, we establish the existence of two intervals of positive real parameters $\lambda$ for which a class of Neumann boundary value equations involving the (p1,...,pn)-Laplacian admits three weak solutions, whose norms are uniformly bounded with respect to $\lambda$ belonging to one of the two intervals. The approach is based on variational methods.

In this paper, we investigate the existence of three generalized solutions for fourth-order Kirchhoff-type problems with a perturbed nonlinear term depending on two real parameters. Our approach is based on variational methods.

Via variational methods, we establish the existence of at least two distinct weak solutions for the Dirichlet problem associated to a possibly degenerate equation in divergence form.

Abstract The well-known Ambrosetti–Prodi theorem considers perturbations of the Dirichlet Laplacian by a nonlinear function whose derivative jumps over the principal eigenvalue of the operator. Various extensions of this landmark result were obtained for self-adjoint operators, in particular by Berger and Podolak, who gave a geometrical description of the solution set. In this text we show that similar theorems are valid for non-self-adjoint operators. In particular, we prove that the semilinear operator is a global fold. As a consequence, we obtain what appears to be the first exact multiplicity result for elliptic equations in non-divergence form. We employ techniques based on the maximum principle. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence and multiplicity results of solutions with prescribed L 2 -norm for a class of Kirchhoff type problems − a + b ∫ R 3 | ∇ u | 2 d x Δ u − λ u = f ( u ) in R 3 , where a , b > 0 are constants, λ ∈ R and f ∈ C ( R , R ) . To obtain such solutions, we look into critical points of the energy functional E b ( u ) = a 2 ∫ R 3 | ∇ u | 2 + b 4 ∫ R 3 | ∇ u | 2 2 − ∫ R 3 F ( u ) constrained on the L 2 -spheres S ( c ) = u ∈ H 1 ( R 3 ) : | | u | | 2 2 = c . Here, c > 0 and F ( s ) ≔ ∫ 0 s f ( t ) d t . Under some mild assumptions on f , we show that critical points of E b unbounded from below on S ( c ) exist for c > 0 . In addition, we establish the existence of infinitely many radial critical points { u n b } of E b on S ( c ) provided that f is odd. Finally, the asymptotic behavior of u n b as b ↘ 0 is analyzed. These conclusions extend some known ones in previous papers. [ABSTRACT FROM AUTHOR]

We prove an Ambrosetti-Prodi type result for a Neumann problem associated to the equation u" + f (x,u(x)) = μ when the nonlinearity has the following form: f (x,u) := a(x)g(u) -- p(x). The assumptions considered generalize the classical one, → (x, u) → + ∞ as |u| → + ∞ without requiring any uniformity condition in x. The multiplicity result which characterizes these kind of problems will be proved by means of the shooting method. [ABSTRACT FROM AUTHOR]

We study the positive subharmonic solutions to the second order nonlinear ordinary differential equation ... where g(u) has superlinear growth both at zero and at infinity, and q(t) is a T-periodic sign-changing weight. Under the sharp mean value condition f0T q(t) dt < 0, combining Mawhin's coincidence degree theory with the Poincare-Birkhoff fixed point theorem, we prove that there exist positive subharmonic solutions of order k for any large integer k. Moreover, when the negative part of q(t) is sufficiently large, using a topological approach still based on coincidence degree theory, we obtain the existence of positive subharmonics of order k for any integer k ≥ 2. [ABSTRACT FROM AUTHOR]

We study the periodic boundary value problem associated with the second order non-linear equation u" + λa+(t) − μa−(t)) g(u) = 0, where g(u) has superlinear growth at zero and sublinear growth at infinity. For λ, μ positive and large, we prove the existence of 3m − 1 positive T-periodic solutions when the weight function a(t) has m positive humps separated by m negative ones (in a T-periodicity interval). As a byproduct of our approach we also provide an abundance of positive subharmonic solutions and symbolic dynamics. The proof is based on coincidence degree theory for locally compact operators on open unbounded sets and also applies to Neumann and Dirichlet boundary conditions. Finally, we deal with radially symmetric positive solutions for the Neumann and the Dirichlet problems associated with elliptic PDEs. [ABSTRACT FROM AUTHOR]