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In this article, we use variational methods to prove the existence of positive solutions for an elliptic equation with two critical Hardy Sobolev exponents at the same pole and a certain nonlinear perturbation. Some parameters play a crucial role in our work. [ABSTRACT FROM AUTHOR]

This work is devoted to study the existence of nontrivial solutions to nonlocal asymmetric problems involving the m -Laplacian. (P) − M ∫ Ω | ∇ u | m d x Δ m u = f (x , u) in  Ω , u = 0 , on  ∂ Ω , where Ω ⊂ R N is a bounded domain with smooth boundary, M is a Kirchhoff function, N ≥ m ≥ 2 and f ∈ C (Ω ¯ × ℝ) is of subcritical polynomial or subcritical exponential growth. Moreover, the existence of nontrivial solutions for the above problem is obtained by using variational methods combined with the Moser–Trudinger inequality. Our interest then is to study (P) without the analogue of Ambrosetti–Rabinowitz superquadratic condition ( (AR) γ condition for short) in the positive semi-axis. To the best of our best knowledge, our results are new even in the asymmetric Kirchhoff Laplacian and m -Laplacian cases. [ABSTRACT FROM AUTHOR]

We consider the existence of multiple solutions of the following singular nonlocal elliptic problem: { − M (∫ R N | x | − a p | ∇ u | p) div (| x | − a p | ∇ u | p − 2 ∇ u) = h (x) | u | r − 2 u + H (x) | u | q − 2 u , u (x) → 0 as | x | → ∞ , where x ∈ R N , and M (t) = α + β t . By the variational method we prove that the problem has infinitely many solutions when some conditions are fulfilled. [ABSTRACT FROM AUTHOR]

We are concerned with multiplicity of solutions for differential inclusions involving the p -biharmonic operator, submitted to a general potential multivalued boundary condition which, among others, covers many particular boundary conditions that are frequently invoked in the literature. Our approach is a variational one and relies on non-smooth critical point theory. [ABSTRACT FROM AUTHOR]

In this paper, we consider local Dirichlet problems driven by the (r(u),s(u))-Laplacian operator in the principal part. We prove the existence of nontrivial weak solutions in the case where the variable exponents r,s are real continuous functions and we have dependence on the solution u. The main contributions of this article are obtained in respect of: (i) Carathéodory nonlinearity satisfying standard regularity and polynomial growth assumptions, where in this case, we use geometrical and compactness conditions to establish the existence of the solution to a regularized problem via variational methods and the critical point theory; and (ii) Sobolev nonlinearity, somehow related to the space structure. In this case, we use a priori estimates and asymptotic analysis of regularized auxiliary problems to establish the existence and uniqueness theorems via a fixed-point argument.

We study the existence and multiplicity of solutions of the problem $$\displaylines{ -\Delta u = -\lambda_1 u^- + g(x,u),\quad \text{in } \Omega; \cr u = 0, \quad \text{on } \partial\Omega, }$$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ $(N\geq 2)$, $u^-$ denotes the negative part of $u\colon\Omega\to\mathbb{R}$, $\lambda_1$ is the first eigenvalue of the N-dimensional Laplacian with Dirichlet boundary conditions in $\Omega$, and $g:\Omega\times\mathbb{R}\to\mathbb{R}$ is a continuous function with $g(x,0) =0$ for all $x\in\Omega$. We assume that the nonlinearity g(x,s) has a strong resonant behavior for large negative values of s and is superlinear, but subcritical, for large positive values of s. Because of the lack of compactness in this kind of problem, we establish conditions under which the associated energy functional satisfies the Palais-Smale condition. We prove the existence of three nontrivial solutions of the problem as a consequence of Ekeland's Variational Principle and a variant of the mountain pass theorem due to Pucci and Serrin [14].

In this article, we prove the existence of solutions for some discrete nonlinear difference equations subjected to a potential boundary type condition. We use a variational technique that relies on Szulkin's critical point theory, which ensures the existence of solutions by ground state and mountain pass methods.

In this paper, we introduce a new concept of slope for a set-valued map using a scalarizing function defined with the help of the Hiriart-Urruty signed distance function. It turns out that this slope possesses most properties of the strong slope of a scalar-valued function. We present some applications in set optimization studied with Kuroiwa's set approach. Namely, we obtain criteria for error bounds of a lower level set and the existence of weak optimal solutions under a Palais–Smale type condition. [ABSTRACT FROM AUTHOR]

We establish perturbation results for a Nirenberg type equation involving the fractional Laplacian on a bounded domain of Rn; n ≥ 2. Our method is based on the critical points at infinity theory of [6]. [ABSTRACT FROM AUTHOR]

We establish perturbation results for a Nirenberg type equation involving the fractional Laplacian on a bounded domain of Rn; n ≥ 2. Our method is based on the critical points at infinity theory of [6]. [ABSTRACT FROM AUTHOR]

We study the global invertibility of non-smooth, locally Lipschitz maps between infinite-dimensional Banach spaces, using a kind of Palais-Smale condition. To this end, we consider the Chang version of the weighted Palais-Smale condition for locally Lipschitz functionals in terms of the Clarke subdifferential, as well as the notion of pseudo-Jacobians in the infinite-dimensional setting, which are the analog of the pseudo-Jacobian matrices defined by Jeyakumar and Luc. Using these tools, we obtain the existence and uniqueness of solution for certain nonlinear equations defined by locally Lipschitz mappings. Along the way, we also obtain a global surjection theorem for locally Lipschitz maps in terms of pseudo-Jacobians. [ABSTRACT FROM AUTHOR]

The paper contains a further concretization of the variational approach to the theory of systems of conservation laws described in the earlier author's work. This approach involves the development of methods for proving the existence and uniqueness theorems for generalized solutions that are based on the search for critical points of functionals in Banach spaces. A new definition of the generalized solution is proposed and its equivalence to the traditional one for the functions of a simple structure is proved. A new strategy for proving existence and uniqueness theorems is proposed. A number of illustrative theorems outlining the implementation of this strategy are proved. [ABSTRACT FROM AUTHOR]

We prove the existence of at least two positive homoclinic solutions for a discrete boundary value problem of equations driven by the -Laplace operator. The properties of the nonlinearity ensure that the energy functional, corresponding to the problem, satisfies a mountain pass geometry and a Palais-Smale compactness condition. [ABSTRACT FROM AUTHOR]

In this work we consider second order Dirichelet boundary value problem with nonlinearity depending on the derivative. Using a global diffeomorphism theorem we propose a new variational approach leading to the existence and uniqueness result for such problems. [ABSTRACT FROM AUTHOR]

We consider the following perturbed nonlinear elliptic problem with critical growth $$\displaylines{ -\varepsilon^2\Delta u+V(x)u=f(x)|u|^{p-2}u +\frac{\alpha}{\alpha+\beta}K(x)|u|^{\alpha-2}u|v|^\beta,\quad x\in \mathbb{R}^N,\cr -\varepsilon^2\Delta v+V(x)v=g(x)|v|^{p-2}v +\frac{\beta}{\alpha+\beta}K(x)|u|^\alpha|v|^{\beta-2}v,\quad x\in \mathbb{R}^N,\cr u(x),\quad v(x)\to 0 \quad \text{as } |x|\to\infty. }$$ Using variational methods, we prove the existence of positive solutions.

We prove the existence of at least three solutions to a general Lidstone problem using the Morse Theory.

Using the critical point theory for convex, lower semicontinuous perturbations of locally Lipschitz functionals, we prove the solvability of the discontinuous Dirichlet problem involving the operator $u\mapsto\mbox{div} \Big(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\Big)$.

This paper is concerned with the existence of at least one solution of the nonlinear 2k-th order BVP. We use the Mountain Pass Lemma to get an existence result for the problem, whose linear part depends on several parameters.

This article concerns the existence and the multiplicity of solutions for nonhomogeneous elliptic Kirchhoff problems involving the critical Sobolev exponent, defined on a regular bounded domain of $\mathbb{R}^3$. Our approach is essentially based on Ekeland's Variational Principle and the Mountain Pass Lemma.

Abstract A bifurcation about the uniqueness of a solution of a singularly perturbed free boundary problem of phase transition associated with the p -Laplacian, subject to given boundary condition is proved in this paper. We show this phenomenon by proving the existence of a third solution through the Mountain Pass Lemma when the boundary data decreases below a threshold. In the second part, we prove the convergence of an evolution to stable solutions, and show the Mountain Pass solution is unstable in this sense. [ABSTRACT FROM AUTHOR]

Abstract Using the Mountain Pass Theorem, we establish the existence of periodic solution for Euler–Lagrange equation. Lagrangian consists of kinetic part (an anisotropic G-function), potential part K − W and a forcing term. We consider two situations: G satisfying Δ 2 ∩ ∇ 2 at infinity and globally. We give conditions on the growth of the potential near zero for both situations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonhomogeneous singular elliptic equation involving a critical Ca¤arelli-Kohn-Nirenberg exponent. By using the Nehari manifold we establish the existence of a ground state solution. [ABSTRACT FROM AUTHOR]

This paper deals with a class of singular Kirchhoff problem involving a critical nonlinearity. The existence and multiplicity of solutions for this problem are obtained by the variational methods. [ABSTRACT FROM AUTHOR]

We study a nonlinear eigenvalue problem which is perturbed by a term with double-phase growth. The main result establishes the existence of at least two nontrivial weak solutions in the case of high perturbations of the parameter. The proof combines variational tools with the Pucci-Serrin three critical points theorem. [ABSTRACT FROM AUTHOR]

In this paper, we study the following fractional Schrödinger‐Poisson system involving competing potential functions ϵ2s(−Δ)su+V(x)u+φu=K(x)f(u)+Q(x)|u|2s∗−2u,inR3,ϵ2t(−Δ)tφ=u2,inR3, where ϵ > 0 is a small parameter, f is a function of C1 class, superlinear and subcritical nonlinearity, 2s∗=63−2s, s>34, t ∈ (0,1), V(x), K(x), and Q(x) are positive continuous functions. Under some suitable assumptions on V, K, and Q, we prove that there is a family of positive ground state solutions with polynomial growth for sufficiently small ϵ > 0, of which it is concentrating on the set of minimal points of V(x) and the sets of maximal points of K(x) and Q(x). [ABSTRACT FROM AUTHOR]

We study a class of nonlinear elliptic problems with Navier boundary condition and involving the Laplace and the biharmonic operators. The main result of this paper establishes a sufficient condition for the existence of nontrivial weak solutions, in relationship with the values of a certain real parameter with respect to the principal eigenvalue of the Laplace operator. [ABSTRACT FROM PUBLISHER]

This paper is concerned with the existence of at least one solution of the nonlinear 2k-th order BVP. We use the Mountain Pass Lemma to get an existence result for the problem, whose linear part depends on several parameters. [ABSTRACT FROM AUTHOR]

We consider a nonlinear Neumann elliptic equation driven by a \(p\)-Laplacian-type operator which is not homogeneous in general. For such an equation the energy functional does not need to be coercive, and we use suitable variational methods to show that the problem has at least two distinct, nontrivial smooth solutions. Our formulation incorporates strongly resonant equations.

Using the global diffeomorphism theorem we consider the existence of solutions to the following Dirichlet problem,, whereis a jointly continuous function subject to some further growth conditions. Together with a given problem we consider the family of its discretizations and as a result we prove the existence of a non-spurious solution. [ABSTRACT FROM PUBLISHER]

This paper is concerned with the existence of positive homoclinic solutions for the second-order differential equation where $$c\ge 0$$ is a constant and the functions a and f are continuous and not necessarily periodic in t. Under other suitable assumptions on a and f, we obtain the existence of positive homoclinic solutions in both cases sub-quadratic and super-quadratic by using critical point theorems. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new framework to deal with the eigenvalue problems. As an application, we show that there exist positive and negative solutions to the semilinear elliptic eigenvalue problem with supercritical exponent. This is a generalization of previous results on the subcritical and critical exponents. [ABSTRACT FROM AUTHOR]

We study the existence and multiplicity of solutions of the problem -Δu = λ1u- + g (x, u) in Ω; u = 0, on φΩ. where is a smooth bounded domain in RN (N ≥ 2), u- denotes the negative part of u: Ω R, ≥1 is the first eigenvalue of the N-dimensional Laplacian with Dirichlet boundary conditions in, and g : Ω R × R is a continuous function with g(x, 0) = 0 for all ∊ Ω. We assume that the nonlinearity g(x, s) has a strong resonant behavior for large negative values of s and is superlinear, but subcritical, for large positive values of s. Because of the lack of compactness in this kind of problem, we establish conditions under which the associated energy functional satisfies the Palais-Smale condition. We prove the existence of three nontrivial solutions of problem (1) as a consequence of Ekeland's Variational Principle and a variant of the mountain pass theorem due to Pucci and Serrin [14]. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence and the stability of the solution of the quasilinear elliptic equation -Δpu-cup-1/ǀxǀp = um +tf in Ω where 0 < c < cN,p = (N-p/p )p, 1 < p < N, m > p-1; t > 0 and f is nonnegative measurable function. [ABSTRACT FROM AUTHOR]

In this article, we prove the existence of solutions for some discrete nonlinear difference equations subjected to a potential boundary type condition. We use a variational technique that relies on Szulkin's critical point theory, which ensures the existence of solutions by ground state and mountain pass methods. [ABSTRACT FROM AUTHOR]

Aim of this paper is to study the following elliptic equation driven by a general non-local integrodifferential operator such that in , in , where , is an open bounded set of ℝn, , with Lipschitz boundary, λ is a positive real parameter, is a fractional critical Sobolev exponent, while is the non-local integrodifferential operator

In this article we study the existence of solutions to eigenvalue problem $$displaylines{ -hbox{div} (|abla u|^{p-2}abla u)-lambda |u|^{p-2}umu=f quad hbox{in }Omega,cr u=0quadhbox{on }partialOmega }$$ where $Omega$ is a bounded domain in $mathbb{R}^N$ and $mu$ is a nonnegative Radon measure.

In this paper we study the nonlinear elliptic problem with \(p(x)\)-Laplacian (hemivariational inequality).We prove the existence of a nontrivial solution. Our approach is based on critical point theory for locally Lipschitz functionals due to Chang [J. Math. Anal. Appl. 80 (1981), 102–129].

In this article, we study the connection between the u-distance and a new Palais-Smale condition of compactness. We compare this Palais-Smale condition with the coercivity.

In this work we give a compactness result which allows us to prove the point-wise convergence of the gradients of a sequence of solutions to a quasilinear inequality and for an arbitrary open set. This result suggests solutions to many problems, notably nonlinear elliptic problems with critical exponent.

We prove the existence and multiplicity of solutions for a nonhomogeneous elliptic equation involving decaying cylindrical potential and critical exponent.

We consider the singular semilinear elliptic equation $$ -Delta u-frac{mu }{| x| ^2}u-lambda u=f(x)| u| ^{2^{ast }-1} $$ in $Omega $, $u=0$ on $partial Omega $, where $Omega $ is a smooth bounded domain, in $mathbb{R}^N$, $Ngeq 4$, $2^{ast }:=frac{2N}{N-2}$ is the critical Sobolev exponent, $f:mathbb{R} ^No mathbb{R}$ is a continuous function, $0

We consider a nonlinear Neumann problem driven by the p-Laplacian differential operator with a nonsmooth potential (hemivariational inequality). Using variational techniques based on the smooth critical point theory and the second deformation theorem, we prove an existence theorem and a multiplicity theorem, under hypothesis that in general do not imply the coercivity of the Euler functional.

In this article we establish the existence of at least two distinct solutions to singular elliptic equations involving a concave term and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions.