Symmetric Mountain Pass Lemma Research Articles

New results concerning a singular biharmonic equations with p-Laplacian and Hardy potential

This paper deals with a singular biharmonic equations with p-Laplacian and Hardy potential. Using Mountain Pass Theorem and Fountain Theorem with Cerami condition, we obtain the existence and multiplicity of sign-changing high-energy solutions under some suitable conditions on the nonlinear term f ( x , u ) . In addition, by applying an abstract critical point theorem of Kajikiya in [A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations. J Funct Anal. 2005;225(2):352–370], a sequence of sign-changing solutions converging to zero for the biharmonic equations with sublinear nonlinearities is also obtained.

Infinitely many low- and high-energy solutions for double phase problems with nonstandard growth

The aim of this paper is the study a class of double phase problems with variable exponent. Using the Clark’s theorem and the symmetric mountain pass lemma, we prove the existence of infinitely many small solutions and infinitely many large solutions, respectively.

Infinitely many solutions for p-fractional Choquard type equations involving general nonlocal nonlinearities with critical growth via the concentration compactness method

We prove the existence of infinitely many solutions to a fractional Choquard type equation(−Δ)psu+V(x)|u|p−2u=(K⁎g(u))g′(u)+εWW(x)f′(u)in RN involving the fractional p-Laplacian and a general convolution term with critical growth. In order to obtain infinitely many solutions, we use a type of the symmetric mountain pass lemma which gives sequences of critical values converging to zero for even functionals. To assure the (PS)c conditions, we also use a nonlocal version of the concentration compactness lemma.

Multiple solutions for a nonlocal problem

This paper is devoted to study a new class Kirchhoff equation. By using the symmetric mountain pass lemma, we establish the existence of infinitely many solutions.

Existence and multiplicity of solutions for a fourth-order differential system with instantaneous and non-instantaneous impulses

Abstract The main purpose is to establish the variational structure of a fourth-order ordinary differential system with both instantaneous and non-instantaneous impulses. Then by using the mountain pass lemma and symmetric mountain pass lemma, the existence and multiplicity results of non-trivial weak solutions are obtained.

Existence and multiplicity of solutions for a new p(x)-Kirchhoff problem with variable exponents

Abstract In this article, we study a class of new p(x)-Kirchhoff problem without satisfying the Ambrosetti-Rabinowitz type growth condition. Under some suitable superliner conditions, we introduce new methods to show the boundedness of Cerami sequences. By using the mountain pass lemma and the symmetric mountain pass lemma, we prove that the p(x)-Kirchhoff problem has a nontrivial weak solution and infinitely many solutions.

Infinitely Many Weak Solutions for Problems Involving Both p(x)-Laplacian and p(x)-Biharmonic Operators

In this paper, we study the existence of infinitely many weak solutions for problems involving both p(x)-Laplacian and p(x)- Biharmonic operators. In the proof of our main result, we use variational methods and the known symmetric mountain pass lemma.

Sign-Changing Solutions for Biharmonic Equations with Navier Boundary Conditions

This work is concerned with sign-changing solutions to the biharmonic equation with Navier boundary conditions. By using the method of invariant sets of descending flow and symmetric mountain pass lemma, we obtain the existence of multiple sign-changing solutions when the nonlinear term is odd.

Multiple Solutions for a Class of System of (p, q)-Kirchhoff Equations in $\mathbb {R}^{N}$

This paper is concerned with the (p, q)-Kirchhoff type equations, for which the existence of infinitely many high energy solutions is proved by employing the symmetric mountain pass lemma. Furthermore, an $L^{\infty }$ estimate for the solutions is given by applying the Moser iteration technique.

Existence of Solutions for Fractional-Choquard Equation with a Critical Exponential Growth in $${\mathbb {R}}^N$$

In this work, we study the existence of solutions to the fractional-Choquard equation 0.1 $$\begin{aligned} (-\Delta )^s_{\nu }u+V(x)|u|^{\nu -2}u=(I_{\alpha }*|u|^q)|u|^{q-2}u+h(u),\quad x\in {{\mathbb {R}}}^N, \end{aligned}$$ where $$\nu =\frac{N}{s}, 0<s<1, N\ge 2$$ , V(x) is a positive and bounded function in $${{\mathbb {R}}}^N$$ , $$I_{\alpha }$$ is the Riesz potential, $$qs>N$$ and the continuous function h(u) behaves like $$\exp (\alpha _0|u|^{\frac{N}{N-s}})$$ growth. Using the symmetric rearrangement method with some special techniques and symmetric mountain pass lemma, we prove the existence of infinitely many solutions for (0.1) in $$W^{s,\nu }({\mathbb {R}}^N)$$ .

Infinitely many solutions for problems in fractional Orlicz–Sobolev spaces

We use a symmetric mountain pass lemma of Kajikiya to prove the existence of infinitely many weak solutions for the Schrödinger Φ-Laplace equation ( − Δ ) Φ u + V ( x ) φ ( u ) = ξ ( x ) f ( u ) in ℝ d , where Φ(t)= ∫ 0tφ(s)ds is an N-function, ΔΦ is the Φ-Laplacian operator, V:ℝd→ℝ is a continuous function, ξ is a function with sign-changing on ℝd and the nonlinearity f is sublinear as |u|→∞. During the study of our problem, we deal with a new compact embedding theorem for the Orlicz–Sobolev spaces. We also study the existence and multiplicity of solutions to the general fractional Φ-Laplacian equations of Kirchhoff type M ( ∫ ℝ 2 d Φ ( u ( x ) − u ( y ) K ( | x − y | ) ) d x d y N ( | x − y | ) ) ( − Δ ) Φ K , N u = f ( x , u ) in Ω , u = 0 in ℝ d ∖ Ω , where Ω is an open bounded subset of ℝd with smooth boundary ∂Ω, d>2, and M:ℝ0+→ℝ+ is a continuous function and f:Ω×ℝ→ℝ is a Carathéodory function. The proofs rely essentially on the fountain theorem and the genus theory.

INFINITELY MANY LOW- AND HIGH-ENERGY SOLUTIONS FOR A CLASS OF ELLIPTIC EQUATIONS WITH VARIABLE EXPONENT

This paper is concerned with the $ p(x) $-Laplacian equation of the form $\left\{ \begin{array}{l} - {\Delta _{p(x)}}u = Q(x)|u{|^{r(x) - 2}}u,{\rm{ in}}\;\Omega ,\\ {\rm{u = 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; on}}\;\partial \Omega , \end{array} \right.$ where $ \Omega\subset{\mathbb{R}}^N $ is a smooth bounded domain, $ 1 p^+ $ and $ Q: \overline{\Omega}\to{\mathbb{R}} $ is a nonnegative continuous function. We prove that (0.1) has infinitely many small solutions and infinitely many large solutions by using the Clark's theorem and the symmetric mountain pass lemma.

Infinitely Many Solutions for a Non-homogeneous Differential Inclusion with Lack of Compactness

Abstract In this paper, we consider the following class of differential inclusion problems in ℝ N {\mathbb{R}^{N}} involving the p ⁢ ( x ) {p(x)} -Laplacian: - Δ p ⁢ ( x ) ⁢ u + V ⁢ ( x ) ⁢ | u | p ⁢ ( x ) - 2 ⁢ u ∈ a ⁢ ( x ) ⁢ ∂ ⁡ F ⁢ ( x , u ) in ⁢ ℝ N . -\Delta_{p(x)}u+V(x)\lvert u\rvert^{p(x)-2}u\in a(x)\partial F(x,u)\quad\text{% in}\ \mathbb{R}^{N}. We are concerned with a multiplicity property, and our arguments combine the variational principle for locally Lipschitz functions with the properties of the generalized Lebesgue–Sobolev space. Applying the nonsmooth symmetric mountain pass lemma and the fountain theorem, we establish conditions such that the associated energy functional possesses infinitely many critical points, and then we obtain infinitely many solutions.

Infinitely many solutions for Kirchhoff equations with Hardy–Littlewood–Sobolev critical nonlinearity

In this paper we study a class of Kirchhoff equations with Hardy–Littlewood–Sobolev critical nonlinearity. By using the second concentration compactness principle together with Kajikiya’s new version of the symmetric mountain pass lemma, we obtain the existence of infinitely many solutions which tend to zero under a suitable value of $$\lambda $$. To our best knowledge, our result is new for this problem.

Multiple Solutions to a Caffarelli–Kohn–Nirenberg-Type Equation in Finsler Manifold

This paper deals with the existence of infinitely many solutions for a generalized form of Caffarelli–Kohn–Nirenberg-type equation in a Finsler manifold, with asymptotically linear term at infinity. This kind of nonlinear term creates some difficulties in the application of the symmetric mountain pass lemma because of the lack of Ambrosetti–Rabinowitz-type condition; hence, it is difficult to verify the classical (P S)c condition. To overcome this difficulty, we use an equivalent version of Cerami’s condition (C)c.

Infinitely many solutions for the p-fractional Kirchhoff equations with electromagnetic fields and critical nonlinearity

In this paper, we consider the fractional Kirchhoff equations with electromagnetic fields and critical nonlinearity. By means of the concentration-compactness principle in fractional Sobolev space and the Kajikiya's new version of the symmetric mountain pass lemma, we obtain the existence of infinitely many solutions, which tend to zero for suitable positive parameters.

On a Degenerate p-Fractional Kirchhoff Equations Involving Critical Sobolev–Hardy Nonlinearities

In this paper, we study a class of degenerate p-fractional Kirchhoff equations with critical Hardy–Sobolev nonlinearities. By means of the Kajikiya’s new version of the symmetric mountain pass lemma, we obtain the existence of infinitely many solutions which tend to zero under a suitable value of \(\lambda \). The main feature and difficulty of our equations is the fact that the Kirchhoff term M could be zero at zero, that is the equation is degenerate. To our best knowledge, our results are new even in the Laplacian and p-Laplacian cases.

Infinitely Many Solutions for a Class of Fractional Impulsive Coupled Systems with (p,q)-Laplacian

By using the symmetric mountain pass lemma, we investigate the problem of existence of infinitely many solutions for a class of fractional impulsive coupled systems with (p,q)-Laplacian, which possesses mixed type nonlinearities, and the nonlinearities do not need to satisfy the well-known Ambrosetti-Rabinowitz condition.

Infinitely Many Solutions for Critical Degenerate Kirchhoff Type Equations Involving the Fractional p–Laplacian

In this paper we study a class of critical Kirchhoff type equations involving the fractional p–Laplacian operator, that is $$\begin{aligned} \begin{array}{ll} \displaystyle M\left( \iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right) (-\Delta )_p^{s} u {=} \lambda w(x)|u|^{q-2}u + |u|^{ p_s^{*}-2 }u,\quad x\in {\mathbb {R}}^N, \end{array} \end{aligned}$$ where $$(-\,\Delta )^s_p$$ is the fractional p–Laplacian operator with $$0<s<1<p<\infty $$ , dimension $$N>ps$$ , $$1<q<p^{*}_{s}$$ , $$p^{*}_s$$ is the critical exponent of the fractional Sobolev space $$W^{s,p}({\mathbb {R}}^N)$$ , $$\lambda $$ is a positive parameter, M is a non-negative function while w is a positive weight. By exploiting Kajikiya’s new version of the symmetric mountain pass lemma, we establish the existence of infinitely many solutions which tend to zero under a suitable value of $$\lambda $$ . The main feature and difficulty of our equations is the fact that the Kirchhoff term M is zero at zero, that is the equation is degenerate. To our best knowledge, our results are new even in the Laplacian and p–Laplacian cases.

Existence of infinitely many solutions for degenerate p-fractional Kirchhoff equations with critical Sobolev–Hardy nonlinearities

In this paper, we study a class of degenerate p-fractional Kirchhoff equations with critical Hardy–Sobolev nonlinearities $$\begin{aligned} M\left( \iint _{\mathbb {R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\mathrm{d}x\mathrm{d}y\right) (-\Delta )_p^s u = \lambda w(x)|u|^{q-2}u + \frac{|u|^{p_s^*(\alpha )-2}u}{|x|^\alpha },\quad x\in \mathbb {R}^N, \end{aligned}$$ where $$(-\Delta )^s_p$$ is the fractional p-Laplacian operator with $$0<s<1<p<\infty $$ , dimension $$N>ps$$ , $$1<q<p^{*}_{s}(\alpha )$$ , $$p^*_s(\alpha ) = p(N-\alpha )/(N-ps)$$ is the critical exponent of the fractional Hardy–Sobolev exponent with $$\alpha \in [0,ps)$$ , $$\lambda $$ is a positive parameter, M is a nonnegative function, while w is a positive weight. By means of the Kajikiya’s new version of the symmetric mountain pass lemma, we obtain the existence of infinitely many solutions which tend to zero under a suitable value of $$\lambda $$ . The main feature and difficulty of our equations is the fact that the Kirchhoff term M could be zero at zero, that is the equation is degenerate. To our best knowledge, our results are new even in the Laplacian and p-Laplacian cases.