Nontrivial Solutions of the Kirchhoff-Type Fractional p-Laplacian Dirichlet Problem by Taiyong Chen, Wenbin Liu, Hua Jin

“…In this article, we consider the new results for the Kirchhoff-type p-Laplacian Dirichlet problem containing the Riemann-Liouville fractional derivative operators. By using the mountain pass theorem and the genus properties in the critical point theory, we get some new results on the existence and multiplicity of nontrivial weak solutions for such Dirichlet problem.…”
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Multiplicity of Solutions for a Class of Fourth-Order Elliptic Problems with Asymptotically Linear Term by Qiong Liu, Dengfeng Lü

“…Using an equivalent version of Cerami's condition and the symmetric mountain pass lemma, we obtain the existence of multiple solutions for the equations.…”
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Existence of Homoclinic Orbits for a Class of Asymptotically p-Linear Difference Systems with p-Laplacian by Qiongfen Zhang, X. H. Tang

“…By applying a variant version of Mountain Pass Theorem in critical point theory, we prove the existence of homoclinic solutions for the following asymptotically p-linear difference system with p-Laplacian Δ(|Δu(n-1)|p-2Δu(n-1))+∇[-K(n,u(n))+W(n,u(n))]=0, where p∈(1,+∞), n∈ℤ, u∈ℝN, K,W:ℤ×ℝN→ℝ are not periodic in n, and W is asymptotically p-linear at infinity.…”
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Existence and Multiplicity of Solutions for a Class of Anisotropic Double Phase Problems by Jie Yang, Haibo Chen, Senli Liu

“…We consider the following double phase problem with variable exponents: −div∇upx−2∇u+ax∇uqx−2∇u=λfx,u in Ω,u=0, on ∂Ω. By using the mountain pass theorem, we get the existence results of weak solutions for the aforementioned problem under some assumptions. …”
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Multiplicity Results for Variable-Order Nonlinear Fractional Magnetic Schrödinger Equation with Variable Growth by Jianwen Zhou, Bianxiang Zhou, Yanning Wang

“…Under appropriate assumptions, firstly, we prove that the system has at least two different solutions by applying the mountain pass theorem and Ekeland’s variational principle. …”
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Multiple Sign-Changing Solutions for Kirchhoff-Type Equations by Xingping Li, Xiumei He

“…Under some suitable conditions, we prove that the equation has three solutions of mountain pass type: one positive, one negative, and sign-changing. …”
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Positive Solutions for a Nonhomogeneous Kirchhoff Equation with the Asymptotical Nonlinearity in R3 by Ling Ding, Lin Li, Jin-Ling Zhang

“…Under appropriate assumptions on k,f, and h, existence of two positive solutions is proved by using the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory.…”
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Multiple Solutions for a Class of N-Laplacian Equations with Critical Growth and Indefinite Weight by Guoqing Zhang, Ziyan Yao

“…Using the suitable Trudinger-Moser inequality and the Mountain Pass Theorem, we prove the existence of multiple solutions for a class of N-Laplacian equations with critical growth and indefinite weight -div∇uN-2∇u+VxuN-2u=λuN-2u/xβ+fx,u/xβ+ɛhx, x∈ℝN, u≠0, x∈ℝN, where 0<β<N, V(x) is an indefinite weight, f:ℝN×ℝ→ℝ behaves like exp⁡αuN/N-1 and does not satisfy the Ambrosetti-Rabinowitz condition, and h∈(W1,N(ℝN))*.…”
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Multiple Solutions for Second-Order Sturm–Liouville Boundary Value Problems with Subquadratic Potentials at Zero by Dan Liu, Xuejun Zhang, Mingliang Song

“…We deal with the following Sturm–Liouville boundary value problem: −Ptx′t′+Btxt=λ∇xVt,x,  a.e. t∈0,1x0cos  α−P0x′0sin  α=0x1cos  β−P1x′1sin  β=0 Under the subquadratic condition at zero, we obtain the existence of two nontrivial solutions and infinitely many solutions by means of the linking theorem of Schechter and the symmetric mountain pass theorem of Kajikiya. Applying the results to Sturm–Liouville equations satisfying the mixed boundary value conditions or the Neumann boundary value conditions, we obtain some new theorems and give some examples to illustrate the validity of our results.…”
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Infinitely Many Solutions for a Superlinear Fractional p-Kirchhoff-Type Problem without the (AR) Condition by Xiangsheng Ren, Jiabin Zuo, Zhenhua Qiao, Lisa Zhu

“…We only consider the non-Ambrosetti-Rabinowitz condition to prove our results by using the symmetric mountain pass theorem.…”
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Two Types of Solutions to a Class of (p,q)-Laplacian Systems with Critical Sobolev Exponents in RN by Jing Li, Caisheng Chen

“…Relying on concentration-compactness principle, mountain pass lemma, and genus theory, the existence of solutions to the elliptic system with η∈(q,p⁎) or η∈(1,p) will be established.…”
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Nonlinear eigenvalue problems in Sobolev spaces with variable exponent by Teodora-Liliana Dinu

“…Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ℤ2-symmetric version for even functionals of the Mountain Pass Lemma and some adequate variational methods.…”
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Multiplicity of Solutions for a Class of Elliptic Problem of p-Laplacian Type with a p-Gradient Term by Zakariya Chaouai, Soufiane Maatouk

“…For this purpose, we use the Mountain Pass theorem, on an equivalent problem to (P) with variational structure. …”
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TRANSFORMATION OF PREHISTORIC TO HISTORIC LANDSCAPE: THE EXAMPLE OF CIVITAS LOPSICA by Vedrana Glavaš, Miroslav Glavičić

“…The favorable strategic and transportation position and a busy harbor encouraged the development of trade which took place over the mountain passes with Iapodes inland. Changes that occurred in the landscape during the Iron Age and early antiquity can still be seen today.…”
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