Abstract We prove the existence of a ground state positive solution Schrödinger–Poisson systems in plane form \[ -\Delta u + V(x)u \frac{\gamma}{2\pi} \left(\log|\cdot| \ast u^2 \right)u = b |u|^{p-2}u \quad\text{in}\ \mathbb{R}^2, \] where $p\ge 4$ , $\gamma,b>0$ and potential $V$ is assumed to be unbounded at infinity. On we do not require any symmetry or periodicity assumption, it suppose...

In the last decade or so, variational gluing methods have been widely used to construct homoclinic and heteroclinic type solutions of nonlinear elliptic equations and Hamiltonian systems. This note is concerned with the procedure of gluing mountain-pass type solutions. The rst procedure to glue mountain-pass type solutions was developed through the work of S er e, and Coti Zelati-Rabinowitz. Th...

In this paper, a system of elliptic equations is investigated, which involves multiple critical Sobolev exponents and singular points. The best Sobelev constant related to the system is studied, which is verified to be independent of the location of singular points. By a variant of the concentration compactness principle and the mountain-pass argument, the existence of positive solutions to the...

this study concerns the existence and multiplicity of positive weak solutions for a class ofsemilinear elliptic systems with nonlinear boundary conditions. our results is depending onthe local minimization method on the nehari manifold and some variational techniques. alsoby using mountain pass lemma, we establish the existence of at least one solution withpositive energy.

In this paper, we continue the mathematical study of adiabatic chemical reactions, started in a previous work (Ann. Henri Poincaré, 5: 477–521, 2004). We consider a molecule with one free atom, the latter having two distinct possible stable positions. We then look for a mountain pass point between these two local minima in the nonrelativistic Schrödinger framework. We prove the existence of a m...

In this paper, we study the following nonlinear elliptic problem -div(a(x) ?u) = f (x,u), x ? u H10(?) (P) where is a regular bounded domain in RN, N 2, a(x) positive function and reaction source strongly asymptotically linear sense lim t?+? f(x,t)/t q(x) uniformly ?. We use variant version of Mountain Pass Theorem to prove that has solution for large class (x,t) q(x). Here, existence proved wi...