In this paper we study the existence of multi-bump solutions for the following Choquard equationwhere µ ∈ (0, 3), p ∈ (2, 6 − µ), λ is a positive parameter and the nonnegative continuous function a(x) has a potential well Ω := int(a −1 (0)) which possesses k disjoint bounded components Ω := ∪ k j=1 Ωj . We prove that if the parameter λ is large enough, then the equation has at least 2 k − 1 multi-bump solutions.Mathematics Subject Classifications (2010): 35J20, 35J65

Using dual method we establish the existence of nodal ground state solution for the following class of problemswhere ∆ 2 is the biharmonic operator, B = ∆ or B = ∂ ∂ν and f is a C 1 − function having subcritical growth. (2010): 35J20, 35J65 Mathematics Subject Classifications

This paper concerns with the study of some bifurcation properties for the following class of nonlocal problems trueright90.0pt{(−Δ)su=λffalse(xfalse)false(u+h(u)false),indouble-struckRN,ufalse(xfalse)>0,forallx∈double-struckRN,falseprefixlim|x|→∞ufalse(xfalse)=0,where N>2s, s∈(0,1), λ>0, f:double-struckRN→R is a positive continuous function, h:R→R is a bounded continuous function and (−Δ)su is the fractional Laplacian. The main tools used are the Leray–Shauder degree theory and the global bifurcation result due to Rabinowitz.

Using variational methods, we establish existence of multi-bump solutions for the following class of problemswhere N ≥ 1, ∆ 2 is the biharmonic operator, f is a continuous function with subcritical growth and V : R N → R is a continuous function verifying some conditions. Mathematics Subject Classifications (2010): 35J20, 35J65

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